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1 This article was downloaded by:[otto-von-guericke-universitaet] On: 0 September 007 Access Details: [subscription number 77067] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, 7-4 Mortimer Street, London WT JH, UK Journal of Thermal Stresses Publication details, including instructions for authors and subscription information: Numerical and Analytical Approaches for Calculating the Effective Thermo-Mechanical Properties of Three-Phase Composites Online Publication Date: 0 August 007 To cite this Article: Berger, H., Kurukuri, S., Kari, S., Gabbert, U., Rodriguez-Ramos, R., Bravo-Castillero, J. and Guinovart-Diaz, R. (007) 'Numerical and Analytical Approaches for Calculating the Effective Thermo-Mechanical Properties of Three-Phase Composites', Journal of Thermal Stresses, 0:8, To link to this article: DOI: 0.080/ URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Taylor and Francis 007

2 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September 007 Journal of Thermal Stresses, 0: 80 87, 007 Copyright Taylor & Francis Group, LLC ISSN: print/5-074x online DOI: 0.080/ NUMERICAL AND ANALYTICAL APPROACHES FOR CALCULATING THE EFFECTIVE THERMO-MECHANICAL PROPERTIES OF THREE-PHASE COMPOSITES H. Berger, S. Kurukuri, S. Kari, and U. Gabbert Otto-von-Guericke-University of Magdeburg, Faculty of Mechanical Engineering, Institute of Mechanics, Magdeburg, Germany R. Rodriguez-Ramos and J. Bravo-Castillero Facultad de Matemática y Computación, Universidad de La Habana, Vedado, Habana 4, Cuba R. Guinovart-Diaz Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Estado de México, Carretera Lago de Guadalupe, Estado de México, México The aim of this article is to envisage the effective thermo-mechanical properties of three phase composites made up of coated unidirectional cylindrical fibers using homogenization techniques. The main focus is on square arrangements of cylindrical fibers in composite. The numerical approach is based on the micro-mechanical unit cell modeling technique using finite element method (FEM) with appropriate boundary conditions and it allows the extension to composites with arbitrary geometrical inclusion configurations, providing a powerful tool for fast calculation of their effective properties. The results obtained from the numerical technique are compared with those obtained by means of the analytical asymptotic homogenization method (AHM) for different fiber volume fractions. In order to analyze the interphase effect, the effective properties are compared with the results obtained from some theoretical approaches reported in the literature. Keywords: composite. Effective thermo-mechanical properties; FEM; Homogenization; Interphase; Three-phase INTRODUCTION Thermo-elastic composites constitute an important class of materials with a wide variety of applications ranging from aerospace structures and electronic printed circuit boards to recreational and commercial equipment. Some of the most important and useful properties of these composites are lightweight, high Received November 006; accepted March 007. This work has been supported by DFG Graduiertenkolleg 88 Micro-Macro Interactions in Structured Media and Particle Systems at the University of Magdeburg. Thanks also to the project PNCB IBMFQ and FENOMEC, PAPIIT, DGAPA, UNAM-Mexico, under grant IN0705. These supports are greatly acknowledged. Address correspondence to H. Berger, University of Magdeburg, Institute of Mechanics, Universitätsplatz, Magdeburg, D-906, Germany. berger@mb.uni-magdeburg.de 80

