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1 MICROMECHANICS OF COMPOSITES USING A TWO PARAM E TER MODEL AND RESPONSE SURFACE PREDICTION OF ITS PROPERTIES By ASHESH SHARMA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013
2 2013 Ashesh Sharma
3 To my parents, thank you both for supporting me in every possible way to help me achieve my goals and fulfill my dreams.
4 ACKNOWLEDGMENTS Firstly, I would like to thank my parents for their encouragement, blessings and never ending support which has h elped me achieve one of the mo st cherished dreams of my life, i.e. to study in the USA which has allowed me to participate in research work of such a magnitude I would like to thank Dr. Bhavani V. Sankar for taking me on as a e a chance to work on this wonderful project. I would also like to thank the members of my supervisory committee, Dr. Raphael T. Hatka and Dr. Ashok Kumar. I am grateful for their willingness to serve on my committee, and for constructive criticism they h ave provided. I would also like to thank my lab colleague, Sayan Banarjee for providing valuable guidance at many stages of this research work A v ery special thanks to Dr Raphael Haftka for being ever so patient while helping me out with doubts concerning surrogate modeling o ver my progress during this project. It has truly been w onderful working with everyone.
5 TABLE OF CONTENTS P age ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 11 1.1 Scope of Work ................................ ................................ ................................ .. 11 1.2 Thesis Outline ................................ ................................ ................................ ... 12 2 BACKGROUND AND LITERATURE REVIEW ................................ ....................... 13 2.1 History of Microme chanics ................................ ................................ ................ 13 2.2.1 Rule of Mixtures Voigt and Reuss Models ................................ .......... 15 ................................ ................................ ...... 17 2.2.3 Self Consistent Theory (1965) and Halpin Tsai Equations (1969) ........ 20 2.2.4 Mori Tanaka Method (1973) ................................ ................................ .. 23 2.2.5 Method of Cells (1989) ................................ ................................ .......... 24 2.3 Use of Finite Element Based Analysis ................................ .............................. 25 2.4 Introduction to Surrogate Modeling ................................ ................................ ... 26 3 TWO PARAMETER BASED APPROACH USING MICROMECHANICS ............... 30 3.1 Concept of Homogenization ................................ ................................ .............. 30 3.1.1 Concept of Average Stresses ................................ ................................ 31 3.1.2 Concept of Average Strains ................................ ................................ ... 32 3.2 Transverse Isotropy ................................ ................................ .......................... 33 3.3 The Two Parameter Model ................................ ................................ ............... 39 3.3.1 Finite element analysis of representative volume element .................... 45 3.3.2 Completing the T matrix ................................ ................................ ........ 55 4 RESPONSE SURFACE PREDICTION OF THE PARAMETERS T 22 AND T 32 .... 57 4.1 Surrogate Modeling Using Polynomial Response Surfaces .............................. 57 4.2 Constructing the Response Surface ................................ ................................ 61 4.2.1 Selection of Variables And Generation of Design Po ints ....................... 61 4.2.2 Constructing and Choosing an Accurate Response Surface ................. 63 4.3 Result Analysis ................................ ................................ ................................ 65 4.3.1 Adopting a Reliable Estimate for the Halpin Tsai Parameter, ............. 66 4.3.2 Result Analysis apropos T 300 Epoxy ................................ ................... 68
6 4.3.3 Result Analysis apropos Graphite Epoxy ................................ .............. 69 4.3.4 Result Analysis apropos Kevlar Epoxy ................................ .................. 70 4.3.5 Result Analysis apropos S glass Epoxy ................................ ................ 71 4.3.6 Result Analysis apropos E glass Epoxy ................................ ................ 72 4.3.7 Result Analysis apropos Boron Epoxy ................................ ................... 73 5 PREDICTING THE LONGITUDNAL SHEAR MODULUS ................................ ....... 74 5.1 Extracting Using Abaqus ................................ ................................ ........... 74 5.1.1 Analysis of the Hexagonal Finite Element Model ................................ .. 74 5.1.2 Using Displacement along Z axis to Calculate ............................... 76 5.2 Predicting Using Polynomial Response Surface ................................ ....... 81 6 CONCL USION ................................ ................................ ................................ ........ 83 6.1 Summary ................................ ................................ ................................ .......... 83 6.2 Future Scope of Work ................................ ................................ ....................... 85 APPENDIX : RESPONSE SURFACE COEFFICENTS ................................ .................. 86 A.1 For The Prediction Of And ................................ ................................ 86 A.2 For The Prediction Of ................................ ................................ ............... 88 REFERENCES ................................ ................................ ................................ .............. 89 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 93
7 LIST OF TABLES Table page 3 1 Results from unit strain analysis along the transverse directions. ...................... 54 4 1 Accuracy and predictive capability characteristics of various response surfaces. ................................ ................................ ................................ ............. 64 4 2 Fiber and matrix properties used for accuracy check of the response surfaces [42, 45]. ................................ ................................ ................................ 65 4 3 Accuracy and predictive capability characteristics of various response surfaces for the prediction of ................................ ................................ ....... 82 A 1 Variable interpretation of numbers ................................ ................................ ...... 86 A 2 Coefficients corresponding to the 3 rd degree full response surface for ........ 86 A 3 Coefficients corresponding to th e 4 th degree reduced response surface for ................................ ................................ ................................ ..................... 87 A 4 Variable interpretation of numbers ................................ ................................ ...... 88 A 5 Coefficients corresponding to the 4 th degree full response surface for ........ 88
8 LIST OF FIGURES Figure page 2 1 the problem. ................................ ................................ ................................ ........ 19 2 2 Arrangement of fiber and matrix ................................ ................................ ......... 22 2 3 Original Method of Cells model with representative volume element (RVE) consisting of 1 fiber sub cell and 3 matrix sub cells. ................................ ........... 25 2 4 A function f(x) trying to mimic the underlying behavior corresponding to a set of selected data points. The space between two data points may or ma y not reflect the true nature of the process that is being modeled. .............................. 28 3 1 Cross section of a hexagonal unit cell lying in the 2 3 plane, before and after a 60 degree rotation about the longitudinal axis. ................................ ................ 36 3 2 A unidirectional fiber surrounded by matrix ................................ ........................ 40 3 3 Representative volume elements for composites ................................ ............... 46 3 4 Axis configuration ................................ ................................ ............................... 47 3 5 1/4 th representative volume element ................................ ................................ ... 49 3 6 Boundary conditions for RVE analysis ................................ ................................ 50 3 7 Schematic of the procedure followed to obtain and .............. 52 3 8 Hexagonal RVE model undergoing unit strain deformation ................................ 53 4 1 Variation of with the fiber volume fraction, in models with isotropic fibers ....... 67 4 2 Comparison between the response surface, and rule of mixtures with respect to finite element analysis of T 300 epoxy using Abaqus ................................ ..... 68 4 3 Comparison between the response surface, and rule of mixtures with respect to finite element analysis of Graphite epoxy using Abaqus ................................ 69 4 4 Comparison between the response surface, and rule of mixtures with respect to finite element analysis of Kevlar epoxy using Abaqus ................................ .... 70 4 5 Comparison between the response surface, and rule of mixtures with respect to finite element analysis of S glass epoxy using Abaqus ................................ .. 71
9 4 6 Comparison between the response surface, and rule of mixtures with respect to finite element analysis of E glass epoxy using Abaqus ................................ .. 72 4 7 Comparison between the response surface, and rule of mixtures with respect to finite element anal ysis of Boron epoxy using Abaqus ................................ ..... 73 5 1 Boundary conditions for shear strain analysis. ................................ ................... 75 5 2 Hexagonal RVE model undergoing unit shear strain deformation in the x z plane. ................................ ................................ ................................ .................. 76 5 4 Quadrilateral equivalent of a triangular reference element. ................................ 79 5 5 Step by step schematic, for calculating the longitudinal shear modulus of the composite using the displacement of the finite element model along the z axis. ................................ ................................ ................................ .................... 80 5 6 Variation of fiber longitudinal shear modulus with respect to the volume fiber ratio ................................ ................................ ................................ ....... 81
10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Master of Science MICROMECHANICS OF COMPOSITES USING A TWO PARAM E TER MODEL AND RESPONSE SURFACE PREDICTION OF ITS PROPERTIES By Ashesh Sharma May 2013 Chair: Bhavani V. Sankar Major: Mechanical Engineering A novel method involving the use of micro mechanical analysis to predict elastic constants of a unidirectional fiber reinforced composite is proposed. The method revolves around a model called the two parameter model. The model, similar to alculation of four out of five elastic constants of a transversely isotropic composite material. The two parameters are obtained by performing one micromechanical analysis of the composite unit cell along any of its transverse directions. Although the theo ry is shown to work for both, square and hexagonal unit cells, the thesis focuses on hexagonal unit cells. The method involves construction of surrogate models that act as response surface for the prediction of the two parameters corresponding to any fiber reinforced composite design that uses a transversely isotropic fiber and an isotropic matrix. The fifth elastic constant, longitudinal shear modulus is predicted using a separate response surface based on finite element simulations in Abaqus. The results indicate that the two parameter model in conjunction with the response surface makes accurate prediction of composite elastic constants. Based on the results modified Halpin Tsai equations are proposed for transverse properties of composites with hexagonal unit cells.
11 CHAPTER 1 INTRODUCTION 1.1 Scope of Work Micromechanics has been a widely adopted approach for the prediction of effective properties of a composite material. Most methods in micromechanics of materials are based on continuum mechanics rather than on atomistic approaches such as molecular dynami cs For more than a century, researchers have made use of continuum mechanics to come up with various mathematical expressions including rules of mixtures (Voigt (1887) and Reuss (1929)), and mean field methods (e.g. Eshelby(1957)) leading to the derivatio n of semi empirical formulae (Halpin and Tsai (1969)), as models for the calculation of approximate effective properties. One problem with semi empirical formulae is that they are not based on observations and experiments, but on theory combined with anal ytics While these models have been shown to be more or less accurate in most of the cases, a need for an entirely empirical expression is felt such as that of a surrogate model, which is constructed solely on the basis of data collected using simulations One such attempt has been made through the medium of the research presented in this thesis. A less expensive way of developing such expressions is through the construction of polynomial response surfaces based on a micromechanical model, where the surroga te models have been built using numerous data points obtained after performing finite element based analysis of designs corresponding to each data point. The essence of this approach is a two parameter model which is essentially a mean field method, simila r to Eshelby's solution for an ellipsoidal inclusion in an infinite solid Although the two parameter model is found to be efficient in predicting the
12 effective longitudinal moduli of composites, this model is more relevant for the prediction of effective transverse moduli and is shown to be slightly more accurate than the Halpin Tsai equations. 1.2 Thesis O utline The primary objective of the dissertation was to come up with a set of expressions that will help in the accurate prediction of the effective composite properties (particularly the elastic moduli) of a unidirectional fiber reinforced composite. Composites with only transversely isotropic fiber or completely isotropic fiber, combined with an isotropic matrix only, have been taken into account. A significant part of coming up with these expressions is the derivation of the two theory for inhomogeneous material. The model although valid for both square and hexagonal un it cells, is implemented using a response surface, for only hexagonal unit cells. The elastic constants predicted using the two parameter model are A separate response surface is constructed for the prediction of the longitudinal shear modulus, All the response surfaces were constructed using the surrogate toolbox developed by Felipe A. C.Viana .
