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Sandwich Panels with Functionally Graded Core


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SANDWICH PANELS WITH FUNCTIONALLY GRADED CORE By NICOLETA ALINA APETRE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Nicoleta Alina Apetre

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This document is dedicated to my loving fam ily: my mother, Elisabeta Apetre, my sister, Dana and my brother, Nicu and to my devoted boyfriend Laurence as they have offered their support and love throughout this endeavor. No words said can express how indebted I am to them for what I am today and fo r what I shall accomplish in the future.

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ACKNOWLEDGMENTS I am profoundly grateful to my advisor, Dr. Bhavani Sankar, for scientific guidance, moral support and endless patience. I would like to thank Dr. Peter Ifju for his help and ideas throughout this enterprise. I would also like to thank my friend, mentor and committee member, Dr. Oana Cazacu, for her guidance and support on my research and my life. Her comments and advice made this dissertation a much more useful document. At the same time, I wish to thank Dr. Raphael T. Haftka and Dr. Bjorn Birgisson for serving on my committee and for their assistance and guidance. I express sincere appreciation to a friend and mentor, Dr. Satchi Venkataraman, for his continuous scientific guidance and moral support during the years. I would like to thank all my professors from whom I learned so much, especially the late Dr. E. Soos, who showed me the path I am walking now. Special thanks go to all my friends, colleagues and family for their continual love, support, and encouragement throughout my time in graduate school and my life. iv

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TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................iv LIST OF TABLES ............................................................................................................vii LIST OF FIGURES .........................................................................................................viii ABSTRACT ......................................................................................................................xii CHAPTER 1 INTRODUCTION AND BACKGROUND.................................................................1 Introduction...................................................................................................................1 Literature Review.........................................................................................................7 Sandwich Plate Theories.......................................................................................7 Contact on Composite Materials.........................................................................14 Cellular Solids and Functionally Graded Materials............................................28 Objectives and Approaches........................................................................................36 2 ANALYTICAL MODELING OF SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE........................................................................38 Exact Method for Sandwich Structures with Functionally Graded Core...................38 Fourier Galerkin Method for Sandwich Structures with Functionally Graded Core.........................................................................................................................42 Analytical Models for Sandwich Structures with Functionally Graded Core............47 An Equivalent Single-Layer First-Order Shear Deformation Theory.................47 An Equivalent Single-Layer Third-Order Shear Deformation Theory...............50 A Higher-Order Shear Deformation Theory.......................................................52 Results for Sandwich Beam with Functionally Graded Core under a Distributed Load........................................................................................................................57 Summary and Conclusions.........................................................................................67 3 CONTACT AND LOW-VELOCITY IMPACT Of SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE........................................................................69 Assumed Stress Distribution Method.........................................................................71 Method of Point Matching..........................................................................................74 v

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Quasi-Static Impact of a Sandwich Beam with FG Core...........................................76 Results.........................................................................................................................78 Summary and Conclusions.......................................................................................103 4 MASS OPTIMIZATION FOR A SANDWICH BEAM WITH FUNCTIONALLY GRADED CORE......................................................................................................105 Linear Model............................................................................................................105 Case 1: Same Flexural Rigidity.........................................................................109 Case 2: Same Mass and Flexural Rigidity.........................................................112 Quadratic Model.......................................................................................................118 Case 1: Same Flexural Rigidity.........................................................................121 Case 2: Same Total Mass..................................................................................130 Summary and Conclusions.......................................................................................138 5 SUMMARY AND CONCLUSIONS.......................................................................140 LIST OF REFERENCES.................................................................................................144 BIOGRAPHICAL SKETCH...........................................................................................153 vi

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LIST OF TABLES Table page 3-1 Maximum normal and shear strains for a given impact energy...............................94 3-2 Core thicknesses for different materials with same flexural stiffness, D 11 and same global stiffness................................................................................................97 4-1 Variations of Youngs modulus, density, mass and flexural rigidity.....................107 4-2 Total mass of sandwich structures with different density variations and different core thicknesses......................................................................................................111 4-3 Maximum normal and shear strains for a given impact energy.............................117 4-4 Variations of Youngs modulus, density, mass and flexural rigidity.....................120 4-5 Total mass of sandwich structures with same geometry and different density variations................................................................................................................123 4-6 Maximum normal and shear strains normal and interfacial shear stresses for a given impact energy...............................................................................................127 4-7 Flexural rigidity of sandwich structures with same geometry and different density variations....................................................................................................132 4-8 Maximum normal and shear strains and stresses for a given impact energy.........134 vii

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LIST OF FIGURES Figure page 1.1 The behavior of (a) flexible face sheet and (b) rigid face sheet.................................4 2.1 Sandwich beam with functionally graded core with schematic of the analysis elements....................................................................................................................39 2.2 Traction forces and displacements at the interfaces of each element in the FGM sandwich beam.........................................................................................................40 2.3 The beam geometry..................................................................................................48 2.4 The beam geometry..................................................................................................53 2.5 Through-the-thickness variations of core modulus..................................................58 2.6 Variation of transverse shear stresses in the sandwich structure with FGM core....59 2.7 Variation of zz stresses in the sandwich structure with FGM core.........................60 2.8 Variation of bending stresses through the thickness of the FGM core....................61 2.9 Variation of u displacements through the thickness of the sandwich structure with FGM core.........................................................................................................61 2.10 Non-dimensional core modulus...............................................................................62 2.11 Comparison of deflections.......................................................................................64 2.12 Comparison of longitudinal displacement................................................................64 2.13 Comparison of axial stress in the core.....................................................................65 2.14 Comparison of shear strain in the core.....................................................................66 2.15 Comparison of shear stress in the core.....................................................................67 3.1 Dimensions of the sandwich panel and the contact length, 2c.................................72 3.2 The stress distribution under the indenter................................................................72 viii

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3.3 Illustration of relation between w deflection and radius of curvature of the deformed surface......................................................................................................73 3.4 Discretization of contact load for the method of point matching.............................74 3.5 Low-velocity impact model.....................................................................................77 3.6 Through the thickness variations of core modulus...................................................80 3.7 Variation of the horizontal displacement.................................................................81 3.8 Deflection, w, in the second half of the sandwich beam..........................................81 3.9 Variation of the deflection........................................................................................82 3.10 Contour plot of normal strain in the functionally graded asymmetric core.............83 3.11 Contour plot of normal strain in the functionally graded symmetric core...............83 3.12 Contour plot of normal strain in the homogeneous core..........................................83 3.13 Contour plot of shear strain in the functionally graded asymmetric core................84 3.14 Contour plot of shear strain in the functionally graded symmetric core..................84 3.15 Contour plot of shear strain in the homogeneous core.............................................85 3.16 Variation of the axial stress......................................................................................85 3.17 Variation of the axial stress through the thickness in functionally graded core.......86 3.18 Variation of the normal (compressive) stress...........................................................87 3.19 Face sheet thickness influence on variation of the normal (compressive) stress.....88 3.20 Indentor radius influence on variation of the normal (compressive) stress.............89 3.21 Face sheet Youngs modulus influence on variation of the normal (compressive) stress.........................................................................................................................89 3.22 Shear stress in functionally graded asymmetric core...............................................90 3.23 Contour plot of shear stress in linear asymmetric core............................................91 3.24 Contour plot of shear stress in linear symmetric core..............................................91 3.25 Contour plot of shear stress in homogeneous core...................................................91 3.26 Variation of contact force with indentation depth in functionally graded beams....92 ix

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3.27 Relation between contact force and the vertical displacement.................................93 3.28 Contact force vs. maximum normal strain...............................................................95 3.29 Contact force vs. maximum shear strain..................................................................96 3.30 Variation of contact force with global deflection.....................................................97 3.31 Comparison of maximum normal strains for different core materials.....................99 3.32 Comparison of maximum shear strains for different core materials........................99 3.33 Finite element simulation for half of the beam......................................................100 3.34 Finite element simulation of the contact between the functionally graded sandwich beam and the rigid spherical indentor....................................................101 3.35 Contact pressure under indenter and total load for three models...........................102 3.36 Deflections at core top face sheet interface for same contact length and different contact load..............................................................................................102 3.37 Deflections at core top face sheet interface for different contact length and same contact load...................................................................................................103 4.1 Geometry and notations for the three cases investigated.......................................108 4.2 Through the thickness non-dimensional variations of core modulus and density.110 4.3 Through the thickness non-dimensional variations of core modulus and density.113 4.4 Relation between contact force and vertical displacement....................................113 4.5 Variation of contact force with indentation depth..................................................114 4.6 Variation of contact force with contact length.......................................................115 4.7 Variation of contact force with contact length.......................................................115 4.8 Contact force vs. maximum normal strain.............................................................116 4.9 Contact force vs. maximum shear strain................................................................117 4.10 Youngs modulus variation....................................................................................119 4.11 Core Youngs modulus...........................................................................................123 4.12 Core densities variation..........................................................................................124 x

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4.13 Relation between contact force and vertical displacement....................................124 4.14 Relation between contact force and contact length................................................125 4.15 Relation between contact force and contact length................................................125 4.16 Relation between indentation (core compression) and contact length...................126 4.17 Contact force vs. core maximum normal strain.....................................................128 4.18 Contact force vs. core maximum shear strain........................................................129 4.19 Contact force vs. core maximum normal stress.....................................................129 4.20 Contact force vs. maximum interfacial shear stress...............................................130 4.21 Core Youngs modulus...........................................................................................132 4.22 Core densities variation..........................................................................................133 4.23 Relation between contact force and vertical displacement....................................134 4.24 Relation between contact force and contact length................................................135 4.25 Contact force vs. core maximum normal strain.....................................................136 4.26 Contact force vs. core maximum shear strain........................................................136 4.27 Contact force vs. core maximum normal stress.....................................................137 4.28 Contact force vs. maximum interfacial shear stress...............................................137 xi

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SANDWICH PANELS WITH FUNCTIONALLY GRADED CORE By Nicoleta Alina Apetre December 2005 Chair: Bhavani V. Sankar Major Department: Mechanical and Aerospace Engineering Although sandwich structure offer advantages over other types of structures, it is important to develop new types of materials in order to obtain the absolute minimum weight for given conditions (e.g., structural geometry and loadings). One alternative is represented by functionally graded materials (defined as materials with properties that vary with location within the material in order to optimize a prescribed function). This study presents different analytical and finite element models for sandwich structures with functionally graded core. The trade-off between weight and stiffness as well as a comparison between these structures and sandwich structures with homogenous core is also presented. The problem of low-velocity impact between a sandwich structure with functionally graded core and a rigid spherical indentor is solved. A few advantages and disadvantages of these types of structure are presented. With the new developments in manufacturing methods these materials can be used for a large number of applications ranging from implant teeth to rocket frames. xii

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CHAPTER 1 INTRODUCTION AND BACKGROUND Introduction A sandwich structure consists of two thin, stiff, and strong face sheets connected by a thick, light and low-modulus core using adhesive joints in order to obtain very efficient lightweight structure ( Zenkert, 1997 ; Vinson, 2001 ). In most of the cases the faces carry the loading, both in-plane and bending, while the core resists transverse shear loads. A sandwich operates in the same way as an I-beam with the difference that the core of a sandwich is of a different material and is stretched out as a continuous support for the face sheets. The main advantage of a sandwich structure is its exceptionally high flexural stiffness-to-weight ratio compared to other architectures. As a consequence, sandwich construction results in lower lateral deformations, higher buckling resistance, and higher natural frequencies than do other structures. Thus, for a given set of mechanical and environmental loads, sandwich construction often results in a lower structural weight than do other configurations. Few of the drawbacks of sandwich structures are: manufacturing methods, quality control and joining difficulties. The idea to use two co-operating faces separated by a distance emerged at the beginning of 19 th century. The concept was first applied commercially during World War II in airplanes. Some of the first theoretical and experimental works on sandwich constructions were published in the late 1940s ( Williams et al., 1941 and March, 1948 ). Since then, the use of sandwich structures has increased rapidly. The need for high-performance, low-weight structures assures that the sandwich construction will continue 1

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2 to be in demand. In view of increasing the use of sandwich structures in the aerospace industry, critical issues such as impact resistance, optimization, fatigue and fracture should be addressed. Lately, due to the difficulty to obtain closed-form solutions for the governing differential equations, finite element method has become the most used design tool for panel analysis. The core material is perhaps the most important component of a sandwich structure. The following properties are required for the core material: low density in order to add as little as possible to the total weight; Youngs modulus perpendicular to the face sheet should be fairly high to prevent excessive deformation in the thickness direction and therefore a rapid decrease in flexural rigidity. Even though the transverse force creating normal stresses perpendicular to the core are usually low, even a small decrease in core thickness would create a large decrease in flexural rigidity. This is crucial in the case of impact loads wherein the contact force acts over a small area. The core is mainly subjected to transverse shear stresses. The core shear strains contribute to global deformation and core shear stresses. Thus, a core must be chosen such that it would not fail under the applied transverse load and with a shear modulus high enough to give the required shear stiffness. Other functions such as thermal and acoustical insulation depend mainly on the core material and its thickness. The core materials have been fashioned in various forms and developed for a range of applications. Few examples are: foam or solid cores (where properties can be tailored by orienting the cell structure), honeycomb cores, web cores and truss cores. Honeycomb cores offer the greatest shear strength and stiffness to weight ratio but require care in ensuring adequate bonding to the face-sheets. Also the core is highly anisotropic and

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3 forming complex curved shapes is difficult. High-density cellular thermoplastic foams, such as poly (vinyl chloride) (PVC) and polyurethane (PUR) do not have the same high stiffness to weight ratio as honeycomb materials but they have more advantages: they are less expensive, they are continuous at the macroscopic level, the foam surface is easier to bond, and they ensure high thermal and acoustic insulation (cellular cores have a very low thermal conductivity, e.g., Gibson and Ashby, 1997 ). A metallic foam core sandwich structure offers also a number of advantages. They can be made with integral skin, eliminating the need for adhesive layer, they can be formed in curved shapes, and their properties are nearly isotropic ( Gibson and Ashby, 1997 ). Foreign objects impact (such as tool drops, hail, bird strikes, and runway debris) can introduce damage and significantly reduce the strength of the sandwich structure. Unlike their solid metallic counterparts, predictions of the effects of low-velocity impact damage on sandwich structures are difficult because significant internal damage is achieved at impact energy levels lower than those required to create visible damage at the surface. Low-velocity impact, one of the topics of present research is described by low projectile velocity; therefore the damage can be analyzed by assuming that the structure is under a quasi-static loading. Impact/contact produces both overall deformation and local deformation. Usually, if the load is applied on top face sheet, the bottom face sheet experience only global deformation so many models assume a clamped bottom face sheet in order to eliminate the global deflection and focus the research on local effects.

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4 There are different factors that can influence the local effect of indentation stress field: face sheet elastic modulus and its thickness, the core density and its thickness. One of the factors that influence the structure behavior under an indentor is the face sheet stiffness: for flexible face-sheets there is a local deformation under the indentor/load whereas the very rigid face-sheets spread the load (Figure 1.1). Usually the local failure starts in the core and results in core crushing, delamination and therefore significant reduction of the sandwich strength. (a) (b) Figure 1.1: The behavior of (a) flexible face sheet and (b) rigid face sheet Contact laws (defined as relation between the contact force and indentation or core compression) or relative displacement of the indentor and the target for the indentation of a sandwich structure are significantly different from those for monolithic laminates. They are usually nonlinear. With sandwich structures the indentation is dominated by the behavior of the core material, which crushes as the shear stress and compressive stress become large. Compressive stress in transverse direction is not uniform through the thickness and is not the only stress component to be considered in order to determine the onset of core failure. Few of the contact problem difficulties are: forced surface displacement, initially unknown contact length/surface and pressure distribution.

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5 Although sandwich structure offer advantages over other types of structures, it is important to develop new types of materials in order to obtain the absolute minimum weight for given conditions (e.g., structural geometry and loadings). These new sandwiches should be compared with other sandwich construction and with alternative structures in order to select the best configuration. One of the new alternatives is a sandwich structure with functionally graded core. Functionally graded materials (FGM) possess properties that vary gradually with location within the material; consequently they are inhomogeneous materials on the macroscopic scale. New developments in manufacturing methods offer the designer the opportunity to tailor the materials microstructure in order to enhance their structural performance. The variation of the properties can be achieved by gradually changing the volume fraction of the constituents materials, by using reinforcements with different properties, sizes and shapes, by interchanging the roles of reinforcement and matrix phases in a continuous manner as well as by adjusting the cell structure. The result is a microstructure that produces continuously changing thermal and mechanical properties at the macroscopic or continuum level. FGMs differ from composites wherein the volume fraction of the reinforcing inclusion is uniform throughout the composite. The closest analogy of FGM is a laminated composite face sheets, but the latter possess distinct interfaces across which properties change abruptly. FGMs have a large spectrum of applications. They can be used as thermal barrier coatings ( Koizumi, 1997 ), interfacial zones to improve the bonding strength and to reduce the residual stresses ( Hsueh and Lee, 2003 ). Cellular materials with continuously varying density or pore size (porosity) that can be used as a

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6 core for a sandwich structure is an example of new FGMs. Although FGMs are highly heterogeneous, it will be useful to idealize them as continua with properties changing smoothly with respect to the spatial coordinate. This will enable obtaining closed-form solutions to some fundamental solid mechanics problems, and also will help in developing finite element models of structures made of FGMs. The major objective of this dissertation is to develop new analytical models for sandwich structures with functionally graded core. Particular attention will be paid to contact and impact problems involving sandwich structures. This chapter provides a framework and background for sandwich structures and functionally graded materials in order to give a perspective for present research. Chapter 2 introduces analytical models for a sandwich beam with functionally graded core and compares them with analytical models presented in the literature. The governing equations for sandwich structures with FG cores are solved for two types of cores by six different methods: exact solutions are presented for the exponential variation of core Young Modulus; a combination of Fourier series and Galerkin method for a polynomial variation of core Young Modulus; two equivalent single-layer theories based on assumed displacements, a higher-order theory and a finite-element analysis. Chapter 3 solves the problems of contact and low-velocity impact between a rigid cylindrical indentor and a sandwich beam with functionally graded core. A comparison of the total mass of different panels is discussed in Chapter 4, where different optimization problems for sandwich structure with FG core based on two models presented in the literature. They are the linear relation between relative density and Youngs modulus ( Choi and Sankar, 2005 ) and quadratic relation between relative density and Youngs modulus ( Gibson and Ashby, 1997 ). The results indicate that the

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7 functionally graded cores can be used to mitigate or prevent impact damage in sandwich composites, so it is shown that materials with gradients in mechanical properties offer better opportunities than traditional homogenous materials. Literature Review The equations describing the behavior of sandwich structures are similar to the shear deformable plate theory for composite laminates (e.g., Reddy, 2003 ) if each of the core, face sheets and adhesive layers is modeled as a continuum. The main difference between the traditional and modern sandwich models is the transverse flexibility of the core. A flexible/soft core is a core for which the ratio of Youngs moduli of the face sheets to the core is high (for structural panels this ratio can be between 500 and 1000). If a transversely flexible core is used a higher order theory is necessary. Typically sandwich composites are subjected to out of plane loadings where the primary loads are applied perpendicular to the panel surface. Localized loads are one of the major causes of failure in sandwich structures, because in most of the cases the face sheets are thin. The load, transmitted through the face sheet causes the core to be subjected to significant deformations locally, which can cause high shear and normal stresses that can exceed the allowable stress for the flexible, weak core material. In order to give a framework to the present research, this chapter presents a literature review on three fields: composite plates theories, contact on composite plates models and functionally graded materials. Sandwich Plate Theories A considerable amount of literature exists on sandwich panels as they are used in large number of applications varying form high-performance composites in aerospace to low-cost materials for building constructions. The literature includes monographs,

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8 conference proceedings, books and bibliographies. Noor et al. (1996) provide an excellent literature review on computational models for sandwich structures that includes over 800 references. The paper reviews the literature, classify the models and identify future direction for research. Similar, Vinson (2001) offers a general introduction to the structural mechanics involved in the field of sandwich structures and a sufficient number of references for further investigation. A large number of books on composite materials were published: they range from physical and mathematical analyses or mechanics of structural components composed of composite materials ( Vinson and Chou, 1975 ; Kollar and Springer, 2003 ; Vinson and Sierakowski, 1986 ), finite element modelling ( Matthews et al., 2001 ) to reference for practical engineering and designing ( Reddy and Miravete, 1995 ). The limitations of classical plate theory in describing complex problems (e.g. contact/impact problems, cutouts or laminate plates) necessitated the development of higher-order theories. The higher-order refers at the level of truncation of terms in a power series expansion about the thickness coordinate for displacements. The models investigated here can be classified into: single-layer theories (where the assumed displacement components represent the weight-average through the thickness of sandwich panel), layer-wise theories (where separate assumptions for displacements in each layer are made) and exact theories (equilibrium equations are solved without displacements assumptions). Although discrete-layer theories are more representative for sandwich construction than the single-layer theories, they experience computations difficulties from a large number of field variables in proportion with number of layers.

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9 First, a literature review on equivalent single-layer theory is presented in order to give details of the field development. Reissner (1945) and Midlin (1951) were the first to propose a theory that included the effect of shear deformation. The displacements were assumed as: 000(,,)(,)(,)(,,)(,)(,)(,,)(,)xyuxyzuxyzxyvxyzvxyzxywxyzwxy (1.1) where z is the coordinate normal to the middle plane, u and v are in-plane displacements and w is out-of-plane displacement and where 0,,xyw are weighted averages. Midlin introduced the correction factor into the shear stress to account for the fact that (1.1)predicts a uniform shear stress through the thickness of the plate. Considering the terms in (1.1) as first terms in a power series expansion in z, it is seen that the classical theory and the shear theory are of the same order of approximation. Yang et al. (1966) extended Mindlins theory for homogeneous plate to laminates consisting of an arbitrary number of bonded layers. Yang et al. (1966) used the theory to solve the frequency equations for propagation of harmonic waves in a two-layer isotropic plate of infinite extent. Based on the same model (Mindlins theory), Whitney and Pagano (1970) developed a theory for anisotropic laminates plates consisting of an arbitrary number of bonded anisotropic layers that includes shear deformation and rotary inertia. Displacement field is assumed to be linear in thickness coordinate. The next order theory was proposed by Essenburg (1975) and was based on the following assumptions:

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10 0020(,,)(,)(,)(,,)(,)(,)(,,)(,)(,)(,)xyzzuxyzuxyzxyvxyzvxyzxywxyzwxyzxyzxy (1.2) Essenburg (1975) presented the advantages of the above expansion for the contact problem. In order to analyze the bending of a plate with a hole, Reissner (1975) included third order terms in the in-plane displacements z-expansion: (1.3) 3320(,,)(,)(,)(,,)(,)(,)(,,)(,)(,)xxyyzuxyzzxyzxyvxyzzxyzxywxyzwxyzxy It should be noted that above assumptions neglect the contribution of in-plane modes of deformation. Lo et al. (1977) theory included both in-plane and out-of-plane modes of deformation was based on: (1.4) 23023020(,,)(,)(,)(,)(,)(,,)(,)(,)(,)(,)(,,)(,)(,)(,)xxxyyyzzuxyzuxyzxyzxyzxyvxyzvxyzxyzxyzxywxyzwxyzxyzxy The principle of potential energy is used to derive the governing equations and boundary conditions and the theory is compared with lower-order plate theories and with exact elasticity solution in order to assess the importance of the displacements expansions. The theory was developed to model the behavior of infinite laminated plates. Chomkwak and Avula (1990) modified previous theory in order to be applied to finite rectangular laminate composite plates. The Lagrange multipliers are used in order to constrain the displacement functions to satisfy the stress boundary conditions.

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11 Reddy (1984) developed a third-order Shear Deformation theory (TSDT) for composite laminates, based on the following assumed displacement fields: (1.5) 2302300(,,)(,)(,)(,)(,)(,,)(,)(,)(,)(,)(,,)(,)xxxyyyuxyzuxyzxyzxyzxyvxyzvxyzxyzxyzxywxyzwxy Functions ,,,xxyy are eliminated based on the condition that the transverse shear stresses vanish on the plate top and bottom surfaces. As in the previous model, the principle of virtual displacements is used to derive the equilibrium equations. Reddy (1990) reviewed a number of other third-order theories and showed that they are special cases of his theory. Reddy (2000) and Reddy and Cheng (2001) expanded TSDT for analysis of functionally graded plates. Tessler (1993) developed a two-dimensional laminate plate theory for the linear elastostatic analysis of thin and thick laminated composite plates, which utilize independent assumptions for the displacements, transverse shear strain and transverse normal stress. The displacements expansions are similar with those assumed by Essenburg (1975) The transverse shear strains and transverse normal stress are assumed to be quadratic and cubic respectively through the thickness; they are expressed in terms of the kinematic variables of the theory by means of a least-squares compatibility requirements for the transverse strains and explicit enforcements of exact traction boundary conditions on top and bottom plate surfaces. The theory was extended to orthotropic shells by Tessler et al. (1995) Barut et al. (2002) developed a higher-order theory based on cubic expansion for in-plane displacements and quadratic expansion for out-of-plane displacement and used Reissners definition for kinematics of thick plates, to approximate the plate

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12 displacements with weighted-average quantities that are functions of the in-plane coordinates. Principle of virtual displacement is employed to write the equilibrium equations and boundary conditions and Fourier series expansion is used to determine the solutions. The accuracy of their model is demonstrated by comparison with exact solution and previous single-layer theories. Discrete-modeling (or layer wise) approach is based on separate assumptions for displacements in each layer: typically classical plates bending for the face sheets and shear deformation resistance for the core. Pagano (1970) provides an exact solution to the problem of a rectangular orthotropic laminated subjected to a laterally distributed load. Governing field equations for each layer are written in terms of displacement components and an assumed exponential-trigonometric solution is substituted into the equations. This provides a set of equations with six unknowns that are tractable. The solution unknowns are determined by the boundary conditions and the interface-continuity conditions. Ogorkiewicz and Sayigh (1973) calculated and compared the central deflection of the sandwich beam with a foam core using three basic mathematical models: one is obtained by transforming beam actual section into an equivalent section of one material; (the core is transformed into a thin web of the same material as the face sheet); second model is based on the strain energy approach; third is based on stress functions. They compared the analytical result with experiments and concluded that the third model is the closest with the experiment observations. Burton and Noor (1994) developed three-dimensional analytical thermoelasticity solutions for static problems of simply supported sandwich panels and cylindrical shells

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13 subjected to mechanical and thermal loads. Each of the layers is modeled as a three-dimensional continuum and double Fourier series expansions are used for displacements and stresses. Frostig et al. (1990, 1992) developed a high-order theory for behavior of a sandwich beam with transversely flexible core based on variational principles. The main feature of the method is the higher order displacement fields in the thickness coordinate. The longitudinal and the transverse deformations of the core determined with the aid of constitutive equations of an isotropic material and the compatibility conditions at the interfaces consist of non-linear expressions in the thickness coordinate. Their formulation is based on the beam theory for face sheets and a two-dimensional elasticity theory for the core. The sandwich behavior is presented in terms of internal resultants and displacements in face sheets, peeling (i.e. normal stresses between the faces and the core that do not impose severe restrains on the performance of the sandwich; e.g. a load applied above the support) and shear stresses at face sheet-core interfaces, and stresses and displacements in the core, even in the vicinity of concentrated loads. Their model can be applied to sandwich structures with honeycomb or foam cores. The model was applied to the vibration analysis ( Frostig and Baruch, 1994 ) and to sandwich structures with nonparallel skins ( Peled and Frostig, 1994 ). Sokolinsky and Nutt (2004) improved Frostig model by accounting for nonlinear acceleration fields in the transverse flexible core essential for the analysis because the inertia loads exerted on the sandwich beams are nonuniformly distributed along the span. A discretized formulation based on an implicit finite difference scheme is presented and validated.

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14 Hohe and Librescu (2003) presented a higher-order geometrically nonlinear sandwich shell theory for sandwich panels that accounts for the transverse compressibility of the core and initial geometric imperfections. Tangential displacement is approximate by a second order polynomial and transverse displacement is approximated by a linear polynomial in thickness coordinate. Among the large number of sandwich structures models present in the literature, the models reviewed here can be summarized as: assumed displacements methods (single-layer or layer-wise) and exact models (when the displacements are not assumed but derived). Few of the above models are modified for a sandwich structure with functionally graded core and the results are presented in Chapter 2. Contact on Composite Materials Foreign objects impact (such as tool drops, hail, bird strikes, and runway debris) can introduce damage and significantly reduce the strength of the sandwich structure. Unlike their solid metallic equivalent, predictions of the effects of impact damage on sandwich structures are difficult because significant internal damage can be achieved at impact energy levels lower than those required to create visible damage at the surface. The first step in understanding the impact dynamics is to study, qualitatively and quantitatively, the contact problem, that describes the contact force contact area relation. In this section two important research paths are reviewed: hertzian and non-hertzian contact problem. From the contact area point of view, contact models can be classified into: Conforming contacts (e.g. flat punch) when the contact area is a constant and is independent of the load. Nonconforming contacts (e.g. sphere) when the contact area is not constant and is depending on the load.

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15 The first contact analysis is due to Heinrich Hertz (1881) who investigated the quasi-static impact of spheres as a theoretical background for the localized deformation of the surface of two glass lenses due to the contact pressure between them. Hertz formulated the following boundary conditions for the contact surface ( Johnson, 1985 ): The displacements and stresses must satisfy the differential equations of equilibrium, and the stresses must vanish at a great distance from the contact. Therefore, the radii of curvature of the contacting bodies need to be large compared with the radius of the contact surface (each surface is treated as an elastic half-space). The bodies are in frictionless contact (only normal pressure is transmitted between the indenter and the specimen). At the bodies surfaces, the normal pressure is zero outside and equal and opposite inside the surface of contact. The distance between the surfaces of the two bodies is zero inside and greater than zero outside the contact surface. The integral of the pressure distribution within the contact surface with respect to the area of the contact surface gives the force acting between the two bodies. Hertz (1881) represented the contact surface by a quadratic function, and assumed a semi-elliptical distribution of pressure: 1/220()1rprpa (1.6) where a is the radius of the contact area, and r is the coordinate in radial direction. For the contact between a rigid sphere and a flat surface, Hertz derived a relation among the radius of the circle of the contact a, the indentor total load F, the indenter radius R and the elastic properties: 343kFRaE (1.7) where

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16 291116EkE 2 (1.8) and where E, and E, are the Youngs modulus and Poissons ratio for the specimen and indentor, respectively. Also Hertz found that the maximum tensile stress: max2(12)2Fa (1.9) is the radial stress that occurs at very edge of the contact and is the stress responsible for producing the Hertz cone cracks formed in a brittle material. Willis (1966) derived formulas for the contact between a rigid sphere and a transversely isotropic half-space. During the loading phase, the modified Hertz law (also known as Meyers law) that relates the total load F and the indentation as: nFk (1.10) where n = 3/2 for elastic contact and n = 1 for a fully plastic contact and where: 1/212211243111 111 kERRRREEE (1.11) and where R 1 and R 2 are the radius of curvature of the indentor and target respectively and E 1 1 are the elastic modulus and Poissons ratio of the indentor and E 2 is the Youngs modulus of the target. For the case where the indentor is much stiffer than the target, the above formula for E can be simplified as: 211EE (1.12)

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17 Even for a small amount of load there could be a significant permanent indentation so the unloading path is different from loading path. For the unloading phase, Crook (1952) proposed the following contact law: 00qmmFF (1.13) where q = 2.5, F m is the contact load at which unloading begins, m is indentation corresponding to F m and 0 is the permanent indentation. The permanent indentation, 0 is zero when the maximum indentation m remains below a critical value cr during the loading phase, so the permanent indentations is given by: 2/501 when crmm mcr (1.14) Problems concerning the contact between bodies have been solved since the work of Hertz in 1880s. A large number of books have been devoted to the mathematical tools used to solve contact problems. Gladwell (1980) summarized mathematical methods (based on complex variable, integral transformation and elliptic functions) used to find the solution for the contact between elastic bodies. An extensive review of the Russian literature for a large number of contact problems is detailed. Johnson (1985) derived expressions for stresses and displacements based on Hertz theory and generalized the theory to include different indentor profiles, interfacial friction, anisotropic and inhomogeneous materials or layered plates. Khludnev and Sokolowski (1997) monograph covers qualitative properties of solutions, optimization problem (e.g., finding the constructions which are of maximal strength and satisfy weight limitations), determination of crack shapes by optimization methods, existence theorems and

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18 sensitivity analysis of solutions. Fischer-Cripps (2000) presented a review of displacements and stresses for different types of indenters (uniform, spherical, flat punch, roller, rigid cone) in order to develop a practical tool for investigating the mechanical properties of engineering materials. The mathematical modeling, variational analysis, and numerical solutions of contact problems involving viscoelastic and viscoplastic materials as well frictional boundary conditions, which lead to time dependent models was presented by Han sand Sofonea (2002) Alexandrov and Pozharskii (2001) developed analytical and numerical methods for nonclasical three-dimensional linear elasticity contact problems. Their methods focused on integral equations include fracture mechanics. When a composite laminate with finite dimensions is in contact with a rigid indentor, the behavior may drastically deviate from that predicted for a half-space. Various methods were developed to describe the contact problem for isotropic, orthotropic, monolithic or laminated beams and plates under different indentors profiles. Most problems are based on a combination between a solution describing the local contact phenomenon and a beam/plate solution for the global response. Different models for predicting the elastic response of a beam/plate under a general load were presented in the previous section. Here a literature review on contact and low-velocity impact problems is presented. Abrate (1998) studied the contact between a sandwich beam and a cylindrical indentor. He assumed that a rigid plate supported the bottom face sheet. Treating the top face-sheet as a beam of rigidity EI on an elastic foundation (as long as compressive stress

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19 in the core does not exceed a maximum value), the following linear contact law is derived: Fk (1.15) where the contact stiffness is given by: 3/43/21/42(ccEbkh )EI (1.16) where E c is the core modulus in the transverse direction, h c is the core thickness and b is the beam width. Lee and Tsotsis (2000) studied the indentation failure behavior of honeycomb core sandwich panels by examining the effects of skin, core and indenter size on load transfer from the top skin to the core. They treat the problem as a plate (top face sheet) on an elastic foundation (core). A good agreement was found between the theoretical and FE model. Also comparison of maximum shear and normal stress with experimental data showed that the core shear failure dominates the onset of indentation failure. Yang and Sun (1982) and Tan and Sun (1985) conducted static indentation tests and finite elements models on laminated plates to validate the contact laws (1.10) (1.13)They found a good agreement with the above laws. The study of the effects of foreign object impact on sandwich structures has been a subject for a large number of researches in recent years. Understanding the mechanic of contact between the impactor and the panel is very important in the study of the impact dynamics in order to accurately predict the contact-force history and to predict how the damage will develop. Many solutions deals with statically, low-velocity impact when the duration of the impact (the interval of time from the first contact to the complete

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20 detachment) is much larger than the time required by the elastic wave, generated at the first point of contact, to propagate through the whole body. The use of static load-deflection behavior of the sandwich beam in the impact analysis needs some justification. In general the wave propagation effects, especially through the thickness of the core, should be considered in impact response of sandwich panels. This will be crucial when spalling type damage occurs in the panels. However, a study by Sankar (1992) showed that for very large impactor mass compared to that of the target plate and for very low impact velocities compared to the wave velocity in the target medium, quasi-static assumptions yield sufficiently accurate results for impact force history and ensuing stresses in the impacted plate. Ambur and Cruz (1995) model the sandwich plate using the first order shear deformation theory (FSDT) and combining the equation of motion of the plate with the equation of motion of the projectile (given by Newtons law) a set of differential equations were obtained. The solution is obtained by numerical integration. A highly nonlinear contact law is derived. Lee et al. (1993) developed a theory based on the assumption that each face sheet deforms as a first order shear deformation plate and the core deformation is obtained by the difference between the transverse displacements of the skins. Sun and Wu (1991) theory is based on the same assumption for the face sheets (first order shear deformation plate) but the core is model by using springs to simulate the transverse and shear rigidities. Meyer-Piening (1989) modeled the sandwich plate as a three-layer plate with independent linear in plane displacement in each layer.

