UFDC Home  myUFDC Home  Help 



Full Text  
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 SingleLayer FirstOrder Shear Deformation Theory .................47 An Equivalent SingleLayer ThirdOrder Shear Deformation Theory ..............50 A HigherOrder 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 LOWVELOCITY IMPACT Of SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE ............................................. ............... 69 Assum ed Stress Distribution M ethod ...................................................................... 71 M ethod of Point M atching......................................................... ............... 74 QuasiStatic 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 31 Maximum normal and shear strains for a given impact energy ............................94 32 Core thicknesses for different materials with same flexural stiffness, D11 and sam e global stiffness ...................... ...... ............ ............................ 97 41 Variations of Young's modulus, density, mass and flexural rigidity..................... 107 42 Total mass of sandwich structures with different density variations and different core th ick n esses .................................................................................... 1 1 1 43 Maximum normal and shear strains for a given impact energy .............................117 44 Variations of Young's modulus, density, mass and flexural rigidity................. 120 45 Total mass of sandwich structures with same geometry and different density variations ..................................... ................................. ......... 123 46 Maximum normal and shear strains normal and interfacial shear stresses for a given im pact energy ......................... ....... ..... .. ...... .............. 127 47 Flexural rigidity of sandwich structures with same geometry and different density variations......... ............................................................ ... ....... ....... 132 48 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 Throughthethickness 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 ondim 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 Lowvelocity 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 nondimensional variations of core modulus and density .110 4.3 Through the thickness nondimensional 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 tradeoff between weight and stiffness as well as a comparison between these structures and sandwich structures with homogenous core is also presented. The problem of lowvelocity 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 lowmodulus 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 inplane and bending, while the core resists transverse shear loads. A sandwich operates in the same way as an Ibeam 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 stiffnesstoweight 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 cooperating 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, lowweight 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 closedform 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 facesheets. Also the core is highly anisotropic and forming complex curved shapes is difficult. Highdensity 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 lowvelocity 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. Lowvelocity 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 quasistatic 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 facesheets there is a local deformation under the indentor/load whereas the very rigid facesheets 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 closedform 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 singlelayer theories based on assumed displacements, a higherorder theory and a finiteelement analysis. Chapter 3 solves the problems of contact and lowvelocity 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 highperformance composites in aerospace to lowcost 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 higherorder theories. The higherorder 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: singlelayer theories (where the assumed displacement components represent the weightaverage through the thickness of sandwich panel), layerwise theories (where separate assumptions for displacements in each layer are made) and exact theories (equilibrium equations are solved without displacements assumptions). Although discretelayer theories are more representative for sandwich construction than the singlelayer theories, they experience computations difficulties from a large number of field variables in proportion with number of layers. First, a literature review on equivalent singlelayer 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 inplane displacements and w is outofplane 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 twolayer 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 inplane displacements zexpansion: 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 inplane modes of deformation. Lo et al. (1977) theory included both inplane and outofplane 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 lowerorder 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 thirdorder 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 thirdorder 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 twodimensional 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 leastsquares 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 higherorder theory based on cubic expansion for inplane displacements and quadratic expansion for outofplane displacement and used Reissner's definition for kinematics of thick plates, to approximate the plate displacements with weightedaverage quantities that are functions of the inplane 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 singlelayer theories. Discretemodeling (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 exponentialtrigonometric 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 interfacecontinuity 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 threedimensional 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 highorder 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 nonlinear expressions in the thickness coordinate. Their formulation is based on the beam theory for face sheets and a twodimensional 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 sheetcore 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 higherorder 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 (singlelayer or layerwise) 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 quasistatic 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 semielliptical distribution of pressure: (r = p, 1r (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 (1V)+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 halfspace. 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 1v2 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. FischerCripps (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 threedimensional 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 halfspace. 