13 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW 2.1 History of Micromechanics Composite materials are engineered or naturally occurring materials made from two or more constituent materials with significant ly different physical or chemical properties which remain separate and distinct with in the finished structure. Engineers have constantly been exploiting the field of composite s as a source for stronger and stiffer, yet lighter materials. Other properties that can be improved by forming a composite material include thermal conductivity, fa tigue life, wear resistance, temperature resistance and thermal properties such as thermal conductivity and coefficie nt of thermal expansion, etc  The primary objective here is to develop a material that has only the characteristics which are needed t o perform the design task in question. Composite materials are usually orthotropic or transversely isotropic . This property makes it difficult for the analyst to analyze their behavior which is necessary to the design process of any structure involvin g the materials. The properties of the composite itself are not known, especially if the constituent materials themselves have been changed. Therefore, it is required, that extensive testing of the composite is performed before deeming it usable. That is, the knowledge of the nature of directionality (isotropic or anisotropic) of the composite is not enough. In addition, an appropriate failure criteria along with a numerical method (such as the Finite Element Method) to solve the boundary value problem shou ld be known for the complete analysis of a laminate of the concerned composite material.
14 Generally, this heterogeneous nature of composite materials is studied from two points of view macromechanics and micromechanics. Many composite analyses are performed using macromechanical analysis, wherein material is assumed to be homogenous and the effects of the constituent materials are treated as averaged apparent macroscopic properties of the composite material. However, the true nature of a composite material is that of a randomly space anisotropic material (called the fiber) in an isotropic me dium (called the matrix). In contrast to macromechanics, micromechanics examine the interaction of the constituent materials at a microscopic scale, and study their effect on the overall properties of the composite. Consequently, investigation of micromech anics is an approach based solely on the constituent materials of a composite and how various proportions and arrangements of the constituent materials may help in achieving the desired properties. In doing so, a micromechanical approach may provide a bett er insight into the interactions between the fiber and the matrix, thus providing with a more accurate model for analyzing the behavior of the composite. In some instances the constituents include the interface between the fiber and the matrix, where the t ransition of properties occur. This interface (at times also termed as interphase) plays a crucial role in the behavior of the composite An important advantage posed by a micromechanical model over a macromechanical model is the ability of micromechanics to capture the physical deformation and damage at a more fundamental scale. This allows for easier grasping of in situ multi axial stress and strain rates Not to mention, micromechanics utilizes simpler isotropic constituent constitutive models and failur e criteria . It is important to
15 note that failure tends to occur mostly at the micromechanical level, making it difficult to be captured at the macroscopic level. P roperties that can be predicted using micromechanics include thermo elastic constants, vi scoelastic properties, strength properties, yield function, fracture toughness, thermal conductivities and electromagnetic properties . It should be pointed out that micromechanics is still an approximation of any real composite model, most significantl y the approximation of the fiber geometry within a composite. Over the next few sub sections, various micromechanical models developed over the years are discussed. Excluding the rule of mixtures (2.2.1) all the analytical approaches described in the foll owing sub By definition, mean field methods assume that the s in the matrix and the fiber, are adequately represented by their volume averaged values, though they differ in the way they account for the elastic interaction between the phases These methods were later extended to include plasticity . 2.2.1 Rule of Mixture s Voigt and Reuss Models Generally, t for some scalar effective physical phase composite take s the form, (2 1) where (i) and (m) denote inhomogeneities (mostly fiber) and matrix respectively. The The Voigt model (1887) assumes constant strain throughout the composite (iso strain model) . The effective composite stiffness is calculated as a combination of the individual fiber and matrix stiffness weighted by their respective volume fractions. The
16 result, thus, is a ru le of mixtures formulation for the stiffness components of a composite ( = 1 in Eq. 2 1). (2 2 ) Here is the equivalent stiffness for the composite. and are the stiffness components for the fiber and matrix elements re spectively. is the fiber volume ratio for the composite material. Reuss (1929) proposed a rule of mixture model where stress was assumed to be constant throughout the composite (iso stress model) . This model calculates the effective composite comp liance as a combination of the individual fiber and matrix compliance weighted by their respective volume fractions. The result, thus, is a rule of mixtures formulation for the compliance components of a composite ( = 1 in Eq. 2 1). (2 3 ) Here is the equivalent compliance for the composite. and are the compliance components for the fiber and matrix elements respectively. It is suggested that Voigt type expressions (Eq. 2.2), correspond to full strain coupling of t he phases (springs in parallel arithmetic averages) "whereas, the Reuss type expressions (Eq. 2.3) correspond to full stress coupling of the phases in series harmonic averages )" . The practical use of these models however, depends on the microt opology and material properties of the composite in discussion The models are clos ely related to the Hill bounds , with the Voigt model providing the upper bound and the Reuss model providing the lower bound .There are a few cases where the predict ion of the Voigt and Reuss models is known to lie outside the Hashin Shtrikman bounds. Hashin Shtrikman provided bounds on the elastic moduli
17 and tensors of transversally isotropic composites  (reinforced by aligned continuous fibers) and isotropic com posites  (reinforced by randomly positioned particles) Another approach closely related to the rule of mixtures is the Vanishing Fiber Diameter Model (VFD). Developed by Dvorak and Bahei el din in 1982, the model depicts combination of average stres s and strain assumptions visualized as each fiber having a vanishing diameter yet finite volume within a matrix [ 6, 12]. 2.2.2 usefulness in the field of continuum micromechanics. The problem is isotropic elastic medium. The inclusion undergoes deformation which, if not constrained by the surroundings (referred to as the matrix), would be equal to an arbitrarily applied homogeneous strain. The question posed by this problem, requires the determination of the elastic state of the inclusion and the matrix. Es helby suggested that the problem could be solved by removing the inclusion from the infinite medium and allowing it to deform. Now, surface tractions were applied to bring the inclusion back to its original form, after which the inclusion was placed back i n the medium. Although the matrix is now stress free, the inclusion is under a state of constant stress. The surface tractions applied to the inclusion will manifest as a layer of body forces spread over the interface between the inclusion and the matrix [ 5] Since these forces were not present in the original problem, they have to be removed by applying equal but opposite body forces, which will induce stresses in the matrix as well as the inclusion. Since the matrix and inclusion are made of the same mate rial this problem could be solved using the
18 solution will yield the correct stresses for the matrix, but for the inclusion the uniform state of compressive stresses shou ld be added to complete the solution. Up until this point, nothing was assumed regarding the shape of the inclusion. However, if an ellipsoidal shape is assumed, it was proved that the stress within the inclusion is uniform . The solution to the transf ormation problem provides pathway for the solution to a different from that of the rest of the medium. Eshelby , studied the effect of this inhomogeneity on remote stresses, and concluded that knowledge of the uniform strain within the ellipsoid was sufficient to analyze most aspects of the inhomogeneous model (Figure 2 1). To solve the i nhomogeneity problem, let us assume the inclusion to be a f and an elasticity tensor C f m and elasticity tensor C m Let the single fiber matrix sys 0 with respect to a local coordinate fiber element as given by Rahman and Chakraborty in  are, (2 4 ) (2 5) : and
19 point tensor. is the fourth ranke Figure 2 1. with the problem.
20 It is to be mentioned here that the Eshelby model is valid only for a single inclusion in an infinite medium. Hence, if more than one inclusion is taken into account, it is important to make sure that the inclusions are far apart. The self consistent method developed by Hill (2.2.3) and the Mori Tanaka method (2.2.4), are effective medium medium. 2.2.3 Self Consistent Theory (1965) and Halpin Tsai Equations (1969) idal problem to replace the matrix with a material having the effective properties, of the composite to be developed. That is, (in Eqs. (1 4) (1 5)) is replaced by (unknown effective stiffness matrix of the composite). Also known as the Self Cons istent Model (SCM), the theory assumes a composite model in which the embedded phase consists of continuous and perfectly aligned cylindrical fibers (Figure 2 2 A ). The inclusion and the medium, both were assumed to be homogeneous and elastically transvers ely isotropic about the fiber direction . An iterative type approach can be used for solving the SCM. Here, it is assumed that the infinite medium in the Eshelby model is made up of the matrix material as before and C is calculated. The matrix is then replaced by the composite material with stiffness C and new C is calculated The iterations are continued until the elastic constants of the composite converge [5 ]. Hermans  generalized the self consistent model, wherein the composite is treated by as suming each fiber to behave as though it were surrounded by a cylinder of pure matrix. Outside this cylinder lies a body with the properties of the composite in discussion (Figure 2 2 B ).