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21 Koissin et al. (2004) determined the closed-form solution of the elastic response of a foamed-core sandwich structure under a local loading, using static Lame equations for the core and thin plate Kickoff-Love dynamic theory for the faces. The integral transformation technique was used to solve the differential equations. A good agreement was found between their analytical solution, experimental results and FE analysis. Choi and Hong (1994) derived a method for prediction the impact force history on composite subjected to low-velocity impact, applicable to isotropic and orthotropic plates with unknown contact laws. The first order shear deformation theory and von Karmans large deflection theory were used to describe the dynamic response of the plate and Yang and Sun (1982) modified Hertz law to describe contact problem. The impact duration is computed from the eigenvalue analysis of the lumped mass system, and the impulse-momentum conservation law is used with the concept of spring-mass model to predict the impact force history. A good agreement was found between their analytical solution and experimental results. A model for small mass impact on monolithic, transversely isotropic Midlin plate with a Hertzian contact law was given by Mittal (1987) An approximate analytical model for small mass impact on specially-orthotropic Kirchhoff plates based on Hertzian contact law was given by Olsson (1992) It is shown that transversal Youngs modulus has a small influence on the contact force, whereas the impact energy, impactor radius and plate thickness have a significant influence. The results are compared with numerical and experimental analyses. Olsson and Mc Manus (1996) included core crushing and large face-sheet deflections in a theory for contact indentation of sandwich panels. Olsson

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22 (1997) used Olsson and Mc Manus (1996) to modify Mittal (1987) for specially-orthotropic sandwich panels with linear load-indentation relation. Fischer-Cripps (1999) used the finite-element method to compute the radius of curvature of the contact surface for both elastic and elastic-plastic contacts. It is shown that indentors involving elastic-plastic deformations are equivalent with a perfectly rigid spherical whose radius is somewhat smaller than calculated using the Hertz equation for elastic contact. An experiment is used to validate his results. The range of applicability of Hertzs theory was found to be limited to very small indentors which also mean low loads. Many non-Hertzian contact problems do not permit analytical solutions in a closed form. This fact leads to various numerical models in order to determine the distribution of normal and tangential tractions, which satisfy the normal and tangential boundary conditions at the interface, inside and outside the contact area. The continuous distribution of traction is replaced by a discrete set of tractions elements: and the boundary conditions are satisfied at a discrete number of points called the matching points. Few examples are: Array of concentrated forces (gives an infinite surface displacements); Stepwise distribution given by uniform tractions acting on discrete segments/areas of the surface (the displacements are finite but the displacements gradients are infinite between the adjacent elements), e.g., Singh and Paul (1974) and Wu and Yen (1994) ; Superposition of a finite number of symmetrical rectangular loadings of unknown magnitudes distributed over known lengths (e.g., Sankar and Sun, 1985 ); Piecewise-linear distribution of tractions given by superposition of overlapping triangular elements (for a three dimensional problem, regular pyramid on a hexagonal base), e.g., Johnson, 1985 In order to find the values of the traction elements, which best satisfy the boundary conditions two different methods were developed:

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23 Direct/matrix inversion in which the boundary conditions are satisfied exactly at the matching points Variational method in which the tractions are chosen such that to minimize an appropriate energy function. Abrate (1997, 1994, 1991) classified the impact models into: spring-mass models (that accounts for the dynamics of the structure in a simplified manner); energy-balance models (when the structure behaves quasi-statically); assumed force distribution and complete models (when the dynamics of the structure is fully modeled). Shivakumar et al. (1985) developed spring-mass and energy-balance models to calculated impact force and duration during low-velocity impact of transversely isotropic circular plates. Energy-balance model is based on equilibrium between the kinetic energy of the impactor and the sum of strain energy due to contact, bending, and transverse shear and membrane deformation at maximum deflection. The solution obtained using numerical methods gives only the maximum force so a spring-mass model is developed to describe the force history. The impactor and sandwich were represented by two rigid masses and their deformations were represented by springs (the general case includes four springs: two to represent linear stiffness of the structure (both bending and shear), one for the nonlinear membrane stiffness and the forth for the nonlinear contact stiffness). The analyses were verified by experimental results. Contact problems for finite thickness layers can be solved using numerical methods. Sankar and Sun (1983) used two types of numerical methods, point matching technique and assumed stress distribution method. The point matching method is essentially a numerical technique to solve the integral equations of the contact problem. However, this method fails when the contact area is too small because of numerical difficulties. In the assumed contact stress distribution method, the contact stresses are

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24 assumed to be of Hertzian form, i.e., similar to that of contact of a half-plane. The contact stresses take the shape of a semi-ellipse. A contact length is assumed and the contact stresses are expressed in terms of only one unknown, the peak contact stress. Requiring that the deflections beneath the contact region match the indentor profile one can solve for the peak contact stress. Another popular method for beams was to use a Greens function and integral equation to correlate the applied force and the beam response. Sankar (1989) superimposed and orthotropic half-space solution on a beam solution to construct an approximate Greens function. To avoid the limitation of previous method, Wu and Yen (1994) developed an analytical method, based on exact Greens function derived from three-dimensional linear anisotropic elasticity theory. Effects of changing the material properties, stacking sequence, span and thickness of the plate and indentor size on the force-indentation relation are studied. Anderson (2003) presented an analytical three-dimensional elasticity solution for a sandwich plate with functionally graded core subjected to transverse loading by a rigid spherical indenter. Governing equations are derived based on Reissners functional and are solved by enforcing continuity of tractions and displacement between the adjacent layers. The sphere unknown contact area and pressure distribution are obtained based on iterative solution method (developed by Singh and Paul, 1974 and modified by Chen and Frederick, 1992 ): a large initial contact region is divided into a number of rectangular patches with assumed constant pressure. The unknown pressure is discretized based on conforming the contact region to the indentor surface. The contact region is also determined on an iterative process: tensile patches are removed from the contact region

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25 and the procedure is repeated until only compressive pressure patches remain. Results of this analysis demonstrate that interfacial normal and transverse shear stresses are not necessary reduced by the incorporation of functionally graded core. Abot et al. (2002) presented a combined experimental and analytical study of the contact behavior of sandwich structures with foam cores. The experimental results were used to model the load-deflection curves that include a linear range (derived from Winkler foundation theory) followed by a nonlinear portion (derived from fitting experiments). Sburlati (2002) determined the indentation produced by rigid indentors (flat punch and spherical) on sandwich plate with high-density closed cellular foam based on a three-layer classical sandwich theory and a distribution of surface pressure reproducing the contact law. The relevant influence of boundary condition on the elastic response of the sandwich plate is shown (a 20% difference in contact laws for simply supported plate and clamped plate was found). Another conclusion of the paper is that the assumption of distribution of surface pressure reproducing the contact of a rigid indentor over a half-space is correct also for a sandwich beam. McCormack et al. (2001) estimated the initial failure load, corresponding to the first deviation from linearity in the load-deflection curves as well as peak load for several failure modes (face yielding, face wrinkling, core yielding and core indentation) of sandwich beam with metallic foam cores. They used finite-element method to compute the critical load in the core indentation case. The results agree with three-point bending experimental results.

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26 Liao (2001) investigated the smooth contact between a rigid cylindrical indentor and a composite laminate with arbitrary ply-orientation (homogenized as an orthotropic medium). An exact Greens function for the surface displacement of an anisotropic beam under cylindrical bending is derived. By matching the displacements of the top surface of the plate and those of the rigid indentor within the contact region, the contact stress distribution and the indentation is obtained. Swanson (2004) calculated the core compressive stress in a sandwich beam with orthotropic face sheets based on elasticity equations for transverse loading of layered orthotropic materials. By systematically varying the contact pressure and comparing the computed surface displacements with the indentor profile the contact pressure distribution was determined. The results showed that the pressure distribution for an orthotropic half-space is applicable to a sandwich beam over a large range of variables. Petras and Sutcliffe (1999) defined a spreading length to consider the way in which different wavelengths of sinusoidal pressure loading on top face sheet are transmitted to the core. This length characterizes the length over which a load is spread out by the face sheet. For a very flexible skin there is a large deformation under the load which can lead to core failure. For very rigid skins indentation failure will be relative hard, as the skins spread out the load. Their solution is obtained based on a combination between the higher order theory developed by Frostig et al. (1992) and contact overlapping triangular elements developed by Johnson (1985) Anderson and Madenci (2000) conducted impact tests in order to characterize the type and extent of the damage observed in a variety of sandwich configurations with graphite/epoxy face sheets and foam or honeycomb cores. They found that the surfaces of

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27 both the honeycomb and foam samples revealed very little damage at levels of impact energy that produced significant core damage. As the impact energy was increased, the samples experienced one of two types of damage: a tear or crack from the center of the laminate to the edge, or significant damage consisting of a dent localized in the region of impact. Wang and Yew (1989) studied the damage produced by low velocity impact in graphite fiber composite plates based on the principle of virtual work. They took into account the time history of impact force, the dynamic deformation of the system and the distribution of damage in the target plate. The material was considered damaged when its designed strength was reduced by the failure of the constituents (matrix cracking, fiber breakages or/and delamination). It was demonstrated that fibers in the consecutive layers must be at an angle with respect to each other in order to achieve an effective containment of the impact induced damage. In order to give a research perspective to the work on impact on sandwich plates with functionally graded core a literature review was presented. Among the large number of analytical models presented in the literature, the models reviewed here can be summarized as: plate on elastic foundation ( Abrate, 1998 ; Lee and Tsotsis, 2000 ), laminate plates ( Yang and Sun, 1982 ; Tan and Sun, 1985 ) and sandwich plates (e.g. Ambur and Cruz, 1995 ; Sun and Wu, 1991 ), non-hertzian contact/numerical methods (e.g. Sankar and Sun, 1983 ) and experiments on impact damage (e.g., Anderson and Madenci, 2000 ). Two numerical methods and a finite element solution for impact on sandwich plates with functionally graded core are presented in Chapter 3.

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28 Cellular Solids and Functionally Graded Materials Cellular solids appear widely in nature in different shapes and with different functions. They can be found in leaves and stalks of plants, in corals and woods or in any types of bones. They can be made of closed cells, regular or non-regular, or can be made of open network of struts. They can be isotropic or can have cells oriented in a particular direction. Recently, engineered cellular materials are made using polymers, metals, and ceramics and it is believed that any solids can be foamed. They are widely used as thermal, and acoustic insulations, as absorbers of kinetic energy from impacts. The cellular/foamed materials are qualitatively and quantitatively investigated in a large number of publications. Among those, two books worth to be mentioned: Ashby et al (2000) design guide for metal foams (contains processing techniques, characterization methods, design and applications) and Gibson and Ashby (1997) monograph of cellular solids. Functionally graded materials (FGMs) are materials or structures in which the material properties vary with location in such a way as to optimize some function of the overall FGM. Nature provides many examples of functionally graded materials. In many of the cases the nature functionally graded structures were evolved based on some mechanical function: bones give a light, stiff frame to the body, wood support the tree under environmental loadings, leaves transport fluids. Bamboo is one of the examples of a structurally smart plant ( Amada et. al. 1997, 2001 ). Bamboo structure, which resembles that of a unidirectional, fiber-reinforced composite, is described by a macroscopically graded geometry that is adapted to environmental wind loads while the fiber distribution exhibits a microscopically graded architecture, which leads to smart properties of

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29 bamboo. Amada et al. (1997, 2001) demonstrated experimentally and analytically how the functionally graded microstructure of a bamboo has been optimized through evolution to maximize its load-bearing capabilities under severe environmental loading conditions. An interesting application of FGMs is dental implants ( Watari et al, 1997 ). It is confirmed that the fabrication technique can be used to produce controlled functionally graded microstructures, and that the biocompatibility of the titanium/ hydroxyapatite implants is superior to that of pure titanium implants. The variation of the properties of FGMs can be achieved by gradually changing the volume fraction of the constituents materials, by using reinforcements with different properties, sizes and shapes, by interchanging the roles of reinforcement and matrix phases in a continuous manner as well as by adjusting the cell structure or the material density. The result is a microstructure that produces continuously or discretely changing thermal and mechanical properties at the macroscopic or continuum scale. The concept of functionally graded materials was proposed in the eighties by materials scientists in Japan as a way to create thermal barrier materials. Koizumi (1997) summarize the first projects in this field. The idea proposed was to combine in a gradual manner heat-resistance ceramics and tough metals with high thermal conductivity. The result is a panel with a high heat-resistance on the high-temperature side and high mechanical strength on the other side. The new materials were obtained by four methods: Chemical Vapor Deposition (CVD), powder metallurgy, plasma sprays, and self-propagating combustion synthesis (SHS). As the use of FGMs increases, for example, in aerospace, automotive, telecommunications and biomedical applications, new methodologies have to be

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30 developed to characterize FGMs, and also to design and analyze structural components made of these materials. Although fabrication technology of FGMs is in its infancy, they offer many advantages. Few of the manufacturing methods are presented here. Fukui et al. (1991, 1997) developed a high-speed centrifugal casting method, in which the layers are formed in the radial direction due to different mass densities. An melted Al-Ni alloy cast into a thick-walled tube was rotated at a speed such that the molten metal experienced an acceleration, thereby producing two kinds of composition gradient, (i.e. phase gradient). The volume fraction of the A13Ni phase was determined by quantitative optical microscopy. Poly (ethylene co-carbon monoxide) (ECO) was chosen to make the FGMs by ultraviolet radiations ( Lambros et. al., 1999 ), because of its rapid degradation under UV light. Irradiated ECO becomes stiffer, stronger and more brittle with increasing time. The exposure time and, thus, the properties or the FGM can be adjusted truly continuous based on the relationship between modulus of elasticity and irradiation time. Another manufacturing method is electrophoretic deposition ( Sarkar, 1997 ) EPD is a combination of two processes, electrophoresis and deposition. Electrophoresis is the motion of charged particles in a suspension under the influence of an electric field gradient. The second process is deposition, i.e. the coagulation of particles into a dense mass. El-Hadek and Tippur (2003) developed functionally graded syntactic foam sheets by dispersing micro balloons (with linear graded volume fraction) in epoxy. They determined Youngs modulus and the density by using the wave speed and density for syntactic foams having homogenous dispersion of the micro balloons. The resulting foam

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31 sheets have a nearly constant Poisons ratio. Other FGM manufacturing methods presented in literature are powder metallurgy, plasma sprays and self-propagating combustion synthesis. Suresh and Mortensen (1998) provide an excellent introduction to FGMs. Suresh (2001) presents a motivation for the use of graded materials and also enumerates few challenges in the field. Literature contains many analytical models: mechanical and thermal loads, fracture and optimization solutions for different structures made of FGM were published. Here, few of the significant papers are reviewed. When using analytical methods to solve problems involving FGMs, a functional form for the variation of thermo-elastic constants has to be assumed. For example, Aboudi et al. (1995a, 1995b, 1999) assumed a simple polynomial approximation for the elastic constants. Another useful approximation is the exponential variation, where the elastic constants vary according to formulas of the type 0zijijcce Many researchers have found this functional form of property variation to be convenient in solving elasticity problems. For example, Delale and Erdogan (1983) derived the crack-tip stress fields for an inhomogeneous cracked body with constant Poisson ratio and a shear modulus variation given by ()0xye Giannakopoulos and Suresh (1997a, 1997b) present two analytical models for indentation of a half-space with gradient in elastic properties, for a point force and an axisymmetric indentor. The analytical solutions, based on elasticity methods for axisymmetric equilibrium problems are compared with finite element simulations. It is found that the influence of the Poissons ratio is strong whenever the elastic modulus is increasing with depth, and is weak whenever the elastic modulus is decreasing with

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32 depth. Also a decreasing elastic modulus with depth results in a spreading of stresses towards the surface rather than to the interior, whereas an increasing elastic modulus results in diffusing the stresses towards the interior of the half-space. Markworth et al. (1995) review analytical models for microstructure-dependent thermo physical properties, and models for design, processing of FGMs. They also present few recommendations relative to areas in which additional work is needed. Functionally graded materials as composite materials are generally described from two points of view: micromechanics and macromechanics. Micromechanics analysis recognizes the basic constituents of FGM but does not consider the internal structure of the constituents. Micro-constituents are treated as homogeneous continua. Although the material at the macro-scale is heterogeneous, it is assumed that the micro-heterogeneities are distributed in such a way such that the material volumes beyond some representative minimum have comparable macroscopic or overall properties. The hypothesis concerning the existence of an representative volume element (RVE) is fundamental: it means that there exist a relatively small sample of the heterogeneous material that is structurally typical of the whole mixture on average and contains sufficient numbers of inclusions, defects, pores such that the overall behavior (the behavior of an equivalent homogeneous material) would not depend on the surface values of constraints and loading as long as those are homogeneous. Properties are prescribed pointwise in the spirit of the mechanics of continua. The transition from the micro to the macroscale consists in finding the relationship between the macroscopic fields (stress, strain) and the volume averages of the same micro fields over RVEs. In mixture-type Continuum-Macromechanics analyses, the material properties are expressed in terms of volume fraction and individual

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33 properties of constituent phases. Macromechanics models can be subdivided in: direct (closed-form analytical solution for the averaged properties), variational (lower and upper bound for the overall properties in terms of the phase volume fraction) and approximation methods. Aboudi et al. (1994a, 1994b 1999) developed a higher order (HOTFGM), temperature-dependent micro-mechanical theory for thermo-elastic response of functionally graded composites. A higher-order representation of the temperature and displacement fields is necessary in order to capture the local effects created by the thermomechanical field gradients, the microstructure of the composite and the finite dimensions in the functionally graded directions. Their model based on the concept of a representative volume element explicitly couples local (microstructural) and global (macrostructural) effects. RVE is not unique in the presence of continuously changing properties due to non-uniform inclusion spacing. Pindera and Dunn (1995) evaluated the higher order theory by performing a detailed finite element analysis of the FGM. They found that the HOTFGM results agreed well with the FE results. Abid Mian and Spencer (1998) present an exact solution for 3D elasticity equations for isotropic linearly elastic, inhomogeneous materials. Lame elastic moduli and the expansion coefficient and the temperature T depend in an arbitrary specified manner on the coordinate z. The 3D elasticity solutions are generalized from solutions for stretching and bending of symmetrically inhomogeneous plates. In order to completely describe these solutions an equivalent plate is introduced. This equivalent plate is a homogeneous plate (in the sense of a classical thin plate) with the same overall geometry as the inhomogeneous plate and elastic moduli that are suitable weighted averages of the

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34 moduli of the inhomogeneous plate. It is shown that the exact three-dimensional solutions are generated by two-dimensional solutions of the thin-plate equations for the equivalent plate. Reddy et al. (2000, 2001) use an asymptotic method to determine three-dimensional thermomechanical deformations of FG rectangular plates. The locally material properties are estimated by the Mori-Tanaka homogenization scheme for which the volume fraction of the ceramic phase is of the power law type. Reddy et al. (2003) developed an FGM beam finite element by deriving the approximation functions from the exact general solution to the static part of the governing equations. These solutions are then used to construct accurate shape functions which result in exact stiffness matrix and a mass matrix that captures mass distribution more accurately compared to any other existing beam finite elements. Thus, the element is an efficient tool for modeling structural systems to study wave propagation phenomena that results due to high frequency and low duration forcing (impact loading). Rooney and Ferrari (1999) developed solutions for tension, bending and flexure of an isotropic prismatic bar with elastic moduli varying across the cross-section. It is demonstrated that the elastic moduli are convex functions of the volume fraction. If a structural member is required to be just in tension, any grading of the phases will result in a improved in the performance. Vel and Batra (2002, 2003) present an exact solution for simply supported functionally graded rectangular thick or thin plates. The material has a power-law through-the-thickness variation of the volume fractions of the constituents. Suitable displacement functions that identically satisfy boundary conditions are used to reduce

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35 equations governing steady state vibrations of a plate to a set of coupled ordinary differential equations, which are then solved by employing the power series method. Woo and Meguid (2000) provide an analytical solution for the coupled large deflection of plates and shallow shells made of FGMs under transverse mechanical load and temperature field. The material properties of the shell are assumed to vary continuously through the thickness of the shell, according to a power law of volume fraction of the constituents. The equations obtained using von Karman theory for large transverse deflection, are solved by Fourier series method. In a series of papers Sankar and his coworkers ( Sankar and Tzeng, 2002 ; Sankar, 2001 ; Venkataraman and Sankar, 2001 ; Apetre, et al. 2002 ) reported analytical methods for the thermo-mechanical and contact analysis of FG beams and also sandwich beams with FG cores. In these studies the thermo-mechanical properties of the FGM were assumed to vary through the thickness in an exponential fashion, e.g., E(z)=E 0 e z The material was assumed to be isotropic and the Poissons ratio was assumed to be constant. The exponential variation of elastic stiffness coefficients allows exact elasticity solution via Fourier transform methods. Later Apetre et al. (2003) and Zhu and Sankar (2004) used Galerkin method to analyze cores with polynomial variation of mechanical properties. Although a relatively new field, the study of functionally graded materials contains a large number of research areas: publications on fabrication techniques, experiments methods, analytical and finite element solutions were widely published later. Developments in manufacturing methods now allow controlled spatial variation in material properties so analytical results can be validated by comparisons with

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36 experimental results. In order to give a research perspective to the present work this section summarized few of the developments in this area. The main objective of present work is to develop analytical methods for sandwich structure with FG cores and to validate them by comparison with finite element solutions. Objectives and Approaches Learning from nature, engineers created new manufacturing methods to produce functionally graded materials. Few of them are presented in previous section and are a good research motivation for the present work. Analytical methods have to be developed in order to enable obtaining closed-form solutions to some fundamental solid mechanics problems, and also to extend finite element models to structures made of FGMs. Impact surfaces and interfaces between different materials where impact damage occur are regions of interest. The main objectives of this dissertation can be summarized as: 1. Develop analytical models for sandwich structures with functionally graded core. 2. Solve contact and impact problems involving sandwich structures with FG core and compare the trade-off between using a functionally graded core as opposed to the conventional sandwich design. 3. Compare the trade-off between the total mass and stiffness in functionally graded materials and homogenous materials by solving optimizations problems. Even with the sandwich structure advantages over the other structures, it is important to develop new types of materials in order to obtain the absolute minimum weight for given conditions (e.g., structural geometry and loadings). These new sandwiches should be compared with other sandwich constructions and with alternative structures in order to select the best configuration. One of the new alternatives is a sandwich structure with functionally graded core. A sandwich beam with soft,

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37 transversely flexible core must be approached with a higher-order theory rather than the classical theory. The core gets compressed and the thickness of the beam is changed under concentrated loads. Plane sections do not remain plane after deformation, so displacement field is not a linear function of the thickness coordinate. Conditions imposed on one face sheet do not necessary hold for the other. These are few of the reasons that make higher-order theory necessary for a sandwich panel with FG core even if this theory involves more computational effort. This research presents analytical solutions based on different models for sandwich structures with FG core when the core Youngs modulus is given by exponential and polynomial variation. A finite element model validates analytical solutions. Low-velocity impact on sandwich structures with FG core based on springs model is also solved. A comparison of the total mass of different panels based on different relations between Youngs modulus and density is present. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.

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CHAPTER 2 ANALYTICAL MODELING OF SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE Although FGMs are highly heterogeneous, it will be useful to idealize them as continua with properties that change smoothly with respect to spatial coordinates. This will enable closed-form solutions for some fundamental solid mechanics problems, and will aid the development of finite element models for structures made of FGMs. This chapter investigates different analytical models available in literature for a sandwich beam and applies them to a sandwich beam with functionally graded core. In the first two sections, the governing equations for sandwich structures with FG cores are solved for two types of core Young Modulus by two different methods: exact solutions are presented for the exponential variation of core Young Modulus and a combination of Fourier series and Galerkin method for a polynomial variation of core Young Modulus. Those methods are compared with two equivalent single-layer theories based on assumed displacements, a higher-order theory and a finite-element analysis. A very good agreement among the Fourier series-Galerkin method, the higher-order theory and the finite-element analysis is found. Exact Method for Sandwich Structures with Functionally Graded Core Venkataraman and Sankar (2001) derived an elasticity solution for stresses in a sandwich beam with a functionally graded core. They used Euler-Bernoulli beam theory for analysis of face sheets and plane elasticity equations for the core. In the present work, 38

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39 the solution of the sandwich problem was improved by using elasticity equation for face sheets also. The dimensions of the sandwich beam are shown in Figure 2.1.The length of the beam is L, the core thickness is h and the face sheet thicknesses are h f The beam is divided into 4 parts or elements: the top face sheet, top half of the core, bottom half of the core and the bottom face sheet. Element 3: Bottom half of sandwich coreElement 1: Top facesheetElement 4: Bottom facesheetElement 2: Top half of sandwich core xz Node (1)Node 3Node (2) hfLh Element 3: Bottom half of sandwich coreElement 1: Top facesheetElement 4: Bottom facesheetElement 2: Top half of sandwich core xz xz Node (1)Node 3Node (2) hfLh Figure 2.1: Sandwich beam with functionally graded core with schematic of the analysis elements. In general, this model can be applied for sandwich structures with core and face sheets orthotropic materials at every point and the principal material directions coincide with the x and z-axes. Consequently, the constitutive relations for each layer are: 4321 element, for ,00005533131311,,,iicccccthixzxxxxiixzxxxx (1.17) or )(zc (1.18) The face sheets are assumed to be homogeneous and isotropic. The core is functionally graded but symmetric about the mid-plane given by z=0. The elastic coefficients (c ij ) of the core are assumed to vary according to:

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40 zijijecc0 (1.19) This exponential variation of elastic stiffness coefficients allows exact elasticity solution. The tractions and displacements at the interface between each element are shown in Figure 2.2. Each element has its own coordinate systems. The coordinate systems of each element are chosen at the interface because displacements and traction compatibility between elements will have to be enforced at these nodes. x z To p face sheet p a p 2 W 2 p 2 W 2 t 3 U 3 x z Top half of sandwich t 4 U 4 x z Bottom face sheet t 3 U 3 p 3 ,W 3 p 3 ,W 3 z x t 4 ,U 4 p 4 ,W 4 p 4 W 4 t 1 ,U 1 p 1 W 1 t 2 ,U 2 Bottom half of sandwich core t 5 ,U 5 p 5 W 5 Figure 2.2: Traction forces and displacements at the interfaces of each element in the FGM sandwich beam The governing equations are formulated separately for each element, and compatibility of displacements and continuity of tractions are enforced at each interface (node) to obtain the displacement and stress field in the sandwich beam. This procedure is analogous to assembling element stiffness matrices to obtain global stiffness matrix in finite element analysis. The top face sheet is subjected to normal tractions such that:

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41 )sin()0,(xpxazz (1.20) where ,....5,3,1 ,nLn (1.21) and p a is known. Since n is assumed to be odd, the loading is symmetric about the center of the beam. The loading given by equation (1.20) is of practical significance because any arbitrary loading can be expressed as a Fourier series involving terms of the same type. The displacement field for each layer is assumed of the form: )sin()(),(4,3,2,1cos)(),(xzWzxwixzUzxuiiii (1.22) where u is horizontal displacement and w is vertical displacement and where it is assumed that: )exp(),(),(zbaWUiiiii (1.23) where a i b i i are constants to be determined. In order to satisfy equilibrium the contributions of the different tractions at each interface should sum to zero. Enforcing the force balance and the compatibility of force and displacements at the interfaces enables us to assemble the stiffness matrices of the four elements to obtain a global stiffness matrix K: 1122334455000000000TTaUWUWUWUWUWpK (1.24) where K is the global stiffness matrix, a summation of the stiffness matrices for each element. The displacements U 1 W 1 W 5 are obtained by solving (1.24). The obtained displacement field along with the constitutive relations is used to obtain the stress field in each element.

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42 Fourier Galerkin Method for Sandwich Structures with Functionally Graded Core Zhu and Sankar (2004) derived an analytical model for a FG beam with Youngs modulus expressed as a polynomial in thickness coordinate using a combined Fourier Series-Galerkin method. In the present work, the model is applied to a sandwich beam with FG core. The geometry (Figure 2.1), the load (1.20), and the constitutive relations (1.17) are the same as in previous model. In this section, a brief description of the procedures to obtain the stiffness matrix of top half of the core is provided. The derivation of stiffness matrices of other elements follows the same procedures. The differential equations of equilibrium for the top half of the core are: 00 zxzxzzxzxzxx (1.25) The variation of Youngs modulus, E in the thickness direction is given by a polynomial in z. e.g., 1)(42332410hzahzahzahzaEzE (1.26) where E 0 is the Youngs modulus at z=0 and a 1 a 2 a 3 and a 4 are material constants. The thickness in y direction is large and plain strain assumption can be used. The elasticity matrix [C] is related to the material constants by:

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43 10()10(1)(12)12002Ez C (1.27) The following assumptions are made for displacements: xzWzxwxzUzxu sin)(),(cos)(),( (1.28) Substituting equation (1.28) into (1.27), the following constitutive relation is obtained: (1.29) xWUxWxUGccccxzzzxxcos'sin'sin000033131311 A prime () after a variable denotes differentiation with respect to z. Boundary conditions of the beam at x=0 and x=L are w(0,z)=w(L,z)=0, and 0),(),0( zLzxxxx which corresponds to simple support conditions in the context of beam theory. Equations (1.29) can be written as: xTxSSzxzzxzzxxcossin (1.30) where )'('33131311WUGzTWUccccSSzx (1.31)

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44 Substituting for xzzzxx ,, from equations (1.31) into equilibrium equations (1.25), a set of ordinary differential equations in U(z) and W(z) are obtained: 0'0'zzzxTSTS (1.32) In order to solve equations (1.32) the Galerkin Method is employed: solutions in the form of polynomials in z are assumed: )()()()()()()()()()()()(zbzbzbzbzbzWzczczczczczU55443322115544332211 (1.33) where s are basis functions, and bs and cs are coefficients to be determined. For simplicity 1, z, z 2 z 3 z 4 are chosen as basis functions: 453423211zzzzzzzzz)(;)(;)(;)(;)( (1.34) Substituting the approximate solution in the governing differential equations, the residuals are obtained. The residuals are minimized by equating their weighted averages to zero: 5,1,0)(5,1,0)('0'0idzzTSidzzTShizzhizx (1.35) Using integration by parts (1.35) can be rewritten as: '00'00()()(0)(0)0()()(0)(0)0,1,5hhixzizizihhziziziziSdzThhTTdzSdzTdzShhSi (1.36)

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45 Substituting for S x (z), S z (z) and T z (z) from equations (1.31) into (1.36) and using the approximate solution for U(z) and W(z) in (1.33) it is obtained: )2()1()4()3()2()1(iiijijijijffcbKKKK (1.37) or (1)(2)iibfcfK (1.38) where: (1.39) )()()()()()()()(,)()('')('')('')('')(hShSfhThTfdzGdzcKidzcdzGKdzcdzGKdzGdzcKziziiziziihjijihijjihjihijhjijhiijjhijhiij000051210013403302301120200131 (1.40) 5432154321cccccbbbbbcbT Let U 2 W 2 U 3 and W 3 be the displacements at top and bottom surface of top half of the element (top half of the core). Evaluating the expressions for U(z) and W(z) at the top and bottom surfaces and equating them to the surface displacements results in the expression:

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46 23422342330000011000000000100001000000000UhhhhWbhhhhUcW 0 (1.41) This can be compactly expressed as: 212335......UbWUWcA (1.42) The tractions T 2 P 2 T 3 and P 3 acting on the surface can be related to the functions f i as follows: 332255443322115544332211)2(5)2(4)2(3)2(2)2(1)1(4)1(4)1(3)1(2)1(1000000000000000000000000000000STSThhhhhhhhhhffffffffff (1.43) or (1)2123(2)35......TfSTSfB (1.44) From (1.37), (1.42) and (1.44) follows:

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47 211121322322UTTWSSUTWSS*AKBK 1T2 (1.45) Finally, the stiffness matrix of the top half of the FGM core [S (2) ] that relates the surface tractions to the surface displacements is obtained as: 221121332332UUTWWSUUTWWS*(2)KS (1.46) In order to satisfy equilibrium, the contributions of the different tractions at each interface should sum to zero. Enforcing the balance and the compatibility of force and displacements at the interfaces enables us to assemble the stiffness matrices of the four elements to obtain a global stiffness matrix [S]: 1122334455000000000TTaUWUWUWUWUWp S (1.47) The displacements U 1 W 1 W 5 are obtained by solving equation (1.47). The displacement field along with the constitutive relations is used to obtain the stress field in each element. Analytical Models for Sandwich Structures with Functionally Graded Core An Equivalent Single-Layer First-Order Shear Deformation Theory The simplest model investigated in the present study is an equivalent single-layer model that includes a transverse shear deformation (First-order Shear Deformation theory, FSDT). The following kinematics assumptions are made:

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48 )(),()()(),(00xwzxwxzxuzxu (1.48) where u is displacement in horizontal direction, x and w is displacement in vertical direction, z. u 0 w 0 are unknown functions to be determined using the equilibrium equations of the first order theory. The dimensions of the sandwich beam are shown in Figure 2.3. The length of the beam is L, the core thickness is h, the top face sheet thickness is h t and the bottom face sheet thickness is h b Figure 2.3: The beam geometry x z 0 h t h/2 h b h/2 z If the core material is orthotropic at every point and the principal material directions coincide with the x and z-axes, the plane strain constitutive relations are: 1113133355()()010()()()010(1)(12)00()12002xxxxxxzzzzzzxzxzxzczczEzczczcz (1.49) or ()zc (1.50) The variation of Youngs modulus E in the thickness direction is given by a polynomial in z. e.g.,

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49 1)(42332410hzahzahzahzaEzE (1.51) where E 0 is the Youngs modulus at z=0 and a 1 a 2 a 3 and a 4 are material constants. In order to calculate the flexural rigidity of the cross-section, the position of the neutral axis z 0 must be found. It is given by the coordinate system for which the first moment of area is zero when integrated over the entire cross-section: 02/)2/(11110)(00zdzzzcBzhhzhhtb (1.52) Equilibrium equations of the first order theory take the following form: 011110xxzduNAdxdMDdxdwVSdx (1.53) where A 11 is the axial rigidity, D 11 is the bending stiffness and S is the extensional stiffness: 000000/21111(/2)/221111(/2)/255(/2)()()()tbtbhhzhhzhhzhhzhzhzAczdDzczSczdz zdz (1.54) For a given set of external loads and boundary conditions, axial force resultant N x bending moment resultant M x and shear resultant V z can be calculated. Then using system (1.53) the displacements are obtained.