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 lowvelocity 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 facesheet 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 contactforce history and to predict how the damage will develop. Many solutions deals with statically, lowvelocity 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 loaddeflection 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, quasistatic 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. MeyerPiening (1989) modeled the sandwich plate as a threelayer plate with independent linear in plane displacement in each layer. Koissin et al. (2004) determined the closedform solution of the elastic response of a foamedcore sandwich structure under a local loading, using static Lame equations for the core and thin plate KickoffLove 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 lowvelocity 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 springmass 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 speciallyorthotropic 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 facesheet 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 loadindentation relation. FischerCripps (1999) used the finiteelement method to compute the radius of curvature of the contact surface for both elastic and elasticplastic contacts. It is shown that indentors involving elasticplastic 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 nonHertzian 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); * Piecewiselinear 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: springmass models (that accounts for the dynamics of the structure in a simplified manner); energybalance models (when the structure behaves quasistatically); assumed force distribution and complete models (when the dynamics of the structure is fully modeled). Shivakumar et al. (1985) developed springmass and energybalance models to calculated impact force and duration during lowvelocity impact of transversely isotropic circular plates. Energybalance 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 springmass 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 halfplane. The contact stresses take the shape of a semiellipse. 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 halfspace 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 threedimensional linear anisotropic elasticity theory. Effects of changing the material properties, stacking sequence, span and thickness of the plate and indentor size on the forceindentation relation are studied. Anderson (2003) presented an analytical threedimensional 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 loaddeflection 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 highdensity 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 loaddeflection 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 finiteelement method to compute the critical load in the core indentation case. The results agree with threepoint bending experimental results. Liao (2001) investigated the smooth contact between a rigid cylindrical indentor and a composite laminate with arbitrary plyorientation (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 halfspace 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), nonhertzian 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 nonregular, 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, fiberreinforced 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 loadbearing 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 heatresistance ceramics and tough metals with high thermal conductivity. The result is a panel with a high heatresistance on the hightemperature 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 highspeed centrifugal casting method, in which the layers are formed in the radial direction due to different mass densities. An melted AlNi alloy cast into a thickwalled 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 cocarbon 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. ElHadek 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 selfpropagating 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 thermoelastic 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 cracktip 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 halfspace 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 halfspace. Markworth et al. (1995) review analytical models for microstructuredependent 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. Microconstituents are treated as homogeneous continue. Although the material at the macroscale is heterogeneous, it is assumed that the microheterogeneities 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 mixturetype ContinuumMacromechanics analyses, the material properties are expressed in terms of volume fraction and individual properties of constituent phases. Macromechanics models can be subdivided in: direct (closedform 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), temperaturedependent micromechanical theory for thermoelastic response of functionally graded composites. A higherorder 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 nonuniform 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 threedimensional solutions are generated by twodimensional solutions of the thinplate 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 MoriTanaka 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 crosssection. 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 powerlaw throughthethickness 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 thermomechanical and contact analysis of FG beams and also sandwich beams with FG cores. In these studies the thermomechanical 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 closedform 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 tradeoff between using a functionally graded core as opposed to the conventional sandwich design. 3. Compare the tradeoff 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 higherorder 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 higherorder 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. Lowvelocity 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 closedform 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 singlelayer theories based on assumed displacements, a higherorder theory and a finiteelement analysis. A very good agreement among the Fourier seriesGalerkin method, the higherorder theory and the finiteelement 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 EulerBernoulli 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 zaxes. 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) Zr 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 midplane 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: ozz (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 SeriesGalerkin 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: lv v 0 E(z) [C]= v 1 v 0 (1.27) (1 + v)( 2v) 12v 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 SingleLayer FirstOrder Shear Deformation Theory The simplest model investigated in the present study is an equivalent singlelayer model that includes a transverse shear deformation (Firstorder 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 zaxes, the plane strain constitutive relations are: F. c I0(z) C13 (Z) 0 ex 1v v 0 ,z = c13 () C33(z) 0 (z) v 1v 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 crosssection, 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 crosssection: 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 SingleLayer ThirdOrder Shear Deformation Theory Reddy (1984) developed a thirdorder 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 thirdorder 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 thirdorder equivalent singlelayer 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+ 22zhhb 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 HigherOrder Shear Deformation Theory Frostig et al. (1992) developed a