21 elf consistent model abbreviating it to an approximate form that can be used to obtain the transverse d the transverse Poison's ratio using the semi empirical formula, (2 6 ) where, (2 7 ) Here, is an effective composite modulus, is the corresponding fiber modulus, is the corresponding matrix modulus, is the fiber volume fraction, and is a measure of the reinforcement geometry which is dependent on the loading conditions. Equation ( 2 6 ) provides reasonable estimates for the equivalent longitudinal shear modulus the equivalent transverse Young's modulus and the equiva lent transverse Poison's ratio Reliable estimates of have been obtained by comparison of equations (2 6 ) and (2 7 ) with the numerical micromechanics solution employing formal elasticity theory. Halpin and Tsai, found that the value gave an excellent fit to the finite differenc e elasticity solution of Adams and Doner  for of composite model of square array with circular fibers. For the same material and fiber volume fraction, a value of was found to be in good agreement for . The Halpin Tsai formula also cl aims to provide a good correlation with Foye's computations  for hexagonal array of circular fibers with volu me fiber fraction of up to 65%. However, exact elasticity ca lculations predict the moduli to rise faster with increasing volume fraction of reinforcement above a volume fiber fraction of 0.70  as compared to the
22 Halpin Tsai equation. To incorporate this effect, Hewitt and de Malherbe  suggested the following values of that were dependent on ( A ) ( B ) Figure 2 2 Arrangement of fiber and matrix in A ) Hi ll's self consistent model B ) Hermans' modified self consistent model
23 2.2.4 Mori Tanaka Method (1973) Until now, none of the methods discussed, take into account the interaction between inhomogeneities. One way of introducing collective interactions amongst inhomogeneities consists of approximating the stresses acting on an inhomogeneity, which may be view ed as disturbance stresses (that are caused due to the presence of other inhomogeneities ) superimposed on the already present far field stress Here f ourth order tensor relates average inclusi on strain to average matrix strain and approximately accounts for fiber interaction effects. An important feature of this model was the consideration of internal elastic energy due to the presence of interaction effects between the inclusions along with the presence of a free boundary. Mori and Tanaka presen ted a method for calculating the average stress, within the matrix of a material containing inclusions, with transformation strain. They showed that the average stress in the matrix is uniform throughout the material and independent of the position of the domain where the average treatment is carried out. They also proved that the actual stress in the matrix is equal to the sum of the average stress and, the locally fluctuating stress (disturbance stresses) the average of which was shown to vanish in the m atrix body of the composite  Mori Tanaka model describe composites consisting of aligned ellipsoidal inhomogeneities embedded in a matrix, i.e., "inhomogeneous materials with a distinct matrix inclusion microtopology" . Ponte Castaneda and Willis [ 26] showed that the Mori Tanaka method could be treated as a special case of the Hashin Shtrikman variational estimates  where the spatial arrangement of the inhomogeneities follows an ellipsoidal distribution which has the same aspect ratio and orient ation as that of the inhomogeneities themselves, as portrayed in 
24 The Mori Tanaka approach has been used in the development of a recently created micromechanical model, the Bridging model . The bridging model requires the knowledge of only the constituent fiber and matrix properties along with the fiber volume ratio of the composite. Here, a f ourth order tensor relates average fiber stress to average matrix stress and thermal residual constituent stresses can be predicted. 2.2.5 Method of Cells (1989) The method of cells is a micromechanical model developed by J. Aboudi , which has been shown to accurately predict the overall behavior of various types of composites from the knowledge of the constituent properties. This method explicitly yields effective constitutive equations for the inelastic beh avior of metal matrix composites The overall behavior of inelastic multi phase unidirectional fibrous composites predicted by the method of cells is portrayed in terms of the effective elas tic moduli effective coefficients of thermal expansion effective thermal conductivities and the effective stress strain resp onse in the inelastic region [29 ]. The inclusion geometry consists of doubly periodic array of fibers  embedded within the m atrix with t he unit cell further divided into four sub cells, one for the fiber and three for the matrix as shown in Figure 2 3 Being a mean field method, the effective stiffness matrix is obtained by relating the average stresses to the average strains i nside the sub cells, and consequently average over the complete volume of the unit cell. Consequently Aboudi went on to propose a generalized formulation (Generalized Method of Cells Model) wherein the repeating unit cell is subdivided into an arbitrary n umber of sub cells. The purpose of this generalization, was to extend the modeling capability of the original method of cells to further include the thermo mechanical
25 response of multi phase metal matrix composites, modeling of variable fiber shapes, analy sis of different fiber arrays, modeling or porosities and damage, and modeling of interfacial degradation . This generalization of the original method of cells allowed for micromechanical analysis of much more complicated periodic array of cells. Figure 2 3 Original Method of Cells model with representative volume element (RVE) consisting of 1 fiber sub cell and 3 matrix sub cells. 2.3 Use of Finite Element Based Analysis The finite element method (FEM) is a numerical technique for finding approximate solutions to part ial differential equations and their systems, as well as integral equations  In simple terms, FEM is a method for dividing up a very complicated problem into small e lements that can be solved in relation to each other. With the onset of an era
26 dominated by computers, finite element based analysis has become one of the most widely accepted approach for engineers, in the field of design. Most micromechanical methods use periodic homogenization That is, the composite material is c haracterized by a periodic unit cell (or a representative volume element). The overall elastic behavior of such a composite is now governed by the behavior of an individual unit cell which can be described using a combination of boundary conditions, stress fields and strain fields. Finite element formulations can prove to be a very efficient tool in accurately capturing the mechanical behavior of such heterogeneous materials. Thus, due to its high flexibility and efficiency, finite element methods, at pres ent are one of the most widely used numerical tool in the field of continuum micromechanics. The presented research was carried out using Abaqus/CAE or C omplete A baqus E nvironment". It is a software application used for both, the modeling and analysis of mechanical components and assemblies (pre process ing) and visualizing the finite element analysis result. A baqus provides the analyst with the ability to add to the material and element libraries through the use of user sub ro utines coded in FORTRAN. For th is research, a student edition of A baqus was used, which limits the maximum number of mesh nodes to 1000. This is the primar y reason as to why only quarter of the complete unit cell has been used for analysis. 2 .4 Introductio n to Surrogate Modeling For many real world problems, a single si mulation can take huge amounts of time, thus increasing the overall cost of the design process A cheaper way of going about this process is by constructing approximation models, known as surrogate models The phenomenon of surrogate modeling can be explained as the process of developing a
27 continuous function f(x) (based on a set of carefully chosen design variables) when only limited amount of data is available  This function helps i n filling the underlying gaps between the simulated data points ( Figure 2 4). These models mimic the behavior of the simulation model as closely as possible while at the same time being computationally cheaper to evaluate. In surrogate model based optimiza tion, the surrogate is tuned to mimic the underlying model as closely as needed over the complete design space. Such surrogates are a useful and cheap way to gain insight into the global behavior of the system  The process comprises of four major step s: Sample selection also known as selection of design of experiments (also known as data points) Simulations at the selected design of experiments Construction of the surrogate model Estimation of the accuracy of the surrogate. For example, a helicopte r design engineer may make use of a surrogate model to simulate the fall time of a helicopter using autorotation. To do so, data from practical tests will have to be collected at selected design points. Using this data a surrogate model will be constructed based on design variables such as the rotor length and rotor width. Depending on the accuracy of the model, the engineer may now use the model to s imulate the fall time of the helicopter at points other than the data points. The accuracy of the surrogate depends on the number and location of samples in the design space, i.e. how evenly the samples have been located within the design space. Various techniques are available to select the design of experiments (DOE). However, at times, it is almost inevitable to filter errors, in particular errors due to noise in the data or errors due to an improper surrogate model.
28 Figure 2 4 A function f(x) trying to mimic the underlying behavior corresponding to a set of selected data points. The space between two data points may or may not reflect the true nature of the process that is being modeled. Some of the most popular surrogate models are Polynomial Response Surfaces (PRS), Kriging Support Vector Regressions (SVR) and Radial Basis Neural Networks (RBNN). For most problems, the nature of true function is not known and so it is not clear which surroga te model will be most accurate.
29 For the work presented in this thesis polynomial response surface methodology was adopted to d evelop surrogate models to mimic the behavior of underlying composite designs. The accuracy of the surrogates is measured with the help of various error es timates. Detailed discussions on polynomial response surface methodology have been presented under Ch apter 4
30 CHAPTER 3 TWO PARAMETER BASED APPROACH USING MICROMECHANICS 3.1 Concept of Homogenization reinforced (UD) composite is essentially a collection of m atrix scale, a distinct geometrical and material structure can be described. But, since designing at this level of observation can prove to be extremely expensive, in most engineering applications, the composite is assumed to b e one statistically homogeneous material with a certain material symmetry. These are the fundamental assumptions in the theory of composite homogenization  Composite homogenization is essentially a mechan ics based modeling technique which helps transf orm a media consisting of heterogeneous material into a constitutively equivalent media comprising of homogeneous continuum. This is a statistically averaging process involving the presence of certain properties (e.g. average stress and average displacemen t) that control the deformation of the continuum media. The concept of periodic homogenization assumes that the region of the composite that we are interested in ( i.e. the structural scale) is a few order of magnitude s greater than the scale of the inclus ions (fibers). For exampl e, when the fiber diameter is of the order of 10 microns the region over which the properties are to be averaged should be of the order of around 100 microns To put it explicitly, there should be sufficient number of unit cells p resent in the structure for the homogeneous properties to be valid. B y homogenization, the composite is voided of its physical microstructure The process of homogenization also assumes that the macro stresses and macro strains are uniform in the region. If there are significant stress gradients present within
31 the region of homogenization, their effects will be to change the effective properties of the composite. For example, in textile structural composites the unit cells tend to be much larger than that of unidirectional composites. The unit cell in a textile composite can be in the order of millimeters, and they approach the structural thickness in many cases. There may be fewer than 5 unit cells present through the thickness of the structure, and the stre ss gradient effects will be very pronounced in the stiffness and strength properties  The micromechanical analysis of a unidirectional fiber composite is performed by analyzing the representative volume element (RVE) of the composite. Here, it would be appropriate to point out the difference between a unit cell and an RVE. There is a slight distinction between RVE and unit cell. When the unit cell is used for homogenization, the assumption is that it repeats itself without any variations. On the other hand an RVE will have several unit cells ranging from a few to hundreds, and it allo ws for minor variations in the shape of the unit cells present within the RVE A n RVE can be thought of as a bigger unit cell com prising of several smaller unit cells with minor variations which repeats itself throughout the composite 3.1.1 Concept of A verage S tresses Before discussing the derivation leading to the two parameter micromechanical model, it would be useful to present some basic concepts related to homogenization. The macroscopic stresses or macro stresses are defined as the average of the element str esses acting over the representative volume element, (3 1 )
32 w here is the stress in the constituent phases of the representative volume element and V is the volume of that element. This is the actual stress in the matrix or the inclusi on (or the fiber) and is referred to as microscopic stress or micro stress. Thus the expression in equation (3 1) can be expressed as a combination of the stresses in the inclusion phase ( ) and the stresses in the matrix phase ( ). (3 2 ) While computing the effective properties, the composite is assumed to be in a state of homogeneous or uniform macroscopic stress. However, the micro stresses, i.e., the stresses in the fiber and matrix, will not be uniform and t h ey vary over corresponding mean values. Further, the micro stresses will be identical at identical locations in two different unit cells. Thus the micro stresses are periodic in space and hence the term periodic homogenization. Similar is the case with st rains. The macroscopic stresses can be related to the traction acting on the surfac es of the RVE as giv e below and as described in . (3 3 ) H ere is the boundary surface of the unit cell, are the surface tractions associated with the representative volume element. 3.1. 2 Concept of Average Strains Similar to the relation between the macro stresses and the trac tions acting on the surfaces of the representative volume element we can rel at e the macro strains to the representative volume element surface displacements. The macro strains are strains
33 that have been averaged over the volume of the representative volume element and are defined as (3 3 ) (3 4 ) H ere is the strain in the constituent phases (fiber and the matrix) of the representative volume element. This is the actual strain in the matrix or the inclusion and is referred to as microscopic strain or micro strains. and are the displacement gradients associated with the composite on the macro scale. U sing the relation in equation (3 4 ) and applying Gauss' Theorem to it, we can find a relation between the volume average displacement gradients and the boundary surface of t he representative volume element as described in . (3 5 ) 3.2 Transverse Isotrop y Before proceeding any further tow ards deriving the two parameter micromechanical model, we shall review some concepts involving the nature of directionality which is highly common and extremely significant among st fiber reinforced composites. The compliance and stiffness matrix of an y anisotropic material (no form of isotropy is present) has 36 components, which are expressed in terms of the engineering constants of the respective material. Using concepts of strain energy, it can be shown that the matrix is symmetric, and only 21 of those components are independent . Along these lines, th e compliance and stiffness matrix corresponding
34 to an orthotropic ma terial can be expressed in terms of 9 independent engineering constants as shown equation in (3 6) (3 6 ) This orthotropic nature of the material can be explained by the presence of two orthogonal planes of material property symmetry. As a result of which, symmetry exists relative to a third mutually orthogonal plane. As mention ed in sub section 1.2, this thesis focuses solely on fiber reinforced composites wherein the fibers are present either in transversely isotropic or completely isotropic configurations and the matrix is always isotropic Hence, it is of great imp ortance, that we discuss the se configurations in detail. The stiffness matrix of a transversely isotropic material can be expressed in terms of only 5 independent engineering constants. T ransverse isotropy is a type of orthotropy wherein, at every point o f a material there is one plane in which the mechanical properties are equal in all the directions  For example, i f we consider a fiber composite wherein the fiber axis lies perpendicular to the 2 3 plane, then the subscripts 2 and 3, on the engineeri ng constants are interchangeable That is, the 2 3 plane is the symmetry plane in which the engineering constants are isotropic and in the 1 direction (direction transverse to the symmetry plane), the engineering constants are different.