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50 This model is applied for a sandwich beam with functionally graded core. The main drawback of the model is given by the fact that the transverse shear strain is constant through the thickness of the beam. Results and discussions are presented in the last section. An Equivalent Single-Layer Third-Order Shear Deformation Theory Reddy (1984) developed a third-order Shear Deformation theory (TSDT) for composite laminates, based on assumed displacement fields and using the principle of virtual displacements. Reddy (1990) reviewed a number of other third-order theories and showed that they are special cases of his theory. Reddy (2000) and Reddy and Cheng (2001) expanded TSDT for analysis of functionally graded plates. Here, a third-order equivalent single-layer model based on Reddys assumption of vanishing of transverse shear stresses on the bounding planes is investigated. The displacement field is assumed to be: )(),()()()()(),(0320xwzxwxzxzxzxuzxu (1.55) where u 0 and w 0 are displacements along middle axis, z=0. Functions and are eliminated using the assumption of zero shear stress at top and bottom: dxdwxhxdxdwxhhhxbt0032 (1.56) where 222hhhhhhhhbtbt (1.57)

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51 If the top face sheet thickness h t equals the bottom face sheet thickness h b then (x)=0. Axial force resultant, bending moment resultant and shear resultant are calculated as follows: 222222,,,tbtbtbhhxxxhhhhxxxhhhhzxzhhNbxzdz M bzxzVbxzdz dz (1.58) where b is the beam width, dxdu00 and constants A N through C V are given by: 222311112222311222231111222323ttbbtbttbbhhhhtbNNhhhhhhtbNhhhhhhtbMMhhhhhhAbczdzBbczzzzdzhhhhCbczzzdzhhhhAbzczdzBbzczzzzh 2231122 323tbhhtbMhhdzhhhCbzczzzdzhh

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52 225522255221222tbtbhhtbVhhhhtbVhhhhBbczzzdzhhhhCbczzzdzhh (1.59) The following system is obtained: 0()()0NNNxMMMxVVzduxdxABCNxdxABCMxdx B CVdwxdx x (1.60) where c 1 and c 2 are constants to be determined from boundary conditions. For a given set of external loads and boundary conditions axial force resultant N x bending moment resultant M x and shear resultant V z can be calculated. Then using system (1.60) the displacements are obtained. This model is applied for a sandwich beam with functionally graded core. The main drawback of the model is given by the fact that the transverse shear strain has a quadratic variation with respect to the thickness coordinate. Results and discussions are presented in the last section. A Higher-Order Shear Deformation Theory Frostig et al. (1992) developed a higher-order theory for a sandwich beam with a transversely flexible core that uses a beam theory for the face-sheets and two-dimensional elasticity equations for the core. Swanson (1999) addressed details of implementation of Frostig model and presented solutions for several cases. The main feature of the method

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53 is the higher order displacement fields in the thickness coordinate: second order for vertical displacement and third order for the longitudinal displacement. Another advantage of this model, as a model based on variational principle is that the boundary conditions are obtained uniquely as a part of derivation. The main difference between this model and the previous one is that now, the higher order displacements are derived and not assumed. The equations developed by Frostig et al. (1992) and modified for a functionally graded core are briefly presented here. The dimensions of the sandwich beam and the coordinate systems are shown in Figure 2.4. The length of the beam is L, the core thickness is h, the top face sheet thickness is h t and the bottom face sheet thickness is h b h b x h z b x u ob z w c x u c z t w t h t Figure 2.4: The beam geometry Constitutive relations (which assume isotropic elastic behavior) for the face-sheets and for the transversely flexible core (i.e. zero longitudinal stress) are given by: 110,110,11,11,,Top face-sheet: Bottom face-sheet: Core: ,,(1),,(1)(12)ttbbxxtxxxbxttbbxxtxxxxbxxcczzczNAuNAu M DwMDwxzGzxzEzxzwxz (1.61)

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54 where are the resultant axial forces in the face-sheets, are the bending moments in the face-sheets, are axial and respectively flexural rigidities for the face-sheets, u xxiN xxiM iiDA1111 and 0i w i are face sheets longitudinal and vertical displacements at the centroid (i=t for top face-sheet and i=b for bottom face-sheet): 2/2/112112/2/1111 )( )(iiiihhiihhiidzzczDdzzcA (1.62) Linear variation for Youngs modulus and shear modulus are assumed: 11()()EzazbGzazb (1.63) Governing differential equations, boundary conditions and continuity conditions are derived based on variational principle. Using the equilibrium equations for the core: 00,,,, xxzzzzxzxx (1.64) the compatibility of the displacement at the core-face-sheets interfaces and the assumption that for a transversely flexible core (i.e. made of a material with much lower modulus relative to the face-sheets), the shear stress is nearly constant through the core the following core displacement fields are derived:

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55 21,,11,,,,,,,11,ln122ln1ln1ln1,ln1tcxxtxbxtxxxcxbxtxxxthauxzxzxzwxzuababazzzhabwwxaahbazzhbwxzxwwxwaaahb ot (1.65) where u 0i and w i are longitudinal and vertical displacements of the centroid of each face-sheet (i=t for top face-sheet and i=b for bottom face-sheet); is core Poissons ratio. In the above expression a linear core Youngs modulus was assumed: E c (z)=az+b. Similar relations can be obtained for any Youngs modulus variation expressed by a differentiable function. Governing equations, written in terms of transverse displacements w t and w b of the top and bottom face-sheets and shear stress of the core, are: xnAxnAxxxxwxwxmxqxwxwxxwxmxqxwxwxxwbbttxxxxxxxxxbxxxtxbbtbxxxxxbxtttbxxxxxt11113,3,3,3,3,22,2,2,11,1,111 (1.66) where q i m i and n i are distributed pressures, moments and axial forces applied on top (i=t) and bottom (i=b) face-sheets and the coefficients are given by:

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56 11111121122232ln1ln1ln12ln1ln1ln1ln12ttbbthhbadadDdaaahhbbhhbadadDdhaaahhbbbahhhabh ahbahb 32133111111ln1 2ln1ln1ln111ln12ln1btbbahhhhhabaahhbbbahhhahabhhaaaabAAhb 311d (1.67) where d is the beam width. In order to obtain the homogeneous solution for (1.66), the following characteristic equation is derived: 41111422223342333330 (1.68) Denoting with j j = 1,,12, the roots of characteristic equation, the following rescaling of constants needs to be done in order to avoid numerical difficulties (given by the case when Real( j ) > 0 exp()jxLx ): Re()Re()() If Re()0solution = () If Re()0solution = jjjjjxjjLLxxjjLiceeiicecee j (1.69)

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57 Forstigs theory is clear but needs the above described rescaling of constants in order to avoid numerical difficulties. For a given set of external loads and boundary conditions the differential governing system (1.66) can be solved. Results and discussions are presented in the next section. Results for Sandwich Beam with Functionally Graded Core under a Distributed Load In this section includes two objectives: validate analytical methods for sandwich structure with FG core and use these methods to compare FG and homogenous cores. The results of the modified Venkataraman and Sankar (2001) and Zhu and Sankar (2004) methods were compared (Figures 2.5 2.9) and found to be identical. A one dimensional sandwich plate of length 0.2 m, core height 2010 -3 m and face-sheet thickness 0.310 -3 m is considered as an example to investigate the effects of varying core properties through the thickness. The face sheet elastic modulus E f is chosen as 10 GPa. The sandwich core modulus E 0 at the mid-plane is kept fixed at E f /1000 while the core modulus at the face sheet interface (E h ) is varied. The ratio of E h /E 0 varies from 1 to 100. When E h /E 0 =1, core properties are constant through the thickness, and hence the problem is identical to that of a conventional sandwich plate. The value of E h is gradually increased until it reaches the value of the face sheet, for E h /E 0 =100. The different profiles of the elastic modulus variations in the sandwich core are shown in Figure 2.5. They display the profile of Youngs modulus for FG materials with E h /E 0 =1 (a 1 =0, a 2 =0, a 3 =0, a 4 =0, a 5 =1), E h /E 0 =10 (a 1 = 3.9372, a 2 = -.94217, a 3 = 3.8705, a 4 = 2.1215, a 5 =1) and E h /E 0 =100 (a 1 =238.50, a 2 =-258.92, a 3 =132.18, a 4 -14.327, a 5 =1.6335), respectively. Poissons ratios for the core and face-sheet, respectively, were 0.35 and 0.25. The results

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58 are restricted to the case where the plate is loaded in the transverse direction by a sinusoidal load given by )/sin(Lxpa with p a =1. An interesting result from analysis of the FGM sandwich beam is the transverse shear stress in the sandwich core at the face-sheet/core interface. Conventional design of sandwich panels restricts the shear stress at the core/ face sheet interface to the bond (adhesive) shear strength, which is typically lower than the shear strength of the core material. Therefore, the core material is not fully utilized. It is hence desirable to reduce the interfacial shear stress while carrying a high shear stress in the core. It appears that this is possible with a functionally graded core. The shear stress variations in the core, given by both methods (exact and Fourier-Galerkin) are plotted in Figure 2.6. The interface shear stress reduces as E h /E 0 ratio is increased. Figure 2.5: Through-the-thickness variations of core modulus considered for the functionally graded sandwich panel (the load is ()sin(/)a p xpxL )

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59 Figure 2.6: Variation of transverse shear stresses (given by Fourier-Galerkin method) in the sandwich structure with FGM core for different ratios (E h /E 0 ) and E f /E 0 = 1000 at x = L/4 (the load is ()sin(/)a p xpxL ). The normal stress zz in the core (Figure 2.7) varies linearly and is independent of the variation in core properties. The value of normal stress varies linearly (approximately) from the applied surface load at the top core to zero at the bottom of the core. This is an interesting result, because it simplifies the calculations required in order to analyze the core-crushing problem. It must be noted that the example considered here, used a smoothly varying (sinusoidal) surface pressure load.

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60 Figure 2.7: Variation of zz stresses (given by Fourier-Galerkin method) in the sandwich structure with FGM core for different values of E h / E 0 and E f /E 0 = 1000 at x = L/4 (the load is ()sin(/)a p xpxL ). The bending stress variation (Figure 2.8) in the core is as expected. The linear variation in strains results in small levels of bending stress in the core near the mid-plane. The stresses increase near the face sheet. This is particularly pronounced as the value of the core modulus is increased to match the value of the face sheet thickness. The u-displacement variation through the thickness is linear (plane sections remain plane) when the gradient in the core properties is small. However, for highly graded cores significant warping of the cross section occurs.

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61 Figure 2.8: Variation of bending stresses (given by Fourier-Galerkin method) through the thickness of the FGM core for different values of E h /E 0 and E f /E 0 = 1000 at x = L/4 (the load is ()sin(/)a p xpxL ). Figure 2.9: Variation of u displacements (given by Fourier-Galerkin method) through the thickness of the sandwich structure with FGM core for different ratios (E h /E 0 ) and E f /E 0 = 1000 at x = L/4 (the load is ()sin(/)a p xpxL ).

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62 Next step is to compare the five models: the equivalent single-layer first-order and third-order shear deformation theories, Fourier series-Galerkin method, Frostig model and a finite-element model. A simply supported sandwich beam, with length L = 0.3 m, core thickness h = 2010 -3 m and face-sheets thickness h f = 0.310 -3 m is considered to investigate the effects of varying core properties through the thickness. The face-sheet Youngs modulus was chosen as 50 GPa. The face-sheets Poissons ratio is f = 0.25 and the core Poissons ratio is = 0.35. (a) E (z) E (z) p a sin ( x/L ) (b) Figure 2.10: (i) Non-dimensional core modulus and (ii) loading for: (a) symmetric core about the centerline and (b) asymmetric core about the centerline Two cases are considered: (a) symmetric core about the centerline under uniform distributed load p = 1 N/m 2 and (b) asymmetric core about the centerline under a distributed load given by ()sin() p xx L N/m 2 Figure 2.10 presents core elastic moduli: for (a), the core Youngs modulus E has a linear symmetric variation with respect to thickness coordinate, z: E = 50 MPa at the middle core and E = 5010 MPa at the core-top face-sheet interface (as well as at the core-bottom face-sheet interface); for

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63 (b), the core Youngs modulus E has a linear asymmetric variation with respect to thickness coordinate, z: E = 50 MPa at the core-bottom face-sheet interface and E=5010 MPa at the core-top face-sheet interface. Using the commercial finite element software ABAQUS a 2D finite element model was created to model problem (a). The FG core was partitioned through the thickness into twenty strips with constant properties. Four elements were considered through the thickness of each strip and two elements were considered through the thickness of the face-sheets. The elements considered were 2D, quadratic, plane strain elements. Boundary conditions assume (0,)(,)0wzwLz Figures 2.11 2.15 present a comparison for five models: the equivalent single-layer first-order and third order shear deformation theories, Fourier series-Galerkin method, Frostig model and an ABAQUS finite-element model. For problem (a), which is the symmetric core under a uniform distributed load, a very good agreement among the Fourier series-Galerkin method and the ABAQUS model was found. Because the core is symmetric about the mid-plane, the variation of displacements, strains and stresses are symmetric with respect to the thickness coordinate. Figure 2.11 presents deflections at bottom face sheet core interface. The FSDT and the TSDT beam are stiffer than the Galerkin beam.

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64 (a) (b) Figure 2.11: Comparison of deflections: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m 2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by ()sin() p xx L N/m 2 Comparison of longitudinal displacements in the core at the same cross-section (L/4) for the two cases are presented in Figures 2.12 (a) and (b). A perfect agreement was found between Galerkin model and the finite element model for (a) and between Galerkin model and the Frostig model for (b). For both problems FSDT gives a linear variation as expected. (a) (b) Figure 2.12: Comparison of longitudinal displacement in the core at x = L/4: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m 2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by ()sin() p xx L N/m 2

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65 Bending stress (Fig. 2.13 (a) and (b)) in the core at the same cross-section (x=L/4) is almost the same for all models. For the asymmetric core, a larger compression value is found at the top core where the load in applied and where Youngs modulus value is larger. (a) (b) Figure 2.13: Comparison of axial stress in the core at x = L/4: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m 2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by ()sin() p xx L N/m 2 Comparison of shear strain in the core at the same cross-section (L/4) is presented in Figure 2.14 (a) and (b). Fourier-Galerkin method, Frostig model and the finite-element solution present a 1/z type variation for the core shear strain. The first order shear deformation theory gives a constant shear strain while the third order shear deformation theory gives a quadratic shear strain with respect to the thickness coordinate. The same conclusion was reached for the shear stress in the core at the same cross-section (Figure 2.15): Fourier-Galerkin method, Frostig model and the finite-element solution present an almost constant shear stress whereas the first order shear deformation theory gives a linear shear stress while the third order shear deformation theory gives a

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66 third order shear stress with respect to the thickness coordinate. The equivalent single-layer theories are not accurate for shear because they are not based on two dimensional equilibrium equations (1.64). In order to obtain accuracy for shear (strain and stress) in single-layer theories (both first order and third order) the shear stress was obtained using equilibrium equations (1.64) for the core and the bending stress previously derived (Figure 2.13): ,,,00(,)(,)(zxxxxzzxzxxxxz ,0) x zxd x (1.70) Figure 2.15 includes both shear stresses: obtained from single-layer theories and obtained from equilibrium equations (1.70). The latest are identical with those obtained based on Fourier-Galerkin and finite element model. (a) (b) Figure 2.14: Comparison of shear strain in the core at x = L/4: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m 2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by ()sin() p xx L N/m 2

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67 (a) (b) Figure 2.15: Comparison of shear stress in the core at x = L/4: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m 2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by ()sin() p xx L N/m 2 Summary and Conclusions In this chapter, analytical models for sandwich structures with FG core are introduced, validated by comparison with finite element solutions and used to compare homogeneous and FG core sandwich plates. The study presents analytical models for sandwich structures with two types of core Youngs modulus variation through the thickness: exact solutions are presented for the exponential variation and a combination of Fourier series and Galerkin method for a polynomial variation of core Youngs modulus. Those methods are compared with models presented in literature and modified for this type of structure: two equivalent single-layer theories based on assumed displacements and a higher-order theory. A finite-element analysis is presented in order to validate the models. A very good agreement among the Fourier Galerkin method, the higher-order theory and the finite-element analysis is found.

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68 A comparison among homogenous and FG cores is presented: sandwiches with same geometry, same sinusoidal distributed load and different core properties are analyzed. As the core becomes stiffer at the core-top face-sheet interface, the interface shear stress reduces as E h /E 0 ratio is increased. The chapter deals only with distributed loads (uniform and sinusoidal). Next chapter introduces a more common load, the localized contact between a rigid indentor and a sandwich plate. The low-velocity impact solution is based on the solution developed here.

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CHAPTER 3 CONTACT AND LOW-VELOCITY IMPACT OF SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE One of the important problems in sandwich structures is damage due to low velocity impact. The interfacial shear stresses due to contact forces can be large enough to cause debonding of the face sheet from the core. Unlike for their solid metallic counterparts, predictions of the effects of low-velocity impact damage in sandwich structures are difficult because significant internal damage is achieved at impact energy levels lower than those required to create visible damage at the surface. One way of reducing the shear stresses is to use functionally graded core so that the abrupt change in stiffness between the face sheet and the core can be eliminated or minimized. The stresses that arise due to low-velocity impact can be easily understood by analyzing the static contact between the impactor and the structure ( Sankar and Sun, 1985 ). The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Youngs modulus is represented by a polynomial in the thickness coordinate, but the Poissons ratio is kept constant. In the previous chapter, the two-dimensional elasticity equations for the plane sandwich structure were solved using different methods. Here, the contact problem is solved using two methods: the assumed 69

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70 contact stress distribution method and method of point matching; and the results are compared with a finite element model. For the impact problem a simple dynamic model based on quasi-static behavior of the panel was used the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites. Contact problems for finite thickness layers can be solved using numerical methods. Sankar and Sun (1983) used two types of numerical methods, point matching technique and assumed stress distribution method. The main difference between the two methods is the way the contact load under the indenter is estimated: assumed stress distribution method is based on the assumption of a semi-elliptical contact stress distribution whereas the method of point matching attempts to capture the actual contact stress distribution approximating it as a superposition of several uniformly distributed loads. The point matching method is essentially a numerical technique to solve the integral equations of the contact problem. However this method fails when the contact area is too small because of numerical difficulties. In the assumed contact stress

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71 distribution method, the contact stresses are assumed to be of Hertzian form, i.e., similar to that of contact between a rigid cylinder and a half-plane. The contact stresses take the shape of a semi-ellipse. A contact length is assumed and the contact stresses are expressed in terms of only one unknown, the peak contact stress. Requiring that the deflections beneath the contact region match the indenter profile, one can solve for the peak contact stress. Here, the two methods are used for a sandwich beam with functionally graded core. Assumed Stress Distribution Method Previous chapter presented analytical models for sandwich structures with FG cores as a tool to be used for different problems. This section presents one of the possible applications of the above-discussed models. Indentation of a surface with an object (sharp, blunt or spherical) is a common engineering problem. If the properties of the material under the indentor are known, the analysis of the contact load versus indentation can provide valuable information about the contact damage. It is shown that spatially grading mechanical properties (e.g. Youngs modulus) it is possible to minimize the contact damage. Sankar and Sun (1985) solved the problem of smooth indentation of a beam by a rigid circular cylindrical indenter using the method of assumed stress distribution. The method is used here for indentation of a sandwich beam with functionally graded core. The indenter is a cylinder of radius R 0 and unit length. The contact length 2c is considered as known and the other quantities (stresses, deflection) are calculated for a given contact length. Indentation is defined as the difference in the vertical displacement of the indenter and the corresponding point on the bottom side of the beam. The

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72 dimensions of the sandwich beam are shown in Figure 3.1. The length of the beam is L, the core thickness is h and the face sheet thickness is h f 2h Figure 3.1: Dimensions of the sandwich panel and the contact length, 2c. The stress distribution under the indenter (Figure 3.2) is assumed to be of the semi-elliptical form: 22max1)(cxpxp (3.1) where p max is maximum value of the stress at center (unknown). pmax R0 c Figure 3.2: The stress distribution under the indenter. A similar stress distribution with some arbitrary p max, say p M is assumed. Writing p(x) in the form of a Fourier sine series, it is obtained: 1,3...()sinnnnxpxpL (3.2) where

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73 LcnJnnppMn12sin2 (3.3) and J 1 is the Bessel function of first order. The vertical displacements of the points in the contact zone are computed using the Fourier series Galerkin Method described in the previous chapter. From geometrical considerations (Figure 3.3), for contact lengths smaller in comparison with the radius of indenter, the following approximate relation is used: )(2)()()(20xRxxwxwx (3.4) where w(x 0 ) is vertical displacement at the middle point (see Figure 3.3). In (3.4) R' is the radius of curvature at x. It is found that R is almost constant (usually, more then 70% of R distribution lies within one standard deviation of the mean value). Then the average radius of curvature of the deformed top face sheet can be derived as: dxxRcRc0)(1 (3.5) R (x) x w(x 0 ) (x) w(x) Figure 3.3: Illustration of relation between w deflection and radius of curvature of the deformed surface.

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74 Generally R is different from the radius of the intender R 0 But, the displacements vary linearly with the load and hence the peak stress p max required producing a radius R 0 is given by: 0maxRRppM (3.6) Once p max is known, vertical displacements and the indentation can be calculated. Plots for several examples are presented and discussed in the Results section. Method of Point Matching The main difference between method of point matching and the method of assumed stress distribution is the assumption of the load. If, for the previous method the load is of a semi-elliptical form, here it is assumed that the stress distribution under the indenter is a superposition of a finite number of symmetrical rectangular loadings of unknown magnitudes (denoted by q j j=1,m) distributed over known lengths (denoted by x j j=1,m) as depicted in Figure 3.4. q j 2jcxjm x 0 Figure 3.4: Discretization of contact load for the method of point matching So, the j th uniform distributed load over the span 2x j is:

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75 0,0,22(),,22220,,22jjjjjxLx j x xLLpxqxxLxL (3.7) Using Fourier series the above load can be written as: 1()sinNjjkkk p xpL x (3.8) where 4sinsin22jjkqkk j p xkL (3.9) For each coefficient jk p the vertical displacements of the points in the contact zone are computed using the Fourier series Galerkin Method described in the previous chapter: is the vertical displacement of j jiw th reference point due to i th uniform distribute load of unit magnitude; is the vertical displacement of mid point reference point due to i 0iw th uniform distribute load of unit magnitude; is the vertical displacement of j jv th reference point due to indentation; is the vertical displacement of mid point due to indentation. Using the same geometrical relation described by (3.4) and writing the displacements at each point as superposition of displacements due to each rectangular load, it is obtained: 0v 00011()()miiimjjjiivxvwqvxvwq i (3.10)

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76 Then, the following algebraic system is obtained: 201,1,2mjijiiixwwqjmR (3.11) From (3.11) the unknown load magnitudes q i are obtained. The total load is obtained as a summation of all rectangular loadings over the contact area. This method was applied for a sandwich beam with functionally graded core and the results are presented in the Results Section. Quasi-Static Impact of a Sandwich Beam with FG Core After the static contact problem was solved, the method is applied to the problem of low-velocity impact of functionally graded sandwich panels. Solving the static contact problem first and combining the solution with the dynamic response of the sandwich panel obtained via simple spring-mass models (quasi-static assumption) accomplish this. The use of static load-deflection behavior of the sandwich beam in the impact analysis needs some justification. In general the wave propagation effects, especially through the thickness of the core, should be considered in impact response of sandwich panels. This will be crucial when spalling type damage occurs in the panels. However, a study by Sankar (1992) showed that for very large impactor mass compared to that of the target plate and for very low impact velocities compared to the wave velocity in the target medium, quasi-static assumptions yield sufficiently accurate results for impact force history and ensuing stresses in the impacted plate. Because the problem deals with the low-velocity impact (applied force vary slowly in time, therefore the system response is relatively slow so that the inertial terms in the equations of motion can be neglected; the structure responds quasistatically), the sandwich beam is modeled as a combination of two springs ( Shivakumar et al. 1985 ), a

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77 linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects as depicted in Figure 3.5. Also, the problem assumes only a rigid indentor (wave effects in the impactor can be neglected) with negligible dynamics. F ki nonlinear ks linear Figure 3.5: Low-velocity impact model Using the numerical results from the contact problem the spring constants k i and k s and the exponent n are determined such that: nikF (3.12) bswkF (3.13) where F is the total load, is the core indentation, w b is the vertical displacement of the core at the at bottom face sheet interface. The displacement of the impactor is calculated as the sum of indentation depth (core compression) and the global deflection of the sandwich beam: nisbkFkFww/1 (3.14) The work done by the impactor during the impact event can be expressed as: ninsFnisFwkFnkFdFkFkFFwwdFFwFdwW/1/1120/1002112 (3.15)

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78 The kinetic energy stored in the target is assumed negligible. The maximum deflection is reached when the velocity of the projectile becomes zero. At that time, the initial kinetic energy of the projectile has been converted into two parts: strain energy stored into the target by global (bending and shear) deformation and the energy used to create local deformations in the contact region. Hence, equating the impactor kinetic energy to the work done or the strain energy stored in the springs, the maximum contact force can be calculated from: 20/1/11max22112maxmvknFkFnins (3.16) where m and v 0 are, respectively, the mass and impact velocity of the impactor. The results of the static contact problem are used to determine the constants that described the stiffness and compression of FG sandwich beam ((3.12) and (3.13)). Using these material properties and the quasi-static model the maximum contact force in the case of low-velocity impact of FG sandwich beam is determined. Using this maximum value the maximum normal and shear strains in the core were determined and compared. Results for several examples are presented in the next section. Results A sandwich beam, with length L = 0.2 m, core thickness h = 2010 -3 m and face sheet thickness h f = 0.310 -3 m is considered to investigate the effects of varying core properties through the thickness. The face-sheet Youngs modulus was chosen as 50 GPa. Although these methods can be applied to a general form of E, in the present work, for core Youngs modulus, two cases are considered (Figure 3.6): linear symmetric about midplane and linear asymmetric. The variation of E with respect to z for the two cases is given below and also depicted in Figure 3.6.

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79 hhzhzEEEEEsymsymsymhsymsym,if1000 (3.17) hhzEEEhzEEEEEasymasymhasymasymasymasymhasymasym,if2200000 (3.18) where 2h is the core thickness; E 0 sym is the Youngs modulus at the midplane for symmetric case; E 0 asym is the Youngs modulus at the bottom surface for asymmetric case. E h sym is the sandwich core Youngs modulus at face sheet interfaces for symmetric case; E h asym is the sandwich core Youngs modulus at top face sheet interface for asymmetric case. Three different variations, such that E h = E 0 (1, 5, 10) are considered. E h is the core modulus at face sheet interfaces for symmetric case and is the sandwich core modulus at top face sheet interfaces for asymmetric case. Poissons ratios are = 0.35 for the core and f = 0.25 for the face-sheet material. The intender is a cylinder with a radius of 10 -3 m. The width of the cylinder and the width of the sandwich panel in y-direction are assumed to be unity. The cylinder is made of steel and its mass is m=15.7 kg. The impact velocity of the impactor is v 0 =6 m/s and the kinetic energy is K=282.2 J.

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80 Figure 3.6: Through the thickness variations of core modulus considered for the functionally graded sandwich beam. (H.C denotes homogeneous core; S.C. denotes symmetric core, A.C. denotes asymmetric core) Figures 3.7 through 3.33 present different displacements, strains and stresses based on assumed stress distribution method and provide a comparison of different sandwich structures. This method is compared with the second numerical method and with the finite element solution at the end of the chapter (Figures 3.33 3.37) Figures 3.7 thought 3.25 present displacements, strains and stresses in sandwich beam with linear symmetric and asymmetric functionally core. For the two cases (symmetric and asymmetric), same Youngs modulus ratio, E h /E 0 of 10 and same semi contact length c = 3 mm are considered. Figure 3.7 presents horizontal displacement in the second half of sandwich structure with asymmetric core. It can be noticed that the displacement is zero at the middle cross-section and has a higher order variation with respect to the thickness coordinate. The half-beam deflection is plotted for the same asymmetric core beam in Figure 3.8, whereas Figure 3.9 shows deflection at core-top face sheet interface and core

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81 bottom face sheet interface near the contact in order to indicate the localized core compression. Figure 3.7: Variation of the horizontal displacement, u, in the second half of the sandwich beam (Linear asymmetric core with E h /E 0 = 10) Figure 3.8: Deflection, w, in the second half of the sandwich beam (Linear asymmetric core with E h /E 0 = 10)

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82 Figure 3.9: Variation of the deflection, w, in the second half of the sandwich beam at core top face sheets interfaces, in the vicinity of contact (Linear asymmetric core with E h /E 0 = 10) In order to visualize the location of the core maximum strains and to compare symmetric and asymmetric core maximum strains with the homogenous core maximum strains, contour plots of the strains (both normal and shear) for the half beam are presented in Figures 3.10 through 3.15. In the same figures the location of semi contact length (c = 3 mm ) is plotted. Maximum normal strain occurs at the middle core (x = L/2 = 100 mm) and slightly above the region with small Youngs modulus (bottom half of the beam for asymmetric core Figure 3.10 and top half of beam for symmetric beam Figure 3.11). For the homogenous core maximum normal strain (Figure 3.12) occurs at the middle core (x = L/2 = 100 mm) and close to the core top face sheet interface.

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83 Figure 3.10: Contour plot of normal strain in the functionally graded asymmetric core (with E h /E 0 = 10) normalized with respect to the maximum normal strain in the core, xx max = 0.071 m/m, (contact length, c = 3 mm). Figure 3.11: Contour plot of normal strain in the functionally graded symmetric core (with E h /E 0 = 10) normalized with respect to the maximum normal strain in the core, xx max = 0.070, (contact length, c = 3 mm). Figure 3.12: Contour plot of normal strain in the homogeneous core (with E h /E 0 = 1) normalized with respect to the maximum normal strain in the core, xx max = 0.052, (contact length, c = 3 mm).

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84 As presented in Figure 2.14, FG core shear strain has a 1/z variation. Maximum shear strain in FG core is spread out along the beam where Youngs modulus is small (core bottom face sheet interface for asymmetric core Figure 3.13 and middle beam for symmetric beam Figure 3.14) and away of contact region. Maximum shear strain in the homogenous core is closed to the contact region, below the core top face sheet interface (Figure 3.15). Maximum stains are used in order to compare different cores. Figure 3.13: Contour plot of shear strain in the functionally graded asymmetric core normalized with respect to the maximum shear strain in the core, xz max = 0.44 (contact length, c = 3 mm). Figure 3.14: Contour plot of shear strain in the functionally graded symmetric core normalized with respect to the maximum shear strain in the core, xz max = 0.37. (contact length, c = 3 mm)

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85 Figure 3.15: Contour plot of shear strain in the homogeneous core normalized with respect to the maximum shear strain in the core, xz max = 0.14 (contact length, c = 3 mm). Figure 3.16 confirms the fact that axial stresses are larger in the face sheets than in the core (an assumption used in Chapter 2 for solving equilibrium equations). Also it can be noted that the axial stress xx is higher on the topside of the core where contact occurs (Figure 3.17). The stress concentration is not only due to contact but also due to the fact that the core Youngs modulus is higher near the interface. Figure 3.16: Variation of the axial stress through the thickness in sandwich beam with functionally graded core at different cross-sections (Linear asymmetric core with E h /E 0 = 10)

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86 Figure 3.17: Variation of the axial stress through the thickness in functionally graded core (Linear asymmetric core with E h /E 0 = 10) The normal stress zz in the core for both asymmetric core and symmetric core is plotted in Figure 3.18 ((a) and respectively (b)). It can be seen that, although the maximum normal stress is slightly larger in the asymmetric core, in both cases it has same region of influence, very closed to the contact region (contact length is c = 3 mm). These stresses are useful in determining if the core will get crushed due to impact or not. Since the core density is higher in the contact region, the FG core will be able to withstand higher compressive stresses compared to uniform density core. In this case the skins are flexible so the load remains localized (it has large values for near the contact and is zero away of contact).

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87 (a) (b) Figure 3.18: Variation of the normal (compressive) stress in (a) asymmetric and (b) symmetric functionally graded core (with E h /E 0 = 10) in the vicinity on the contact (contact length, c = 3 mm) Usually the local failure starts in the core and results in core crushing, delamination and therefore significant reduction of the sandwich strength. There are different factors that can influence the local effect of indentation stress field. Here, the face sheets thickness, h f and indentor radius, R and the face sheet Youngs modulus, E f are varied in order to study their influence on contact. The normal (compressive) stress in asymmetric functionally graded core (with E h /E 0 = 10) in the vicinity of the contact, at core top face sheet interface is plotted for all cases (different h f, R and E f ). Figure 3.19 presents the face sheet thickness influence on impact: for the same asymmetric functionally graded core (with E h /E 0 = 10) and same contact length, c, three cases with respect to the face sheet thickness are considered: h f = (0.3, 1, 2) mm. It can be noted that as the face sheet thickness increases not only the maximum compressive stress is increased, but because the face sheet becomes rigid, it spreads out the load. The variation of normal stress with indentor radius is plotted in Figure 3.20: for the same asymmetric functionally graded core (with E h /E 0 = 10) and same contact length, c, three cases with respect to the indentor radius are considered: R= (5, 10, 20) mm. In this case the maximum normal stress

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88 decreases as the radius increases, but the non-zero stress region remains the same for different R-values. Figure 3.19: Face sheet thickness influence on variation of the normal (compressive) stress in asymmetric functionally graded core (with E h /E 0 = 10) in the vicinity on the contact (c = 0.003 m), at core top face sheet interface (indenter radius, R = 10 mm). The same type of collusions can be reached for the face sheet Youngs modulus influence on impact. Keeping the same geometry, indenter and the same type of functionally graded core and increasing the face sheet Youngs modulus results in an increase in the maximum compressive stress but the non-zero region remains the same (Figure 3.21)

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89 Figure 3.20: Indentor radius influence on variation of the normal (compressive) stress in asymmetric functionally graded core (with E h /E 0 = 10) in the vicinity on the contact (c = 0.003 m), at core top face sheet interface (face sheet thickness, h f = 0.3 mm). Figure 3.21: Face sheet Youngs modulus influence on variation of the normal (compressive) stress in asymmetric functionally graded core (with E h /E 0 = 10) in the vicinity on the contact (c = 0.003 m), at core top face sheet interface (face sheet thickness, h f = 0.3 mm).

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90 The transverse shear stresses in the core are plotted in Figures 3.22-3.23. Figure 3.22 gives a three dimensional variation in the asymmetric core whereas Figures 3.23 3.25 give contour plots for symmetric, asymmetric and homogenous core, respectively, in order to visualize the location of maximum shear stress. It is very interesting to see that the maximum shear stress occurs slightly below the core top face sheet interface, and because of contact there is a severe stress concentration in the vicinity of contact (for graded core the maximum shear stress is at the end of the contact interval, Figures 3.23 and 3.24 whereas for homogenous core the maximum shear stress is outside the contact interval, Figure 3.25). These stresses will be useful in predicting the shear failure of the core due to impact. Figure 3.22: Shear stress in functionally graded asymmetric core (with E h /E 0 = 10).

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91 Figure 3.23: Contour plot of shear stress (MPa) in linear asymmetric (with E h /E 0 = 10) core in the contact vicinity (c = 3 mm). Figure 3.24: Contour plot of shear stress (MPa) in linear symmetric core (with E h /E 0 = 10) in the contact vicinity (c = 3 mm). Figure 3.25: Contour plot of shear stress (MPa) in homogeneous core (with E h /E 0 = 1) in the contact vicinity (c = 3 mm).