higherorder theory for a sandwich beam with a transversely flexible core that uses a beam theory for the facesheets and twodimensional 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 facesheets and for the transversely flexible core (i.e. zero longitudinal stress) are given by: Top facesheet: N' = A',UOtx Bottom facesheet: 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 facesheets, M' are the bending moments in the facesheets, Ai and D1' are axial and respectively flexural rigidities for the facesheets, uo,, w, are face sheets longitudinal and vertical displacements at the centroid (i=t for top facesheet and i=b for bottom facesheet): 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 corefacesheets interfaces and the assumption that for a transversely flexible core (i.e. made of a material with much lower modulus relative to the facesheets), 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 facesheets 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 facesheet and i=b for bottom facesheet); 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 facesheets 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) facesheets 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+lh 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 facesheet thickness 0.3x103 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 midplane 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, a414.327, a5=1.6335), respectively. Poisson's ratios for the core and facesheet, 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 facesheet/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 FourierGalerkin) 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: Throughthethickness 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,, ha 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 FourierGalerkin 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 corecrushing 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 FourierGalerkin 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 midplane. 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 udisplacement 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 FourierGalerkin 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 II I I I I 3 2 1 0 1 2 3 9 x 10 Figure 2.9: Variation of u displacements (given by FourierGalerkin 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 singlelayer firstorder and thirdorder shear deformation theories, Fourier seriesGalerkin method, Frostig model and a finiteelement model. A simply supported sandwich beam, with length L = 0.3 m, core thickness h = 20x103 m and facesheets thickness hf= 0.3x103 m is considered to investigate the effects of varying core properties through the thickness. The facesheet Young's modulus was chosen as 50 GPa. The facesheets 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) Nondimensional 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 coretop facesheet interface (as well as at the corebottom facesheet 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 corebottom facesheet interface and E=50x10 MPa at the coretop facesheet 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 facesheets. 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 firstorder and third order shear deformation theories, Fourier seriesGalerkin method, Frostig model and an ABAQUS finiteelement model. For problem (a), which is the symmetric core under a uniform distributed load, a very good agreement among the Fourier seriesGalerkin method and the ABAQUS model was found. Because the core is symmetric about the midplane, 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 crosssection (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 crosssection (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 crosssection (L/4) is presented in Figure 2.14 (a) and (b). FourierGalerkin method, Frostig model and the finiteelement 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): FourierGalerkin method, Frostig model and the finiteelement 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 singlelayer 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 singlelayer theories and obtained from equilibrium equations (1.70). The latest are identical with those obtained based on FourierGalerkin 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. EOIFEM E  FSDT I 6 FSDT+pqull TSDT , TSDT+equil a SankarN SFST FSDTaqull %,.. iw 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 singlelayer theories based on assumed displacements and a higherorder theory. A finite element analysis is presented in order to validate the models. A very good agreement among the Fourier Galerkin method, the higherorder theory and the finiteelement 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 coretop facesheet 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 lowvelocity impact solution is based on the solution developed here. CHAPTER 3 CONTACT AND LOWVELOCITY 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 lowvelocity 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 lowvelocity impact can be easily understood by analyzing the static contact between the impactor and the structure (Sankar and Sun, 1985). The problem of lowspeed impact of a onedimensional 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 springmass model (quasistatic 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 quasistatic 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 semielliptical 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 halfplane. The contact stresses take the shape of a semiellipse. 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 abovediscussed 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 semielliptical 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. QuasiStatic Impact of a Sandwich Beam with FG Core After the static contact problem was solved, the method is applied to the problem of lowvelocity 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 springmass models (quasistatic assumption) accomplish this. The use of static loaddeflection 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, quasistatic assumptions yield sufficiently accurate results for impact force history and ensuing stresses in the impacted plate. Because the problem deals with the lowvelocity 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: Lowvelocity 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 quasistatic model the maximum contact force in the case of lowvelocity 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 103 m and face sheet thickness hf 0.3x 103 m is considered to investigate the effects of varying core properties through the thickness. The facesheet 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 facesheet 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 ydirection 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 crosssection and has a higher order variation with respect to the thickness coordinate. The halfbeam deflection is plotted for the same asymmetric core beam in Figure 3.8, whereas Figure 3.9 shows deflection at coretop 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) 045 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 oxx 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 crosssections (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 nonzero stress region remains the same for different Rvalues. x 107 xO 0  hf = 1 mm 2 hh = 2 mm / hf = 0.3 mm 4 S6 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 nonzero region remains the same (Figure 3.21) 