35 Using the formula the stress strain relations in coordinates aligned with principal material directions can thus be expressed by (3 7 ) where, (3 8 ) The relation in equation (3 8) has not been discussed widely for a hexagonal unit cell belonging to a transversely isotropic material (which is our primary focus) Hence, the mathematical calculations leading to the equation have been presented in the literature ensuing Let us consider the transverse cross section of a hexagonal unit cell lyingin the 2 3 plane with 1 being the fiber axis. Now, if the unit cell, a long with the axes, is rotated by 60 degrees in the anti clockwise direction about the longitudinal axis (axis 1) due to its hexagonal geometry, the new position of the unit cell will overlap perfectly with the original position of the unit cell ( Fi g. 3 1) However, the current material coordinate system ha s been visibly displaced /rotated in the 2 3 plane with respect to the initial
36 material coordinate system T he current transformed axes are Note that the position of the longitudinal axis remains unchanged. Figure 3 1 Cross section of a hexagonal unit cell lying in the 2 3 plane, before and after a 60 degree rotation about the longitudinal axis. The stiffness m atrix of the unit cell is similar to the mat rix presented in equation (3 7) except, for the purpose of derivation we will treat as an independent constant too. The compliance matrix in equation (3 7) is invert ed to provide us with the stiffness matrix for the unit cell when in its initial position (3 9 ) where, (3 10 )
37 There upon, t he stress strain relations, corresponding to the initial unit cell configuration in the 2 3 plane are given by (3 11 ) In order to find a relation between the stiffness matrix of the unit cell rotated by an angle and the original unit cell, we first bring into consideration, the transformation of stress and strain matri ces The stress and strain matrix for the current coordinate system are hence given by, (3 12 ) (3 13 ) Here and are the stress and strain transformation matri ces, defined by equations (3 14) and (3 15) respectively  (3 14 ) (3 15 ) Equation (3 11), is equivalent of the constitutive for continuous media and can be written as Thus, substituting for and from equations (3 12) and (3 13), we have, (3 16 ) Using equations (3 14) and (3 15), the strain transformation matrix can be expressed in terms of the stress transformation matrix such that . (3 17 ) Comparing equation (3
38 (3 18 ) From figure 3 1, it is evident that the material properties will remain unchanged after rotation since the 2 3 plane is a plane of isotropy. Hence, it can be deduced that the stiffness matrix remains unchanged after rotation. Therefore, Making use of this property and substituting for and from equations (3 11) and (3 14) respectively, for equation (3 18) can be written as, (3 19 ) Multiplying the matrices on the right h and side of the equation (3 19), and equating ( from equation (3 9), ) from the left hand side matrix to its corresponding element on the right hand side, we have (3 20 ) Substituting the elements of the stiffness matrix in terms of engineering consta nts as displayed in equation (3 9), (3 21 ) which brings us to the exact r elation depicted in equation (3 8). Thus, we may now conclude that the transverse shear modulus ( ) in a hexagonal unit cell of a trans versely isotropic material can also be calculated in terms of the transverse elastic modulus ( ) and the transverse poisons ratio ( ) as is done the case of a square unit cell
39 T he stiffness matrix for an isotropic material can be expressed in terms of only 2 independent engineering constants (equation (3 22)) due to the presence of material symmetry in each of the principal planes. (3 22 ) On observing equation s (3 7) and (3 22) it is noted that there is no expectation of any sort of interaction between the normal stresses and the shearing strains Similarly, there will be no interaction between the normal strains and the shear stresses 3.3 The Two Parameter Model For the derivation of the two parameter micromechanical model for a finite medium fibers are assumed of circular cross section pa cked in a square/hexagonal array. No sort of isotropy is associated with the fiber or the matrix throughout the derivation. We shall consider a unidirectional fiber reinforced composite surrounded with matrix to comprise a square /hexagonal unit cell (Figur e 3 2 ) for the derivation of our constitutive equation. As discussed under sub section 1.2, we hope to obtain the elastic properties of the composite by performing only one axial micromechanical analysis. Thus, the unit cell is expected to undergo deformation only along the axial directions and no bending or
40 tw isting will be involved. This unit cell is assumed to be under a state of uniform average strain at the macroscopic level called macroscale strains or macrostrains, and the corresponding axial stresses will be referred to as macrostresses. However, the mic rostresses, whic h are the actual stresses in the constituent phases of the representative volume element, will have spatial variation. The surface tractions are assumed to be continuous at the fiber matrix interface. ( A ) ( B ) Figure 3 2 A unidirectional fiber s urrounded by matrix comprising A ) square unit cel l B ) hexagonal unit cell The effective composite macrostress can be expressed explicitly in terms of the stress present in the fiber and the matrix phas es, and is given by equation (3 2). Using the constitutive stress strain relationship, the equation can be rewritten as,
41 (3 23 ) Here constitute the unknown microstrains in the fiber or the matrix phase s a nd respectively represent the fiber and matrix phase volumes ( ). Since there is no macroscale shear deformation,it should be noted that t he respective st rain vectors and their corresponding stiffness matrices ( and ) present in equation (3 23 ) will be characterized by only the normal stress/strain terms and are void of any of the shear terms as shown in equations (3 24 ) and (3 25 ). (3 24 ) (3 25 ) The relations presented in equations (3 24 ) and (3 25 ) are for the effective composite stiffness matrix and the effective strain composite strain vectors. Similar expressions hold true for the corresponding fiber and matrix phases (where effective elastic properties of the composite are replaced with the corresponding fiber and matrix properties). Using equation (3 23 ), we are able to manipulate the expression and modify it by a dding and subtracting from in the first term. The expression can thus be written as:
42 (3 26 ) We now multiply and divide the first term in equation. (3 26 ) with the fiber volume and express the second term in equation (3 26 ) as a function of effective composite macrostrains (Eq. (3 4)) s uch that: (3 27 ) The ratio / in equation (3 27 ) is equal to the fiber volume ratio ( ) of the composite. Thus, substituting for the fiber volume ratio we have, (3 28 ) The term inside the integral is similar t o the expression in equation (3 4) and can be defined as the average strain in the fiber phase such that, (3 29 ) where, is the average strai n in the fiber phase. To express the average fiber strain in terms of the effective composite macrostrain, we shift our focus to the unit cell s displayed in figure 3 2 wherein, the unidirectional fiber has its longi tudinal axis along direction '1'. We stretch the unit cell along the axial direction '1' and keep it constrained along the remaining two directions so that deformation is observed only along axis '1'. Now, referring to equation (3 4) t he
43 average strain (m acrostrain) of the unit cell along direction 1 'can be expressed in terms of the displacement gradient along the respective direction, and is given by: (3 30 ) Substituting from equation ( 3 5 ) in the above equation, the strain can be expressed in term of the total displacement along the axial direction '1'. (3 31 ) The total displacement 'u' along axis '1' can be expressed as the difference between the displacements undergone by any point on face 'a' and face 'b'. Let the displacement of a point on face 'a', along '1' axis, be given by u a and displacement of a corresponding point on face 'b' be given by u b The respective displacement vectors are now given by and Along these lines, the total axial displacement 'u' can be expressed as 'u a u b '. (3 32 ) Multi plying and dividing equation (3 32 ), with the surface area perpendicular to the major fiber axis, we have, (3 32 ) where L1, L2 and L3 are the unit cell dimensions along the respective axes as shown in figure 3 2 Solv ing the integral in equation (3 33 ) and expressing the equation in terms of displacement vectors and the macrostrain along direction '1' is given by,
44 (3 34 ) On examining equation (3 34 ) and figure 3 1, one may note the length L1 of the unit cell is equal to the fiber length. Also since the front and rear face of the fiber lie on face 'a' and face 'b' of the unit cell respectively, it can be said that the average strain of the unit cell is equal to the average fiber strain along '1' axis (the longitudinal axis of the fiber). (3 3 5 ) Making use of this property the average fiber strain vector may now be portrayed in terms of the average composite strain vector such that, (3 36 ) where, T is called the volume averaged fiber strain concentration matrix and is defined as, (3 37 ) Us ing the relation in equation (3 36 ) and substituting for in eq u ation (3 2 9 ) we have, (3 38 ) It is noted that since the macroscopic strain appears in each term of the equation (3 38 ), it can be said that the equation holds true for all values of macroscopic strain which bring us to a constitutive equation relating the effective composite stiffness matrix with the constituent fiber and matrix stiffness matrices.
45 (3 39 ) It can thus be concluded, that the effective stiffness matrix (given by equation ( 3 24 ) ) of any composite can be obtained using the above equation provided one is able t o calculate each element of the volume averaged fiber strain concentration matrix (i.e. the T matrix) displayed in Eq. (3 36 ). Calculating the T Matrix Although equation (3 39) holds true for all types of fiber reinforced materials, we are only interested in composites with transversely isotropic or isotropic fiber material. For the calculation of the T matrix, a procedure was developed keeping in mind the transverse isotropy nat ure of the fibers. Ergo the elements constituting the T matrix, for composites with isotropic fibers, are calculated in an exactly similar fashion. Equation (3 37) has 6 unknown parameters. There are two phases to calculating these parameters. The former stage which involves finite element analysis of the RVE (representative volume element), give us four of the six parameters. T he latter stage involves some basic algebra to help deduce the remaining two parameters. 3.3.1 Finite element analysis of repres entative volume element A representative volume element repeats itself throughout th e composite geometry just like the unit cell would For composites with hexagonal unit cell, a rectangular RVE has been used which effectively encompasses two complete he xagonal unit cells (Fig. 3 3 A ). For composites with square unit cell, the unit cell is used as the RVE (Fig. 3 3 B )
46 ( A ) (B) Figure 3 3 Representative volume elements for composites with A) hexagonal unit cells B ) square unit cells.