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92 Figure 3.26 depicts the contact load-indentation (beam thickness compression) relations for various beams. It may be noted that the contact stiffness is higher when the core density is also higher near the face sheet. For an indentation of 0.2 mm the contact force increases by a factor of about 8 when the core modulus at the interface increases by a factor of 10. Evidently there is a shielding effect due to the stiff face sheet and that is why the increase in contact force is not in the same order as the core stiffness. The load-indentation relations are needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam. Figure 3.26: Variation of contact force with indentation depth in functionally graded beams. (H.C.: homogeneous core; S.C.: symmetric core, A.C.: asymmetric core) The central deflection of the beam as a function of applied force is plotted in Figure 3.27. The relations are approximately linear. The FG core affects the stiffness of the beam as the high-density core near the face sheet contributes significantly to the flexural stiffness of the beam. Also, it can be noted that the asymmetric beam is stiffer than the

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93 symmetric one because near the top face sheet (where the contact occurs) the Youngs modulus for the asymmetric case is larger than that for symmetric case. Using these results the contact stiffness that describe the relation between the contact force and indentation (3.12) and the global stiffness that relates the contact force to the vertical displacement (3.13) were determined. These constants are needed for the quasi-static model. The total loads and maximum vertical displacements for various contact lengths are presented in Figure 3.27 in order to present the trend. Although the range of deflection in Figure 3.27 are much larger (~40 mm) for small deflection theory, the impactor mass and velocity in the example impact problems are taken such that the maximum deflection (~ 8 mm) does not exceed the limits for linear elastic approach. Figure 3.27: Relation between contact force and the vertical displacement at bottom midpoint in functionally graded sandwich beams. (H.C.: homogeneous core; S.C.: symmetric core, A.C.: asymmetric core) In order to determine if the core can withstand the contact loads due to impact not only the maximum stresses in the core needs to be computed but also it needs to be compared with corresponding strength values. Since the strength of a FG core is expected

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94 to vary with location and also we do not have sufficient data on strength, it was decided to calculate the maximum strains. We assume that the maximum strain theory will hold well at all densities and thus strains can be used to determine the efficacy of FG cores compared to uniform cores. The variation of maximum normal strain xx and maximum shear strain xz in the core for a given contact force are plotted in the Figures 3.28 and 3.29, respectively. Using the maximum contact force values for a given impact energy (282 J) in various panels we determined the maximum strains for that impact event. These results are summarized in Table 3-1. Table 3-1: Maximum normal and shear strains for a given impact energy of 282 J. FG denotes functionally graded core. The % reduction in strain in FG cores is with respect to the maximum strain in uniform core. Core type E h /E 0 F max (N) x xz Maximum % Reduction Maximum % Reduction Uniform 1 5.45 4 0.0300 0.0978 FG, Symmetric 5 7.0310 4 0.0257 14.3 0.0830 15.1 FG, Symmetric 10 7.8910 4 0.0194 35.3 0.0700 28.4 FG, Asymmetric 5 7.3910 4 0.0195 35.0 0.0500 48.9 FG, Asymmetric 10 8.3110 4 0.0176 41.3 0.0368 62.3 From Table 3-1 one can notice that maximum normal strain corresponding to the maximum impact load decreases by approximately 40% as the Youngs modulus of material at the top of the core increases by a factor of 10 for symmetric case, and by approximately 35% for the asymmetric cases. Also the maximum shear strain corresponding to the maximum impact load decreases by approximately 60% as the Youngs modulus of the asymmetric material at top of the core is increased by a factor of 10. For the symmetric core case the reduction is with approximately 30% when Youngs

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95 modulus of the symmetric material at top of the core is increased by a factor of 10. An interesting conclusion is that the maximum strain corresponding to the maximum impact load for asymmetric core is smaller than maximum strain for symmetric core. The reason for this is that the asymmetric core is stiffer in a larger region in the vicinity of contact compared to the symmetric core. Figure 3.28: Contact force vs. maximum normal strain for beams with FGM core and homogeneous core. (H.C denotes homogeneous core; S.C. denotes symmetric core, A.S denotes asymmetric core)

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96 Figure 3.29: Contact force vs. maximum shear strain for beams with FGM core and homogeneous core. (H.C.: homogeneous core; S.C.: symmetric core, A.C.: asymmetric core) From Figure 3.27 one can notice that the FG core affects the stiffness of the beam as the high-density core near the face sheet contributes significantly to the flexural stiffness of the beam. Hence, to maintain the same flexural stiffness for different core materials, the core thickness is varied as described below. The flexural stiffness is defined as: ffhhhhijijdzzCD222 (3.19) where C ij are stiffness coefficients of the constituents, face sheets and the core, h is the core thickness and h f is the face sheets thickness. The core thicknesses of the FG cores were chosen such that the flexural stiffness of the sandwich beam will be equal to that of the beam with homogeneous core. That is (3.20) hom1111DDFG

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97 Table 3-2: Core thicknesses for different materials with same flexural stiffness, D 11 and same global stiffness. H.C.: homogeneous core; S.C.: symmetric core, A.S.: asymmetric core. For constant D 11 For constant global stiffness Core type E h /E 0 Core thickness, h (mm) Global stiffness, k s (MN/m) Core thickness, h (mm) Global stiffness, k s (MN/m) Uniform 1 20 5.5 20 5.65 Symmetric 5 12.58 4.4 14.44 5.65 Symmetric 10 10.08 3.67 12.88 5.65 Asymmetric 5 13.82 5.5 13.82 5.65 Asymmetric 10 11.28 4.78 12.32 5.65 The results for core thickness obtained using this method are presented in Table 3-2. All the thicknesses for FGM obtained in this way are smaller than that for homogeneous core. Figure 3.30: Variation of contact force with global deflection at bottom midpoint in functionally graded beams with different core thickness but the same flexural stiffness D 11 (H.C: homogeneous core; S.C.: symmetric core, A.S.: symmetric core) The load-deflection behavior of the sandwich beams having the same flexural stiffness is presented in Figure 3.30 and in Table 3-2. The results indicate that the beam

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98 stiffness is not the same for all beams since the shear deformation effects wasnt considered in Eq. (3.19). Since the beam stiffness include both flexural and shear stiffness, an approximate method was used to obtain the thicknesses of FGM cores such that these beams exhibit the same stiffness as the homogenous core beam: the thicknesses of FGM were chosen such that the slopes of load-deflection curves to be the same as the slope of load-deflection curve for the homogeneous core. The results are presented in Table 3-2. One can notice that the values for the core thicknesses in the case of same global stiffness are slightly larger than those for the case of same flexural stiffness. The impact analysis was repeated for the two sets of beams: having the same flexural stiffness and having the same total stiffness, and the results for maximum normal and shear strains in the core for all cases, for a given impact energy are presented in Figs. 3.31 and 3.32. From those plots some important results can be inferred. For both symmetric and asymmetric material as the ratio E h /E 0 increases the material become stiffer and the maximum strains decrease. Also the maximum strains corresponding to the maximum impact load for asymmetric core are smaller than maximum strains for symmetric core as the asymmetric core material is stiffer in the region of contact. For the same material (symmetric or asymmetric, with same ratio E h /E 0 ) as the core thickness decreases, the total stiffness also decreases and the maximum shear strain corresponding to the maximum impact load increases. Overall the most graded (with the largest ratio E h /E 0 ) asymmetric core gives the smaller maximum strains. As a final conclusion, the results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.

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99 Figure 3.31: Comparison of maximum normal strains for different core materials.E h /E 0 =1 represents the homogeneous core. Figure 3.32: Comparison of maximum shear strains for different core materials. E h /E 0 =1 represents the homogeneous core. All the above results are derived and plotted based on assumed distribution method. In what follows three contact methods are compared: the two numerical methods

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100 (assumed stress distribution and method of point matching) are compared with a finite element solution. Using the commercial finite element software ABAQUS a 2D finite element model was created to model the problem of contact between a rigid indentor and a sandwich structure with functionally graded core (Figures 3.33 and 3.34). The FG core was partitioned through the thickness into a number of strips with constant properties, each strip containing more elements through the thickness. The elements considered were 2D, quadratic, plane strain elements. Boundary conditions assume in order to be identical with those from Fourier series Galerkin method. The load was applied by moving the rigid indenter with a specified distance. The contact length was calculated based on contact pressure at surface nodes: the contact length is the distance with non-zero contact pressure. In order to compare the analytical models with the finite element solution, the finite element contact length is calculated first. Then, the analytical solutions are obtained using this contact length. (0,)(,)0wzwLz Figure 3.33: Finite element simulation for half of the beam

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101 Figure 3.34: Finite element simulation of the contact between the functionally graded sandwich beam and the rigid spherical indentor The contact pressure under indenter and the total load for the three models are calculated and plotted in Figure 3.35. As assumed, the contact pressure for the assumed stress distribution has a semi elliptical form. A striking feature of the contact pressure distribution for both methods of point matching and finite element is the sharp peaks in pressure at the edges of the contact. As noted by Johnson (1985) these peaks are not transmitted thought the skins as they are of short wave length than the spreading length scales of the beam. Another important observation is that the finite element gives a smaller contact force, although it has the same shape. Figure 3.36 presents the deflections at core top face sheet interface for the same contact length and different total forces (given in Figure 3.33). Because the total load differs among the models, the displacements differ also. For a better comparison the contact length was changed such that to obtained the same total load, and the displacements are plotted in Figure 3.37. As expected, they are matching.

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102 Figure 3.35: Contact pressure under indenter for a given contact length c = 3 mm and total load for three models Figure 3.36: Deflections at core top face sheet interface for same contact length and different contact load in three models

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103 Figure 3.37: Deflections at core top face sheet interface for different contact length and same contact load in three models Summary and Conclusions Unlike for their solid metallic counterparts, predictions of the effects of low-velocity impact damage on sandwich structures are difficult because significant internal damage is achieved at impact energy levels lower than those required to create visible damage at the surface. So, it is important to understand the evolution of strains and stresses created inside a sandwich structure with functionally graded core in order to predict the impact damage. The first step in solving the dynamic impact is to understand the effects of a static contact. This chapter solves the problems of contact and low-velocity impact between a rigid spherical indenter and a sandwich structure with functionally graded core. Two numerical methods are compared with a finite element solution. For the impact problem a simple dynamic model based on quasi-static behavior of the panel was used the

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104 sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. The beam geometry and face sheet properties influence on the contact are studied. It is shown that changing the indenter radius or the face sheet Youngs modulus results in a change of maximum normal stress in the core top face sheet interface over the same region whereas changing the face sheet thickness results in spreading the load over a larger region in the top core. Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. Overall the most graded (with the largest ratio E h /E 0 ) asymmetric core gives the smaller maximum strains. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.

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CHAPTER 4 MASS OPTIMIZATION FOR A SANDWICH BEAM WITH FUNCTIONALLY GRADED CORE For a better understanding of the advantages of sandwich panels with FG cores over those with uniform cores, a comparison of the total mass of different panels is also needed. This chapter discuses three optimization problems for sandwich structure with FG core based on two models presented in the literature: Choi and Sankars (2005) linear relation between relative density and Youngs modulus and Gibson and Ashbys (1997) quadratic relation between relative density and Youngs modulus. The core dimensions and properties are changed such that to optimize the sandwich beam total mass and/or its flexural rigidity. The results show that sandwich panels with FG cores are more efficient than the one with uniform core. Linear Model Choi and Sankar (2005) assumed a cellular medium consisting of struts of square cross-section in a rectangular array. Based on mechanics of materials type calculation, and considering that the cellular solid is a homogeneous orthotropic material, they derived the following linear relation between the relative density of the foam material and its Youngs modulus: 3 s sEE (4.1) 105

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106 where is the foam density, s is the solid material density, E* is foam Youngs modulus and E s is the solid material Youngs modulus. Their results were verified by finite element models. Using the above formula, a relation between the density and Youngs modulus as functions of thickness coordinate, z can be derived: )()()()()(3)()(3)(0000zEzEzzzEEzzEEzssss (4.2) where z 0 is the thickness coordinate of a given/particular sandwich core point. As presented in Table 4-1 and also in Figure 4.1, three Youngs modulus variations are investigated: homogeneous core, linear asymmetric and linear symmetric about the core middle line. Based on equation (4.2), the linearity and the symmetry/asymmetry of the cores are preserved for the density also. The first step is to choose a Youngs modulus profile and a particular point z 0 Then using equation (4.2) a formula for the densities is derived and then the sandwich total mass and sandwich flexural rigidities are calculated, using the formulas: /200/2/22/2ffffhhLbVhhhhhh M dVzdxdydzDzEzdz (4.3)

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Table 4-1: Variations of Youngs modulus, density, mass and flexural rigidity for the three cases investigated. 107 Homogeneous core Linear asymmetric core Linear symmetric core Core Youngs modulus 0)(EzEH ()2/22tbtbAbbAbEEEEzEzEEhE ()12/2hmSmmSzEEEzEEh Core density 0)( zH ()2/22tbtbAbbAb E EEEzzEhE ()12/2hmSmmSzEEzEh Total mass 02 H Hff M Lbhh 122tAAbffbEh M LbhE 122hSSmffmEh M LbhE Flexural rigidity 30232HHfhDED 332btA A fEEh D D 3362hmSSfEEhDD

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108 In the Table 4-1, L is the beam length, b is the beam width, h s are the core thicknesses, h f is the face-sheet thickness, E s and s are, respectively the Youngs modulus and density of core for the three cases (see Figure 4.1), and D f is the face-sheets flexural rigidity (same for all the cases): 223222223fffffhhDEhhh (4.4) 0 x z h f 0 x z (c) Symmetric core: E S (-h S /2)= E S (h S /2)=E h E S (0)=E m S (-h S /2)=(h S /2)= h S (0)= m h f 0 x z h f h S h A h H (a) Homogeneous core: E H (z) = E 0 H ( z ) = 0 (b) Asymmetric core: E A (-h A /2) = E b E A (h A /2) = E t A (-h A /2) = b A (h A /2) = t Figure 4.1: Geometry and notations for the three cases investigated: (a) sandwich beam with homogeneous core; (b) sandwich beam with linear asymmetric core and (c) sandwich beam with linear symmetric core. (The linearity and symmetry/asymmetry is referred at both elastic modulus and density.) Because the three sandwich structures have identical face-sheets, only the cores contribute to the difference in mass. In order to better investigate the cores mass, the following ratios between the core masses are derived:

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109 00/210(0)222/21/2(/2)222SSScoreSSScoreHHAAAcoreAAAAcore H HEhhEMhMEhhEhMhhM (4.5) where S denotes asymmetric core, A denotes asymmetric core and H denotes homogeneous core. The right hand sides of equations (4.5) contain all the parameters that can be changed in order to optimize the total mass and flexural rigidities of sandwich beams. The following two problems were solved: 4. Keeping Youngs moduli and densities fixed, determine the core thicknesses such that to obtain the same flexural rigidity. 5. Keeping core thicknesses fixed, determine Youngs moduli and densities such that to obtain the same mass and the same flexural rigidity. Case 1: Same Flexural Rigidity A sandwich beam with length L = 0.3 m, unit width b = 1 m, and facesheet thickness h f = 0.310 -3 m is considered to investigate the effects of varying core properties through the thickness. The face-sheet Youngs modulus was chosen as 50 GPa and its density as 1,400 kg/m 3 The non-dimensional variations of core Youngs modulus with respect to z, as well as the non-dimensional variations of the core density given in Table 4-1 are plotted in Figure 4.2 (as described by equation (4.2) they are equal). The core Youngs modulus is varied from 50 MPa at bottom core (asymmetric) or middle core (symmetric) to 50 10 MPa at core top face-sheet interface. In the same trend the core density is varied from

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110 60 kg/m 3 at bottom core (asymmetric) or middle core (symmetric) core to 60 10 kg/m 3 at core top face-sheet interface. Figure 4.2: Through the thickness non-dimensional variations of core modulus and density considered for the homogeneous and functionally graded sandwich beam with same core thicknesses. As described in Chapter 3, FG core affects the stiffness of the beam as the high-density core near the face sheet contributes significantly to the flexural stiffness of the beam. Hence, to maintain the same flexural stiffness for different core materials, the core thickness needs to be varied. For each of the three cases (homogeneous, linear symmetric and linear asymmetric) three cases from core thickness point of view were compared: 6. Same core thickness, h = 2010 -3 m; 7. The core thicknesses of the FG cores were chosen such that the flexural stiffness, D 11 of the sandwich beam will be equal to that of the beam with homogeneous core. 8. The core thicknesses of the FG cores were chosen such that the global stiffness of the sandwich beam will be equal to that of the beam with homogeneous core.

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111 The results for the total mass are presented in Table 4-2. As it was expected (based on (4.5)) the total mass of a sandwich structure with functionally graded core that is stiffer/denser at core-top face-sheet interface is larger that the mass of a sandwich structure with homogeneous core. 0(0)(/2)/2/210010/25.55.5SASSAASAAcoreSScoreHHcoreAAcoreHHhEhEhEEhMhMhMhMh (4.6) For the same core thickness, FG core mass is 5.5 larger than the uniform core mass. For different thickness, the homogeneous core is thicker (almost double) and spreads the mass outward to provide maximum moment of inertia (D 11 ). Hence, the FG cores mass for different thicknesses (given in Table 4-2) and same stiffness is around 3.15 larger than the homogenous core. Table 4-2: Total mass of sandwich structures with different density variations and different core thicknesses (face sheets mass, the same for all cases is M f = 0.252 kg) Homogeneous core Symmetric core Asymmetric core Same h = 2010 -3 m M H = 0.612 kg M S = 2.232 kg M A = 2.232 kg h H = 2010 -3 m h S = 10.0810 -3 m h A = 11.2810 -3 m Different h, same flexural rigidity, D 11 M H = 0.612 kg M S = 1.25 kg M A = 1.37 kg h H = 2010 -3 m h S = 12.8810 -3 m h A = 12.3210 -3 m Different h, same global stiffness M H = 0.612 kg M S = 1.52 kg M A = 1.47 kg The results for strains and stresses for low-velocity impact of these nine sandwich panels are included in Chapter 3. For the same core thickness, the FG cores weight increases 5.5 times with respect to the homogenous core but there is a significant reduction (in the range of 28.4 62.3 %) in maximum strains (Table 3-1) of FG cores.

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112 For different core thickness and same stiffness, the FG cores weight increases around 3.15 times with respect to the homogenous core and maximum strains is reduced by 4 52 % (Figures 3.31 and 3.32). Case 2: Same Mass and Flexural Rigidity The next step was to keep the identical geometry for both sandwich structures (with FG core and with homogeneous core) and to change the material properties (Young modulus and density) such that to obtain the same weight, same flexural rigidity, D 11 and same extensional stiffness A 11 For this problem only two cores are compared: homogeneous core (described by E 0 0 ) and asymmetric core (described by E t E b b ) It was found that, for a given linear asymmetric core (as presented in Table 1), the homogeneous density and Youngs modulus given by: 2)(2)(00btHbtHEEEzEz (4.7) are required to obtain the same mass, same flexural rigidity, D 11 and same extensional stiffness A 11 The non-dimensional density and Youngs modulus profiles are presented in Figure 4.3.

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113 0 1 10 -10 -5 0 5 10 Non-dimensional core density, (z)/(z0) andNon-dimensional core elastic modulus, E(z)/E(z0)Thickness Coordinate, mm Homogeneous coreAsymmetric core Figure 4.3: Through the thickness non-dimensional variations of core modulus and density for the homogeneous and linear asymmetric functionally graded sandwich beams with same mass, same flexural rigidity and same extensional stiffness. 0 0.05 0.1 0.15 0.2 0 1 2 3 4 5 6 7 8x 105 Contact load,F (N)Vertical displacement, wbottom (m) Homogeneous coreAsymmetric core Figure 4.4: Relation between contact force and vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams. The impact problem presented in Chapter 3 was applied to both structures described above, and the results are plotted in Figures 4.4 4.9. Figure 4.4 presents

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114 linear relation obtained between contact force and the vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams. The slopes of these curves give the flexural rigidity and as the problem requires they are almost the same (the small difference in the slopes comes from the local indentation effects). Figure 4.5 gives the indentation (or core compression) contact load relation and also the approximation functions needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam. For each case, k i and n are determined, such that: ()niFk (4.8) Also from Figure 4.5, it can be noted that the contact stiffness of the asymmetric FG core is larger than the contact stiffness of the uniform core. 0 2 4 6 8x 10-4 0 2 4 6 8 10x 105 Indentation, (m)Contact load,F (N) Homogeneous coreApproximation function FH()=kinAsymmetric coreApproximation function FA()=kin Figure 4.5: Variation of contact force with indentation depth in functionally graded and homogeneous sandwich beams and the approximation functions.

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115 0 1 2 3 4 5x 10-3 0 1 2 3 4 5 6 7 8x 105 Contact load,F (N)Contact length, c (m) Homogeneous coreAsymmetric core Figure 4.6: Variation of contact force with contact length in functionally graded and homogeneous sandwich beams. Figure 4.7: Variation of contact force with contact length in functionally graded and homogeneous sandwich beams, approximation functions and Hertz contact law. Figures 4.6 and 4.7 present contact length contact load relations: it can be noticed that for the same contact length, the FG core load is larger than the uniform core load.

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116 Also, from Figure 4.7, it can be noted that these relations not applicable for very small contact lengths, when the contact force acts only upon the top face-sheet. Instead, in this case, the Hertz contact law ( Timoshenko and Goodier, 1970 ) should be applied: 24fcEF R (4.9) where E f is face-sheet Youngs modulus and R is the indenter radius. Figure 4.7 presents the approximation functions needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam. For each case, F 0 k c and are determined, such that: 0()cFcFkc (4.10) 0 1 2 3 4 5 6 7 8x 105 0 0.05 0.1 0.15 0.2 Contact load, F(N)xx max Homogeneous core Asymmetric core Figure 4.8: Contact force vs. maximum normal strain for beams with FGM core and homogeneous core.

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117 0 2 4 6 8x 105 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Contact load, F(N)xz max Homogeneous core Asymmetric core Figure 4.9: Contact force vs. maximum shear strain for beams with FGM core and homogeneous core. The variation of maximum normal strain xx and maximum shear strain xz in the core for a given contact force are plotted in the Figures 4.8 and 4.9, respectively. Using the maximum contact force values for a given impact energy (282 J) in various panels we determined the maximum strains for that impact event. These results are summarized in Table 4-3. Table 4-3: Maximum normal and shear strains for a given impact energy of 282 J. FG denotes functionally graded core. The % change in strain in FG core is with respect to the maximum strain in uniform core. Core type E h /E 0 F max (N) x xz Maximum % Change Maximum % Change Uniform 1 5.33 4 0.0125 0.0978 FG, Asymmetric 10 5.1610 4 0.0135 + 7.2 0.0368 23.32 From Table 4-3 one can notice that maximum normal strain corresponding to the maximum impact load of FG asymmetric core increases by approximately 7%. But the maximum shear strain corresponding to the maximum impact load decreases by approximately 23%.

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118 Quadratic Model Gibson and Ashby (1997) presented a very detail derivation for mechanics of foams, based on relative density and the degree to which the cells are open or closed. Their results are compared with, and calibrated against experimental data. The linear elastic behavior of foam made of open equiaxed cell loaded in compression is described by: 21ssECE (4.11) where is the foam density, s is the solid material density, E* is foam Youngs modulus and E s is the solid material Youngs modulus and where C 1 includes all of the geometric constants of proportionality. Experimental data show C 1 1. In this chapter two optimization problems based on this model are solved. The core density vary continuously with the core Youngs modulus: )()()()(00zEzEzz (4.12) where z 0 in the thickness coordinate of a given/particular sandwich core point. From elastic modulus point of view, as presented in Table 4-4 and Figure 4.10, three cases are investigated: uniform core, linear asymmetric and linear symmetric about the core middle line. The first step is to choose a Youngs modulus profile and a particular point z 0 Then using equation (4.12) a formula for the densities is derived and then the sandwich total mass and sandwich flexural rigidities are calculated, using the equations (4.3).

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119 0 x z h f h 0 x z h f 0 x z (c) Symmetric core: E S (-h/2)= E S (h/2)=E h E S (0)=E m h f h h (a) Homogeneous core: E H (z) = E 0 H ( z ) = 0 (b) Asymmetric core: E A (-h/2) = E b E A (h/2) = E t Figure 4.10: Youngs modulus variation for the three cases investigated: (a) sandwich beam with homogeneous core; (b) sandwich beam with linear asymmetric core and (c) sandwich beam with linear symmetric core.

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Table 4-4: Variations of Youngs modulus, density, mass and flexural rigidity 120 Homogeneous core Linear asymmetric core Linear symmetric core Core Youngs modulus 0)(EzEH ()2/22tbtbAbbAbEEEEzEzEEhE ()12/2hmSmmSzEEEzEEh Core density 0)( zH ()2/22tbtbAbbAb E EEEzzEhE ()12/2hmSmmSzEEzEh Total mass 02 H Hf f M Lbhh 3212231tbAbAfftbEEh M LbhEE 3212231hmSmSffhmEEh M LbhEE Flexural rigidity 30232H H fshDE D 332btA A fsEEhDD 3362hmSSfsEEh D D

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In the Table 4-4 L is the beam length, b is the beam width, s are: AASS(-h/2) (h/2) (-h/2) (-h/2) (0) btSShSm (4.13) all E i are described in Figure 4.10 and D fg is the face-sheets flexural rigidity: 322322222fffffshhhhhED (4.14) For a given geometry (that is for a given core thickness, h and for a given face-sheets thickness, h f ) and identical face-sheets two basic optimization problems are solved: 1. Determine core Youngs modulus and density profiles such that to obtain same flexural rigidity but allow different masses; 2. Determine core Youngs modulus and density profiles such that to obtain same mass but allow different flexural rigidity; Case 1: Same Flexural Rigidity Based on Table 4-4, in order to fully prescribe the densities and Youngs moduli of the three sandwich structures, the following eight quantities need to be given: Homogeneous core: E 0 0 Asymmetric core: E t E b b Symmetric core: E h E m m It is assumed that the three sandwiches have identical face-sheets and identical dimensions. It is assumed that the value of Youngs modulus at middle core in the case of a symmetric core (E m ) equals the Youngs modulus value at core-bottom face-sheet interface (E b ) for the asymmetric core. So, given etermine such that to obtain the same flexural rigidity: bmtbEEEE,,d hEE,0 SAHDDD (4.15)

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122 The resulting quantities are: 3220tbhtbEEEEEE (4.16) Then using the formulas given in Table 4-4 the densities profiles are determined. In order to illustrate the above concepts the following numerical examples are considered: three sandwich beams with length L = 0.3 m, unit width b = 1 m, core thickness h = 2010 -3 m and face-sheet thickness h f = 0.310 -3 m. The face-sheet Youngs modulus was chosen as 50 GPa and its density as 1400 kg/m 3 The variation of Youngs modulus for the linear asymmetric core is between E b = 5 MPa at the core-bottom face-sheet interface and E t = 5x10 MPa at the core-top face-sheet interface. The density at the middle core for the symmetric core ( m ) equals the density at the core-bottom face-sheet interface for the asymmetric core ( b = 60 kg/m 3 ). The non-dimensional (with respect to mEE b ) variations of core Youngs modulus with respect to thickness coordinate z, given in Table 4-4 and based on (4.16) are plotted in Figure 4.11. The uniform core Youngs modulus has the average value between the asymmetric top core and bottom core Youngs modulus values. In the same trend the non-dimensional (with respect to m b ) core densities (not linear) are presented in Figure 4.12 The results for the total mass of the three sandwich structures investigated are presented in Table 4-5. In this case, the total mass of a sandwich structure with homogeneous core is larger than the total mass of a sandwich structures with FG cores.

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123 0 2 4 6 8 10 -10 -5 0 5 10 E(z)/EbCore thickness coordinate, z (mm) HomogeneousLinear symmetricLinear asymmetric Figure 4.11: Core Youngs modulus variation for the three cases investigated: sandwich beam with homogeneous core; sandwich beam with linear asymmetric core and sandwich beam with linear symmetric core. Table 4-5: Total mass of sandwich structures with same geometry and different density variations (face sheets mass, the same for all cases is M f = 0.252 kg). The % reduction in the mass in FG core is with respect to the homogenous core mass. D 11 =3109.2 N-m Homogeneous core Symmetric core Asymmetric core Total mass M H = 1.0963 kg M S = 0.9528 kg M A = 1.0686 kg % Reduction -14 -3 The impact problem, described in Chapter 3 was applied to the three structures described above and the results are plotted in Figures 4.13 4.20. Figure 4.13 presents linear relation obtained between contact force and the vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams. The slopes of these curves give the flexural rigidity and as the problem required they are almost the same (the small difference in the slopes comes from the local indentation effects).

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124 -10 -5 0 5 10 1 1.5 2 2.5 3 3.5 Core thickness coordinate, z (mm) (z)/b HomogeneousAsymmetricSymmetric Figure 4.12: Core densities variation for the three cases investigated: sandwich beam with homogeneous core; sandwich beam with linear asymmetric core and sandwich beam with linear symmetric core. 0 0.05 0.1 0.15 0.2 0 1 2 3 4 5 6 7 8x 104 Load,F (N)Vertical displacement, w (m) HomogeneousSymmetricAsymmetric Figure 4.13: Relation between contact force and vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams.

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125 0 1 2 3 4 5x 10-3 0 1 2 3 4 5 6 7 8x 104 Contact length, c (m) Contact load, F (N) HomogeneousSymmetricAsymmetric Figure 4.14: Relation between contact force and contact length in functionally graded and homogeneous sandwich beams. 0 1 2 3 4 5 6x 10-4 0 1 2 3 4 5 6 Contact length, c (m) Contact load, F (kN) Hertz lawHomogeneousSymmetricAsymmetric Figure 4.15: Relation between contact force and contact length in functionally graded and homogeneous sandwich beams and the Hertz contact law. Figures 4.14 and 4.15 present contact length contact load relations: it can be noticed that for the same contact length, the asymmetric FG core contact load is larger than the uniform core contact load but the symmetric FG core load equals the uniform

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126 core contact load. Also, from Figure 4.15, it can be noted that these relations are not applicable for very small contact lengths, when the contact force acts only upon the top face-sheet. Instead, in this case, the Hertz contact law ( Timoshenko and Goodier, 1970 ) should be applied: 24fcEF R (4.17) where E f is face-sheet Youngs modulus and R is the indenter radius. Figure 4.16 gives the indentation (or core compression) contact load relation needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam. For a given contact load the symmetric core produces a larger indentation than the uniform core, but the asymmetric core and the uniform core give almost the same indentation. 0 0.5 1 1.5 2x 10-3 0 1 2 3 4 5 6 7 8x 104 Contact load, F (N)Indentation, (m) HomogeneousSymmetricAsymmetric Figure 4.16: Relation between indentation (core compression) and contact length in functionally graded and homogeneous sandwich beams.

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127 Table 4-6: Maximum normal and shear strains normal and interfacial shear stresses for a given impact energy of 282 J. FG denotes functionally graded core. The % change in strain/stress in FG core is with respect to the maximum strain/stress in uniform core. Core type F max (kN) x xz x (MPa) Interface xz (MPa) Max % Change Max % Change Max % Change Max % Change Uniform 16.91 0.0415 0.1000 2.09x10 2 0.845 FG, Symm 15.78 0.0448 + 7.9 0.1735 + 23.5 1.92x10 2 9.3 1.109 + 31.1 FG, Asymm 16.37 0.0430 + 3.6 0.0735 26.5 2.0x10 2 4.7 1.254 + 48.4 The variation of maximum normal strain xx and maximum shear strain xz in the core for a given contact force are plotted in the Figures 4.17 and 4.18, respectively. For small contact lengths, which mean small contact loads, the maximum strains are almost the same. Using the maximum contact force values for a given impact energy (282 J) in various panels we determined the maximum strains for that impact event. These results are summarized in Table 4-6. From Table 4-6 one can notice that maximum normal strain corresponding to the maximum impact load slightly increases for FG cores. The maximum shear strain corresponding to the maximum impact load decreases by approximately 27% as the Youngs modulus of the asymmetric material at top of the core is increased by a factor of 1.8. But the maximum shear strain corresponding to the maximum impact load increases by approximately 24% as the Youngs modulus of the symmetric material at top of the core is increased by a factor of 1.3. An interesting conclusion is that the maximum strains (both normal and shear) corresponding to the maximum impact load for asymmetric core is smaller than maximum strains for symmetric core. The reason for this is that the

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128 asymmetric core is stiffer in a larger region in the vicinity of contact compared to the symmetric core. The variation of maximum normal stress xx in the core for a given contact force is plotted in the Figures 4.19. For a given contact load, the maximum normal stress xx in the core is the same for the three cases considered. Table 4-6 presents the maximum normal stresses contact corresponding to the maximum impact load for a given impact energy (282 J) in various panels. The FG core maximum normal stress corresponding to the maximum impact load decreases by less than 10% with respect to the uniform core maximum normal stress. 0 2 4 6 8x 104 0 0.05 0.1 0.15 0.2 Contact load, F (N)xx max HomogeneousSymmetricAsymmetric Figure 4.17: Contact force vs. core maximum normal strain for beams with FGM core and homogeneous core. Figure 4.20 gives the variation of maximum interfacial shear stress xz for a given contact force. There is a significant increase (48% for asymmetric and 31% for symmetric) in maximum interfacial stress stresses corresponding to the maximum impact load for a given impact energy (282 J) in the panels with FG core.

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129 0 1 2 3 4 5 6 7x 104 0 0.1 0.2 0.3 0.4 0.5 Contact load, F (N)xz max HomogeneousSymmetricAsymmetric Figure 4.18: Contact force vs. core maximum shear strain for beams with FGM core and homogeneous core. 0 1 2 3 4 5 6 7 8x 104 0 1 2 3 4 5 6 7 8x 108 Contact load, F (N)xx max (Pa) HomogeneousSymmetricAsymmetric Figure 4.19: Contact force vs. core maximum normal stress for beams with FGM core and homogeneous core.

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130 0 1 2 3 4 5 6 7 8x 104 0 0.5 1 1.5 2 2.5 3x 106 Contact load, F (N)Interfacial xz max (Pa) HomogeneousSymmetricAsymmetric Figure 4.20: Contact force vs. maximum interfacial shear stress for beams with FGM core and homogeneous core. Case 2: Same Total Mass Based on Table 4-4, in order to fully prescribe the densities and Youngs moduli of the three sandwich structures, the following eight quantities need to be given: Homogeneous core: E 0 0 Asymmetric core: E t E b b Symmetric core: E h E m m The above quantities were determined based on the same mass assumption: 32032012311231tbbtbHAHShmmhmEEEEMMMMEEEE (4.18) From the above equations result two sufficient conditions:

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131 mbmhbtEEEE (4.19) For a given b 0 can be determined from (4.19). On the other hand, the relation between density and Youngs modulus is: )()()()(00zEzEzz (4.20) Let z 0 =-h/2 and z=z* such that to obtain the homogeneous core density: 0)( z (4.21) Then, from (4.20) the homogeneous core Youngs modulus is: bbEEzE200)( (4.22) In order to illustrate the above concepts we consider the following numerical examples: three sandwich beams with length L = 0.3 m, unit width b = 1 m, core thickness h = 2010 -3 m and face-sheet thickness h f = 0.310 -3 m. The face-sheet Youngs modulus was chosen as 50 GPa and its density as 1,400 kg/m 3 The variation of Youngs modulus for the linear asymmetric core is between E b = 5 MPa at the core-bottom face-sheet interface and E t = 510 MPa at the core-top face-sheet interface. The variation of Youngs modulus for the linear symmetric core is between E m = 5 MPa at the middle core and E h = 510 MPa at the core-top face-sheet interface. The density at the middle core for the symmetric core ( m ) equals the density at the core-bottom face-sheet interface for the asymmetric core ( b = 60 kg/m 3 ). It is important to note that the values of Youngs modulus at the top core in the 2 FG cores are the same.

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132 The non-dimensional variations of core Youngs modulus with respect to thickness coordinate z, given in Table 4 and based on (4.22) are plotted in Figure 21. For this case, the uniform core Youngs modulus based on (4.22) is E 0 = 25.72 MPa. In the same style the non-dimensional core densities (not linear) are presented in Figure 4.22. For this case, the uniform core density is 0 = 136.01 kg/m 3 0 2 4 6 8 10 -10 -5 0 5 10 E(z)/EbCore thickness coordinate, z (mm) HomogeneousLinear symmetricLinear asymmetric Figure 4.21: Core Youngs modulus variation for the three cases investigated: sandwich beam with homogeneous core; sandwich beam with linear asymmetric core and sandwich beam with linear symmetric core. The results for the flexural rigidities of the three sandwich structures investigated are presented in Table 4-7. They are almost the same because the face-sheets (identical) carry the larger contribution. Table 4-7: Flexural rigidity of sandwich structures with same geometry and different density variations (face sheets flexural rigidity, the same for all cases, is D f = 3090.9 N-m). The total mass is M = 1.0686 kg Homogeneous core Symmetric core Asymmetric core Flexural rigidity D H = 3108.1 N-m D S = 3109.2 N-m D A =3116.7 N-m

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133 -10 -5 0 5 10 1 1.5 2 2.5 3 3.5 Core thickness coordinate, z Non-dimensional density, (z)/b Homogeneous AsymmetricSymmetric Figure 4.22: Core densities variation for the three cases investigated: sandwich beam with homogeneous core; sandwich beam with linear asymmetric core and sandwich beam with linear symmetric core. The impact problem was applied to the structures described above, and the results are plotted in Figures 4.23 4.28. Figure 4.23 presents linear relation obtained between contact force and the vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams. The slopes of these curves give the flexural rigidity and as they are almost the same because the three panels have identical face-sheets which bring the largest contribution in flexural rigidity. Figure 4.24(a) (c) present contact length contact load relations: it can be noticed that for the same contact length, the FG core load is larger than the uniform core load. Figure 4.24 presents also the Hertz law (4.9) used for very small contact lengths, when contact force acts only upon the top face-sheet. Figures 4.24(b) and (c) presents the approximation functions (4.10) needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam.

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134 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 1 2 3 4 5 6 7 8x 104 Contact load, F (N)wbottom (m) HomogeneousSymmetricAsymmetric Figure 4.23: Relation between contact force and the vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams. Figures 4.25-4.28 present the variation of maximum normal strain xx maximum shear strain xz maximum normal stress xx in the core and maximum interfacial shear stress xz for a given contact force. Using the maximum contact force values for a given impact energy (282 J) in various panels the maximum strains and stresses for that impact event were determined. These results are summarized in Table 4-8. Table 4-8: Maximum normal and shear strains and stresses for a given impact energy of 282 J. FG denotes functionally graded core. The % change in strain/stress in FG core is with respect to the maximum strain/stress in uniform core. Core type F max (kN) x xz x (MPa) Interface xz (MPa) Max % Change Max % Change Max % Change Max % Change Uniform 16.78 0.0409 0.1025 212 0.827 FG, Symmetric 16.43 0.0386 5.6 0.1050 + 2.4 197 6.9 1.372 + 65.9 FG, Asymmetric 16.37 0.0429 + 4.8 0.075 27 205 3.4 1.263 + 52.7

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135 0 1 2 3 4 5 6x 10-4 0 2 4 6 8 10 Contact length, c (m) Contact load, F (kN) Hertz lawHomogeneous Symmetric Asymmetric (a) 10-4 10-3 10-2 103 104 105 Contact length, c (m)Contact load, F (N) Imput data -AsymmetricAprox F(c)=F0+kc (b) 10-4 10-3 10-2 103 104 105 Contact length, c (m)Contact load, F (N) Input data HomogeneousAprox F(c)=F0+kic (c) Figure 4.24: Relation between contact force and contact length in functionally graded and homogeneous sandwich beams, the Hertz contact law (a), and approximation functions for asymmetric core (b) and uniform core (c). From Table 4-8 one can notice that maximum normal strain corresponding to the maximum impact load slightly increases for FG asymmetric core and slightly decreases for FG symmetric core. The maximum shear strain corresponding to the maximum impact load for symmetric core almost equals the one for uniform core. But the maximum shear strain corresponding to the maximum impact load decreases by approximately 27% as the Youngs modulus of the asymmetric material at top of the core is increased by a factor of 1.9. An interesting conclusion is that the maximum strain

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136 corresponding to the maximum impact load for asymmetric core is smaller than maximum strain for symmetric core. The reason for this is that the asymmetric core is stiffer in a larger region in the vicinity of contact compared to the symmetric core. 0 1 2 3 4 5 6 7 8x 104 0 0.05 0.1 0.15 0.2 Contact load, F (N)xx max HomogeneousSymmetricAsymmetric Figure 4.25: Contact force vs. core maximum normal strain for beams with FGM core and homogeneous core. 0 1 2 3 4 5 6x 104 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Contact load, F (N)xz max HomogeneousSymmetricAsymmetric Figure 4.26: Contact force vs. core maximum shear strain for beams with FGM core and homogeneous core.

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137 0 2 4 6 8x 104 0 1 2 3 4 5 6 7 8x 108 Contact load, F (N)xx max (Pa) HomogeneousSymmetricAsymmetric Figure 4.27: Contact force vs. core maximum normal stress for beams with FGM core and homogeneous core. 0 2 4 6 8x 104 0 0.5 1 1.5 2 2.5 3x 106 Contact load, F (N)Interfacial xz max (Pa) HomogeneousSymmetricAsymmetric Figure 4.28: Contact force vs. maximum interfacial shear stress for beams with FGM core and homogeneous core. From Table 4-8 one can notice that maximum normal stress corresponding to the maximum impact load slightly decreases for FG cores. But the maximum interfacial shear stress corresponding to the maximum impact load for FG cores almost equals the

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138 one for uniform core. But the maximum shear stress corresponding to the maximum impact load significantly increases (by approximately 66% for symmetric core and by approximately 50% for asymmetric core) as the Youngs modulus of the FG material at top of the core is increased only by a factor of 2. Summary and Conclusions Based on two models found in the literature and based on different relations between relative density and elastic modulus, this chapter solves three optimization problems: the geometry of the sandwich panels is kept constant while the properties (Youngs modulus and density) are changed such that to optimize the total mass and/or the flexural rigidity. Choi and Sankars (2005) linear relation between the relative density and Youngs modulus is used to determine the FG asymmetric core and the uniform core parameters such as to obtain panels with the same mass and same flexural rigidity. The results of the impact problem prove that there is a significant reduction (approximately 24%) in the maximum shear strain corresponding to maximum impact load of FG asymmetric core compared with the uniform core maximum shear strain, while the maximum normal strains corresponding to maximum impact load are almost the same. The second model is based on the quadratic relation between the relative density and Youngs modulus derived by Gibson and Ashby (1997) The Youngs modulus and the density were varied such as to obtain panels with same flexural rigidity or to obtain panels with the same total mass. The results of the impact analysis provided similar results for both problems: among the three sandwich panels (with homogeneous core, symmetric FG core and asymmetric FG core), the one with FG asymmetric core gives the smaller maximum shear strain corresponding to the maximum impact load

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139 (approximately 27% less than the uniform core) while the difference in maximum normal strain corresponding to the maximum impact load is small. Also, when the properties are changed such as to obtain the same flexural rigidity, the sandwich panel with asymmetric FG core gives the smaller total mass. These conclusions based on optimization studies emphasize again the superior capability of sandwich structures with FG cores over the sandwich structures with uniform cores.

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CHAPTER 5 SUMMARY AND CONCLUSIONS New developments in manufacturing methods allow gradation in material composition, structure and properties over a broad range of scales from nanometers to meters. It has been recognized that gradients in materials compositions are more damage-resistant at surface or interface between different materials where the failure usually occurs. These type of materials can be used for a large number of applications ranging from implant teeth to rocket frames. A sandwich structure with functionally graded core is beneficial from two aspects: as a sandwich structure (e.g. exceptionally high flexural stiffness-to-weight ratio compared to other architectures) and as a graded material (e.g. lower impact damage). A sandwich beam with soft, transversely flexible core must be approached with a higher-order theory rather than the classical theory. Higher-order refers to the order of polynomial in thickness coordinate that describes the displacements. Few of the reasons that make higher-order theory necessary for a sandwich panel with FG core even if this theory involves more computational effort are: The core gets compressed and the thickness of the beam is changed under localized loads. Plane sections do not remain plane after deformation, so displacement field is not a linear function of the thickness coordinate. Conditions imposed on one face sheet do not necessary hold for the other. 140

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141 This study presented different analytical and finite element models for sandwich structures with functionally graded core. Two analytical models for sandwich structures with two types of core Young modulus are developed: exact solutions are presented for the exponential variation of core Young modulus and a combination of Fourier series and Galerkin method for a polynomial variation of core Young modulus. In fact, any arbitrary varaiation of properties can be solved using the Fourier-Galerkin method. Using a polynomial approximation for the exponential function it is shown that the results of the two methods coincide. Those methods are also compared with models presented in literature and modified for this type of structure: two equivalent single-layer theories based on assumed displacements, a higher-order theory and a finite-element analysis. A very good agreement among the Fourier series-Galerkin method, the higher-order theory and the finite-element analysis is found. This research solved the problems of contact and low-velocity impact between a rigid cylindrical indenter and a sandwich structure with functionally graded core. Two numerical methods are compared with the finite element solution. For the impact problem a simple dynamic model based on quasi-static behavior of the panel is used the sandwich beam is modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the

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142 flexural stiffness was equal to that of a beam with homogeneous core. Overall the most graded (with the largest difference between the largest and smallest core Youngs moduli) asymmetric core gives the smaller maximum strains. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites. The trade-off between total mass and stiffness as well as a comparison between these structures and sandwich structures with homogenous core are also presented. Based on two models found in the literature that relates Youngs moduli and densities, three optimization problems are solved. The geometry of the sandwich panels is kept constant while the properties (Youngs modulus and density) are changed such that to optimize the total mass and/or the flexural rigidity. Choi and Sankars (2005) linear relation between the relative density and Youngs modulus is used to determine the FG asymmetric core and the uniform core parameters such as to obtain panels with same mass and same flexural rigidity. The results of the impact problem prove that there is a significant reduction (approximately 24%) in the maximum shear strain corresponding to maximum impact load of FG asymmetric core compared with the uniform core maximum shear strain, while the maximum normal strains corresponding to maximum impact load are almost the same. The second model is based on the quadratic relation between the relative density and Youngs modulus derived by Gibson and Ashby (1997) The Youngs modulus and the density were varied such that to obtain panels with same flexural rigidity or to obtain panels with same total mass. The results of the impact analysis provided similar results for both problems: among the three sandwich panels (with homogeneous core, symmetric

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143 FG core and asymmetric FG core), the one with FG asymmetric core gives the smaller maximum shear strain corresponding to the maximum impact load (approximately 27% less than the uniform core) while the difference in maximum normal strain corresponding to the maximum impact load is small. Also, when the properties are changed such as to obtain the same flexural rigidity, the sandwich panel with asymmetric FG core gives the smaller total mass. These conclusions emphasize one more time the superior capability of sandwich structures with FG cores over the sandwich structures with uniform cores. The objective of this research was to developed analytical models of sandwich structure with functionally graded cores and to use them in order to compare these structures with traditional sandwich structure with homogenous cores. For the particular problems solved and for the particular type of materials considered, the results indicate functionally graded cores are superior to uniform core to be used in sandwich construction.

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LIST OF REFERENCES Abid Mian, M., Spencer, A. J. M., 1998. Exact Solutions for Functionally Graded and Laminated Elastic Materials, Journal of the Mechanics and Physics of Solids, 46(12), 2283-2295. Abot, J.L., Daniel, I.M., Gdoutos, E., 2002. Contact Law for Composite Sandwich Beams, Journal of Sandwich Structures and Materials, 4, 157-173. Aboudi, J., Pindera, M. J., Arnold, S.M., 1999. Higher-Order Theory for Functionally Graded Materials, Composites: Part B, 30, 777. Aboudi, J., Pindera, M.J., 1995a. Thermoelastic Theory for the Response of Materials Functionally Graded in Two Directions with Applications to the Free-Edge Problem. NASA TM 106882, Lewis Research Center, Cleveland, OH. Aboudi, J., Pindera, M.J., Arnold, S. M., 1995b. Thermo-Inelastic Response of Functionally Graded Composites, International Journal of Solids Structures 32(12), 1675-1710. Aboudi, J., Arnold, S.M., Pindera, M.J., 1994a. Response of Functionally Graded Composites to Thermal Gradients, Composites Engineering, 4, 1-18. Aboudi, J., Pindera, M.J., Arnold, S.M., 1994b. Elastic Response of Metal Matrix Composites with Tailored Microstructures to Thermal Gradients, International Journal of Solids Structures, 31, 1393-1428. Abrate, S., 1998. Impact on Composite Structures, Cambridge, Cambridge University Press. Abrate, S., 1997. Localized Impact on Sandwich Structures with Laminated Facing, Applied Mechanics Review, 50(2), 69-82. Abrate, S., 1994. Impact on Laminated Composite: Recent Advances, Applied Mechanics Review, 47(11), 517-544. Abrate, S., 1991. Impact on Laminated Composite Materials, Applied Mechanics Review, 44 (4), 155-190. Alexandrov, V.M., Pozharskii, D.A., 2001. Three-Dimensional Contact Problems, Dordrecht, Kluwer Academic Publishers. 144

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145 Amada, S., Untao, S., 2001. Fracture Properties of Bamboo, Composites: Part B, 32, 451-459. Amada, S., Ichikawa, Y., Munekata, T., Nagase, V., Shimizu, H., 1997. Fiber Texture and Mechanical Graded Structure of Bamboo, Composites Part B, 28B, 13-20. Ambur, D. R., Cruz, J., 1995. Low-Speed Impact Response Characteristics of Composite Sandwich Panels, Proceedings of 36 th AIAA/ASME/ASCEAHS/ASS Structures, Structural Dynamics and Material Conference, New Orleans, LA, 2681-2689. Anderson, T.A., 2003. A 3-D Easticity Solution for a Sandwich Composite with Functionally Graded Core Subjected to Transverse Loading by a Rigid Sphere, Composite Structures, 60, 265-274. Anderson,T., Madenci, E., 2000. Experimental Investigation of Low-Velocity Impact Characteristics of Sandwich Composites, Composite Structures, 50, 239-247. Apetre, N.A., Sankar, B.V., Ambur, D.R., 2003. Functionally-Graded Sandwich Core with Arbitrary Variation in Properties, AIAA Paper 2002-1683, Proceedings of the 44 th AIAA Structures, Structural Dynamics and Materials Conference, Norfolk, Virginia, April 7-10. Apetre, N.A., Sankar, B.V., Venkataraman, S., 2002. Indentation of a Sandwich Beam with Functionally Graded Core, AIAA Paper 2002-1683, Proceedings of the 43 rd AIAA Structures, Structural Dynamics and Materials Conference, Denver, Colorado, April 22-25. Ashby, M., Evans, A., Fleck, N., Gibson, L.J., Hutchinson, J.W., Wadley, H.G., 2000. Metal Foams: A Design Guide, Boston, Butterworth Heinemann. Barut, A., Madenici, E., Anderson, T., Tessler, A., 2002. Equivalent Single-Layer Theory for a Complete Stress Field in Sandwich Panels under Arbitrary Distributed Loading, Composite Structures, 58, 483-495. Burton,W.S., Noor, A.K., 1994. Three-Dimensional Solutions for Thermomechanical Stresses In Sandwich Panels and Shells, Journal of Engineering Mechanics, 120(10), 2044-2071. Chen, C-F., Frederick, D., 1992. Contact of Isotropic Square Plates with Rigid Spherical Indentors, International Journal of Solids and Structures, 30(5), 637-650. Choi, I.H., Hong, C.S., 1994. New Approach for Simple Prediction of Impact Force History on Composite Laminates, AIAA Journal, 32(10), 2067-2072. Choi, S., Sankar, B.V., 2005. A Micromechanical Method to Predict the Fracture Toughness of Cellular Materials, International Journal of Solids and Structures, 42(5-6), 1797-1817.

PAGE 158

146 Chomkwak, V., Avula, X.J.R., 1990. A Higher-Order Theory for Laminated Composite Plates Using Lagrange Multiplier Technique, Computers and Structures, 37(5), 845-861. Crook, A. W., 1952, A Study of Some Impacts between Metal Bodies by a Piezoelectric Method, Proceedings of the Royal Society, London, Series A, 212, page 377. Delale, F., Erdogan, F., 1983. The Crack Problem for a Nonhomogeneous Plane, ASME Journal of Applied Mechanics, 50, 609-614. El-Hadek, M., Tippur, H.V., 2003. Dynamic Fracture Parameters and Constraint Effects in Functionally Graded Syntactic Epoxy Foams, International Journal of Solids Structures, 40, 1885-1906. Essenburg, F., 1975. On the Significance of the Inclusion of the Effect of Transverse Normal Strain in Problems Involving Beams with Surface Constraints, ASME Journal of Applied Mechanics, 97, 127-132. Fischer-Cripps, A. C., 2000. Introduction to Contact Mechanics, Berlin, Springer Fischer-Cripps, A. C., 1999. The Hertzian Contact Surface, Journal of Materials Science, 34, 129-137. Frostig, Y., Baruch, M., 1994. Free Vibrations of Sandwich Beams with a Transverse Flexible Core: A High Order Approach, Journal of Sound and Vibration, 176(2), 195-208. Frostig, Y., Baruch. M., Vinay, O., Sheinman, I., 1992. High-Order Theory for Sandwich Beams Behavior with Transversely Flexible Core, Journal of Engineering Mechanics, 118(5), 1026-1043. Frostig, Y., Baruch, M., 1990. Bending of Sandwich Beams with Transversely Flexible Core, AIAA Journal, 28(3), 523-531. Fukui, Y., 1991. Fundamental Investigation of Functionally Graded Materials Manufacturing System Using Centrifugal Force, JSME International Journal Series III, 34(1), 144-148. Fukui, Y., Yamanaka, N., Enokida, Y., 1997. Bending Strength of an AI-AI3Ni Functionally Graded Material, Composites Part B, 28B, 37-43. Giannakopoulos, A.E., Suresh, S., 1997a. Indentation of Solids with Gradients in Elastic Properties: Part I. Point force, International Journal of Solids Structures, 34(19), 2357-2392. Giannakopoulos, A.E., Suresh, S., 1997b. Indentation of Solids with Gradients in Elastic Properties: Part II. Axisymmetric Indentors, International Journal of Solids Structures, 34(19), 2393-242.

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147 Gibson, L.J., Ashby, F., 1997. Cellular Solids: Structure and Properties, Cambridge, Cambridge University Press Gladwell, G. M. L., 1980. Contact Problems in the Classical Theory of Elasticity, Germantown, Sijthoff & Noordhoff Han, W., Sofonea, M., 2002. Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Providence, American Mathematical Society and International press. Hertz, H., 1881. On the Contact of Elastic Solids, J. Reine Angew. Math., 92, 156-171. Translated and reprinted in English in Hertzs Miscellaneous Papers, Macmillian and Co., London. Hohe, J., Librescu, L., 2003. A Higher-Order Geometrically Nonlinear Sandwich Shell Theory Accounting for Transverse Compressibility of the Core, Proceedings at 6 th International Conference on Sandwich Structures, Ft. Lauderdale, Florida, CRC Press, 796-805. Hsueh, C.H., Lee, S., 2003. Modeling of Elastic Thermal Stresses in Two Materials Joined by a Graded Layer, Composites Part B: Engineering, 34, 747-752. Johnson, K. L., 1985. Contact Mechanics, Cambridge, Cambridge University Press. Khludnev, A.M., Sokolowski, J., 1997. Modeling and Control in Solid Mechanics, Basel, Birkhauser Verlag Koissin, V. Skvortsov, V, Krahmalev, S., Shilpsha, A, 2004. The Elastic Response of Sandwich Structures to Local Loading, Composite Structures, 63, 375-385. Koizumi M., 1997. FGM Activities in Japan, Composites Part B, 28B, 1-4. Kollar, L.P., Springer, G.S., 2003. Mechanics of Composite Structures, Cambridge, Cambridge University Press. Lambros, J., Santare, M.H., Li, H., Sapna, G.H., 1999. A Novel Technique for the Fabrication of Laboratory Scale Model of FGM, Experimental Mechanics, 39(3). Lee, S. M., Tsotsis, T. K., 2000. Indentation Failure Behavior of Honeycomb Sandwich Panels, Composite Science and Technology, 60, 1147-1159. Lee, L.J., Huang, K.Y., Fann, Y.J., 1993. Dynamic Response of Composite Sandwich Plates Impacted by a Rigid Ball, Journal of Composite Materials, 27(13), 1238-1256. Liao, W.C., 2001. Smooth Indentation of a Multilayer Laminated by a Cylindrical Indentor, Composite Science and Technology, 61, 1801-1809

PAGE 160

148 Lo, K.H., Christensen, R.M., Wu, E.M., 1977. A Higher-Order Theory of Plate Deformation, Part 1: Homogeneous Plates, Part 2: Laminate Plates, ASME Journal of Applied Mechanics, 44, 663-676. March, H.W., 1948. Effects of Shear Deformation in the Core of a Flat Rectangular Sandwich Panel 1. Buckling under Compressive End Loads, 2. Deflection under Uniform Transverse Load, U.S. Forest Products Laboratory Report 1583. Markworth, A.J., Ramesh, K.S., Parks, W. P., 1995. Modeling Studies Applied to Functionally Ggraded Materials, Journal of Materials Science, 30, 2183-2193. Matthews, F.L., Davies, G.A.O., Hitchings, D., Soutis, C., 2001. Finite Element Modeling of Composite Materials and Structures, Boca Raton, CRC Press. McCormack, T.M., Miller, R., Kesler, O., Gibson, L.J., 2001. Failure of Sandwich Beams with Metallic Cores, International Journal of Solids and Structures, 38, 4901-4920. Meyer-Piening, H.R., 1989. Remarks on Higher Order Sandwich Stress and Deflection Analyses, Proceeding of 1 st International Conference on Sandwich Construction, Stockholm Sweden, 107-127. Midlin, R. D., 1951. Influence of Rotary Inertia and Shear on Flexural Motion of Isotropic Elastic Plates, ASME Journal of Applied Mechanics, 73, 69-77. Mittal, R.K., 1987. A Simplified Analysis of the Effects of Transverse Shear on the Response of Elastic Plates to Impact Landing, International Journal of Solids and Structures, 23(8), 1587-1596. Noor, A. K., Burton, W. S., Bert, C. W., 1996. Computational Models for Sandwich Panels and Shells, Applied Mechanics Review, 49, 155198. Ogorkiewicz, R. M., Sayigh, A.M., 1973. Deflection of Carbon Fibre/Acrylic Foam Sandwich Beams, Composites, Nov. Olsson, R., 1997. Theory for Small Mass Impact on Sandwich Panels, Proceedings of the EUROMECH 360 Colloquium held in Saint-ETIENNE, Fr. Olsson, R., McManus, H. L., 1996. Improved Theory for Contact Indentation of Sandwich Panels, AIAA Journal, 34(6), 1238 1244. Olsson, R., 1992. Impact Response of Orthotropic Composite Plates Predicted from a One-Parameter Differential Equation, AIAA Journal, 30(6), 1587-1596. Pagano, N. J., 1970. Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates, Journal of Composites Materials, 4, 20-34.

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149 Peled, D., Frostig, Y., 1994. High-Order Bending of Sandwich Beams with Transversely Core and Nonparallel Skins, Journal of Engineering Mechanics, 120(6), 1255-1269. Petras, A., Sutcliffe, M.P.F., 1999. Indentation Resistance of Sandwich Beams, Composite Structures, 46, 413-424. Pindera, M. -J., Dunn, P., 1995. An Evaluation of Coupled Microstructural Approach for the Analysis of Functionally Graded Composites via the Finite Element Method, NASA CR 195455. Lewis Research Center, Cleveland, OH. Reddy, J. N., 2003. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, Boca Raton, CRC Press. Reddy, J. N., 2000. Analysis of Functionally Graded Plates, International Journal for Numerical Methods in Engineering, 47, 663-684. Reddy, J. N., 1990. A General Non-linear Third-order Theory of Plates with Moderate Thickness, International Journal for Non-Linear Mechanics, 25, 677-686. Reddy, J. N., 1984. A Simple Higher-order Theory for Laminated Composite Plates, Journal for Applied Mechanics, 51, 745-752. Reddy, J.N., Cheng, Z.Q., 2001. Three-dimensional Thermomechanical Deformations of Functionally Graded Rectangular Plates, European Journal of Mechanics, A/Solids, 20, 841. Reddy, J.N., Miravete, A., 1995. Practical Analysis of Composite Laminates, Boca Raton, CRC Press. Reissner, E., 1975. On Transverse Bending of Plates, Including the Effects of Transverse Shear Deformation, International Journal of Solids and Structures, 11, 569-573. Reissner, E., 1945. The Effects of Transverse Shear Deformation on the Bending of Elastic Plates, ASME Journal of Applied Mechanics, 67, 69-77. Rooney, F. and Ferrari, M, 1999. Tension, Bending and Flexure of Functionally Graded Cylinders, International Journal of Solids and Structures, 38, 413-421. Sankar, B.V., 2001. An elasticity solution for functionally graded beams, Composites Science & Technology, 61, 689-696. Sankar, B.V. 1992. Scaling of Low-velocity Impact for Symmetric Laminates. Journal of Reinforced Plastics and Composites, 11(3), 296-309. Sankar, B.V., 1989. Smooth Indentation of Orthotropic Beams, Composite Science and Technology, 34, 94-111.

PAGE 162

150 Sankar, B.V., Tzeng, J.T., 2002. Thermal Stresses in Functionally Graded Beams, AIAA Journal, 40(6), 1228-1232. Sankar, B.V., Sun, C.T., 1985. Smooth Indentation of an Initially Stressed Orthotropic Beam, International Journal of Solids Structures, 21(2), 161-176. Sankar, B.V., Sun, C.T., 1983. Indentation of a Beam by a Rigid Cylinder, International Journal of Solids and Structures, 19, 293-303. Sarkar, P., Datta, S., Nicholson, P. S., 1997. Functionally Graded Ceramic/Ceramic and Metal/Ceramic Composites by Electrophoretic Deposition, Composites Part B, 28B, 49-56. Sburlati, R., 2002. The Contact Behavior Between a Foam Core Sandwich Plate and a Rigid Indentor, Composites: Part B, 33, 325-332. Shivakumar, K. N., Elber, W., Illg,W., 1985. Prediction of Impact Force and Duration During Low-Velocity Impact on Circular Composite Laminates, Trans. ASME, Journal of Applied Mechanics, 52, 674-680. Singh, K.P., Paul, B., 1974. Numerical Solution of Non-Hertzian Elastic Contact Problems, ASME Journal of Applied Mechanics, 41, 484-490. Sokolinsky, V. S., Nutt, S. R. 2004. Consistent Higher-Order Dynamic Equations for Soft Core Sandwich Beams, AIAA Journal, 42(2). Sun, C. T., Wu, C.L., 1991. Low-velocity Impact on Composite Sandwich Panels, Proceedings of 32 nd AIAA/ASME/ASCEAHS/ASS Structures, Structural Dynamics and Material Conference, Baltimore, MD, 1123-1129. Suresh, S., 2001. Graded Materials for Resistance to Contact Deformation and Damage, Science, 292, 2447-2451. Suresh, S., Mortensen, A., 1998. Fundamentals of Functionally Graded Materials, IOM Communications Limited, London, United Kingdom. Swanson, S. R., 2004. Core compression in Sandwich Beams under Contact Loading, Composite Structures, 64, 389-398 Swanson, S.R., 1999. An Examination of a Higher Order Theory for Sandwich Beam, Composite Structures, 44,169-177. Tan, T.M., Sun, C. T., 1985. Use of Statical Indentation Laws in the Impact Analysis of Laminated Composite Plates. Journal of Applied Mechanics, 52, 6-12. Tessler, A., Tsui, T., Saether, E., 1995. A {1, 2} Order Theory for Elastodynamic Analysis of Thick Orthotropic Shells, International Journal of Solids and Structures, 32(22), 3237-3260.

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151 Tessler, A., 1993. An Improved Plate Theory of {1, 2} Order for Thick Composite Laminates, International Journal of Solids and Structures, 30(7), 981-1000. Timoshenko,S., Goodier.J.,1970. Theory of Elasticity, New York, McGraw-Hill. Vel, S. S., Batra, R. C., 2003. Three-dimensional Analysis of Transient Thermal Stresses in Functionally Graded Plates, International Journal of Solids and Structures, 40, 7181. Vel, S. S., Batra, R. C., 2002. Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates, AIAA Journal, 40(7). Venkataraman, S., Sankar, B.V., 2001. Analysis of Sandwich Beams with Functionally Graded Core, AIAA Paper 2001-1281, Proceedings of the 42nd AIAA Structures, Structural Dynamics and Materials Conference, Seattle, Washington, April 16-19. Vinson, J.R., 2001. Sandwich Structures, Applied Mechanics Review, 54(3), 201-214. Vinson J.R., Sierakowski, R.L., 1986. The behavior of structures composed of composite materials, Dordrecht, Martinus Nijhoff Publisher. Vinson J.R., Chou, T.W., 1975. Composite Materials and Their Use in Structures, John Wiley and Sons, New York, Applied Science Publishers LTD. Wang, G-Y., Yew, C.H., 1989. Impact Damage in Composite Laminates, Computers and Structures, 37(6), 967-982. Watari, F., Yokoyama, A., Saso, F., Uo, M., Kawasaki, T., 1997. Fabrication and Properties of Functionally Graded Dental Implant, Composites: Part B, 28B, 5-11. Whitney, J. M., Pagano, N. J., 1970. Shear Deformation in Heterogeneous Anisotropic Plates, Journal of Applied Mechanics, 92, 1031-1036. Williams D., Leggett D.M.A., Hopkins H. G., 1941. Flat Sandwich Panels under Compressive End Loads, A.R.C., R & M 1987. Willis, J.R., 1966. Hertzian Contact of Anisotropic Bodies, Journal of Mechanics and Physics of Solids, 12, 163-176 Woo, J., Meguid, S.A., 2000. Nonlinear Analysis of Functionally Graded Plates and Shallow Shells. International Journal of Solids and Structures, 38, 7409-7421. Wu, E., Yen, C-S., 1994. The Contact Behavior between Laminated Composite Plates and Rigid Spheres, ASME Journal of Applied Mechanics, 61, 60-66. Yang, P.C., Norris, C.H., and Stavsky, Y., 1966. Elastic Wave Propagation in Heterogeneous Plates, International Journal of Solids and Structures, 2, 665-684.

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152 Yang, S. H, Sun, C. T., 1982. Indentation Law for Composite Laminates, Composite Materials: Testing and Design, ASTM STP 787, American Society of Testing and Materials, 425-449. Zenkert D., 1997. The Handbook of Sandwich Construction, Engineering Materials Advisory Service Ltd. Zhu, H., Sankar, B. V., 2004. A Combined Fourier Series Galerkin Method for the Analysis of Functionally Graded Beams. Journal of Applied Mechanics, 71(3), 421-424.

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BIOGRAPHICAL SKETCH Nicoleta A. Apetre graduated in 1998 with a Bachelor of Science in mathematics-mechanics from University of Bucharest, Romania. In 2000 she received her Master of Science in mathematics-mechanics from the same university. Deciding to pursue a PhD in engineering, she joined the Department of Mechanical and Aerospace Engineering at University of Florida, in January 2001. 153


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Permanent Link: http://ufdc.ufl.edu/UFE0012061/00001

Material Information

Title: Sandwich Panels with Functionally Graded Core
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0012061:00001

Permanent Link: http://ufdc.ufl.edu/UFE0012061/00001

Material Information

Title: Sandwich Panels with Functionally Graded Core
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0012061:00001


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SANDWICH PANELS WITH FUNCTIONALLY GRADED CORE


By

NICOLETA ALINA APETRE



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005





























Copyright 2005

by

Nicoleta Alina Apetre



























This document is dedicated to my loving family: my mother, Elisabeta Apetre, my sister,
Dana and my brother, Nicu and to my devoted boyfriend Laurence as they have offered
their support and love throughout this endeavor. No words said can express how indebted
I am to them for what I am today and for what I shall accomplish in the future.















ACKNOWLEDGMENTS

I am profoundly grateful to my advisor, Dr. Bhavani Sankar, for scientific

guidance, moral support and endless patience. I would like to thank Dr. Peter Ifju for his

help and ideas throughout this enterprise.

I would also like to thank my friend, mentor and committee member, Dr. Oana

Cazacu, for her guidance and support on my research and my life. Her comments and

advice made this dissertation a much more useful document.

At the same time, I wish to thank Dr. Raphael T. Haftka and Dr. Bjorn Birgisson

for serving on my committee and for their assistance and guidance.

I express sincere appreciation to a friend and mentor, Dr. Satchi Venkataraman, for

his continuous scientific guidance and moral support during the years.

I would like to thank all my professors from whom I learned so much, especially

the late Dr. E. Soos, who showed me the path I am walking now.

Special thanks go to all my friends, colleagues and family for their continual love,

support, and encouragement throughout my time in graduate school and my life.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ......... ...... ................................... ....... ............. vii

LIST O F FIG U R E S ......... .......... ............................... .. ................ .. viii

ABSTRACT ........ .............. ............. .. ...... .......... .......... xii

CHAPTER

1 INTRODUCTION AND BACKGROUND ..................................... ...............

In tro d u ctio n ...................................... .................................. ................. .
L literature R review ................. .................................. ........ ........ .......... ...... .
Sandw ich Plate Theories ............................................... ............................ 7
Contact on Composite Materials ................... ...... ............. 14
Cellular Solids and Functionally Graded Materials ........................................28
Objectives and A approaches ......................................................... ............... 36

2 ANALYTICAL MODELING OF SANDWICH BEAMS WITH
FUNCTIONALLY GRADED CORE ............................................. ............... 38

Exact Method for Sandwich Structures with Functionally Graded Core .................38
Fourier Galerkin Method for Sandwich Structures with Functionally Graded
C o re .................. .................. ............. .......... .... ............ ......... ................ 4 2
Analytical Models for Sandwich Structures with Functionally Graded Core............47
An Equivalent Single-Layer First-Order Shear Deformation Theory .................47
An Equivalent Single-Layer Third-Order Shear Deformation Theory ..............50
A Higher-Order Shear Deformation Theory ................... ...........................52
Results for Sandwich Beam with Functionally Graded Core under a Distributed
L o a d .................................................................................................................. 5 7
Sum m ary and C onclu sions .............................................................. .....................67

3 CONTACT AND LOW-VELOCITY IMPACT Of SANDWICH BEAMS WITH
FUNCTIONALLY GRADED CORE ............................................. ............... 69

Assum ed Stress Distribution M ethod ...................................................................... 71
M ethod of Point M atching......................................................... ............... 74









Quasi-Static Impact of a Sandwich Beam with FG Core .......................................76
R e su lts .................. ........................................................ ................ 7 8
Sum m ary and Conclusions ......................................................... .............. 103

4 MASS OPTIMIZATION FOR A SANDWICH BEAM WITH FUNCTIONALLY
GRADED CORE ................................................ ........................... 105

Linear M odel .............. ................... ......... .................. .......... 105
Case 1: Sam e Flexural Rigidity ................................................................... 109
Case 2: Sam e M ass and Flexural Rigidity..................................... ............... 112
Q u adratic M o d el ............ ...... .. ........ .. .. ........................................ .. .... 1 18
Case 1: Sam e Flexural Rigidity...................................................................... 121
Case 2: Sam e Total M ass ............................................................................ 130
Sum m ary and Conclusions ......................................................... .............. 138

5 SUMMARY AND CONCLUSIONS ........... ................................. ...............140

LIST OF REFEREN CES ........................................................... .. ............... 144

BIOGRAPHICAL SKETCH ............................................................. ............... 153
















LIST OF TABLES


Table page

3-1 Maximum normal and shear strains for a given impact energy ............................94

3-2 Core thicknesses for different materials with same flexural stiffness, D11 and
sam e global stiffness ...................... ...... ............ ............................ 97

4-1 Variations of Young's modulus, density, mass and flexural rigidity..................... 107

4-2 Total mass of sandwich structures with different density variations and different
core th ick n esses .................................................................................... 1 1 1

4-3 Maximum normal and shear strains for a given impact energy .............................117

4-4 Variations of Young's modulus, density, mass and flexural rigidity................. 120

4-5 Total mass of sandwich structures with same geometry and different density
variations ..................................... ................................. ......... 123

4-6 Maximum normal and shear strains normal and interfacial shear stresses for a
given im pact energy ......................... ....... ..... .. ...... .............. 127

4-7 Flexural rigidity of sandwich structures with same geometry and different
density variations......... ............................................................ ... ....... ....... 132

4-8 Maximum normal and shear strains and stresses for a given impact energy .........134
















LIST OF FIGURES


Figure page

1.1 The behavior of (a) flexible face sheet and (b) rigid face sheet..............................4

2.1 Sandwich beam with functionally graded core with schematic of the analysis
elem ents....................................................... ................... .... ... .. .... 39

2.2 Traction forces and displacements at the interfaces of each element in the FGM
sa n d w ich b e am ................................................................................................... 4 0

2.3 The beam geom etry .................. ................................... ................. 48

2.4 The beam geom etry .................. ................................... ................. 53

2.5 Through-the-thickness variations of core modulus...............................................58

2.6 Variation of transverse shear stresses in the sandwich structure with FGM core....59

2.7 Variation of o-, stresses in the sandwich structure with FGM core ......................... 60

2.8 Variation of bending stresses through the thickness of the FGM core ....................61

2.9 Variation of u displacements through the thickness of the sandwich structure
w ith F G M core ..................................................... ................ 6 1

2.10 N on-dim ensional core m odulus ........................................ .......................... 62

2.11 C om prison of deflections ............................................... ............................ 64

2.12 Comparison of longitudinal displacement..................................... ............... 64

2.13 Comparison of axial stress in the core. ........................................ ............... 65

2.14 Comparison of shear strain in the core .................. ........ ......... ..... .......... 66

2.15 Comparison of shear stress in the core .................. ........ ......... ..... .......... 67

3.1 Dimensions of the sandwich panel and the contact length, 2c ..............................72

3.2 The stress distribution under the indenter. .................................... .................72









3.3 Illustration of relation between w deflection and radius of curvature of the
deform ed surface. ............................. ............ .............................73

3.4 Discretization of contact load for the method of point matching.............................74

3.5 Low-velocity impact m odel ........................................................................77

3.6 Through the thickness variations of core modulus............................................80

3.7 Variation of the horizontal displacement ..... ......... ........................................81

3.8 Deflection, w, in the second half of the sandwich beam ........................................81

3.9 V ariation of the deflection.............................................................. .... ...........82

3.10 Contour plot of normal strain in the functionally graded asymmetric core ............83

3.11 Contour plot of normal strain in the functionally graded symmetric core...............83

3.12 Contour plot of normal strain in the homogeneous core........................................83

3.13 Contour plot of shear strain in the functionally graded asymmetric core ...............84

3.14 Contour plot of shear strain in the functionally graded symmetric core ................ 84

3.15 Contour plot of shear strain in the homogeneous core.....................................85

3.16 V ariation of the axial stress ........................................................ ............. 85

3.17 Variation of the axial stress through the thickness in functionally graded core.......86

3.18 Variation of the normal compressivee) stress........................ .................... 87

3.19 Face sheet thickness influence on variation of the normal compressivee) stress.....88

3.20 Indentor radius influence on variation of the normal compressivee) stress .............89

3.21 Face sheet Young's modulus influence on variation of the normal compressivee)
str e s s ................................................... ...................... ................ 8 9

3.22 Shear stress in functionally graded asymmetric core................... ....... .........90

3.23 Contour plot of shear stress in linear asymmetric core ........................................91

3.24 Contour plot of shear stress in linear symmetric core ...... ..................................91

3.25 Contour plot of shear stress in homogeneous core .......... ................................... 91

3.26 Variation of contact force with indentation depth in functionally graded beams ....92









3.27 Relation between contact force and the vertical displacement.............................93

3.28 Contact force vs. maximum normal strain .................................... ............... 95

3.29 Contact force vs. maximum shear strain ...................................... ............... 96

3.30 Variation of contact force with global deflection ...............................................97

3.31 Comparison of maximum normal strains for different core materials ...................99

3.32 Comparison of maximum shear strains for different core materials........................99

3.33 Finite element simulation for half of the beam ..................................................... 100

3.34 Finite element simulation of the contact between the functionally graded
sandwich beam and the rigid spherical indentor ...................................................101

3.35 Contact pressure under indenter and total load for three models...........................102

3.36 Deflections at core top face sheet interface for same contact length and
different contact load ......... ...................................................... .. ..... .. 102

3.37 Deflections at core top face sheet interface for different contact length and
sam e contact load ................... .... .......................................... .. ............. 103

4.1 Geometry and notations for the three cases investigated ............... ...... ......... 108

4.2 Through the thickness non-dimensional variations of core modulus and density .110

4.3 Through the thickness non-dimensional variations of core modulus and density .113

4.4 Relation between contact force and vertical displacement ...............................113

4.5 Variation of contact force with indentation depth...............................114

4.6 Variation of contact force with contact length...................................................115

4.7 Variation of contact force with contact length...................................................115

4.8 Contact force vs. maximum normal strain .............................116

4.9 Contact force vs. m axim um shear strain ............................ ......... ..................... 117

4.10 Young's modulus variation ................................. ................................... 119

4.11 C ore Y oung's m odulus................................................. .............................. 123

4.12 C ore densities variation ...................... .. .. ......... .. ...................... ............... 124









4.13 Relation between contact force and vertical displacement ..................................124

4.14 Relation between contact force and contact length...............................................125

4.15 Relation between contact force and contact length...............................................125

4.16 Relation between indentation (core compression) and contact length.................126

4.17 Contact force vs. core maximum normal strain ............................................. 128

4.18 Contact force vs. core maximum shear strain ................................................129

4.19 Contact force vs. core maximum normal stress ............................................. 129

4.20 Contact force vs. maximum interfacial shear stress..............................................130

4.21 C ore Y oung's m odulus................................................. .............................. 132

4.22 Core densities variation .......................................................... ............... 133

4.23 Relation between contact force and vertical displacement ..................................134

4.24 Relation between contact force and contact length............................................135

4.25 Contact force vs. core maximum normal strain ............................................. 136

4.26 Contact force vs. core maximum shear strain. ...............................................136

4.27 Contact force vs. core maximum normal stress ............................................. 137

4.28 Contact force vs. maximum interfacial shear stress..............................................137















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SANDWICH PANELS WITH FUNCTIONALLY GRADED CORE


By

Nicoleta Alina Apetre

December 2005

Chair: Bhavani V. Sankar
Major Department: Mechanical and Aerospace Engineering

Although sandwich structure offer advantages over other types of structures, it is

important to develop new types of materials in order to obtain the absolute minimum

weight for given conditions (e.g., structural geometry and loadings). One alternative is

represented by functionally graded materials (defined as materials with properties that

vary with location within the material in order to optimize a prescribed function).