47 The essence of using a rectangualr RVE instead of the hexagonal unit cell lies in the convinience observed while applying boundary conditions for the finite element analysis of the cell. With a hexagonal unit cell we would have to make use of periodic boundary conditions  whereas with the rectangular RVE, the boundary conditions become very much similar to those for a square unit cell. The boundary conditions are discussed in detail At this point, it would be important to point out that the default axis configuration in Abaqus is not consistentwith the configuration of a principal material coordinate system. This anomoly is portrayed in Fig. 3 4. (A) (B) Figure 3 4. Axis configuration A) used by Abaqus B) a principal material coordinate system. Although Abaqus displays the coordinate system in terms of x, y and z, we have taken the liberty to express the same in terms of 1, 2 and 3 for convenience purposes. As a consequence, utmost care has to be taken while extracting field output values or
48 ente ring material properties especially for the fiber section, when the fiber properties in concern are transversely isotropic. Throughout this document, any field v ariable that has been obtained from Abaqus, has been portrayed in terms of the coordinate syste m used by Abaqus (e.g. S11, E11, etc.). In contrast, all variables associated with the derivation of the two paramter model (such as ) ,and any material constant (e.g. ) have been portrayed using the principal material coordinate system. A 2D planar deformable part type with base feature shell is chosen for the micromechanical analysis since we are interested in the behaviour of the RVE cross section a cross the 2 3 plane. The sections are solid homogenous and a plane strain thickness of 1 unit is assigned to each section of the model. For testing purposes, we keep the volume fiber ratio fixed at 0.6 (Fig. 3 5 ) Due to the 1000 node restriction present in the student version of Abaqus, we are forced to use only qu arter of the representative volume element. It should be n oted that using quarter of the RVE does not affect the axial analysis due to the symmetry of the RVE. This can be confirmed by performing a full RVE analysis and comparing the results with the a nalys is of a quarter of the RVE. Micromechanical analysis is performed along each of the transverse axes. Analysis along the 2 nd axis will provide us with and These terms are pretty much analogous to average strains in the 2 nd and 3 rd direction respectively, due to the deformations along 2 nd direction. Similarly, micromechanical analysis in the 3 rd direction will provide us with and The dimensions of the RVE are evidently based on those for the unit cells. For composites wit h square unit cells, a square of edge length of 1 unit is used. Similarly, for composites with hexagonal unit cells, a hexagon edge
49 length of 1 unit was used. This resulted in a rectangular RVE of edge length of 1.5 units along the 2 nd direction and edge l ength of 0.866 units along the 3 rd direction. (A) (B) Figure 3 5 1/4 th representative volume element A) for composites with hexagonal unit cells B) for composites with square unit cells. The red section s denote the fiber and the green section s denote the matrix.
50 (A) (B) Figure 3 6 Boundary conditions for RVE analysis A) along direction 2 B) along direction 3.
51 Boundary conditions : Boundary conditions applied to the representative volume elements correspond to a unit strain analysis. That is, for a micromechanical analysis along either of the transversere directions, unit strain is applied along the respective direction Fig. 3 6 depicts the boundary conditions for micromechanical analysis of composites with square unit cell. The boundary conditions for composites with hexagonal unit cells using rectangular RVE remain the same except for the fact that the values of L and B are different. Meshing : Meshing of the finite element models was d one using 4 node bilateral plane strain quadilateral elements with provisions for reduced integration. Since there was almost no loss in accuracy, reduced integration was opted for as it helps reduce the computational time. A total of 244 elements were use d for the model representing quarter of the square RVE model (Fig. 3 5 A ) and 612 elements were used for the model representing quarter of the rectangular RVE model (Fig. 3 5 B ). Mesh convergance analysis was carried out for both the models to affirm the accuracy of the results obtained upon the analysis of the models. For instance, in the case of the model for hexagonal unit cells, a model with 3 24 elemets was observed to produce the exact results that were obtained using a model with 612 elements. Field outputs : Field outputs t hat were extracted, comprised the elemental strain in the fiber of the model (E11 and E22, both extracted at the centroid of each element), the elemental stresses S11, S22, S33 for the complete model (also extracted at the centroi d of each element). Additionally the volume of each element in the relevant model was extracted. Fig. 3 7 portrays a schematic that summarizes the whole process of obtaining and via the analysis of the finite element models.
52 Figure 3 7 Schematic of the procedure followed to obtain and
53 Figure 3 8. displays the deformed hexagonal RVE models with displacement contours, upon micromechanical analysis along each of composite transverse direction. (A) (B) Figure 3 8 Hexagonal RVE model undergoing unit strain deformation A) along 2 nd direction B) along 3 r d direction.
54 For an initial testing of the model given by equation ( 3 39 ) the procedure in Fig. 3 7 was applied to both the square and rectangular finite element models. Finite element analysis was performed using two random designs one with a transversely isotrop ic fiber (graphite  ) and the other with a completely isotropic fiber ( variant of e glass  ) Since we are interested only in isotropic materials for the matrix, epoxy was used for the same. On performing micromechanical analysis of the models for both, composites with square and hexagonal unit cells, it was observed that and The relevant results have been tabulated under Table 3 1. Table 3 1. Results from unit strain analysis along the transverse directions. Composi te T 22 T 33 T 32 T 23 Square unit cell Graphite Epoxy 0.612 0.612 0.029 0.029 Glass Epoxy 0.260 0.260 0.015 0.015 Hexagonal unit cell Graphite Epoxy 0.540 0.540 0.033 0.033 Glass Epoxy 0.205 0.205 0.029 0.029 Based on the above results, it can be deduced that there is absolutely no requirement for the researcher to perform micromechanical analysis along both the transverse axes. Instead only one analysis is sufficient to provide us with the values of and The strain concentration matrix T, first introduced in equation ( 3 37 ) now takes the form,
55 (3 40 ) Thus, the number of independent terms in the T matrix has now been reduced to a total of 4 terms. Over the next sub section, we will describe as to how the number of independent terms reduces to 2, thus giving way to the two parameter model. 3.3. 2 Completing the T matrix While performing the initial testing of the two parameter model, it was observed that both, transversely isotropic and isotropic fibers combined with an isotropic matrix resulted in a transversely isotropic composite. Referring to equati on (3 39), it is evident that the terms in the right hand side of the equation shoul d add up to constitute the stiffness matrix of the composite ( axial elements in eq. ( 3 7 ) ). Consequently the resultant matrix obtained in the right han d side (RHS) of the eq uation will be symmetric. (3 41 ) The right hand side in the above equation can be divided into two parts as shown. The first part being the matrix stiffness matrix is inherently symmetric. For the right hand side to be symmetric, the second part should be symmetric too. i.e. the product o f should be a symmetric matrix. Let (3 42 ) To derive expressions for and we apply the property of a symmetric matrix using which elements, (2,1) should be equal (1,2) and,(1,3) be equal to (3,1).
56 (3 43 ) Equation (3 43) presents a simultaneous set of linear equations. To solve the set of equations, we multiply the first equation with and the second equation with which brings us to, (3 44 ) Subtracting the second equation from the first, we get (3 44 ) T he symmetry in a composite with transversely isotropic stiffness matrix thus leads us to, (3 45 ) Hence, making use of equations (3 40 ) and (3 45 ), it is concluded that the T matrix can be computed by performing single micromechanical analysis along any of the material transverse directions That is, each element in the T matrix can be expressed in terms of two parameters either and or and In other words, knowledge of these two parameters along with the material properties of the constituent materials is sufficient enough for t he calculation of the stiffness matrix h ence the name of the model.
57 CHAPTER 4 RESPONSE SURFACE PREDICTION OF THE PARAMETERS T 22 AND T 32 4.1 Surrogate Modeling Using Polynomial Response Surfaces A response surface methodology portrays the relationship between several explanatory (independent) variables and one or more response (dependent) variables. The method was first introduced by G. E. P. Box and K. B. Wi lson in 1951 . The essential thought behind developing the response surface methodology wa s to use a sequence of controlled designed experiments to obtain an optimal response. It is important to acknowledge that this model is only an approximation, but is widely use d because such a model is easy to estimate and apply, even when little is known about the process. In the ensuing paragraphs, basic terminologies an d error measures involved in fitting an approximation to a set of data have been discussed. In its most generic form, a response function y (that needs to be approximated), can be denoted as a combination of (approximated function of a design variable v ector x and a vector of parameters ), and is the vector of errors associated with the curve fit. (4 1 ) One has the data obtained from experiments at design points. If each design point is denoted by then, (4 2 ) Our main objective is to find the set of parameters that will best fit the experimental data. The process of finding the vector which best fits the experimental data, is called regression and is called a response surface.
58 One of the most commonly used measure of the error observed while approximating a response is the root mean square (rms) error and is given by, (4 3 ) The normalized root mean square error is a much more comparable measure of the RMS error as it is measured in percentage. It is calculated by dividing the root mean square error by the range of observed values of the response being predicted. The expression for normalized root mean square error (NRMSE) ha s been portrayed in eq uation (4 4). (4 4 ) Although there are other measures that are used for calculating the error in approximation such as the average absolute error and the maximum error, throughout the thesis we will concentrate only on response surfaces which minimize the overall root mean square of the fit. Apart from the RMS error of the fit, a way for evaluating the accuracy of the fit is by calculating the variation of the data throughout the response surface. The variation of the data from its average is given by The variation of the response s urface from that same average is given by Thus, (4 4 ) (4 5 )
59 The ratio of to denoted by measures the fraction of the variation in data as captured by the response surface. As explained in , increasing the number of coefficients increases the value of This by no means conveys the statement that the prediction capabilities of the res ponse surface have improved. And hence, there is an adjusted form of as depicted in  which is given by equation (4 6) (4 6 ) Thus, the adjusted value of is a better measure of goodness of fit. It is important to note that if this value decreases upon increasing the number of coefficients, it is a sign that we might be fitting the data better but losing out on the predictive capabilities of the response su rface. A much more reliable measure of the predictive capabilities of the response surface is testing it at the data points which have not been used in its construction. Instead of carrying out additional test s or numerical evaluations, the predictive capa bility can be checked by leaving out a few points will not change drastically the quality of the fit This can only be done provided the numbe r of points used for the fit is substantially larger than the number of coefficients We can th us leave out a few points, fit the response surface to the remaining points and then check the error at the points that have been left out. The idea is to continue this procedure with other points till each point ha s been left o ut once. This procedure is commonly known as cr oss validation. Ideally, cross validation is usually done leaving one point at a time. Since we are using the sum of the squares of the prediction errors as a measure of the predictive accuracy of the response surface, the error is termed as PRESS RMS erro r (for pr ediction error sum of squares). We will make use of the normalized PRESS RMS error ( N PRESS RMSE,
60 calculated in a fashion similar to that of NRMSE) which expresses the PRESS RMS error in percentage. As mentioned in section 1.2, the program for con structing a polynomial response surface has been extracted from a surrogate toolbox created by Felipe A.C. Viana  Design of Experiments (DOEs) Selecting the points in the design space where experiments are to be performed is probably the most important part of obtaining good approximations to any response system. These points where experiments are to be performed are known as design of experiments. Design of experiments is inherently a multi objective optimization problem. Points should be sele cted so as maximize the accuracy of the information gathered from the experiments. At the same time, it is advisable to achieve this accuracy through a minimum number of experiments to help reduce the overall optimization cost. Hence, it can be said that o ur primary goal while choosing the design of experiments, is to choose points that will maximize the predictive capability of the model. There are various ways of generating a certain set of design of experiments such as the 2 level full factorial design, central composite design and LHS deign. For creating our DOEs, a design space was generated using the Latin Hypercube Sampling function provided in MATLAB. The technique was first presented by McKay in 1979 . Latin hypercube sampling ( LHS ) is a statistical method for generating a distribution of plausible collections of parameter values from a multidimensional distribution  The sampling method is often applied in uncertainty analysis. In LHS, equal probability intervals with one point at each interval. A big advantage of using LHS design space is that the
61 design points are better distribu ted for e ach variable as depicted in Figure 5 .6 A downfall of the method is that even though the design points are well spread out, there are still some voids left in between the data points. To overcome this problem, we perform multiple iterations in an attempt to maximize the minimum distance between points. This helps to populate the entire design space. A more comprehensive study on the topic of response surface methodo logy has been presents in [ 30 31, 38, 40]. It is important to note that the LHS function in MATLAB produces DOEs in a range of zero to one They have to be scaled to the respective variable ranges in order to render them usable. 4.2 Constructing the Response Surface The knowledge of and is sufficient enough for the prediction of the stiffness matrix of the composite given by equation (3 24) and thus, the predict ion of materials constants As noted previously, transversely isotropic fibers an d isotropic fibers, both individually combined with an isotropic matrix, result in a transversely isotropic composite. Hence, the only material constant that is not obtained using the two parameter model is for which, a separate response surface is d evloped ( chapter 5 ) As discussed in section 1.2, the response surfaces have been constructed only for circular fibers in a hexagonal array. 4.2 .1 Selection of Variables And Generation of Design Points Since we are interested in only isotropic materials forming the matrix, f rom equation (3 39) it is evident that the parameters depend on the, L belonging to the fiber and the matrix respectively.