This study presents different analytical and finite element models for sandwich

structures with functionally graded core. The trade-off between weight and stiffness as

well as a comparison between these structures and sandwich structures with homogenous

core is also presented. The problem of low-velocity impact between a sandwich structure

with functionally graded core and a rigid spherical indentor is solved. A few advantages

and disadvantages of these types of structure are presented.

With the new developments in manufacturing methods these materials can be used

for a large number of applications ranging from implant teeth to rocket frames.














CHAPTER 1
INTRODUCTION AND BACKGROUND

Introduction

A sandwich structure consists of two thin, stiff, and strong face sheets connected by

a thick, light and low-modulus core using adhesive joints in order to obtain very efficient

lightweight structure (Zenkert, 1997; Vinson, 2001). In most of the cases the faces carry

the loading, both in-plane and bending, while the core resists transverse shear loads. A

sandwich operates in the same way as an I-beam with the difference that the core of a

sandwich is of a different material and is stretched out as a continuous support for the

face sheets. The main advantage of a sandwich structure is its exceptionally high flexural

stiffness-to-weight ratio compared to other architectures. As a consequence, sandwich

construction results in lower lateral deformations, higher buckling resistance, and higher

natural frequencies than do other structures. Thus, for a given set of mechanical and

environmental loads, sandwich construction often results in a lower structural weight than

do other configurations. Few of the drawbacks of sandwich structures are: manufacturing

methods, quality control and joining difficulties.

The idea to use two co-operating faces separated by a distance emerged at the

beginning of 19th century. The concept was first applied commercially during World War

II in airplanes. Some of the first theoretical and experimental works on sandwich

constructions were published in the late 1940's (Williams et al., 1941, and March, 1948).

Since then, the use of sandwich structures has increased rapidly. The need for high-

performance, low-weight structures assures that the sandwich construction will continue









to be in demand. In view of increasing the use of sandwich structures in the aerospace

industry, critical issues such as impact resistance, optimization, fatigue and fracture

should be addressed. Lately, due to the difficulty to obtain closed-form solutions for the

governing differential equations, finite element method has become the most used design

tool for panel analysis.

The core material is perhaps the most important component of a sandwich

structure. The following properties are required for the core material: low density in order

to add as little as possible to the total weight; Young's modulus perpendicular to the face

sheet should be fairly high to prevent excessive deformation in the thickness direction

and therefore a rapid decrease in flexural rigidity. Even though the transverse force

creating normal stresses perpendicular to the core are usually low, even a small decrease

in core thickness would create a large decrease in flexural rigidity. This is crucial in the

case of impact loads wherein the contact force acts over a small area. The core is mainly

subjected to transverse shear stresses. The core shear strains contribute to global

deformation and core shear stresses. Thus, a core must be chosen such that it would not

fail under the applied transverse load and with a shear modulus high enough to give the

required shear stiffness. Other functions such as thermal and acoustical insulation depend

mainly on the core material and its thickness.

The core materials have been fashioned in various forms and developed for a range

of applications. Few examples are: foam or solid cores (where properties can be tailored

by orienting the cell structure), honeycomb cores, web cores and truss cores. Honeycomb

cores offer the greatest shear strength and stiffness to weight ratio but require care in

ensuring adequate bonding to the face-sheets. Also the core is highly anisotropic and









forming complex curved shapes is difficult. High-density cellular thermoplastic foams,

such as poly (vinyl chloride) (PVC) and polyurethane (PUR) do not have the same high

stiffness to weight ratio as honeycomb materials but they have more advantages: they are

less expensive, they are continuous at the macroscopic level, the foam surface is easier to

bond, and they ensure high thermal and acoustic insulation (cellular cores have a very

low thermal conductivity, e.g., Gibson and Ashby, 1997). A metallic foam core sandwich

structure offers also a number of advantages. They can be made with integral skin,

eliminating the need for adhesive layer, they can be formed in curved shapes, and their

properties are nearly isotropic (Gibson and Ashby, 1997).

Foreign objects impact (such as tool drops, hail, bird strikes, and runway debris)

can introduce damage and significantly reduce the strength of the sandwich structure.

Unlike their solid metallic counterparts, predictions of the effects of low-velocity impact

damage on sandwich structures are difficult because significant internal damage is

achieved at impact energy levels lower than those required to create visible damage at the

surface.

Low-velocity impact, one of the topics of present research is described by low

projectile velocity; therefore the damage can be analyzed by assuming that the structure

is under a quasi-static loading. Impact/contact produces both overall deformation and

local deformation. Usually, if the load is applied on top face sheet, the bottom face sheet

experience only global deformation so many models assume a clamped bottom face sheet

in order to eliminate the global deflection and focus the research on local effects.









There are different factors that can influence the local effect of indentation stress

field: face sheet elastic modulus and its thickness, the core density and its thickness. One

of the factors that influence the structure behavior under an indentor is the face sheet

stiffness: for flexible face-sheets there is a local deformation under the indentor/load

whereas the very rigid face-sheets spread the load (Figure 1.1). Usually the local failure

starts in the core and results in core crushing, delamination and therefore significant

reduction of the sandwich strength.




A(a)




(b)


Figure 1.1: The behavior of (a) flexible face sheet and (b) rigid face sheet


Contact laws (defined as relation between the contact force and indentation or core

compression) or relative displacement of the indentor and the target for the indentation of

a sandwich structure are significantly different from those for monolithic laminates. They

are usually nonlinear. With sandwich structures the indentation is dominated by the

behavior of the core material, which crushes as the shear stress and compressive stress

become large. Compressive stress in transverse direction is not uniform through the

thickness and is not the only stress component to be considered in order to determine the

onset of core failure. Few of the contact problem difficulties are: forced surface

displacement, initially unknown contact length/surface and pressure distribution.









Although sandwich structure offer advantages over other types of structures, it is

important to develop new types of materials in order to obtain the absolute minimum

weight for given conditions (e.g., structural geometry and loadings). These new

sandwiches should be compared with other sandwich construction and with alternative

structures in order to select the best configuration. One of the new alternatives is a

sandwich structure with functionally graded core.

Functionally graded materials (FGM) possess properties that vary gradually with

location within the material; consequently they are inhomogeneous materials on the

macroscopic scale. New developments in manufacturing methods offer the designer the

opportunity to tailor the material's microstructure in order to enhance their structural

performance. The variation of the properties can be achieved by gradually changing the

volume fraction of the constituents' materials, by using reinforcements with different

properties, sizes and shapes, by interchanging the roles of reinforcement and matrix

phases in a continuous manner as well as by adjusting the cell structure. The result is a

microstructure that produces continuously changing thermal and mechanical properties at

the macroscopic or continuum level.

FGMs differ from composites wherein the volume fraction of the reinforcing

inclusion is uniform throughout the composite. The closest analogy of FGM is a

laminated composite face sheets, but the latter possess distinct interfaces across which

properties change abruptly. FGMs have a large spectrum of applications. They can be

used as thermal barrier coatings (Koizumi, 1997), interfacial zones to improve the

bonding strength and to reduce the residual stresses (Hsueh and Lee, 2003). Cellular

materials with continuously varying density or pore size (porosity) that can be used as a









core for a sandwich structure is an example of new FGMs. Although FGMs are highly

heterogeneous, it will be useful to idealize them as continue with properties changing

smoothly with respect to the spatial coordinate. This will enable obtaining closed-form

solutions to some fundamental solid mechanics problems, and also will help in

developing finite element models of structures made of FGMs.

The major objective of this dissertation is to develop new analytical models for

sandwich structures with functionally graded core. Particular attention will be paid to

contact and impact problems involving sandwich structures. This chapter provides a

framework and background for sandwich structures and functionally graded materials in

order to give a perspective for present research. Chapter 2 introduces analytical models

for a sandwich beam with functionally graded core and compares them with analytical

models presented in the literature. The governing equations for sandwich structures with

FG cores are solved for two types of cores by six different methods: exact solutions are

presented for the exponential variation of core Young Modulus; a combination of Fourier

series and Galerkin method for a polynomial variation of core Young Modulus; two

equivalent single-layer theories based on assumed displacements, a higher-order theory

and a finite-element analysis. Chapter 3 solves the problems of contact and low-velocity

impact between a rigid cylindrical indentor and a sandwich beam with functionally

graded core. A comparison of the total mass of different panels is discussed in Chapter 4,

where different optimization problems for sandwich structure with FG core based on two

models presented in the literature. They are the linear relation between relative density

and Young's modulus (Choi and Sankar, 2005) and quadratic relation between relative

density and Young's modulus (Gibson and Ashby, 1997). The results indicate that the









functionally graded cores can be used to mitigate or prevent impact damage in sandwich

composites, so it is shown that materials with gradients in mechanical properties offer

better opportunities than traditional homogenous materials.

Literature Review

The equations describing the behavior of sandwich structures are similar to the

shear deformable plate theory for composite laminates (e.g., Reddy, 2003) if each of the

core, face sheets and adhesive layers is modeled as a continuum. The main difference

between the traditional and modern sandwich models is the transverse flexibility of the

core. A flexible/soft core is a core for which the ratio of Young's moduli of the face

sheets to the core is high (for structural panels this ratio can be between 500 and 1000). If

a transversely flexible core is used a higher order theory is necessary.

Typically sandwich composites are subjected to out of plane loadings where the

primary loads are applied perpendicular to the panel surface. Localized loads are one of

the major causes of failure in sandwich structures, because in most of the cases the face

sheets are thin. The load, transmitted through the face sheet causes the core to be

subjected to significant deformations locally, which can cause high shear and normal

stresses that can exceed the allowable stress for the flexible, weak core material.

In order to give a framework to the present research, this chapter presents a

literature review on three fields: composite plates theories, contact on composite plates

models and functionally graded materials.

Sandwich Plate Theories

A considerable amount of literature exists on sandwich panels as they are used in

large number of applications varying form high-performance composites in aerospace to

low-cost materials for building constructions. The literature includes monographs,









conference proceedings, books and bibliographies. Noor et al. (1996) provide an

excellent literature review on computational models for sandwich structures that includes

over 800 references. The paper reviews the literature, classify the models and identify

future direction for research. Similar, Vinson (2001) offers a general introduction to the

structural mechanics involved in the field of sandwich structures and a sufficient number

of references for further investigation. A large number of books on composite materials

were published: they range from physical and mathematical analyses or mechanics of

structural components composed of composite materials (Vinson and Chou, 1975; Kollar

and Springer, 2003; Vinson and Sierakowski, 1986), finite element modelling (Matthews

et al., 2001) to reference for practical engineering and designing (Reddy and Miravete,

1995).

The limitations of classical plate theory in describing complex problems (e.g.

contact/impact problems, cutouts or laminate plates) necessitated the development of

higher-order theories. The higher-order refers at the level of truncation of terms in a

power series expansion about the thickness coordinate for displacements.

The models investigated here can be classified into: single-layer theories (where the

assumed displacement components represent the weight-average through the thickness of

sandwich panel), layer-wise theories (where separate assumptions for displacements in

each layer are made) and exact theories (equilibrium equations are solved without

displacements assumptions). Although discrete-layer theories are more representative for

sandwich construction than the single-layer theories, they experience computations

difficulties from a large number of field variables in proportion with number of layers.









First, a literature review on equivalent single-layer theory is presented in order to give

details of the field development.

Reissner (1945) and Midlin (1951) were the first to propose a theory that included

the effect of shear deformation. The displacements were assumed as:

u(x, y, z) = u (x, y) + z{V (x, y)
v(x, Y, z) = Vo (x, y) + zy (x, y) (1.1)
w(x, y,z)= w (x, y)

where z is the coordinate normal to the middle plane, u and v are in-plane displacements

and w is out-of-plane displacement and where y, V/y, w, are weighted averages. Midlin

introduced the correction factor into the shear stress to account for the fact that (1.1)

predicts a uniform shear stress through the thickness of the plate. Considering the terms

in (1.1) as first terms in a power series expansion in z, it is seen that the classical theory

and the shear theory are of the same order of approximation. Yang et al. (1966) extended

Mindlin's theory for homogeneous plate to laminates consisting of an arbitrary number of

bonded layers. Yang et al. (1966) used the theory to solve the frequency equations for

propagation of harmonic waves in a two-layer isotropic plate of infinite extent. Based on

the same model (Mindlin's theory), Whitney and Pagano (1970) developed a theory for

anisotropic laminates plates consisting of an arbitrary number of bonded anisotropic

layers that includes shear deformation and rotary inertia. Displacement field is assumed

to be linear in thickness coordinate.

The next order theory was proposed by Essenburg (1975) and was based on the

following assumptions:









u(x, y, z) = u (x, y) + z ((x, y)
v(x,, z) = (x, y) + zyy (x, y) (1.2)
w(xV Y, z) = Wo (x, y) + zV (x, y) + z24 (x, y)

Essenburg (1975) presented the advantages of the above expansion for the contact

problem.

In order to analyze the bending of a plate with a hole, Reissner (1975) included

third order terms in the in-plane displacements z-expansion:

u(x, y, z) = z{ (x, y) + z3 (x, y)
v(x, y, z) z= y (x, y) + z3( (x,y) (1.3)
w(x, y, Z) = w(x, y)+ z2 (x, y)

It should be noted that above assumptions neglect the contribution of in-plane

modes of deformation.

Lo et al. (1977) theory included both in-plane and out-of-plane modes of

deformation was based on:

u(x,, y,z) = Uo (x, y) + zI (x, y) + z2 (x, y) + z3 (x, y)
v(x, y,z) = v (x, y) + zy/ (x, y) + z2 (x, y) + z3' (x, y) (1.4)
w(x, y, z) = Wo (x, y) + z (x, y) + z2C (x, y)

The principle of potential energy is used to derive the governing equations and

boundary conditions and the theory is compared with lower-order plate theories and with

exact elasticity solution in order to assess the importance of the displacements

expansions. The theory was developed to model the behavior of infinite laminated plates.

Chomkwak and Avula (1990) modified previous theory in order to be applied to finite

rectangular laminate composite plates. The Lagrange multipliers are used in order to

constrain the displacement functions to satisfy the stress boundary conditions.









Reddy (1984) developed a third-order Shear Deformation theory (TSDT) for

composite laminates, based on the following assumed displacement fields:

u(x, y,z) = U (x,y)+ zyx(x, y) + z24 (x, y) + z3 (x, y)
v(x, y,z)) = v(x, y)+ z (x, y) + z2(x, y) + z3 (x, y) (1.5)
w(x, y,z)= w (x, y)

Functions C, y, ,~ are eliminated based on the condition that the transverse

shear stresses vanish on the plate top and bottom surfaces. As in the previous model, the

principle of virtual displacements is used to derive the equilibrium equations. Reddy

(1990) reviewed a number of other third-order theories and showed that they are special

cases of his theory. Reddy (2000) and Reddy and Cheng (2001) expanded TSDT for

analysis of functionally graded plates.

Tessler (1993) developed a two-dimensional laminate plate theory for the linear

elastostatic analysis of thin and thick laminated composite plates, which utilize

independent assumptions for the displacements, transverse shear strain and transverse

normal stress. The displacements expansions are similar with those assumed by

Essenburg (1975). The transverse shear strains and transverse normal stress are assumed

to be quadratic and cubic respectively through the thickness; they are expressed in terms

of the kinematic variables of the theory by means of a least-squares compatibility

requirements for the transverse strains and explicit enforcement of exact traction

boundary conditions on top and bottom plate surfaces. The theory was extended to

orthotropic shells by Tessler et al. (1995).

Barut et al. (2002) developed a higher-order theory based on cubic expansion for

in-plane displacements and quadratic expansion for out-of-plane displacement and used

Reissner's definition for kinematics of thick plates, to approximate the plate









displacements with weighted-average quantities that are functions of the in-plane

coordinates. Principle of virtual displacement is employed to write the equilibrium

equations and boundary conditions and Fourier series expansion is used to determine the

solutions. The accuracy of their model is demonstrated by comparison with exact solution

and previous single-layer theories.

Discrete-modeling (or layer wise) approach is based on separate assumptions for

displacements in each layer: typically classical plates bending for the face sheets and

shear deformation resistance for the core.

Pagano (1970) provides an exact solution to the problem of a rectangular

orthotropic laminated subjected to a laterally distributed load. Governing field equations

for each layer are written in terms of displacement components and an assumed

exponential-trigonometric solution is substituted into the equations. This provides a set of

equations with six unknowns that are tractable. The solution unknowns are determined by

the boundary conditions and the interface-continuity conditions.

Ogorkiewicz and Sayigh (1973) calculated and compared the central deflection of

the sandwich beam with a foam core using three basic mathematical models: one is

obtained by transforming beam actual section into an equivalent section of one material;

(the core is transformed into a thin web of the same material as the face sheet); second

model is based on the strain energy approach; third is based on stress functions. They

compared the analytical result with experiments and concluded that the third model is the

closest with the experiment observations.

Burton and Noor (1994) developed three-dimensional analytical thermoelasticity

solutions for static problems of simply supported sandwich panels and cylindrical shells









subjected to mechanical and thermal loads. Each of the layers is modeled as a three-

dimensional continuum and double Fourier series expansions are used for displacements

and stresses.

Frostig et al. (1990, 1992) developed a high-order theory for behavior of a

sandwich beam with transversely flexible core based on variational principles. The main

feature of the method is the higher order displacement fields in the thickness coordinate.

The longitudinal and the transverse deformations of the core determined with the aid of

constitutive equations of an isotropic material and the compatibility conditions at the

interfaces consist of non-linear expressions in the thickness coordinate. Their formulation

is based on the beam theory for face sheets and a two-dimensional elasticity theory for

the core. The sandwich behavior is presented in terms of internal resultants and

displacements in face sheets, peeling (i.e. normal stresses between the faces and the core

that do not impose severe restrains on the performance of the sandwich; e.g. a load

applied above the support) and shear stresses at face sheet-core interfaces, and stresses

and displacements in the core, even in the vicinity of concentrated loads. Their model can

be applied to sandwich structures with honeycomb or foam cores. The model was

applied to the vibration analysis (Frostig and Baruch, 1994) and to sandwich structures

with nonparallel skins (Peled and Frostig, 1994). Sokolinsky and Nutt (2004) improved

Frostig model by accounting for nonlinear acceleration fields in the transverse flexible

core essential for the analysis because the inertia loads exerted on the sandwich beams

are nonuniformly distributed along the span. A discretized formulation based on an

implicit finite difference scheme is presented and validated.









Hohe and Librescu (2003) presented a higher-order geometrically nonlinear

sandwich shell theory for sandwich panels that accounts for the transverse

compressibility of the core and initial geometric imperfections. Tangential displacement

is approximate by a second order polynomial and transverse displacement is

approximated by a linear polynomial in thickness coordinate.

Among the large number of sandwich structures models present in the literature,

the models reviewed here can be summarized as: assumed displacements methods

(single-layer or layer-wise) and exact models (when the displacements are not assumed

but derived). Few of the above models are modified for a sandwich structure with

functionally graded core and the results are presented in Chapter 2.

Contact on Composite Materials

Foreign objects impact (such as tool drops, hail, bird strikes, and runway debris)

can introduce damage and significantly reduce the strength of the sandwich structure.

Unlike their solid metallic equivalent, predictions of the effects of impact damage on

sandwich structures are difficult because significant internal damage can be achieved at

impact energy levels lower than those required to create visible damage at the surface.

The first step in understanding the impact dynamics is to study, qualitatively and

quantitatively, the contact problem, that describes the contact force contact area

relation. In this section two important research paths are reviewed: hertzian and non-

hertzian contact problem.

From the contact area point of view, contact models can be classified into:

* Conforming contacts (e.g. flat punch) when the contact area is a constant and is
independent of the load.

* Nonconforming contacts (e.g. sphere) when the contact area is not constant and is
depending on the load.









The first contact analysis is due to Heinrich Hertz (1881) who investigated the

quasi-static impact of spheres as a theoretical background for the localized deformation

of the surface of two glass lenses due to the contact pressure between them. Hertz

formulated the following boundary conditions for the contact surface (Johnson, 1985):

* The displacements and stresses must satisfy the differential equations of
equilibrium, and the stresses must vanish at a great distance from the contact.
Therefore, the radii of curvature of the contacting bodies need to be large compared
with the radius of the contact surface (each surface is treated as an elastic half-
space).

* The bodies are in frictionless contact (only normal pressure is transmitted between
the indenter and the specimen).

* At the bodies' surfaces, the normal pressure is zero outside and equal and opposite
inside the surface of contact.

* The distance between the surfaces of the two bodies is zero inside and greater than
zero outside the contact surface.

* The integral of the pressure distribution within the contact surface with respect to
the area of the contact surface gives the force acting between the two bodies.

Hertz (1881) represented the contact surface by a quadratic function, and assumed a

semi-elliptical distribution of pressure:


(r = p, 1-r (1.6)
a

where a is the radius of the contact area, and r is the coordinate in radial direction. For

the contact between a rigid sphere and a flat surface, Hertz derived a relation among the

radius of the circle of the contact a, the indentor total load F, the indenter radius R and

the elastic properties:

4 kFR
a3 -k (1.7)
3E


where










k=-9 (1-V)+-E) (1.8)_12
16 E'

and where E, v and E', v' are the Young's modulus and Poisson's ratio for the specimen

and indentor, respectively.

Also Hertz found that the maximum tensile stress:

F
ma = (1- 2V) (1.9)
2;a2

is the radial stress that occurs at very edge of the contact and is the stress responsible for

producing the Hertz cone cracks formed in a brittle material.

Willis (1966) derived formulas for the contact between a rigid sphere and a

transversely isotropic half-space. During the loading phase, the modified Hertz law (also

known as Meyer's law) that relates the total load F and the indentation a as:

F =ka" (1.10)

where n = 3/2 for elastic contact and n = 1 for a fully plastic contact and where:


k =4ER12
3

-+- (1.11)
R R, R,
1 1-v2 1
+-
E E1 E2

and where R1 and R2 are the radius of curvature of the indentor and target respectively

and El, v, are the elastic modulus and Poisson's ratio of the indentor and E2 is the

Young's modulus of the target. For the case where the indentor is much stiffer than the

target, the above formula for E can be simplified as:

1 1
1 1 (1.12)
E E2









Even for a small amount of load there could be a significant permanent indentation

so the unloading path is different from loading path. For the unloading phase, Crook

(1952) proposed the following contact law:


F U aa )q (1.13)
a,,, a0

where q = 2.5, Fm is the contact load at which unloading begins, am is indentation

corresponding to Fm and ao is the permanent indentation. The permanent indentation, a0

is zero when the maximum indentation am remains below a critical value ac, during the

loading phase, so the permanent indentations is given by:


ao = a 1 ac2 when a > a, (1.14)


Problems concerning the contact between bodies have been solved since the work

of Hertz in 1880's. A large number of books have been devoted to the mathematical tools

used to solve contact problems. Gladwell (1980) summarized mathematical methods

(based on complex variable, integral transformation and elliptic functions) used to find

the solution for the contact between elastic bodies. An extensive review of the Russian

literature for a large number of contact problems is detailed. Johnson (1985) derived

expressions for stresses and displacements based on Hertz theory and generalized the

theory to include different indentor profiles, interfacial friction, anisotropic and

inhomogeneous materials or layered plates. Khludnev and Sokolowski (1997) monograph

covers qualitative properties of solutions, optimization problem (e.g., finding the

constructions which are of maximal strength and satisfy weight limitations),

determination of crack shapes by optimization methods, existence theorems and









sensitivity analysis of solutions. Fischer-Cripps (2000) presented a review of

displacements and stresses for different types of indenters (uniform, spherical, flat punch,

roller, rigid cone) in order to develop a practical tool for investigating the mechanical

properties of engineering materials. The mathematical modeling, variational analysis, and

numerical solutions of contact problems involving viscoelastic and viscoplastic materials

as well frictional boundary conditions, which lead to time dependent models was

presented by Han sand Sofonea (2002). Alexandrov and Pozharskii (2001) developed

analytical and numerical methods for nonclasical three-dimensional linear elasticity

contact problems. Their methods focused on integral equations include fracture

mechanics.

When a composite laminate with finite dimensions is in contact with a rigid

indentor, the behavior may drastically deviate from that predicted for a half-space.

Various methods were developed to describe the contact problem for isotropic,

orthotropic, monolithic or laminated beams and plates under different indentors profiles.

Most problems are based on a combination between a solution describing the local

contact phenomenon and a beam/plate solution for the global response. Different models

for predicting the elastic response of a beam/plate under a general load were presented in

the previous section. Here a literature review on contact and low-velocity impact

problems is presented.

Abrate (1998) studied the contact between a sandwich beam and a cylindrical

indentor. He assumed that a rigid plate supported the bottom face sheet. Treating the top

face-sheet as a beam of rigidity El on an elastic foundation (as long as compressive stress









in the core does not exceed a maximum value), the following linear contact law is

derived:

F =ka (1.15)

where the contact stiffness is given by:

S 3/4
k 23/2 cb (El)1/4 (1.16)


where Ec is the core modulus in the transverse direction, he is the core thickness and b is

the beam width.

Lee and Tsotsis (2000) studied the indentation failure behavior of honeycomb core

sandwich panels by examining the effects of skin, core and indenter size on load transfer

from the top skin to the core. They treat the problem as a plate (top face sheet) on an

elastic foundation (core). A good agreement was found between the theoretical and FE

model. Also comparison of maximum shear and normal stress with experimental data

showed that the core shear failure dominates the onset of indentation failure.

Yang and Sun (1982) and Tan and Sun (1985) conducted static indentation tests

and finite elements models on laminated plates to validate the contact laws (1.10) (1.13)

They found a good agreement with the above laws.

The study of the effects of foreign object impact on sandwich structures has been a

subject for a large number of researches in recent years. Understanding the mechanic of

contact between the impactor and the panel is very important in the study of the impact

dynamics in order to accurately predict the contact-force history and to predict how the

damage will develop. Many solutions deals with statically, low-velocity impact when the

duration of the impact (the interval of time from the first contact to the complete









detachment) is much larger than the time required by the elastic wave, generated at the

first point of contact, to propagate through the whole body.

The use of static load-deflection behavior of the sandwich beam in the impact

analysis needs some justification. In general the wave propagation effects, especially

through the thickness of the core, should be considered in impact response of sandwich

panels. This will be crucial when spelling type damage occurs in the panels. However, a

study by Sankar (1992) showed that for very large impactor mass compared to that of the

target plate and for very low impact velocities compared to the wave velocity in the target

medium, quasi-static assumptions yield sufficiently accurate results for impact force

history and ensuing stresses in the impacted plate.

Ambur and Cruz (1995) model the sandwich plate using the first order shear

deformation theory (FSDT) and combining the equation of motion of the plate with the

equation of motion of the projectile (given by Newton's law) a set of differential

equations were obtained. The solution is obtained by numerical integration. A highly

nonlinear contact law is derived.

Lee et al. (1993) developed a theory based on the assumption that each face sheet

deforms as a first order shear deformation plate and the core deformation is obtained by

the difference between the transverse displacements of the skins.

Sun and Wu (1991) theory is based on the same assumption for the face sheets

(first order shear deformation plate) but the core is model by using springs to simulate the

transverse and shear rigidities.

Meyer-Piening (1989) modeled the sandwich plate as a three-layer plate with

independent linear in plane displacement in each layer.









Koissin et al. (2004) determined the closed-form solution of the elastic response of

a foamed-core sandwich structure under a local loading, using static Lame equations for

the core and thin plate Kickoff-Love dynamic theory for the faces. The integral

transformation technique was used to solve the differential equations. A good agreement

was found between their analytical solution, experimental results and FE analysis.

Choi and Hong (1994) derived a method for prediction the impact force history on

composite subjected to low-velocity impact, applicable to isotropic and orthotropic plates

with unknown contact laws. The first order shear deformation theory and von Karman's

large deflection theory were used to describe the dynamic response of the plate and Yang

and Sun (1982) modified Hertz law to describe contact problem. The impact duration is

computed from the eigenvalue analysis of the lumped mass system, and the impulse-

momentum conservation law is used with the concept of spring-mass model to predict the

impact force history. A good agreement was found between their analytical solution and

experimental results.

A model for small mass impact on monolithic, transversely isotropic Midlin plate

with a Hertzian contact law was given by Mittal (1987). An approximate analytical model

for small mass impact on specially-orthotropic Kirchhoff plates based on Hertzian

contact law was given by Olsson (1992). It is shown that transversal Young's modulus

has a small influence on the contact force, whereas the impact energy, impactor radius

and plate thickness have a significant influence. The results are compared with numerical

and experimental analyses. Olsson and Mc Manus (1996) included core crushing and

large face-sheet deflections in a theory for contact indentation of sandwich panels. Olsson









(1997) used Olsson and Mc Manus (1996) to modify Mittal (1987) for specially-

orthotropic sandwich panels with linear load-indentation relation.

Fischer-Cripps (1999) used the finite-element method to compute the radius of

curvature of the contact surface for both elastic and elastic-plastic contacts. It is shown

that indentors involving elastic-plastic deformations are equivalent with a perfectly rigid

spherical whose radius is somewhat smaller than calculated using the Hertz equation for

elastic contact. An experiment is used to validate his results.

The range of applicability of Hertz's theory was found to be limited to very small

indentors which also mean low loads. Many non-Hertzian contact problems do not permit

analytical solutions in a closed form. This fact leads to various numerical models in order

to determine the distribution of normal and tangential tractions, which satisfy the normal

and tangential boundary conditions at the interface, inside and outside the contact area.

The continuous distribution of traction is replaced by a discrete set of tractions elements:

and the boundary conditions are satisfied at a discrete number of points called the

matching points. Few examples are:

* Array of concentrated forces (gives an infinite surface displacements);

* Stepwise distribution given by uniform tractions acting on discrete segments/areas
of the surface (the displacements are finite but the displacements gradients are
infinite between the adjacent elements), e.g., Singh and Paul (1974) and Wu and
Yen (1994);

* Superposition of a finite number of symmetrical rectangular loadings of unknown
magnitudes distributed over known lengths (e.g., Sankar and Sun, 1985);

* Piecewise-linear distribution of tractions given by superposition of overlapping
triangular elements (for a three dimensional problem, regular pyramid on a
hexagonal base), e.g., Johnson, 1985.

In order to find the values of the traction elements, which best satisfy the boundary

conditions two different methods were developed:









* Direct/matrix inversion in which the boundary conditions are satisfied exactly at
the matching points

* Variational method in which the tractions are chosen such that to minimize an
appropriate energy function.

Abrate (1997, 1994, 1991) classified the impact models into: spring-mass models

(that accounts for the dynamics of the structure in a simplified manner); energy-balance

models (when the structure behaves quasi-statically); assumed force distribution and

complete models (when the dynamics of the structure is fully modeled).

Shivakumar et al. (1985) developed spring-mass and energy-balance models to

calculated impact force and duration during low-velocity impact of transversely isotropic

circular plates. Energy-balance model is based on equilibrium between the kinetic energy

of the impactor and the sum of strain energy due to contact, bending, and transverse shear

and membrane deformation at maximum deflection. The solution obtained using

numerical methods gives only the maximum force so a spring-mass model is developed

to describe the force history. The impactor and sandwich were represented by two rigid

masses and their deformations were represented by springs (the general case includes

four springs: two to represent linear stiffness of the structure (both bending and shear),

one for the nonlinear membrane stiffness and the forth for the nonlinear contact stiffness).

The analyses were verified by experimental results.

Contact problems for finite thickness layers can be solved using numerical

methods. Sankar and Sun (1983) used two types of numerical methods, point matching

technique and assumed stress distribution method. The point matching method is

essentially a numerical technique to solve the integral equations of the contact problem.

However, this method fails when the contact area is too small because of numerical

difficulties. In the assumed contact stress distribution method, the contact stresses are









assumed to be of Hertzian form, i.e., similar to that of contact of a half-plane. The contact

stresses take the shape of a semi-ellipse. A contact length is assumed and the contact

stresses are expressed in terms of only one unknown, the peak contact stress. Requiring

that the deflections beneath the contact region match the indentor profile one can solve

for the peak contact stress.

Another popular method for beams was to use a Green's function and integral

equation to correlate the applied force and the beam response. Sankar (1989)

superimposed and orthotropic half-space solution on a beam solution to construct an

approximate Green's function. To avoid the limitation of previous method, Wu and Yen

(1994) developed an analytical method, based on exact Green's function derived from

three-dimensional linear anisotropic elasticity theory. Effects of changing the material

properties, stacking sequence, span and thickness of the plate and indentor size on the

force-indentation relation are studied.

Anderson (2003) presented an analytical three-dimensional elasticity solution for a

sandwich plate with functionally graded core subjected to transverse loading by a rigid

spherical indenter. Governing equations are derived based on Reissner's functional and

are solved by enforcing continuity of tractions and displacement between the adjacent

layers. The sphere unknown contact area and pressure distribution are obtained based on

iterative solution method (developed by Singh and Paul, 1974 and modified by Chen and

Frederick, 1992): a large initial contact region is divided into a number of rectangular

patches with assumed constant pressure. The unknown pressure is discretized based on

conforming the contact region to the indentor surface. The contact region is also

determined on an iterative process: tensile patches are removed from the contact region









and the procedure is repeated until only compressive pressure patches remain. Results of

this analysis demonstrate that interfacial normal and transverse shear stresses are not

necessary reduced by the incorporation of functionally graded core.

Abot et al. (2002) presented a combined experimental and analytical study of the

contact behavior of sandwich structures with foam cores. The experimental results were

used to model the load-deflection curves that include a linear range (derived from

Winkler foundation theory) followed by a nonlinear portion (derived from fitting

experiments).