62 Transverse belonging to the fiber and the matrix respectively. belonging to the fiber and the matrix respectively. belonging to the fiber and the matrix respectively. Fiber volume fraction The range of the material constants (those listed above along with the longitudinal shear modulus) for both, the fiber and the matrix was chosen based on the data encountered across various sources of literature such that, (4 7 ) Equation (4 7) presents the range of material constants for composites with transversely isotropic fibers. For composites using completely isotropic fibers, the ranges that were taken into account were those of and, It should be noted t hat although the two parameter model does not require the knowledge of the value of longitudinal shear modulus for either of the fiber or matrix, Abaqus as an analysis software needs the complete information for e ach material used in the model.
63 Points were generated based on the variables listed in equation (4 7). For the criterion for minimizing the minimum distance between any two points in the design space. 100 designs were generated for composites with transversely isotropic fiber and 40 designs for composites with isotropic fiber In an effort to make the numerical ranges across all the variables uniform, and make the variable data points more generic, 5 variables wer e chosen such that the final set of variables used in constructing the response surface is given by, (4 8 ) The design points generated usi ng the variables in equation (4 7) were propagated to the set of variables in equation (4 8). Th is generic set of variables allowed for the possible selection of a consistent isotropic design to be used for the matrix, throughout each of the 140 simulations. The isotropic design was accorded material valu es of and 4.2 .2 Constructing and Choosing an Accurate Response Surface Simulations were carried out at each of the orthotro pic and isotropic design points For each simulation a set of parameters and was obtained (i.e. 140 sets of parameters were recorded). Thus, independent response surfaces were constructed for the prediction of and (i.e at a time, a response surface can predict only one of the two parameters) An initi al predictive capab ility test involving various types and degrees of polynomial response surfaces was carried out for six real world composite designs to check the average error in prediction of each fit The volume fiber ratio was maintained at a constant value of 60% for each of the six composite designs and, e poxy
64 was the matrix of choice for all the six fibers, the material properties of which ha ve been listed under Table 4 2. With 140 data points available, response surfaces were constructed and tested for degrees 1 4. Table 4 1 Accuracy and predictive capability char acteristics of various response surfaces. Type N PRESS RMSE (%) b c NRMSE (%) b ,c Adjusted go o dness of fit ( ) b ,c Av g. e rror in prediction (%) d Number of coeff b ,c 1 st Degree Full ( ) 6.73 4.23 0.978 17.5 6 Full ( ) 22.5 8.73 0.845 30.1 6 2 nd Degree Reduced a ( ) 1.09 0.60 0.999 2.45 13 Reduced a ( ) 8.43 2.81 0.983 15.3 14 Full ( ) 1.10 0.60 0.999 2.35 21 Full ( ) 8.26 2.76 0.983 15.9 21 3 rd Degree Reduced a ( ) 0.34 0.11 1.00 0.689 40 Reduced a ( ) 7.54 1.36 0.995 6.85 37 Full ( ) 0.34 0.11 1.00 0.621 56 Full ( ) 7.24 1.32 0.995 7.08 56 4 th Degree Reduced a ( ) 0.96 0.002 1.00 1.22 96 Reduced a ( ) 7.87 0.12 0.999 6.09 89 a A reduced fit removes the coefficients which do not contribute much to the response of the response surface. b The range of varies from 0.025 1.12. To increase the absolute minimum value of the response, the variables are fit against which has a range of 1.60 0.05. Thus, measures are with respect to c The range of varies from 0.02 2 0.071 To increase the absolute minimum value of the response, the variables are fit against which has a range of 0.950 1.18 Thus, measures are with respect to d Average e rror in prediction of and over the 6 composite models analyzed in Abaqus using material properties listed under Table 4 2.
65 Table 4 2 Fiber and matrix properties used for accuracy c heck of the response surfaces [ 42 45 ]. Material Fiber Properties T 300 2.21E+11 1.38E+10 0.200 0.430 8.96E+09 Graphite 2.63E+11 1.90E+10 0.200 0.350 2.76E+10 Kevlar 1.52E+11 4.14E+09 0.350 0.361 2.90E+09 S glass 8.55E+10 8.55E+10 0.200 0.200 3.56E+10 E glass 7.31E+10 7.31E+10 0.220 0.220 3.01E+10 Boron 4.00E+11 4.00E+11 0.200 0.200 1.67E+11 Matrix Properties Epoxy 3.50E+09 3.50E+09 0.350 0.350 1.29 E+ 09 On observing the error measures (Table 4 1), the 3 rd degree full fit for the prediction of and, the 4 th degree reduced fit for the prediction of were deemed the most accurate. Thus we proceed with in depth result analysis using these response surfaces in the following section C oefficients corresponding to these particular fits have bee n listed under Appendix. 4.3 Result Analysis The accuracy of the response surfaces selected in the previous sub section is tested further for their accuracy over a wider range of fiber volume fraction (0.45 0.65) as compared to the uniform 0.6 used during the initial result analysis. To do so, parameters and are predicted using the response surfaces, for the composites slated und er Table 4 ratio are predicted for wach of the six composites. These predictions are cross checked with the val ues obtained using the paramters from the finite element analysis of each of the six composite models. The material constant predictions are further compared with
66 the results obtained using the Halpin Tsai (HT) equation s ( Eqs. (2 6 2 7) for and ) and the rule of mixtures ( Eq. (2 1) for and ). 4.3.1 Adopting a Reliable Estimate for the Halpin Tsai Parameter, I t is critical to determine a value of the parameter to be used i n the Halpin Tsai model (Eq. (2 6)). Although the use of for calculating the transverse elastic modulus is consensus in the world of composites using circular fibers in a square array, circular fibers in a hexagonal array, w r the parameter . To find a reliable estimate of the data was extracted from all the FE based simulations such that there were 140 values available for both and Using backward calculations (proceeding w ith known values of and arriving at a value for ) an value of 0.73 was found to give good results when predicting ( is obtained using Eq. (3 8)) of composites with transversely isotropic fibers. However, this iterative method did not work well for composites havi ng completely isotropic fibers. Instead, it was observed that the value increased with an increase in the value of the volume fiber fr action ( Fig. 4 1A ). A n approximate solution to this problem is given by fitting a curve between and the volume fiber fraction using data points corresponding to composites with only isotropic fibers, such that, (4 9 ) A similar problem was encountered in the process of obtaining a reasonable estimate of for the calculation of (Fig. 4 1B ). The expression obtained by fitting a curve between and the volume fiber fraction in this case is given by, (4 10 )
67 The values of in equations (4 9 4 10) although found using composites with isotropic fibers, were found to fit very well in the Halpin Tsai empirical model for composites with both isotropic and transversely isotropic fibers. (A) (B) Figure 4 1. Variation of with the fiber volume fraction, in models with isot ropic fibers A) corresponding to B) corresponding to
68 4 .3.2 Result Analysis aprop os T 300 Epoxy Fi g ure 4 2 displays the accuracy of three different methods (response surface, Halpin Tsai and Rule of Mixtures) and their comparison with respect to the results obtained using finite element analysis for predicting the material properties of the T 300 epoxy composite Figure 4 2 Comparison between the response surface and rule of mixtures with respect to finite element analysis of T 300 epoxy using Abaqus A) for and C ) for and Halpin Tsai model for B) and D) equations
69 4 .3.3 Result Analysis apropos Graphite Epoxy Fi g ure 4 3 displays the accuracy of three different methods (response surface, Halpin Tsai and Rule of Mixtures) and their comparison with respect to the results obtained using finite element analysis for predicting the material properties of the graphite epoxy comp osite. Figure 4 3. Comparison between the response surface, and rule of mixtures with respect to finite element analysis of Graphite epoxy using Abaqus A) for and C) for and Halpin Tsai model for B) and D) equations.
70 4.3.4 Result Analysis apropos Kevlar Epoxy Fi g ure 4 4 displays the accuracy of three different methods (response surface, Halpin Tsai and Rule of Mixtures) and their comparison with respect to the results obtained using finite element analysis for predicting the materi al properties of the Kevlar epoxy composite. Figure 4 4 Comparison between the response surface, and rule of mixtures with respect to finite element analysis of Kevlar epoxy using Abaqus A) for and C) for and Halpin Tsai model for B) and D) equations.
71 4.3. 5 Result Analysis apropos S glass Epoxy Fi g ure 4 5 displays the accuracy of three different methods (response surface, Halpin Tsai and Rule of Mixtures) and their comparison with respect to the results obtained using finite ele ment analysis for predicting the material properties of the S glass epoxy composite. Figure 4 5. Comparison between the response surface, and rule of mixtures with respect to finite element analysis of S glass epoxy using Abaqus A) for and C) for and Halpin Tsai model for B) and D) equations.
72 4.3. 6 Result Analysis apropos E glass Epoxy Fi g ure 4 6 displays the accuracy of three different methods (response surface, Halpin Tsai and Rule of Mixtures) and their comparison with respect to th e results obtained using finite element analysis for predicting the material properties of the E glass epoxy composite. Figure 4 6. Comparison between the response surface, and rule of mixtures with respect to finite element analysis of E glass epoxy us ing Abaqus A) for and C) for and Halpin Tsai model for B) and D) equations.
73 4.3. 7 Result Analysis apropos Boron Epoxy Fi g ure 4 7 displays the accuracy of three different methods (response surface, Halpin Tsai and Rule of Mixtures) and their comparison with respect to the results obtained using finite element analysis for predicting the material properties of the boron epoxy composite. Figure 4 7. Comparison between the response surface, and rule of mixtures with respect to finite element analysis of Boron epoxy using Abaqus A) for and C) for and Halpin Tsai model for B) and D) equations.
74 CHAPTER 5 PREDICTING THE LONG ITUDNAL SHEAR MODULUS The stiffness matrix of a transversely isotropic composite has 5 indepen dent material constants (Eq. (3 7)). In the previous chapter, we focused on the two parameter models combined with response surfaces for accurate prediction of 4 ( ) of these constants. For t he fifth constant, the longitudinal shear modulus a distinct method result ing in a separate response surface, is adopted This method has been tested only for composites with a hexagonal array of unit cells. 5.1 Extracting Using Abaqus Due to the inconsistency portrayed by the Abaqus coordinate system (Fig. 3 4), it is not possible to calculate directly as a field output. As a result, we will be using Abaqus to calculate (which corresponds to in the material coordinate system). There are two phases that lead to the calculation of The first is the finite element analysis of the hexagonal RVE model to obtain the displacement along the z direction (w). The second phase c onsists of a MATLAB program to calculate using the displacement, w. 5.1.1 Analysis of the Hexagonal Finite Element Model A lthough exactly similar in terms of design, to the hexagonal RVE model (Fig. 3 5 B) is used for the finite element analysis, th is model is constructed in a 3D design space in contrast to the earlier use of a 2D design space. This allows for deformation along the z axis. Homogenous shell type sections are used throughout the model. For the mesh, 4 node shell elements have been us ed. This type of element helps avoid shear lo cking. The essence of using shell elements lies in the fact t hat they can be curved and they carry bending stresses.
75 (A) (B) (C) Figure 5 1. Boundary conditions for shear strain analysis A) Nodes restricted along z direction, B) Nodes allowed movement only along the z axis, C) Nodes displaced along z axis by a value equal to the horizontal length of the model (i.e. = 1.5 units)
76 Boundary Conditions: The boundary conditions have been applied such that the model is under an average shear strain of unity, in the x z plane. The complete set of boundary conditions adopted, have been depicted in Figure 5 1. It should be pointed out, that in Figure 5 1(B), the set of nodes in quest ion do not include any of the nodes along the right or left edges of the model. Field outputs : Field outputs that were extracted, comprised only of the nodal dis placement, w. Fig. 5 2 shows the displacement countours of the finite element model under a st ate of unit shear strain in x z plane. Figure 5 2. Hexagonal RVE model undergoing unit shear strain deformation in the x z plane. 5.1.2 Using Displacement along Z axis to Calculate In this section, we make use of physical to reference element mapping to achieve the desired result. Each element in the finite element model also referred to as the physical eleme n t, can be mapped to a four node iso parametric quadrilateral element (Fig. 5 3 ) also referred to as the reference element
77 Figure 5 3. Four node quadrilateral reference element defined using an s t coordinate system. Numerical value of any physical variable (such as displacement, force, etc.) for each physical element is approximated using numerical integration. This involves calculating the variable at each integration point ( marked as a cross in the reference element ) and adding them up. Our parametric element is a case of 2 point integration in a 2D space, the numerical integration of which is given by [ 1 ], (5 1 ) Where, for a 2 point integration system. For calculating the physical variable, in our case is (explained later in the section). The displacement (w) and coordinate (x, y) of any point in the physical element, in terms of the parametric coordinate system are given by, (5 2 )
78 Here, are shape functions corresponding to e ach node of the reference element, and are given by, (5 3 ) The values corresponding to and at each of the four integration point s in the reference element are given by, (5 4 ) Using chain rule of differentiation in equation (5 3 ), it can be said that (5 5 ) Expressing in matrix form, (5 6 ) Here is the Jacobian matrix and it plays an important role in evaluating the validity of the mapping as well as the quadrilateral element [4 4 ]. The expression in equation (5 6 ) is a function of and and should be multiplied with the determinant of the Jac obian
79 matrix to account for the disparity in volume of the element in real and parametric space. Thus, (5 7 ) To obtain for an element, the matrix in the above equation should be calculated at ever y integration point for the particular element (Eq. (5 1)) Recalling the expression for shear strain from elasticity theory, we have, (5 8 ) Due to restrained movement allowed along x and y axes .Thus, with the knowledge of for each element, equation (5 8) is used to calculate for each element ( depending on the section in which the element falls, each element will have value belonging to the fiber or the matrix material) Summing the value for all elemen ts, and dividing by the volume of the complete model, we obtain the value for the given composite (note that an average shear strain of 1 was applied). When implementing calculations in MATLAB, triangular elements in the physical model are interpreted in terms of an equivalent physical quadrilateral element [ 1 ] Figure 5 4. Quadrilateral equivalent of a triangular reference element.
80 Figure 5 5. Step by step schematic, for calculating the longitudinal shear modulus of the composite using the displacement of the finite element model along the z axis.
81 5.2 Predicting Using Polynomial Response Surface The unit shear strain analysis was performed for each of the 100 transversely isotropic designs generated for the two parameter model. Wholly i sotropic design were not used due to the variables (E q. 5 9) being independent of the directionality of the composite models. (5 9 ) Figure 5 6 depicts the distribution of the fiber longitudinal shear modulus with respect to the volume fiber ratio for the 100 transversely isotropic composite designs generated using Latin h ypercube s ampling (LHS). As there were six variables used during the design point generations, the designs do not appear to be as evenly generated as they would with all the variables in a six dimensional space (Eq. 4 7). Figure 5 6. Variation of fiber longitudinal shear modulus with respect to the volume fiber ratio
82 Various response surfaces were constructed to a find the most accurate fit between the variables and the longitudinal shear modulus of the composite model The high number of data points and low number of variables allowe d for the construction of full response surfaces (i.e. no reduced response surface was constructed) with degrees as high as 4. Table 4 3. Accuracy and predictive capability characteristics of various response surfaces for the prediction of Type N PRESS RMSE (%) NRMSE (%) Adjusted goodness of fit ( ) Av g. e rror in prediction (%) a Number of coeff. 1 st Degree 6.14 5.8 0 0.917 4.10 3 2 nd Degree 1.37 1.21 0.996 1.48 6 3 rd Degree 0.588 0.368 0.999 0.385 10 4 th Degree 0.132 0.066 1.00 0.098 15 a Average error in prediction of and over the 6 composite models analyzed in Abaqus using material properties listed under Table 4 2. On the basis of accuracy characteristics, f or purpose of manual calculations, the 2 nd degree response surface (Eq. 5 10) appears to be very plausible. (5 10 ) The rule of mixtures was found to give an average prediction error of less than 1%. The most accurate amongst all the response surfaces, the 4 th degree fit was found to give average prediction error of less than 0.1%. The coefficients of the 4 th degree fit h ave been listed under Appendix
83 CHAPTER 6 CONCL USION 6.1 Summary We studied the history of microme chanics dating more than a century back. One of trend for modern day micromechanics. It is shown, that for a transversely isotropic composite using a hexagonal array of ce lls, the transverse shear modulus, is a A micromechanical model was derived inhomogeneous material, for the prediction of of a transversely isotropic composite the accuracy of which was confirmed for composites using either a square or a hexagonal array of cells. Using finite element based analysis, it was proved that unit strain analysis along any one of the mate rial transverse directions is sufficient enough for the two parameter model to function accurately. Surrogate modeling of the two parameters (for composites using a hexagonal array of cells only) using polynomial response surfaces was carried out for the a ccurate prediction of the elastic constants The response surfaces were constructed based on the data obtained from 140 individua simulations of composite designs generated using Latin hypercube sampling. The designs were generated keeping in mind a certain range of design variables. An effort was made to come up with an accurate expression for the Halpin Tsai parameter In case of composites with completely isotropic fiber, there was a quadratic trend observed between the parameter and the volume fiber fraction, based on which, a second degree response surface was constructed. The expression
84 obtained, was found to be fairly accurate, when tested for composties with transversely isotropic fibers. In depth result analysis of six real world composite designs was carried out wherein the elastic cons tants obtained using the two paramter model, the Halpin Tsai model and the rule of mixtures, were compared against the results obtained finte element analysis. The rule of mixtures, as expected gave highly accurate results for However, the results wer e not so convincing for the prediction of using the rule of mixtures. The Halpin Tsai model was found to give highly accurate results for the prediction of in the case for composites with isotropic fibers but there was a fair amount of erro r when composites with transversely isotropic fiber were taken into account. This discrepancy was primarily due to the Halpin Tsai parameter being based solely on composites with completely isotropic fiber. For the two paramter model, the prediction error was very low for all four of the elastic constants except for a few instances wherin the value of certain curve fit variables might have exceeded the range based on which the response surfaces were fitted in the first place. A separate response surface, ba sed on shear strain simluations of finite element models in 3D space, was constructed for the prediction of Due to the inconsistency in Abaqus and material coordinate system, the displacement along the z axes was used to arrive at using methods of numerical integration The accuracy of this response surface was tested using the six real world composite designs. Although prediction using a 2 nd degree fit were deemed accurate enough, a further 4 th degree fit was found to give results more accurate that the rule of mixtures for (which was found to give average prediction error of less than 1%).
85 6.2 Future Scope of Work Although a fairly accurate response surface has been provided for the prediction of the elastic constants , the trend for the prediction of the parameter was found to be highly erratic. A solution to this problem might be to try various options i n the field of surrogate modeling that might help in a more accurate and consistent prediction of Another issue with the current response surface is that it may not cater well to the needs of those interested in composites having variable values falling outside the range specified. Another area of concern was the Halpin Tsai parameter which was derived solely on the basis of the data relating that of the composites using only isotropic fibers. And even though, the parameter was found to be agreeable for composites w ith transversely isotropic fibers, the results were not as accurate for the same. Hence, a problem may be posed where in one might find the optimal value of the paramter The proceedings of the research will be put in to setting up a website or a program that will have the response surface data including the coefficient values pre fed so as to give the user the values of the elastic constant with respect to the design variable he/she may enter.