Sburlati (2002) determined the indentation produced by rigid indentors (flat punch

and spherical) on sandwich plate with high-density closed cellular foam based on a three-

layer classical sandwich theory and a distribution of surface pressure reproducing the

contact law. The relevant influence of boundary condition on the elastic response of the

sandwich plate is shown (a 20% difference in contact laws for simply supported plate and

clamped plate was found). Another conclusion of the paper is that the assumption of

distribution of surface pressure reproducing the contact of a rigid indentor over a half-

space is correct also for a sandwich beam.

McCormack et al. (2001) estimated the initial failure load, corresponding to the

first deviation from linearity in the load-deflection curves as well as peak load for several

failure modes (face yielding, face wrinkling, core yielding and core indentation) of

sandwich beam with metallic foam cores. They used finite-element method to compute

the critical load in the core indentation case. The results agree with three-point bending

experimental results.









Liao (2001) investigated the smooth contact between a rigid cylindrical indentor

and a composite laminate with arbitrary ply-orientation (homogenized as an orthotropic

medium). An exact Green's function for the surface displacement of an anisotropic beam

under cylindrical bending is derived. By matching the displacements of the top surface of

the plate and those of the rigid indentor within the contact region, the contact stress

distribution and the indentation is obtained.

Swanson (2004) calculated the core compressive stress in a sandwich beam with

orthotropic face sheets based on elasticity equations for transverse loading of layered

orthotropic materials. By systematically varying the contact pressure and comparing the

computed surface displacements with the indentor profile the contact pressure

distribution was determined. The results showed that the pressure distribution for an

orthotropic half-space is applicable to a sandwich beam over a large range of variables.

Petras and Sutcliffe (1999) defined a spreading length to consider the way in which

different wavelengths of sinusoidal pressure loading on top face sheet are transmitted to

the core. This length characterizes the length over which a load is spread out by the face

sheet. For a very flexible skin there is a large deformation under the load which can lead

to core failure. For very rigid skins indentation failure will be relative hard, as the skins

spread out the load. Their solution is obtained based on a combination between the higher

order theory developed by Frostig et al. (1992) and contact overlapping triangular

elements developed by Johnson (1985).

Anderson and Madenci (2000) conducted impact tests in order to characterize the

type and extent of the damage observed in a variety of sandwich configurations with

graphite/epoxy face sheets and foam or honeycomb cores. They found that the surfaces of









both the honeycomb and foam samples revealed very little damage at levels of impact

energy that produced significant core damage. As the impact energy was increased, the

samples experienced one of two types of damage: a tear or crack from the center of the

laminate to the edge, or significant damage consisting of a dent localized in the region of

impact.

Wang and Yew (1989) studied the damage produced by low velocity impact in

graphite fiber composite plates based on the principle of virtual work. They took into

account the time history of impact force, the dynamic deformation of the system and the

distribution of damage in the target plate. The material was considered damaged when its

designed strength was reduced by the failure of the constituents (matrix cracking, fiber

breakages or/and delamination). It was demonstrated that fibers in the consecutive layers

must be at an angle with respect to each other in order to achieve an effective

containment of the impact induced damage.

In order to give a research perspective to the work on impact on sandwich plates

with functionally graded core a literature review was presented. Among the large number

of analytical models presented in the literature, the models reviewed here can be

summarized as: plate on elastic foundation (Abrate, 1998; Lee and Tsotsis, 2000),

laminate plates (Yang and Sun, 1982; Tan and Sun, 1985) and sandwich plates (e.g.

Ambur and Cruz, 1995; Sun and Wu, 1991), non-hertzian contact/numerical methods

(e.g. Sankar and Sun, 1983) and experiments on impact damage (e.g., Anderson and

Madenci, 2000). Two numerical methods and a finite element solution for impact on

sandwich plates with functionally graded core are presented in Chapter 3.









Cellular Solids and Functionally Graded Materials

Cellular solids appear widely in nature in different shapes and with different

functions. They can be found in leaves and stalks of plants, in corals and woods or in any

types of bones. They can be made of closed cells, regular or non-regular, or can be made

of open network of struts. They can be isotropic or can have cells oriented in a particular

direction. Recently, engineered cellular materials are made using polymers, metals, and

ceramics and it is believed that any solids can be foamed. They are widely used as

thermal, and acoustic insulations, as absorbers of kinetic energy from impacts.

The cellular/foamed materials are qualitatively and quantitatively investigated in a

large number of publications. Among those, two books worth to be mentioned: Ashby et

al (2000) design guide for metal foams (contains processing techniques, characterization

methods, design and applications) and Gibson and Ashby (1997) monograph of cellular

solids.

Functionally graded materials (FGMs) are materials or structures in which the

material properties vary with location in such a way as to optimize some function of the

overall FGM.

Nature provides many examples of functionally graded materials. In many of the

cases the nature functionally graded structures were evolved based on some mechanical

function: bones give a light, stiff frame to the body, wood support the tree under

environmental loadings, leaves transport fluids. Bamboo is one of the examples of a

structurally smart plant (Amada et. al. 1997, 2001). Bamboo structure, which resembles

that of a unidirectional, fiber-reinforced composite, is described by a macroscopically

graded geometry that is adapted to environmental wind loads while the fiber distribution

exhibits a microscopically graded architecture, which leads to smart properties of









bamboo. Amada et al. (1997, 2001) demonstrated experimentally and analytically how

the functionally graded microstructure of a bamboo has been optimized through evolution

to maximize its load-bearing capabilities under severe environmental loading conditions.

An interesting application of FGMs is dental implants (Watari et al, 1997). It is

confirmed that the fabrication technique can be used to produce controlled functionally

graded microstructures, and that the biocompatibility of the titanium/ hydroxyapatite

implants is superior to that of pure titanium implants.

The variation of the properties of FGMs can be achieved by gradually changing the

volume fraction of the constituents' materials, by using reinforcements with different

properties, sizes and shapes, by interchanging the roles of reinforcement and matrix

phases in a continuous manner as well as by adjusting the cell structure or the material

density. The result is a microstructure that produces continuously or discretely changing

thermal and mechanical properties at the macroscopic or continuum scale.

The concept of functionally graded materials was proposed in the eighties by

materials scientists in Japan as a way to create thermal barrier materials. Koizumi (1997)

summarize the first projects in this field. The idea proposed was to combine in a gradual

manner heat-resistance ceramics and tough metals with high thermal conductivity. The

result is a panel with a high heat-resistance on the high-temperature side and high

mechanical strength on the other side. The new materials were obtained by four methods:

Chemical Vapor Deposition (CVD), powder metallurgy, plasma sprays, and self-

propagating combustion synthesis (SHS).

As the use of FGMs increases, for example, in aerospace, automotive,

telecommunications and biomedical applications, new methodologies have to be









developed to characterize FGMs, and also to design and analyze structural components

made of these materials. Although fabrication technology of FGMs is in its infancy, they

offer many advantages. Few of the manufacturing methods are presented here. Fukui et

al. (1991, 1997) developed a high-speed centrifugal casting method, in which the layers

are formed in the radial direction due to different mass densities. An melted Al-Ni alloy

cast into a thick-walled tube was rotated at a speed such that the molten metal

experienced an acceleration, thereby producing two kinds of composition gradient, (i.e.

phase gradient). The volume fraction of the A13Ni phase was determined by quantitative

optical microscopy.

Poly (ethylene co-carbon monoxide) (ECO) was chosen to make the FGMs by

ultraviolet radiations (Lambros et. al., 1999), because of its rapid degradation under UV

light. Irradiated ECO becomes stiffer, stronger and more brittle with increasing time. The

exposure time and, thus, the properties or the FGM can be adjusted truly continuous

based on the relationship between modulus of elasticity and irradiation time.

Another manufacturing method is electrophoretic deposition (Sarkar, 1997) EPD is

a combination of two processes, electrophoresis and deposition. Electrophoresis is the

motion of charged particles in a suspension under the influence of an electric field

gradient. The second process is deposition, i.e. the coagulation of particles into a dense

mass.

El-Hadek and Tippur (2003) developed functionally graded syntactic foam sheets

by dispersing micro balloons (with linear graded volume fraction) in epoxy. They

determined Young's modulus and the density by using the wave speed and density for

syntactic foams having homogenous dispersion of the micro balloons. The resulting foam









sheets have a nearly constant Poison's ratio. Other FGM manufacturing methods

presented in literature are powder metallurgy, plasma sprays and self-propagating

combustion synthesis.

Suresh and Mortensen (1998) provide an excellent introduction to FGMs. Suresh

(2001) presents a motivation for the use of graded materials and also enumerates few

challenges in the field. Literature contains many analytical models: mechanical and

thermal loads, fracture and optimization solutions for different structures made of FGM

were published. Here, few of the significant papers are reviewed.

When using analytical methods to solve problems involving FGMs, a functional

form for the variation of thermo-elastic constants has to be assumed. For example,

Aboudi et al. (1995a, 1995b, 1999) assumed a simple polynomial approximation for the

elastic constants. Another useful approximation is the exponential variation, where the

elastic constants vary according to formulas of the type cj = ce"'. Many researchers

have found this functional form of property variation to be convenient in solving

elasticity problems. For example, Delale and Erdogan (1983) derived the crack-tip stress

fields for an inhomogeneous cracked body with constant Poisson ratio and a shear

modulus variation given by / = /,,e v.

Giannakopoulos and Suresh (1997a, 1997b) present two analytical models for

indentation of a half-space with gradient in elastic properties, for a point force and an

axisymmetric indentor. The analytical solutions, based on elasticity methods for

axisymmetric equilibrium problems are compared with finite element simulations. It is

found that the influence of the Poisson's ratio is strong whenever the elastic modulus is

increasing with depth, and is weak whenever the elastic modulus is decreasing with









depth. Also a decreasing elastic modulus with depth results in a spreading of stresses

towards the surface rather than to the interior, whereas an increasing elastic modulus

results in diffusing the stresses towards the interior of the half-space.

Markworth et al. (1995) review analytical models for microstructure-dependent

thermo physical properties, and models for design, processing of FGMs. They also

present few recommendations relative to areas in which additional work is needed.

Functionally graded materials as composite materials are generally described from

two points of view: micromechanics and macromechanics. Micromechanics analysis

recognizes the basic constituents of FGM but does not consider the internal structure of

the constituents. Micro-constituents are treated as homogeneous continue. Although the

material at the macro-scale is heterogeneous, it is assumed that the micro-heterogeneities

are distributed in such a way such that the material volumes beyond some representative

minimum have comparable macroscopic or overall properties. The hypothesis concerning

the existence of an representative volume element (RVE) is fundamental: it means that

there exist a relatively small sample of the heterogeneous material that is structurally

typical of the whole mixture on average and contains sufficient numbers of inclusions,

defects, pores such that the overall behavior (the behavior of an equivalent homogeneous

material) would not depend on the surface values of constraints and loading as long as

those are homogeneous. Properties are prescribed pointwise in the spirit of the mechanics

of continue. The transition from the micro to the macroscale consists in finding the

relationship between the macroscopic fields (stress, strain) and the volume averages of

the same micro fields over RVEs. In mixture-type Continuum-Macromechanics analyses,

the material properties are expressed in terms of volume fraction and individual









properties of constituent phases. Macromechanics models can be subdivided in: direct

(closed-form analytical solution for the averaged properties), variational (lower and upper

bound for the overall properties in terms of the phase volume fraction) and approximation

methods.

Aboudi et al. (1994a, 1994b 1999) developed a higher order (HOTFGM),

temperature-dependent micro-mechanical theory for thermo-elastic response of

functionally graded composites. A higher-order representation of the temperature and

displacement fields is necessary in order to capture the local effects created by the

thermomechanical field gradients, the microstructure of the composite and the finite

dimensions in the functionally graded directions. Their model based on the concept of a

representative volume element explicitly couples local (microstructural) and global

(macrostructural) effects. RVE is not unique in the presence of continuously changing

properties due to non-uniform inclusion spacing. Pindera and Dunn (1995) evaluated the

higher order theory by performing a detailed finite element analysis of the FGM. They

found that the HOTFGM results agreed well with the FE results.

Abid Mian and Spencer (1998), present an exact solution for 3D elasticity

equations for isotropic linearly elastic, inhomogeneous materials. Lame elastic moduli X

and [t, the expansion coefficient ac and the temperature T depend in an arbitrary specified

manner on the coordinate z. The 3D elasticity solutions are generalized from solutions for

stretching and bending of symmetrically inhomogeneous plates. In order to completely

describe these solutions an equivalent plate is introduced. This equivalent plate is a

homogeneous plate (in the sense of a classical thin plate) with the same overall geometry

as the inhomogeneous plate and elastic moduli that are suitable weighted averages of the









moduli of the inhomogeneous plate. It is shown that the exact three-dimensional solutions

are generated by two-dimensional solutions of the thin-plate equations for the equivalent

plate.

Reddy et al. (2000, 2001) use an asymptotic method to determine three-

dimensional thermomechanical deformations of FG rectangular plates. The locally

material properties are estimated by the Mori-Tanaka homogenization scheme for which

the volume fraction of the ceramic phase is of the power law type. Reddy et al. (2003)

developed an FGM beam finite element by deriving the approximation functions from the

exact general solution to the static part of the governing equations. These solutions are

then used to construct accurate shape functions which result in exact stiffness matrix and

a mass matrix that captures mass distribution more accurately compared to any other

existing beam finite elements. Thus, the element is an efficient tool for modeling

structural systems to study wave propagation phenomena that results due to high

frequency and low duration forcing (impact loading).

Rooney and Ferrari (1999) developed solutions for tension, bending and flexure of

an isotropic prismatic bar with elastic moduli varying across the cross-section. It is

demonstrated that the elastic moduli are convex functions of the volume fraction. If a

structural member is required to be just in tension, any grading of the phases will result in

a improved in the performance.

Vel and Batra (2002, 2003) present an exact solution for simply supported

functionally graded rectangular thick or thin plates. The material has a power-law

through-the-thickness variation of the volume fractions of the constituents. Suitable

displacement functions that identically satisfy boundary conditions are used to reduce









equations governing steady state vibrations of a plate to a set of coupled ordinary

differential equations, which are then solved by employing the power series method.

Woo and Meguid (2000) provide an analytical solution for the coupled large

deflection of plates and shallow shells made of FGMs under transverse mechanical load

and temperature field. The material properties of the shell are assumed to vary

continuously through the thickness of the shell, according to a power law of volume

fraction of the constituents. The equations obtained using von Karman theory for large

transverse deflection, are solved by Fourier series method.

In a series of papers Sankar and his coworkers (Sankar and Tzeng, 2002; Sankar,

2001; Venkataraman and Sankar, 2001; Apetre, et al. 2002) reported analytical methods

for the thermo-mechanical and contact analysis of FG beams and also sandwich beams

with FG cores. In these studies the thermo-mechanical properties of the FGM were

assumed to vary through the thickness in an exponential fashion, e.g., E(z)=EoeZ. The

material was assumed to be isotropic and the Poisson's ratio was assumed to be constant.

The exponential variation of elastic stiffness coefficients allows exact elasticity solution

via Fourier transform methods. Later Apetre et al. (2003) and Zhu and Sankar (2004)

used Galerkin method to analyze cores with polynomial variation of mechanical

properties.

Although a relatively new field, the study of functionally graded materials contains

a large number of research areas: publications on fabrication techniques, experiments

methods, analytical and finite element solutions were widely published later.

Developments in manufacturing methods now allow controlled spatial variation in

material properties so analytical results can be validated by comparisons with









experimental results. In order to give a research perspective to the present work this

section summarized few of the developments in this area. The main objective of present

work is to develop analytical methods for sandwich structure with FG cores and to

validate them by comparison with finite element solutions.

Objectives and Approaches

Learning from nature, engineers created new manufacturing methods to produce

functionally graded materials. Few of them are presented in previous section and are a

good research motivation for the present work. Analytical methods have to be developed

in order to enable obtaining closed-form solutions to some fundamental solid mechanics

problems, and also to extend finite element models to structures made of FGMs. Impact

surfaces and interfaces between different materials where impact damage occur are

regions of interest.

The main objectives of this dissertation can be summarized as:

1. Develop analytical models for sandwich structures with functionally graded core.

2. Solve contact and impact problems involving sandwich structures with FG core and
compare the trade-off between using a functionally graded core as opposed to the
conventional sandwich design.

3. Compare the trade-off between the total mass and stiffness in functionally graded
materials and homogenous materials by solving optimizations problems.

Even with the sandwich structure advantages over the other structures, it is

important to develop new types of materials in order to obtain the absolute minimum

weight for given conditions (e.g., structural geometry and loadings). These new

sandwiches should be compared with other sandwich constructions and with alternative

structures in order to select the best configuration. One of the new alternatives is a

sandwich structure with functionally graded core. A sandwich beam with soft,









transversely flexible core must be approached with a higher-order theory rather than the

classical theory. The core gets compressed and the thickness of the beam is changed

under concentrated loads. Plane sections do not remain plane after deformation, so

displacement field is not a linear function of the thickness coordinate. Conditions

imposed on one face sheet do not necessary hold for the other. These are few of the

reasons that make higher-order theory necessary for a sandwich panel with FG core even

if this theory involves more computational effort.

This research presents analytical solutions based on different models for sandwich

structures with FG core when the core Young's modulus is given by exponential and

polynomial variation. A finite element model validates analytical solutions. Low-velocity

impact on sandwich structures with FG core based on springs' model is also solved. A

comparison of the total mass of different panels based on different relations between

Young's modulus and density is present. The results indicate that functionally graded

cores can be used effectively to mitigate or completely prevent impact damage in

sandwich composites.














CHAPTER 2
ANALYTICAL MODELING OF SANDWICH BEAMS WITH FUNCTIONALLY
GRADED CORE


Although FGMs are highly heterogeneous, it will be useful to idealize them as

continue with properties that change smoothly with respect to spatial coordinates. This

will enable closed-form solutions for some fundamental solid mechanics problems, and

will aid the development of finite element models for structures made of FGMs. This

chapter investigates different analytical models available in literature for a sandwich

beam and applies them to a sandwich beam with functionally graded core.

In the first two sections, the governing equations for sandwich structures with FG

cores are solved for two types of core Young Modulus by two different methods: exact

solutions are presented for the exponential variation of core Young Modulus and a

combination of Fourier series and Galerkin method for a polynomial variation of core

Young Modulus. Those methods are compared with two equivalent single-layer theories

based on assumed displacements, a higher-order theory and a finite-element analysis. A

very good agreement among the Fourier series-Galerkin method, the higher-order theory

and the finite-element analysis is found.

Exact Method for Sandwich Structures with Functionally Graded Core

Venkataraman and Sankar (2001) derived an elasticity solution for stresses in a

sandwich beam with a functionally graded core. They used Euler-Bernoulli beam theory

for analysis of face sheets and plane elasticity equations for the core. In the present work,










the solution of the sandwich problem was improved by using elasticity equation for face

sheets also.

The dimensions of the sandwich beam are shown in Figure 2.1.The length of the

beam is L, the core thickness is h and the face sheet thicknesses are hf. The beam is

divided into 4 parts or elements: the top face sheet, top half of the core, bottom half of the

core and the bottom face sheet.


S hf Element 1: Top facesheet
A I Node (1)
-Element 2: Top half ofsandwich core
h k I Node (2)
Element 3: Bottom half of sandwich core
4 Node 3
L W
--Element 4: Bottom facesheet


Figure 2.1: Sandwich beam with functionally graded core with schematic of the analysis
elements.


In general, this model can be applied for sandwich structures with core and face

sheets orthotropic materials at every point and the principal material directions coincide

with the x and z-axes. Consequently, the constitutive relations for each layer are:

C 11 C13 0
07 = c OI33 x, forth element, i = 1,2,3,4 (1.17)

Z-r 0 c _55 7x= ,

or {)= [c(z)]{}

(1.18)

The face sheets are assumed to be homogeneous and isotropic. The core is

functionally graded but symmetric about the mid-plane given by z=0. The elastic

coefficients (c,,) of the core are assumed to vary according to:





























Topf


Bo
sai


c, = c0 e (1.19)

This exponential variation of elastic stiffness coefficients allows exact elasticity

solution.

The tractions and displacements at the interface between each element are shown in

Figure 2.2. Each element has its own coordinate systems. The coordinate systems of each

element are chosen at the interface because displacements and traction compatibility

between elements will have to be enforced at these nodes.


Pa
facesheet A A A A, A Atl,U1,PI, W

P2, W2
Top half of z p
sandwich

P3 ,W3
)ttom half of t3, U3
ndwich core tIU


P4,W4
Bottom face sheet A A 4. U

t5,U5,P5, W5


Figure 2.2: Traction forces and displacements at the interfaces of each element in the
FGM sandwich beam


The governing equations are formulated separately for each element, and

compatibility of displacements and continuity of tractions are enforced at each interface

(node) to obtain the displacement and stress field in the sandwich beam. This procedure

is analogous to assembling element stiffness matrices to obtain global stiffness matrix in

finite element analysis.

The top face sheet is subjected to normal tractions such that:









o-zz (x,O) = p sin(x) (1.20)

where

nfi7
= ,n = 1,3,5.... (1.21)


andpa is known. Since n is assumed to be odd, the loading is symmetric about the center

of the beam. The loading given by equation (1.20) is of practical significance because any

arbitrary loading can be expressed as a Fourier series involving terms of the same type.

The displacement field for each layer is assumed of the form:

u, (x, z) = U, (z) cos(x) i = 1, 2,3, 4
w (x,z) = W (z)sin(x)

where u is horizontal displacement and w is vertical displacement and where it is

assumed that:

(U,, W )=(a,, b, )exp(a, z) (1.23)

where a,, b,, a, are constants to be determined.

In order to satisfy equilibrium the contributions of the different tractions at each

interface should sum to zero. Enforcing the force balance and the compatibility of force

and displacements at the interfaces enables us to assemble the stiffness matrices of the

four elements to obtain a global stiffness matrix K:

[K]{U1 WI U2 W2 U3 W3 U4 W4 U5 T
T (1.24)
={0 p, 0 0 0 0 0 0 O

where K is the global stiffness matrix, a summation of the stiffness matrices for each

element. The displacements U1, W ... Ws, are obtained by solving (1.24). The obtained

displacement field along with the constitutive relations is used to obtain the stress field in

each element.











Fourier Galerkin Method for Sandwich Structures with Functionally Graded
Core

Zhu and Sankar (2004) derived an analytical model for a FG beam with Young's

modulus expressed as a polynomial in thickness coordinate using a combined Fourier

Series-Galerkin method. In the present work, the model is applied to a sandwich beam

with FG core. The geometry (Figure 2.1), the load (1.20), and the constitutive relations

(1.17) are the same as in previous model.

In this section, a brief description of the procedures to obtain the stiffness matrix of

top half of the core is provided. The derivation of stiffness matrices of other elements

follows the same procedures.

The differential equations of equilibrium for the top half of the core are:


8x 8z
S+ =0
ax aZ (1.25)

+ = 0
8x 8z

The variation of Young's modulus, E in the thickness direction is given by a

polynomial in z. e.g.,

( 4 3 2
E(z) = Eo az + + aj J + +1 (1.26)


where Eo is the Young's modulus at z=0 and a,, a2, as and a4 are material constants. The

thickness iny direction is large and plain strain assumption can be used. The elasticity

matrix [C] is related to the material constants by:










l-v v 0
E(z)
[C]= v 1 v 0 (1.27)
(1 + v)( 2v)
1-2v
2

The following assumptions are made for displacements:

u(x, z) U(z) cos 4x
w(x, z) = W(z) sin x (1.28)

Substituting equation (1.28) into (1.27), the following constitutive relation is

obtained:

CxX C11 c1i 0" Usingx '
zz =c3 C33 0 W'sinx (1.29)
r z 0 0 G) U'+Wcosx)

A prime () after a variable denotes differentiation with respect to z. Boundary

conditions of the beam at x=0 and x=L are w(O,z) w(L,z) 0, and ax (0, z) = ax (L, z) = 0,

which corresponds to simple support conditions in the context of beam theory. Equations

(1.29) can be written as:


Szz Sz (1.30)

Txz = Tz cosx

where

Sx c11 c13 -U
)S C13 C33 c W'
(1.31)
T = G(U'+rW)
z









Substituting for oxx0, o=, zx from equations (1.31) into equilibrium equations

(1.25), a set of ordinary differential equations in U(z) and W(z) are obtained:

S + Tz'= 0
Sz'-Tz= 0 (1.32)

In order to solve equations (1.32) the Galerkin Method is employed: solutions in

the form of polynomials in z are assumed:

U(z) = c11(z) + c22(z) + c303 (z) + c404( ) + c505 (z)
W(z) = bl (z) + b22 (z) + b303(z) + b44(z) + b55 (z) (1.33)

where Os are basis functions, and b 's and c 's are coefficients to be determined. For

simplicity 1, z, z2, z3, z4 are chosen as basis functions:

(1(Z)= ; 02(Z)= ; 03(Z)= 2;
4(z) = 3 5(Z) = 4 (1.34)

Substituting the approximate solution in the governing differential equations, the

residuals are obtained. The residuals are minimized by equating their weighted averages

to zero:


(S + Tz '), (z)dz = 0, i =1,5

(Sz' T (z)dz =0, i=1,5 (1.35)

Using integration by parts (1.35) can be rewritten as:

h h
f Sxdz + (h)T,(h)-T (0)(0)- T dz 0
0 0
h h (1.36)
S ,dz+ Tdz-(S(h)(h)-S(O)(O)=, i=1,5
0 0








Substituting for Sx(z), S (z) and T,(z) from equations (1.31) into (1.36) and using the

approximate solution for U(z) and W(z) in (1.33) it is obtained:

KIi( KS Jb If
K(3) K(4) C /JC (2) (1.37)

or


[K] =(1.38)


where:

h h
K(1) = c1i dz IG jdz
0 0
h h
K(2) = -f G dz i = 1,5



0 0 (1.39)
K4 = c13 jdz Jf G ',dz
0 0
fl) i= (O)Tz(O) O(h)Tz (h)
fi(2) = (O)Sz(O) i (h)Sz (h)


= (b1 b2 b3 b b 5 c1 c2 C3 C4 C5) (1.40)

Let U2, W2, U3 and W3 be the displacements at top and bottom surface of top half of

the element (top half of the core). Evaluating the expressions for U(z) and W(z) at the top

and bottom surfaces and equating them to the surface displacements results in the

expression:












(W 3, 1
U3 f


0 0 0


bC


(1.41)


This can be compactly expressed as:


U b
wU =[A]
W37 LC5


(1.42)


The tractions T2, P2, Ts and Ps acting on the surface can be related to the functions

as follows:


f
f2(1

f4()
f4(l)
f(2)

f(2)
f 2
f(2)
f (2)


1(h)
3 (h)
3 (h)
4 (h)
05 (h)
0
0
0
0
0


0
0
0
0
0
-02(h)
023(h)

03 (h)
-, (h)
0, (h)


p2(0)
S3(0)


4,(0)
53(0)
0
0
0
0
0


0
0
0
0
0



03(0)
4 (0)
0,(0)


S2
T3
\S3


(1.43)










(1.44)


L 1 T2

S2) ] S2
.. T3
f(2) S3,


From (1.37), (1.42) and (1.44) follows:











Uw2 =[A][K] [B] 2 =[K] (1.


Finally, the stiffness matrix of the top half of the FGM core [S(2)] that relates the

surface tractions to the surface displacements is obtained as:


=S7; K =J S, W(1.



In order to satisfy equilibrium, the contributions of the different tractions at each

interface should sum to zero. Enforcing the balance and the compatibility of force and

displacements at the interfaces enables us to assemble the stiffness matrices of the four

elements to obtain a global stiffness matrix [S]:

[s]Lu, w, U2 w2 U3 W3 U4 ,4 U5 w5 =


45)









46)


=Lo p, 0 0 0 0 0 0 0 O

The displacements U1, W1... Ws, are obtained by solving equation (1.47). The

displacement field along with the constitutive relations is used to obtain the stress field in

each element.


Analytical Models for Sandwich Structures with Functionally Graded Core


An Equivalent Single-Layer First-Order Shear Deformation Theory

The simplest model investigated in the present study is an equivalent single-layer

model that includes a transverse shear deformation (First-order Shear Deformation

theory, FSDT). The following kinematics assumptions are made:









u(x, z)= U, (X)+ z r(x)
(1.48)
w(x,z)= (x)

where u is displacement in horizontal direction, x and w is displacement in vertical

direction, z. uo, Wo, y are unknown functions to be determined using the equilibrium

equations of the first order theory. The dimensions of the sandwich beam are shown in

Figure 2.3. The length of the beam is L, the core thickness is h, the top face sheet

thickness is h, and the bottom face sheet thickness is hb.





h/2

x
h/2

Shb
Figure 2.3: The beam geometry


If the core material is orthotropic at every point and the principal material

directions coincide with the x and z-axes, the plane strain constitutive relations are:


F. c I0(z) C13 (Z) 0 ex 1-v v 0
,z = c13 () C33(z) 0 (z) v 1-v 0 EL z
rxz 0 0 c5, (z) yX 0 1- 2v (+)
2
(1.49)

or {} = [c(z)]{}W (1.50)

The variation of Young's modulus E in the thickness direction is given by a

polynomial in z. e.g.,









J(z)4 2z3 z2
E(z)= E, az + + a3 + +a4 +1 (1.51)


where Eo is the Young's modulus at z=0 and al, a2, as and a4 are material constants.

In order to calculate the flexural rigidity of the cross-section, the position of the

neutral axis zo must be found. It is given by the coordinate system for which the first

moment of area is zero when integrated over the entire cross-section:

t+h/2 zo
B,, = zc,,(z)dz=00 > zo (1.52)
(hb+h/2+zo)

Equilibrium equations of the first order theory take the following form:

du,
Nx A 0 All
dx

X D 1 (1.53)


Z + S
dx

where All is the axial rigidity, D1 is the bending stiffness and S is the extensional

stiffness:

,+h/2zo
A4 = J cl(z)dz
(hb+h/2+Zo)
ht+h/2 z,
D, = f z2c1(z)dz (1.54)
(hb+h/2+zo)
h/2zo
S= c55 (z)d
(h 2+Zo)

For a given set of external loads and boundary conditions, axial force resultant Nx,

bending moment resultant Mx, and shear resultant Vz can be calculated. Then using

system (1.53) the displacements are obtained.









This model is applied for a sandwich beam with functionally graded core. The main

drawback of the model is given by the fact that the transverse shear strain is constant

through the thickness of the beam. Results and discussions are presented in the last

section.

An Equivalent Single-Layer Third-Order Shear Deformation Theory

Reddy (1984) developed a third-order Shear Deformation theory (TSDT) for

composite laminates, based on assumed displacement fields and using the principle of

virtual displacements. Reddy (1990) reviewed a number of other third-order theories and

showed that they are special cases of his theory. Reddy (2000) and Reddy and Cheng

(2001) expanded TSDT for analysis of functionally graded plates.

Here, a third-order equivalent single-layer model based on Reddy's assumption of

vanishing of transverse shear stresses on the bounding planes is investigated. The

displacement field is assumed to be:

u(x,z)) = u(x)Z+ z(x)+ z2x(x)++ z3(x)
(1.55)
w(x, Z) = W(x)

where uo and wo are displacements along middle axis, z=0. Functions and 0 are

eliminated using the assumption of zero shear stress at top and bottom:


(x)= h (x)+
x (1.56)
2 (x) dw0
3h* adx

where


h* = 2hhb + hth + h + -
2 (1.57)






51


If the top face sheet thickness ht equals the bottom face sheet thickness hb then

X(x) 0.

Axial force resultant, bending moment resultant and shear resultant are calculated

as follows:


h



2
N =b J"
2 +
h

h
Ms b


ca (x, z)dz


zax (x,z)dz


(1.58)


h

V = b J (x,z)dz

2


where b is the beam width, ,o


h

A,=b J


c1, (z)dz


h
-+4
CN=b f c11(z) z2 h


h
2
Am=b f 11 (z~dz
h[


du- and constants AN through Cv are given by:
dx


BN b ( z + 2 h,-
(h h*


z3 2Jdz
3h*I


BM bzJ z11 +z z -- z
h~hb 3h*1


kh 3 d
h* 3h*]I


z3 dz
3h*I


h


h
2
CM=b J

-('"


zc11 (z) z


2








h
B,=b 2 c55(z)L h z22J

C b J c5(z1+ 22zh-hb z2 dz




The following system is obtained:
duo (x)
S(1.59)(x)

4 Bm < C->= M (x). (1.60)
dx
0 B, C V (x)
B C dw (x) L V)
dx

where cl and c2 are constants to be determined from boundary conditions. For a given set

of external loads and boundary conditions axial force resultant Nx, bending moment

resultant Mx, and shear resultant Vz can be calculated. Then using system (1.60) the

displacements are obtained.

This model is applied for a sandwich beam with functionally graded core. The main

drawback of the model is given by the fact that the transverse shear strain has a quadratic

variation with respect to the thickness coordinate. Results and discussions are presented

in the last section.


A Higher-Order Shear Deformation Theory

Frostig et al. (1992) developed a higher-order theory for a sandwich beam with a

transversely flexible core that uses a beam theory for the face-sheets and two-dimensional

elasticity equations for the core. Swanson (1999) addressed details of implementation of

Frostig model and presented solutions for several cases. The main feature of the method









is the higher order displacement fields in the thickness coordinate: second order for

vertical displacement and third order for the longitudinal displacement. Another

advantage of this model, as a model based on variational principle is that the boundary

conditions are obtained uniquely as a part of derivation. The main difference between this

model and the previous one is that now, the higher order displacements are derived and

not assumed. The equations developed by Frostig et al. (1992) and modified for a

functionally graded core are briefly presented here.

The dimensions of the sandwich beam and the coordinate systems are shown in

Figure 2.4. The length of the beam is L, the core thickness is h, the top face sheet


x, ue
VZ, Wc

h


X, lb b


thickness is ht and the bottom face sheet thickness is hb.

Figure 2.4: The beam geometry


Constitutive relations (which assume isotropic elastic behavior) for the face-sheets

and for the transversely flexible core (i.e. zero longitudinal stress) are given by:

Top face-sheet: N' = A',UOtx Bottom face-sheet: N. = AI lUb,x

M = -D,, M = -D1 b,
Core: r(x,z)=G,(z)y(x,z) (1.61)

(- v)E, (z)w ,z)
(1 + v)( 2v) c









where N' are the resultant axial forces in the face-sheets, M' are the bending

moments in the face-sheets, Ai and D1' are axial and respectively flexural rigidities for

the face-sheets, uo,, w, are face sheets longitudinal and vertical displacements at the

centroid (i=t for top face-sheet and i=b for bottom face-sheet):


A11= cll (z)dz
S(1.62)

D,= z'cI (z)dz
-h/2

Linear variation for Young's modulus and shear modulus are assumed:

E(z) = az + b
G(z) = az + b,

Governing differential equations, boundary conditions and continuity conditions

are derived based on variational principle. Using the equilibrium equations for the core:

(-x,x + rxz,z 0 0
0, (1.64)
0 + =0

the compatibility of the displacement at the core-face-sheets interfaces and the

assumption that for a transversely flexible core (i.e. made of a material with much lower

modulus relative to the face-sheets), the shear stress is nearly constant through the core

the following core displacement fields are derived:









(z1 a, (2 z hn+z 1 +
uc (x,z)= (x) l In az+1 +xx(x)- z2 -t,(x) z+h+uo +
za + b 2 (a z2 +
C b\ (a -
z+ In -z+1 -z
















differentiable function.
a C( ha













top and bottom face-sheets and shear stress of the core, r are:
a22 (xY)=+ -,x (x)+ 2w ()i= qb (x)- b, (x) +(1.66)
SIn h+1 I


where uo, and w, are longitudinal and vertical displacements of the centroid of each face-

sheet (i=t for top face-sheet and i=b for bottom face-sheet); v is core Poisson's ratio. In

the above expression a linear core Young's modulus was assumed: Ec(z)=az+b. Similar

relations can be obtained for any Young's modulus variation expressed by a

differentiable function.

Governing equations, written in terms of transverse displacements wt and wb of the

top and bottom face-sheets and shear stress of the core, r are:


alwe,^xx(x)+ p, (x)+1vb 1X)+ ^tx t) = x)-mtx)

Sw1, x (x)+ swb,(xxx x)+ ^ x)+ tx)= nx) nb(x)
Ait A

where q,, m, and n, are distributed pressures, moments and axial forces applied on top

(i=t) and bottom (i=b) face-sheets and the coefficients are given by:










ah D h b
SIn h+1 h


b hhb h
a, = D, 8 =d -h- + h
2 ah
SIn h+


hh+b )ln(ah+l-h
a 3 -h -+ ()
2 a
2 In ^1h+1


h+ b )ln( ah+l h
3 h2 h
2a a In h+1
(b


ad
n77 +1
In h+1
(b


ad
A ==
In h+1
(b1


b ad
2= -- /'2 =
In a h+1
b

h+ b )ln~ah+l
hb h+Ialn ah+1
Al =lnh
2 a


ad
In h+1
[b

h


where d is the beam width.

In order to obtain the homogeneous solution for (1.66), the following characteristic

equation is derived:


al'Z4 + U1
a'3
a3 V3


a'24 + /72
A 'Z3


'61A
62 A 0
)73A4 + PU3A2 w)3


(1.68)


Denoting with 2j,j = 1,...,12, the roots of characteristic equation, the following

rescaling of constants needs to be done in order to avoid numerical difficulties (given by

the case when Real(A) > 0 -> exp(Ax) oo ):


(i) If Re(A) < 0 > solution = c eRe( )x
e -AL = e (Lx
(ii) If Re(A) > 0 > solution c e = -e =ce
j


(1.69)


(1.67)


p3 = I In a h+1


0)3 -d +
A,, l









Forstig's theory is clear but needs the above described rescaling of constants in

order to avoid numerical difficulties. For a given set of external loads and boundary

conditions the differential governing system (1.66) can be solved. Results and

discussions are presented in the next section.


Results for Sandwich Beam with Functionally Graded Core under a Distributed
Load

In this section includes two objectives: validate analytical methods for sandwich

structure with FG core and use these methods to compare FG and homogenous cores.

The results of the modified Venkataraman and Sankar (2001) and Zhu and Sankar

(2004) methods were compared (Figures 2.5 2.9) and found to be identical. A one

dimensional sandwich plate of length 0.2 m, core height 20x 103 m and face-sheet

thickness 0.3x10-3 m is considered as an example to investigate the effects of varying

core properties through the thickness. The face sheet elastic modulus Ef is chosen as 10

GPa. The sandwich core modulus Eo at the mid-plane is kept fixed at E/1000 while the

core modulus at the face sheet interface (Eh) is varied. The ratio of Eh Eo varies from 1 to

100. When Eh/Eo 1, core properties are constant through the thickness, and hence the

problem is identical to that of a conventional sandwich plate. The value of Eh is gradually

increased until it reaches the value of the face sheet, for EhEo= 100. The different profiles

of the elastic modulus variations in the sandwich core are shown in Figure 2.5. They

display the profile of Young's modulus for FG materials with Eh/Eo=1 (ai=0, a2=0, a3=0,

a4=0, as=l), Eh/Eo=10 (ai= 3.9372, a2= -.94217, a3= 3.8705, a4= 2.1215, as=l) and

Eh/Eo=100 (ai=238.50, a2=-258.92, a3=132.18, a4-14.327, a5=1.6335), respectively.

Poisson's ratios for the core and face-sheet, respectively, were 0.35 and 0.25. The results







58


are restricted to the case where the plate is loaded in the transverse direction by a

sinusoidal load given by pa sin(7c/L) withpa=1.

An interesting result from analysis of the FGM sandwich beam is the transverse

shear stress in the sandwich core at the face-sheet/core interface. Conventional design of

sandwich panels restricts the shear stress at the core/ face sheet interface to the bond

(adhesive) shear strength, which is typically lower than the shear strength of the core

material. Therefore, the core material is not fully utilized. It is hence desirable to reduce

the interfacial shear stress while carrying a high shear stress in the core. It appears that

this is possible with a functionally graded core. The shear stress variations in the core,

given by both methods (exact and Fourier-Galerkin) are plotted in Figure 2.6. The

interface shear stress reduces as Eh/Eo ratio is increased.


10

**** EIEI=1
E E IE = 10
N-
E- h/E0 =100

O 0
0

E -5-



-10
0 20 40 60 80 100 120
E/Eo

Figure 2.5: Through-the-thickness variations of core modulus considered for the
functionally graded sandwich panel (the load is p(x) = pa sin(rx / L))







59



0.01

0.008 ------- ---- --- -- EE-----
:E IE E=10
0.006 ------~~~~~~_~~~~ ------e--------- --------- r--------,--, -h--a-----

0.004 ---------- .. ....... .--------. ------ ------ I ------- I ------- I-. ------ ------

0.00 ------

o - --------- ------ I------ I------.- ------- .------ -------

-0 0 2 -- --- -------------- ------ ------ ------------- ------ ----- _

-0.004

-0.006 ------

-0.008 ------- ----

-0.01
-2.3 -22 -2 1 -2 -1 9 -1 8 -1 7 -1 6 -1.5 -1 4



Figure 2.6: Variation of transverse shear stresses (given by Fourier-Galerkin method) in
the sandwich structure with FGM core for different ratios (Eh/Eo) and E/Eo =
1000 at x = L/4 (the load isp(x) = p sin(;rx/L)).


The normal stress o- in the core (Figure 2.7) varies linearly and is independent of


the variation in core properties. The value of normal stress varies linearly


(approximately) from the applied surface load at the top core to zero at the bottom of the


core. This is an interesting result, because it simplifies the calculations required in order


to analyze the core-crushing problem. It must be noted that the example considered here,


used a smoothly varying sinusoidall) surface pressure load.







60




0 015
E /E=
h 0


0 005 ------- P I- Q - - ---------- ------- ----------------^
001 5h







00 -- - --- --^ ------- -- -- - -- - -
-0 005


-0.01


-0 015 I I I
-0 1 0 01 02 03 0.4 05 06 0.7 0.8


Figure 2.7: Variation of o- stresses (given by Fourier-Galerkin method) in the sandwich
structure with FGM core for different values of Eh Eo and E/Eo = 1000 at x =
L/4 (the load isp(x)= pa sin(;rx/L)).



The bending stress variation (Figure 2.8) in the core is as expected. The linear


variation in strains results in small levels of bending stress in the core near the mid-plane.


The stresses increase near the face sheet. This is particularly pronounced as the value of


the core modulus is increased to match the value of the face sheet thickness.


The u-displacement variation through the thickness is linear (plane sections remain


plane) when the gradient in the core properties is small. However, for highly graded cores


significant warping of the cross section occurs.








61


0.01 "i1

0008 .-E.Oi ---------- ------ ----- ----
EhlE -10
0 006 ....:

0 004 -------- -------- -------- -------- --- ----- -------- -------_




O ---------- ------ ------ --------II--
0 0 004 ---------------- -------- --------. -------- -------- ---------------
0 006 ---


.------- --------r -------- Tr -------- T- -------
-n










-0 008 --------- ------ ------ -- ------ -------- -------- -------- -------
I- -0004-
t -0 0 0 4 .. . . . . . . . .... . - - - - - - . ..-




-0 00
-001 -- I I I I i -
-40 -30 -20 10 0 10 20 30 40



Figure 2.8: Variation of bending stresses (given by Fourier-Galerkin method) through the
thickness of the FGM core for different values of Eh/Eo and E/Eo= 1000 at x

= L/4 (the load isp(x) = pa sin(rx/L)).




u displacement at x=3L/4
0.015 ------------------------ --------------------------------------------------

E IE
0.01 ----EhEO==-10-------------- ^ .










Eh. ----0 1-
N 0.005 ----------- ------------- ------------. ----.y '-- ''-------- ------------



0.0 -- - - -- -r -- - - --- -- -


Q 0 --- --.- ---------- -------- -- ---- ---- ---- ----- -------- -----------






-0.015 I-I I I I I
-3 -2 -1 0 1 2 3
-9
x 10


Figure 2.9: Variation of u displacements (given by Fourier-Galerkin method) through the
thickness of the sandwich structure with FGM core for different ratios (Eh/Eo)

and E//Eo = 1000 at x = L/4 (the load isp(x) = pa sin(rx / L)).









Next step is to compare the five models: the equivalent single-layer first-order and

third-order shear deformation theories, Fourier series-Galerkin method, Frostig model

and a finite-element model.

A simply supported sandwich beam, with length L = 0.3 m, core thickness h =

20x103 m and face-sheets thickness hf= 0.3x10-3 m is considered to investigate the

effects of varying core properties through the thickness. The face-sheet Young's modulus

was chosen as 50 GPa. The face-sheets Poisson's ratio is v/= 0.25 and the core Poisson's

ratio is v = 0.35.


L


..... Symmetric core ... **
-6- Asymmetric core ..
,......... -" -(a)
E..) .E(z)
E 0- *_
S.... ,,sin (rx/L)



D- do (b)
IA E(7.)

Figure 2.10: (i) Non-dimensional core modulus and (ii) loading for: (a) symmetric core
about the centerline and (b) asymmetric core about the centerline


Two cases are considered: (a) symmetric core about the centerline under uniform

distributed loadp = 1 N/m2 and (b) asymmetric core about the centerline under a

distributed load given by p(x) = sin(;rx/L) N/m2. Figure 2.10 presents core elastic

moduli: for (a), the core Young's modulus E has a linear symmetric variation with

respect to thickness coordinate, z: E = 50 MPa at the middle core and E = 50x10 MPa at

the core-top face-sheet interface (as well as at the core-bottom face-sheet interface); for


7









(b), the core Young's modulus E has a linear asymmetric variation with respect to

thickness coordinate, z: E = 50 MPa at the core-bottom face-sheet interface and E=50x10

MPa at the core-top face-sheet interface.

Using the commercial finite element software ABAQUS a 2D finite element model

was created to model problem (a). The FG core was partitioned through the thickness into

twenty strips with constant properties. Four elements were considered through the

thickness of each strip and two elements were considered through the thickness of the

face-sheets. The elements considered were 2D, quadratic, plane strain elements.

Boundary conditions assume w(0, z) = w(L, z) = 0

Figures 2.11 2.15 present a comparison for five models: the equivalent single-

layer first-order and third order shear deformation theories, Fourier series-Galerkin

method, Frostig model and an ABAQUS finite-element model. For problem (a), which is

the symmetric core under a uniform distributed load, a very good agreement among the

Fourier series-Galerkin method and the ABAQUS model was found. Because the core is

symmetric about the mid-plane, the variation of displacements, strains and stresses are

symmetric with respect to the thickness coordinate.

Figure 2.11 presents deflections at bottom face sheet core interface. The FSDT

and the TSDT beam are stiffer than the Galerkin beam.
















E E 115
E -2
-2.5 -2
-3~
-2.5
-3.5
0 50 100 1i0 0 50 100 150
x, mm (a) x, mm (b)


Figure 2.11: Comparison of deflections: (a) symmetric core about the centerline under a
constant uniform distributed loadp = 1 N/m2 and (b) asymmetric core about
the centerline under a sinusoidal distributed load given by p(x) = sin(;rx/L)
N/m2.


Comparison of longitudinal displacements in the core at the same cross-section

(L/4) for the two cases are presented in Figures 2.12 (a) and (b). A perfect agreement was

found between Galerkin model and the finite element model for (a) and between Galerkin

model and the Frostig model for (b). For both problems FSDT gives a linear variation as

expected.







E E
o Eo
E o --- ------...... ....... I- -.---------------------I





-5 TSDT 5 -TSDT
1 I FEM -S. Sankar
-4- Sankar -- FSDT
-- FSDT IFrosllg
-2 1 0 1 -1.5 -1 0.5 0 0.5 1 1.5
u,mm x 10 (a) ",mm Xao (b)


Figure 2.12: Comparison of longitudinal displacement in the core at x = L/4: (a)
symmetric core about the centerline under a constant uniform distributed load
p = 1 N/m2 and (b) asymmetric core about the centerline under a sinusoidal
distributed load given by p(x) = sin(;rx/L) N/m2.











Bending stress (Fig. 2.13 (a) and (b)) in the core at the same cross-section (x=L/4)

is almost the same for all models. For the asymmetric core, a larger compression value is

found at the top core where the load in applied and where Young's modulus value is

larger.


_10




E E


-5- TSDT -......- rSDT
I FEM Sankar
-9- Sankar -- FSDT
-- FSDT -I- Frolilg
-30 -20 -10 0 10 20 30 -14 -12 -10 -4 -Z 0 2
x, Pa (a) o., Pa (b)

Figure 2.13: Comparison of axial stress in the core at x = L/4: (a) symmetric core about
the centerline under a constant uniform distributed loadp = 1 N/m2 and (b)
asymmetric core about the centerline under a sinusoidal distributed load given
by p(x) = sin(;rx/L) N/m2.


Comparison of shear strain in the core at the same cross-section (L/4) is presented

in Figure 2.14 (a) and (b). Fourier-Galerkin method, Frostig model and the finite-element

solution present a 1/z type variation for the core shear strain. The first order shear

deformation theory gives a constant shear strain while the third order shear deformation

theory gives a quadratic shear strain with respect to the thickness coordinate.

The same conclusion was reached for the shear stress in the core at the same cross-

section (Figure 2.15): Fourier-Galerkin method, Frostig model and the finite-element

solution present an almost constant shear stress whereas the first order shear deformation

theory gives a linear shear stress while the third order shear deformation theory gives a










third order shear stress with respect to the thickness coordinate. The equivalent single-

layer theories are not accurate for shear because they are not based on two dimensional

equilibrium equations (1.64). In order to obtain accuracy for shear (strain and stress) in

single-layer theories (both first order and third order) the shear stress was obtained using

equilibrium equations (1.64) for the core and the bending stress previously derived

(Figure 2.13):


-xx,x + Zz,z = 0 rxz (x, z) = o .x (x, )d + (x, ) (1.70)
0

Figure 2.15 includes both shear stresses: obtained from single-layer theories and obtained

from equilibrium equations (1.70). The latest are identical with those obtained based on

Fourier-Galerkin and finite element model.


10-. a j 10 I .
-TSDT -TSDT I
-- Sankar -0- Sarikar
-- FSDT -- FSDT '
I I FEM 5/ 5 n- Frostig

E E


-5 -5 I


15 11) -5 -2 -1.5 -1 -0.5 0
mm xi o4 (a) Y,,, mrm x1 (b)


Figure 2.14: Comparison of shear strain in the core at x = L/4: (a) symmetric core about
the centerline under a constant uniform distributed loadp = 1 N/m2 and (b)
asymmetric core about the centerline under a sinusoidal distributed load given
by p(x)= sin(rx/L) N/m2.
















EO-IFEM E
-- FSDT I
-6- FSDT+pqull TSDT
-, TSDT+equil
-a- SankarN
S--FST
FSDTaqull %,..
i--w Frn;,llg N %
-8 -6 4 -2 0 -7 -6 .5 -4 3 .2 -1 0
l Pa (a) ,"Pa (b)


Figure 2.15: Comparison of shear stress in the core at x = L/4: (a) symmetric core about
the centerline under a constant uniform distributed loadp = 1 N/m2 and (b)
asymmetric core about the centerline under a sinusoidal distributed load given
by p(x) = sin(;rx/L) N/m2.




Summary and Conclusions

In this chapter, analytical models for sandwich structures with FG core are

introduced, validated by comparison with finite element solutions and used to compare

homogeneous and FG core sandwich plates.

The study presents analytical models for sandwich structures with two types of core

Young's modulus variation through the thickness: exact solutions are presented for the

exponential variation and a combination of Fourier series and Galerkin method for a

polynomial variation of core Young's modulus. Those methods are compared with

models presented in literature and modified for this type of structure: two equivalent

single-layer theories based on assumed displacements and a higher-order theory. A finite-

element analysis is presented in order to validate the models. A very good agreement

among the Fourier Galerkin method, the higher-order theory and the finite-element

analysis is found.









A comparison among homogenous and FG cores is presented: sandwiches with

same geometry, same sinusoidal distributed load and different core properties are

analyzed. As the core becomes stiffer at the core-top face-sheet interface, the interface

shear stress reduces as Eh/Eo ratio is increased.

The chapter deals only with distributed loads (uniform and sinusoidal). Next

chapter introduces a more common load, the localized contact between a rigid indentor

and a sandwich plate. The low-velocity impact solution is based on the solution

developed here.














CHAPTER 3
CONTACT AND LOW-VELOCITY IMPACT OF SANDWICH BEAMS WITH
FUNCTIONALLY GRADED CORE


One of the important problems in sandwich structures is damage due to low

velocity impact. The interfacial shear stresses due to contact forces can be large enough

to cause debonding of the face sheet from the core. Unlike for their solid metallic

counterparts, predictions of the effects of low-velocity impact damage in sandwich

structures are difficult because significant internal damage is achieved at impact energy

levels lower than those required to create visible damage at the surface. One way of

reducing the shear stresses is to use functionally graded core so that the abrupt change in

stiffness between the face sheet and the core can be eliminated or minimized. The

stresses that arise due to low-velocity impact can be easily understood by analyzing the

static contact between the impactor and the structure (Sankar and Sun, 1985).

The problem of low-speed impact of a one-dimensional sandwich panel by a rigid

cylindrical projectile is considered. The core of the sandwich panel is functionally graded

such that the density, and hence its stiffness, vary through the thickness. The problem is a

combination of static contact problem and dynamic response of the sandwich panel

obtained via a simple nonlinear spring-mass model (quasi-static approximation). The

variation of core Young's modulus is represented by a polynomial in the thickness

coordinate, but the Poisson's ratio is kept constant. In the previous chapter, the two-

dimensional elasticity equations for the plane sandwich structure were solved using

different methods. Here, the contact problem is solved using two methods: the assumed









contact stress distribution method and method of point matching; and the results are

compared with a finite element model. For the impact problem a simple dynamic model

based on quasi-static behavior of the panel was used the sandwich beam was modeled

as a combination of two springs, a linear spring to account for the global deflection and a

nonlinear spring to represent the local indentation effects.

Results indicate that the contact stiffness of the beam with graded core increases

causing the contact stresses and other stress components in the vicinity of contact to

increase. However, the values of maximum strains corresponding to the maximum impact

load are reduced considerably due to grading of the core properties. For a better

comparison, the thickness of the functionally graded cores was chosen such that the

flexural stiffness was equal to that of a beam with homogeneous core. The results

indicate that functionally graded cores can be used effectively to mitigate or completely

prevent impact damage in sandwich composites.

Contact problems for finite thickness layers can be solved using numerical

methods. Sankar and Sun (1983) used two types of numerical methods, point matching

technique and assumed stress distribution method. The main difference between the two

methods is the way the contact load under the indenter is estimated: assumed stress

distribution method is based on the assumption of a semi-elliptical contact stress

distribution whereas the method of point matching attempts to capture the actual contact

stress distribution approximating it as a superposition of several uniformly distributed

loads. The point matching method is essentially a numerical technique to solve the

integral equations of the contact problem. However this method fails when the contact

area is too small because of numerical difficulties. In the assumed contact stress









distribution method, the contact stresses are assumed to be of Hertzian form, i.e., similar

to that of contact between a rigid cylinder and a half-plane. The contact stresses take the

shape of a semi-ellipse. A contact length is assumed and the contact stresses are

expressed in terms of only one unknown, the peak contact stress. Requiring that the

deflections beneath the contact region match the indenter profile, one can solve for the

peak contact stress. Here, the two methods are used for a sandwich beam with

functionally graded core.

Assumed Stress Distribution Method

Previous chapter presented analytical models for sandwich structures with FG cores

as a tool to be used for different problems. This section presents one of the possible

applications of the above-discussed models. Indentation of a surface with an object

(sharp, blunt or spherical) is a common engineering problem. If the properties of the

material under the indentor are known, the analysis of the contact load versus indentation

can provide valuable information about the contact damage. It is shown that spatially

grading mechanical properties (e.g. Young's modulus) it is possible to minimize the

contact damage.

Sankar and Sun (1985) solved the problem of smooth indentation of a beam by a

rigid circular cylindrical indenter using the method of assumed stress distribution. The

method is used here for indentation of a sandwich beam with functionally graded core.

The indenter is a cylinder of radius Ro and unit length. The contact length 2c is

considered as known and the other quantities (stresses, deflection) are calculated for a

given contact length. Indentation is defined as the difference in the vertical displacement

of the indenter and the corresponding point on the bottom side of the beam. The







72


dimensions of the sandwich beam are shown in Figure 3.1. The length of the beam is L,

the core thickness is h and the face sheet thickness is hf.






2h 2c

.......
L



Figure 3.1: Dimensions of the sandwich panel and the contact length, 2c.


The stress distribution under the indenter (Figure 3.2) is assumed to be of the semi-

elliptical form:

2
p() = Pmax (3.1)


where pm, is maximum value of the stress at center (unknown).







c



Figure 3.2: The stress distribution under the indenter.


A similar stress distribution with some arbitrary pm, say pm is assumed. Writing

p(x) in the form of a Fourier sine series, it is obtained:


p(x)= p sin nflx (3.2)
n=1,3 L


where










2pm i (n1rj (nrc (33
Pn= sin JI (3.3)
n 2 L

and J1 is the Bessel function of first order. The vertical displacements of the points in the

contact zone are computed using the Fourier series Galerkin Method described in the

previous chapter. From geometrical considerations (Figure 3.3), for contact lengths

smaller in comparison with the radius of indenter, the following approximate relation is

used:

2
05(x) = w(x) w(xo) = (3.4)
2R'(x)

where w(xo) is vertical displacement at the middle point (see Figure 3.3). In (3.4) R' is the

radius of curvature at x. It is found that R' is almost constant (usually, more then 70% of

R' distribution lies within one standard deviation of the mean value). Then the average

radius of curvature of the deformed top face sheet can be derived as:

1 C
R = R'(x)dx (3.5)
Co






R '(x)



x


w(x)


Figure 3.3: Illustration of relation between w deflection and radius of curvature of the
deformed surface.









Generally R is different from the radius of the intended Ro. But, the displacements

vary linearly with the load and hence the peak stress pmax required producing a radius Ro

is given by:

R
Pmax = PM (3.6)


Once pmax is known, vertical displacements and the indentation can be calculated.

Plots for several examples are presented and discussed in the Results section.

Method of Point Matching

The main difference between method of point matching and the method of assumed

stress distribution is the assumption of the load. If, for the previous method the load is of

a semi-elliptical form, here it is assumed that the stress distribution under the indenter is a

superposition of a finite number of symmetrical rectangular loadings of unknown

magnitudes (denoted by qj,j=l,m) distributed over known lengths (denoted by xj,j=l,m)

as depicted in Figure 3.4.


So, thejth uniform distributed load over the span 2x, is:


Figure 3.4: Discretization of contact load for the method of point matching










L xi
O, x 1 0,


L x)= L x (3.7)
pj (x)= qj x e '2 ,- + (3.7)


0 xe r+ L ,L


Using Fourier series the above load can be written as:

N k >
pj (x)= p~ sin x (3.8)
k= \L

where

4q k (kr
P = sin sin x (3.9)
k 2 2 y2L )

For each coefficient p the vertical displacements of the points in the contact zone

are computed using the Fourier series Galerkin Method described in the previous

chapter: w, is the vertical displacement ofjth reference point due to ith uniform distribute


load of unit magnitude; wo, is the vertical displacement of mid point reference point due

to ith uniform distribute load of unit magnitude; vi is the vertical displacement ofjth

reference point due to indentation; v0 is the vertical displacement of mid point due to

indentation. Using the same geometrical relation described by (3.4) and writing the

displacements at each point as superposition of displacements due to each rectangular

load, it is obtained:

m
v(xO) = v = O Wo, ,
m=1 (3.10)
m

1=1









Then, the following algebraic system is obtained:

m x2
i(W -wj,)q= -- j 1,m (3.11)
1=1 2R

From (3.11) the unknown load magnitudes q, are obtained. The total load is

obtained as a summation of all rectangular loadings over the contact area. This method

was applied for a sandwich beam with functionally graded core and the results are

presented in the Results Section.

Quasi-Static Impact of a Sandwich Beam with FG Core

After the static contact problem was solved, the method is applied to the problem of

low-velocity impact of functionally graded sandwich panels. Solving the static contact

problem first and combining the solution with the dynamic response of the sandwich

panel obtained via simple spring-mass models (quasi-static assumption) accomplish this.

The use of static load-deflection behavior of the sandwich beam in the impact analysis

needs some justification. In general the wave propagation effects, especially through the

thickness of the core, should be considered in impact response of sandwich panels. This

will be crucial when spelling type damage occurs in the panels. However, a study by

Sankar (1992) showed that for very large impactor mass compared to that of the target

plate and for very low impact velocities compared to the wave velocity in the target

medium, quasi-static assumptions yield sufficiently accurate results for impact force

history and ensuing stresses in the impacted plate.

Because the problem deals with the low-velocity impact (applied force vary slowly

in time, therefore the system response is relatively slow so that the inertial terms in the

equations of motion can be neglected; the structure responds quasistatically), the

sandwich beam is modeled as a combination of two springs (Shivakumar et al. 1985), a









linear spring to account for the global deflection and a nonlinear spring to represent the

local indentation effects as depicted in Figure 3.5. Also, the problem assumes only a rigid

indentor (wave effects in the impactor can be neglected) with negligible dynamics.

SF



k, nonlinear

ks linear



Figure 3.5: Low-velocity impact model


Using the numerical results from the contact problem the spring constants k, and k,

and the exponent n are determined such that:

F = k,a" (3.12)

F = kwb (3.13)

where F is the total load, a is the core indentation, wb is the vertical displacement of the

core at the at bottom face sheet interface.

The displacement of the impactor is calculated as the sum of indentation depth

(core compression) and the global deflection of the sandwich beam:

1 / n
F F
w = a + wb + (3.14)
k k,


The work done by the impactor during the impact event can be expressed as:

w F F fF\ F2 1 F "lln
W= FFdw=Fw wdFFw- -+ ndF=- + (3.15)
0 0 0 k I 2k, n +I 2ki1 "









The kinetic energy stored in the target is assumed negligible. The maximum

deflection is reached when the velocity of the projectile becomes zero. At that time, the

initial kinetic energy of the projectile has been converted into two parts: strain energy

stored into the target by global (bending and shear) deformation and the energy used to

create local deformations in the contact region. Hence, equating the impactor kinetic

energy to the work done or the strain energy stored in the springs, the maximum contact

force can be calculated from:

F2 Fml+l 1
mx max = -mv (3.16)
2k, (n + )k1 2

where m and vo are, respectively, the mass and impact velocity of the impactor.

The results of the static contact problem are used to determine the constants that

described the stiffness and compression of FG sandwich beam ((3.12) and (3.13)). Using

these material properties and the quasi-static model the maximum contact force in the

case of low-velocity impact of FG sandwich beam is determined. Using this maximum

value the maximum normal and shear strains in the core were determined and compared.

Results for several examples are presented in the next section.

Results

A sandwich beam, with length L = 0.2 m, core thickness h = 20x 10-3 m and face

sheet thickness hf 0.3x 103 m is considered to investigate the effects of varying core

properties through the thickness. The face-sheet Young's modulus was chosen as 50 GPa.

Although these methods can be applied to a general form of E, in the present work,

for core Young's modulus, two cases are considered (Figure 3.6): linear symmetric about

midplane and linear asymmetric. The variation of E with respect to z for the two cases is

given below and also depicted in Figure 3.6.










Em = yE -Eoymh Iz + if ze h,h] (3.17)


E aEhasym -'asym asym asym
Eaym = Eym o z-+ 0 if ze [-h,h] (3.18)
2E"O"Q h 2E"o )

where 2h is the core thickness; Eos"" is the Young's modulus at the midplane for

symmetric case; Eoa"sy is the Young's modulus at the bottom surface for asymmetric

case. Ehsym is the sandwich core Young's modulus at face sheet interfaces for symmetric

case; Eha"ym is the sandwich core Young's modulus at top face sheet interface for

asymmetric case. Three different variations, such that Eh= Eox(l, 5, 10) are considered.

Eh is the core modulus at face sheet interfaces for symmetric case and is the sandwich

core modulus at top face sheet interfaces for asymmetric case.

Poisson's ratios are v= 0.35 for the core and vf= 0.25 for the face-sheet material.

The intended is a cylinder with a radius of 10 x 103 m. The width of the cylinder and the

width of the sandwich panel in y-direction are assumed to be unity. The cylinder is made

of steel and its mass is m=15.7 kg. The impact velocity of the impactor is vo=6 m/s and

the kinetic energy is K=282.2 J.














E 5 / .
E .. .
.C / ~...
s, ... / EhlEo=1, H. C.
0 0- hh
u. -- EhIE =5, S. C.
S.' ... ....... EhIE =10, S. C.
S.... EhIE =5, A. C.
... .. EhlEO=10, A. C.



50 250 500
Elastic Modulus of the Sandwich Core, MPa

Figure 3.6: Through the thickness variations of core modulus considered for the
functionally graded sandwich beam. (H.C denotes homogeneous core; S.C.
denotes symmetric core, A.C. denotes asymmetric core)


Figures 3.7 through 3.33 present different displacements, strains and stresses based

on assumed stress distribution method and provide a comparison of different sandwich

structures. This method is compared with the second numerical method and with the

finite element solution at the end of the chapter (Figures 3.33 3.37)

Figures 3.7 thought 3.25 present displacements, strains and stresses in sandwich

beam with linear symmetric and asymmetric functionally core. For the two cases

(symmetric and asymmetric), same Young's modulus ratio, Eh/Eo of 10 and same semi

contact length c = 3 mm are considered.

Figure 3.7 presents horizontal displacement in the second half of sandwich

structure with asymmetric core. It can be noticed that the displacement is zero at the

middle cross-section and has a higher order variation with respect to the thickness

coordinate. The half-beam deflection is plotted for the same asymmetric core beam in

Figure 3.8, whereas Figure 3.9 shows deflection at core-top face sheet interface and core-






81


bottom face sheet interface near the contact in order to indicate the localized core

compression.



u, mm


- 200


-10 100


150


Figure 3.7: Variation of the horizontal displacement, u, in the second half of the sandwich
beam (Linear asymmetric core with Eh/Eo = 10)


w, mm


.51
-10-
-15-
-20-
-25-


0
\t'z


/
-t:
C


150
)1 I


Figure 3.8: Deflection, w, in the second half of the sandwich beam (Linear asymmetric
core with Eh/Eo = 10)


Srnr













-24-


-25-




-27 core top face sheet interface
-7-- core bottom face sheet interface
100 105 110 mm 115 120 125


Figure 3.9: Variation of the deflection, w, in the second half of the sandwich beam at core
top face sheets interfaces, in the vicinity of contact (Linear asymmetric core
with Eh/Eo = 10)


In order to visualize the location of the core maximum strains and to compare

symmetric and asymmetric core maximum strains with the homogenous core maximum

strains, contour plots of the strains (both normal and shear) for the half beam are

presented in Figures 3.10 through 3.15. In the same figures the location of semi contact

length (c = 3 mm ) is plotted. Maximum normal strain occurs at the middle core (x = L/2

= 100 mm) and slightly above the region with small Young's modulus (bottom half of the

beam for asymmetric core Figure 3.10 and top half of beam for symmetric beam -

Figure 3.11). For the homogenous core maximum normal strain (Figure 3.12) occurs at

the middle core (x = L/2 = 100 mm) and close to the core top face sheet interface.


























Figure 3.10: Contour plot of normal strain in the functionally graded asymmetric core
(with Eh/Eo= 10) normalized with respect to the maximum normal strain in
the core, xx max = 0.071 m/m, (contact length, c = 3 mm).


200


Figure 3.11: Contour plot of normal strain in the functionally graded symmetric core
(with Eh/Eo= 10) normalized with respect to the maximum normal strain in
the core, xx max = 0.070, (contact length, c = 3 mm).


200


x, mm


Figure 3.12: Contour plot of normal strain in the homogeneous core (with Eh/Eo = 1)
normalized with respect to the maximum normal strain in the core, e, max
0.052, (contact length, c = 3 mm).


-0 2


-0 2










As presented in Figure 2.14, FG core shear strain has a 1/z variation. Maximum shear

strain in FG core is spread out along the beam where Young's modulus is small (core

bottom face sheet interface for asymmetric core Figure 3.13 and middle beam for

symmetric beam Figure 3.14) and away of contact region. Maximum shear strain in the

homogenous core is closed to the contact region, below the core top face sheet interface

(Figure 3.15). Maximum stains are used in order to compare different cores.


1U 10U 14U 160 180 200UU


Figure 3.13: Contour plot of shear strain in the functionally graded asymmetric core
normalized with respect to the maximum shear strain in the core, Yxz max
0.44 (contact length, c = 3 mm).



1 0




E
E 0 __ n 0.5400


-CH x, mm

Figure 3.14: Contour plot of shear strain in the functionally graded symmetric core
normalized with respect to the maximum shear strain in the core, Yxz max
0.37. (contact length, c = 3 mm)












0-45

5 07
0.73

E 0





0 1
0 120 140 160 180 200
Oc) x, mm


Figure 3.15: Contour plot of shear strain in the homogeneous core normalized with
respect to the maximum shear strain in the core, yz max = 0.14 (contact length,
c = 3 mm).



Figure 3.16 confirms the fact that axial stresses are larger in the face sheets than in


the core (an assumption used in Chapter 2 for solving equilibrium equations). Also it can


be noted that the axial stress o-xx is higher on the topside of the core where contact occurs


(Figure 3.17). The stress concentration is not only due to contact but also due to the fact


that the core Young's modulus is higher near the interface.


10


5

o






-10
-3000 -2000 -1000 0 1000 2000
Normal stress, _xx, MPa


Figure 3.16: Variation of the axial stress through the thickness in sandwich beam with
functionally graded core at different cross-sections (Linear asymmetric core
with Eh/Eo= 10)











oxx, MPa 10


0
0-

-20 -
-20

-40 -- -30

-60 -40

100 150. "200 -


-10 100, "


Figure 3.17: Variation of the axial stress through the thickness in functionally graded core
(Linear asymmetric core with Eh/Eo = 10)


The normal stress ozz in the core for both asymmetric core and symmetric core is

plotted in Figure 3.18 ((a) and respectively (b)). It can be seen that, although the

maximum normal stress is slightly larger in the asymmetric core, in both cases it has

same region of influence, very closed to the contact region (contact length is c = 3 mm).

These stresses are useful in determining if the core will get crushed due to impact or not.

Since the core density is higher in the contact region, the FG core will be able to

withstand higher compressive stresses compared to uniform density core. In this case the

skins are flexible so the load remains localized (it has large values for near the contact

and is zero away of contact).







87


MP a MPa
10
0- 10
-20-
480-





(a) (b)


Figure 3.18: Variation of the normal compressivee) stress in (a) asymmetric and (b)
symmetric functionally graded core (with Eh/Eo= 10) in the vicinity on the
contact (contact length, c = 3 mm)


Usually the local failure starts in the core and results in core crushing, delamination

and therefore significant reduction of the sandwich strength. There are different factors

that can influence the local effect of indentation stress field. Here, the face sheets

thickness, hf and indentor radius, R and the face sheet Young's modulus, Efare varied in

order to study their influence on contact. The normal compressivee) stress in asymmetric

functionally graded core (with Eh/Eo= 10) in the vicinity of the contact, at core top face

sheet interface is plotted for all cases (different hfR and Ef). Figure 3.19 presents the face

sheet thickness influence on impact: for the same asymmetric functionally graded core

(with Eh/Eo= 10) and same contact length, c, three cases with respect to the face sheet

thickness are considered: hf = (0.3, 1, 2) mm. It can be noted that as the face sheet

thickness increases not only the maximum compressive stress is increased, but because

the face sheet becomes rigid, it spreads out the load. The variation of normal stress with

indentor radius is plotted in Figure 3.20: for the same asymmetric functionally graded

core (with Eh/Eo= 10) and same contact length, c, three cases with respect to the indentor

radius are considered: R= (5, 10, 20) mm. In this case the maximum normal stress











decreases as the radius increases, but the non-zero stress region remains the same for

different R-values.


x 107
xO-
0
-- hf = 1 mm
-2 hh = 2 mm
---/- hf = 0.3 mm

-4-

S-6-

-8_


-10


0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135
c x, m


Figure 3.19: Face sheet thickness influence on variation of the normal compressivee)
stress in asymmetric functionally graded core (with Eh/Eo= 10) in the vicinity
on the contact (c = 0.003 m), at core top face sheet interface (indenter radius,
R = 10 mm).



The same type of collusions can be reached for the face sheet Young's modulus

influence on impact. Keeping the same geometry, indenter and the same type of

functionally graded core and increasing the face sheet Young's modulus results in an

increase in the maximum compressive stress but the non-zero region remains the same

(Figure 3.21)