86 APPENDIX RESPONSE SURFACE COEFFICENTS A.1 For The Predictio n Of And Table A 1. Variable interpretation of numbers Number Variable Interpretation 1 2 3 4 5 Table A 2.Coefficients corresponding to the 3 rd degree full response surface for Variable Coefficient Variable Coefficient '0' a 0.2446 '1.2.4' 0.0080 '1' 0.3373 '1.2.5' 0.0161 '2' 0.6991 '1.3.3' 0.1458 '3' 1.3014 '1.3.4' 0.1555 '4' 1.0992 '1.3.5' 0.1637 '5' 1.0437 '1.4.4' 0.1501 '1.1' 0.1916 '1.4.5' 0.2382 '1.2' 0.0384 '1.5.5' 0.0026 '1.3' 0.6110 '2.2.2' 0.0301 '1.4' 0.4968 '2.2.3' 0.1353 '1.5' 0.0298 '2.2.4' 0.0555 '2.2' 0.1216 '2.2.5' 0.2281 '2.3' 0.3600 '2.3.3' 0.0143 '2.4' 0.4207 '2.3.4' 0.0700 '2.5' 0.1160 '2.3.5' 0.0777 '3.3' 0.5120 '2.4.4' 0.2551 '3.4' 0.6964 '2.4.5' 0.2562 '3.5' 0.6904 '2.5.5' 0.8349 '4.4' 0.8543 '3.3.3' 0.0220 '4.5' 0.7302 '3.3.4' 0.3315 '5.5' 1.4123 '3.3.5' 0.1803 '1.1.1' 0.0495 '3.4.4' 0.1039 '1.1.2' 0.0699 '3.4.5' 0.7498 '1.1.3' 0.0043 '3.5.5' 0.0790 '1.1.4' 0.0470 '4.4.4' 0.2857 '1.1.5' 0.0313 '4.4.5' 0.8290 '1.2.2' 0.0568 '4.5.5' 0.1743 '1.2.3' 0.2119 '5.5.5' 0.7691 a denotes a constant
87 Table A 3.Coefficients corresponding to the 4 th degree reduced response surface for Variable Coefficient Variable Coefficient Variable Coefficient '2' 0.8316 '18.104.22.168' 1.4626 '22.214.171.124' 1.7475 '3' 4.1193 '126.96.36.199' 1.7163 '188.8.131.52' 3.3953 '4' 1.1897 '184.108.40.206' 0.6647 '220.127.116.11' 1.0424 '1.1' 0.7633 '18.104.22.168' 1.3339 '22.214.171.124' 4.7409 '1.2' 1.9012 '126.96.36.199' 2.1957 '188.8.131.52' 2.1549 '2.2' 0.5355 '184.108.40.206' 0.1523 '2.3' 0.6014 '220.127.116.11' 0.1107 '2.4' 1.6434 '18.104.22.168' 0.4974 '3.3' 6.3157 '22.214.171.124' 0.2337 '3.5' 1.6184 '126.96.36.199' 0.3869 '4.5' 5.2575 '188.8.131.52' 0.7952 '5.5' 5.5794 '184.108.40.206' 1.4564 '1.1.3' 1.5266 '220.127.116.11' 0.9085 '1.1.4' 2.0022 '18.104.22.168' 2.2496 '1.1.5' 1.6830 '22.214.171.124' 1.4947 '1.2.2' 0.6352 '126.96.36.199' 4.1968 '1.2.3' 1.3136 '188.8.131.52' 0.7365 '1.2.4' 3.4136 '184.108.40.206' 3.2355 '1.2.5' 4.2079 '220.127.116.11' 1.9654 '1.3.3' 3.3372 '18.104.22.168' 2.4127 '1.4.5' 7.4153 '22.214.171.124' 1.6781 '1.5.5' 4.3661 '126.96.36.199' 1.7371 '2.2.2' 0.7191 '188.8.131.52' 0.0279 '2.2.5' 2.8193 '184.108.40.206' 0.0459 '2.3.3' 1.4052 '220.127.116.11' 0.3333 '2.3.4' 4.4400 '18.104.22.168' 0.1759 '2.3.5' 1.2850 '22.214.171.124' 0.3213 '2.4.4' 1.2555 '126.96.36.199' 0.3264 '2.4.5' 4.1133 '188.8.131.52' 0.4446 '2.5.5' 1.9102 '184.108.40.206' 0.1606 '3.3.4' 11.1961 '220.127.116.11' 1.1492 '3.3.5' 9.4417 '18.104.22.168' 0.7042 '3.4.4' 9.0760 '22.214.171.124' 1.4510 '3.5.5' 9.1636 '126.96.36.199' 3.5528 '4.4.4' 2.2582 '188.8.131.52' 1.7623 '184.108.40.206' 0.1587 '220.127.116.11' 3.5257 '18.104.22.168' 0.3544 '22.214.171.124' 3.0227 '126.96.36.199' 0.2171 '188.8.131.52' 0.8456 '184.108.40.206' 0.0638 '220.127.116.11' 2.3383 '18.104.22.168' 0.5307 '22.214.171.124' 0.4344 '126.96.36.199' 1.0502 '188.8.131.52' 1.4218 '184.108.40.206' 0.5503 '220.127.116.11' 3.7301
88 A.2 For The Prediction Of Table A 4. Variable interpretation of numbers Number Variable Interpretation 1 2 Table A 5.Coefficients corresponding to the 4 th degree full response surface for Variable Coefficient '0' 2.902817 '1' 0.97243 '2' 18.675 '1.1' 0.61719 '1.2' 9.346358 '2.2' 48.94534 '1.1.1' 0.169068 '1.1.2' 1.30883 '1.2.2' 20.2252 '2.2.2' 54.2172 '18.104.22.168' 0.111066 '22.214.171.124' 1.30674 '126.96.36.199' 1.970565 '188.8.131.52' 13.69299 a denotes a constant
89 REFERENCES [1 ] Kumar, A. V., Course material for E ML5526Finite Element Analysis Uni versity of Florida, Gainesville [2 ] Viana, Felipe A. C., Multipe s urrogate for p rediction and o ptimization PhD Thesis, University of Florida, 2011. [3 ] Gardener, Jeffery P., Micromechanical modeling of composite materials in finite element analysis using an embedded cell approach MS Thesis, MIT, 1994. [4 ] Aboudi Jacob, Arnold Steven M., Bednarcyk Brett A. Micromechanics of Composite Materials: A Generalized Multiscale Analysis approach Butterworth Heinemann 2012 [5 ] Sankar, B.V., Course material for EAS 6242 Advanced Composites, University of Florida, Gainesville. [6 ] Helmut J. B., Course mater ial for Composites Engineering II Continuum Micromechanics Vienna University of Technology, Austria. [7 ] Voigt, Abhandlungen der KniglichenGesellschaft der Wissenschaften, Gttingen, Vol. 34, 1887, pp. 3 51. [8 ] Berechnung der Fliegrenze von Mischkristallen auf Grund der Journal of Applied Mathematics and Mechanics Vol. 9, pp. 49 58. [9 ] Hill In proceedings of the physical society section A Vol. 65, 1952, pp. 349 354. [10 ] Mechanics of composite materials: A unified micromechanical approach Studies in applied mechanics Elsevier Science limited Amsterdam V ol 29 1991. [11 ] Hashin, Z., Shtrikman, S. "Note on a Variational Approach to the Theory of Composite Elastic Materials". Journal of the Frahk lin Institute vol. 271, 1961 pp. 336 341. [12 ] Dvorak, G.J., Bahei el Din, Y.A. (1982)."Plasticity Analysis of Fibrous Journal of Applied Mechancis Vol. 49, pp. 327 335. [13 ] Hashin, Z., Shtrikman, S. (1963). "A Variational Approach to the Theory of the Elastic Behavior of Multiphase Materials". Journal of the Mechanics and Physics of Solids Vol. 11, pp. 127 140.
90 [14 ] Seguardo J. Llorca J. Gonzalez On the accuracy of mean field approaches to simulate the plastic deformation of composites Scripta Material Vol. 46, 2002, pp. 525 529. [15 ] f the Elastic Field of an Ellipsoidal Inclusion, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences Vol. 241, 1957, pp. 376 396. [16 ] Rahman S. Chakrabarthy A. A stochastic micromechanical model for elastic propertiesof functionally graded materials Mechanics of Materials Vol. 39, 2007, pp. 548 563. [17 ] Halpin, J.C., Kardos, J. L., Polymer Engineering and Science Vol.16, 1976, pp. 344 352. [18 ] Hil l Journal of the Mechanics and Physics of Solids Vol. 13, 1965, pp.213 222. [19 ] Halpin, J. C., Effects of Environmental Factors on Composite Materials Technical Report AFML TR 67 423, Ohio, 1969 [20 ] Hermans, The Elastic Properties of Fiber Reinforced Materials when the Fibers are Aligned Proceedings of the KoninklijkeNederlandse Akademie van Weten schappen Amsterdam, Series B, Vol 70, Number 1, 1967 pp. 1 9. [21 ] Adams, D. F., Doner, D. R., Journal of Composite Materials Vol. 1, 1967, pp. 152 164 [22 ] Adams, D. F. Doner, D. R., Journal of Composite Materials Vol. 1,1 967 pp. 4 17 [23 ] Foye, R.L. SAMPE: Advanced Fibrous Reinforced Composites 10th National Symposium & Exhibit Vol. 10, 1966, pp. G31 G42. [24 ] He 4 no. 2, 1970, pp. 280 282. [25 ] Energy of ACTA Metallurgica Vol. 21, pp. 571 574. [26 ] Casta neda P. P., The effect of spatial distribution on the effectivebehavior of compos Journal of the Mechanics and Physics of Solids Vol. 43, 1995, pp.1919 1951.
91 [27 ] Yan, C, On Homogenization And De Homogenization Of Composite Material PhD Thesis, Drexel University, 2003. [28 ] Aboudi, J., "Micromechanical Analysis of Composites by the Method of Cells," Applied M echanics Reviews 1989, Vol. 42, pp. 193 221. [29 ] Aboudi, J., Pindera, M., Micromechanics of Metal Matrix Composites Using the NASA Contractor Report 190756, 1992. [30 ] Progress in Aerospace Sciences Vol. 41, 2005, pp. 1 28. [31 ] Gorissen, Dirk, Heterogeneous Evolution of Surrogate ModelsI, Thesis, KatholiekeUniversiteit Leuven, 2007. [32 ] Huang Zheng Ming, Zhou Ye Xin, Strength of Fibrous Composites Springer, 2012. [33 ] Pelosi, Giuseppe "The finite element method, Part I: R. L. Courant: Histori cal Antennas and Propagation Magazine, IEEE Vol. 49. Issue 2, 2007, pp. 180 182. [34 ] Jones Robert M., Mechanics of composite materials, Taylor & Francis, 1998. [35 ] Box, On the Experimental Attainment of Opti mum Conditions Journal of the Royal Statistical Society S eries B Vol. 13 No. 1 195 0 pp. 1 45. [36 ] Sankar, B.V., Course material for EAS 6939 Aerospace Composites 1, University of Florida, Gainesville. [37 ] Finite Element Method Base Annual Technical Conference of the American Society for Composites Arlington, Texas, 2012 [38 ] Haftka, R. T., Course material for EAS 6939 Approximation and Optimization in Design University of Florida, Gainesville. [39 ] McKay, M.D., Be ckman, R.J., Conover, W.J. (May 1979). "A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code" Technometrics ,Vol. 21 No. 2, pp. 239 245. [40 ] Khuri, A. I., Response surface methodology and related topics World Scientific Publishing, 2006. [41 ] Hirsa, Ali, Computational Methods in Finance Chapman & Hall, 2012.
92 [42 ] Stamblewski, Christopher, Sankar, B. V., Zenkert, Analysis of Three Dimensional Quadratic Failure Criteria for Thick Composites using the Direct Micromechanics Method Journal of Composite Materials Vol. 42 No. 7, 2008, pp. 635 654. [43 ] mposite Laminates Journal of Composite Materials Vol. 40 No. 12, 2005, pp. 1077 1091. [44 ] Kim, Nam Ho, Sankar, B.V., Introduction to Finite Element Analysis and Design Wiley, 2008. [45 ] Gibson, Ronald F., Principles of Compos ite Material Mechanics, Taylor & Francis, 2007
93 BIOGRAPHICAL SKETCH Ashesh Sharma was b orn in Jabalpur, India, in 1989 He received a Bachelor of Engineering in the field of mechatronics from Manipal University, India He would often go on office tours with his father and marvel at how computers could control operations with help of actuators at oil rigs situated miles away. The revolution that compu ters combined with mechanics could bring to the society was apparent to him His mother a nuclear physicist, loves science and discussions with her over everyday scientific matters drew him even closer to the concepts. His decision to pursue a graduate degree was, indeed heavily inspired by his parents, but also by an incessant passion to k now more and better understand innovation in the context of emerging markets He has previously carried out a 6 month research internship at Siemens AG in Karlsruhe, Germany. Working on integration of pressure sensors into an electro pneumatic positioner, h e was introduced to how engineering is practiced in another country and, of course, had the opportunity to be exposed to the German culture, which unlike the American culture, is largely unfamiliar to folks in India. His comfort with assimilation into di verse circumstances also stems from growing up in Kolkata and Mumbai, India. While Kolkata is one of the most conservative cities in the country, the cosmopolitan city of Mumbai embraces a heterogeneous mix of people who spea k three distinct languages and pr actice widely distinct customs. His future plans include carrying out doctoral studies in inter disciplinary areas concerning opt imization and mechanics.