3 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September H. BERGER ET AL. strength and stiffness, excellent frictional properties, good resistance to fatigue and retention of these properties at high temperatures. The combination of these properties has placed thermo-elastic composites at first rank among materials used for heat shields, leading edges, re-entry tips, rocket nozzles and brakes for military and advanced civilian aircrafts. The effective thermo-mechanical properties of the composite depend upon properties of the constituents and the fiber volume fraction. Many authors have developed techniques to study the elastic behavior of fibrous composites. They take into account the existence of an intermediate layer between the matrix and the fiber. These thin layers are called interphases or interfacial zones between fiber and matrix. The effective utilization of the fiber reinforced composites depends on efficient load transfer from the matrix to fibers through these interphases. These interphases are formed due to, for example, chemical reaction between the matrix and fiber materials or the use of protective coatings on the fiber during manufacturing. Although small in thickness, interphases can significantly affect the overall mechanical properties of the fiber-reinforced composites. It is the weakest link in the load path, and consequently most failures in fiber reinforced composites, such as debonding, fiber pullout, and matrix cracking, occur in or near this region. Thus, it is crucial to fully understand the mechanism and effects of the interphases on the overall material properties of fiber reinforced composites. Without seeking to make a deep revision of the results reached in the past two decades, we will show a summary of some of the many reviewed articles, some of which possess results that can be used in comparisons with the predictions obtained in our model. Several authors have developed techniques to study the elastic behavior of multiphase fibrous composites. Hill [] and Hashin s bounds [, ] provide widely established benchmarks for validating the predicted effective properties of multiphase fiber reinforced composites with arbitrary phase geometry. It was shown by Walpole [4] that the thin coating on an inclusion has a pronounced effect on the fields just outside the inclusion. The stress field in a coated continuous fiber composite subjected to thermo-mechanical loading has been considered by Mikata et al. [5]. Theocaris et al. [6] described a model to predict the influence of the interphase in fibrous composites. Their model was based on a correct version of Kerner s model, which is conveniently modified by introducing an interphase layer between the fiber and the matrix. Two models to approximate the thermo-elastic response of a composite body reinforced by coated fibers oriented in various directions were developed in Pagano et al. [7, 8]. The fundamental representative volume element (RVE) is a three-phase concentric circular cylinder under prescribed displacement components and surface tractions. The analysis leads to estimating how a coating applied to the fiber influences the effective thermo-elastic properties of fiber reinforced composites. Hashin s [9] imperfect interface conditions are defined in terms of linear relations between interface tractions and displacement jumps. All the thermoelastic properties of unidirectional fiber composites with such interface conditions are evaluated on the basis of a generalized self-consistent scheme model. The effect of interphase on the transverse properties of elasto-plastic composites is studied employing the finite element method in Yeh [0]. In Sutcu [] a simple recursive algorithm is presented that considers only two concentric cylinders at a time in order to calculate five effective elastic constants and two linear thermal expansion coefficients for a uni-axially aligned composite that contains an arbitrary number of coatings on its fibers.

4 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September 007 THERMO-MECHANICAL PROPERTIES OF THREE-PHASE COMPOSITES 80 In Dasgupta and Bhandarkar [] a method to obtain the transversely isotropic effective thermo-mechanical properties of unidirectional composites reinforced with coated cylindrical fibers is discussed. In this work, the method proposed in Benveniste et al. [] is extended to composites with multiply-coated cylindrical reinforcements, and a generalized self-consistent scheme is proposed for obtaining the transverse shear properties. Chu and Rokhlin [4] reported a method for the inverse determination of effective elastic moduli of mesophases using a multiphase generalized self-consistent model. Lagache et al. [5] determined numerically the effect of a mesophase using a finite element formulation in order to solve the local problems derived from the homogenization method. The effect of an interphase on the behavior of a glass/epoxy composite based on the Theocaris interphase model and taking a hexagonal cell for the three-phase composite was discussed in Chouchaoui and Benzeggagh [6]. Agbossou and Pastor [7] developed a thermal self-consistent model for n-layered fiber composites subjected to a uniform temperature. They propose analytical and semi-analytical models for determining the thermal behavior of composites. Liu et al. [8] have modeled interphases in fiber reinforced composites under transverse loading using the boundary element method. The interphases are regarded as elastic layers between the fibers and the matrix. Hashin [9] reported that the imperfect interphase conditions are equivalent to the effect of a thin elastic interphase, and high accuracy of the method is proved by comparison of solutions of several problems in terms of the explicit presence of the interphase as a third phase. For two-phase composites, Rodriguez-Ramos et al. [0] and Guinovart-Diaz et al. [], using the asymptotic homogenization method (AHM), obtained analytical formulae for the global elastic constants of a binary fiber composite with perfect interfaces (which means that tractions and displacements are continuous across the interface) in a periodic structure, particularly with square and hexagonal distributions of fibers in the matrix. There are some studies using the finite element approach for the micromechanical unit cell modeling to investigate more complex structures, for instance, Golanski et al. [], Terada and Kikuchi [], Berger et al. [4], Kari et al. [5] and Xia et al. [6]. However, in many cases of interest the perfect interphase is not an adequate model, and it is necessary to include in unit cell modeling one or more interphases separating the reinforcement inclusion phase from the host matrix phase. In the present study, the previously stated experiences of the authors for twophase composites are extended, considering now a third phase between the fiber and matrix. In this work, effective thermo-mechanical properties of three-phase composites for different volume fractions of reinforced fiber material are predicted using micro-mechanical unit cell modeling. The unit cell is analyzed using the finite element method and appropriate periodic boundary conditions for different loading conditions. In this model the perfect adhesion between the phases and the matrix is considered. BASIC EQUATIONS: HOMOGENIZATION METHOD IN THERMOELASTICITY Many industrial and engineering materials as well as the majority of natural materials are inhomogeneous, i.e., they consist of dissimilar constituents (or phases ) that are distinguishable at some (small) length scale. Each constituent

5 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September H. BERGER ET AL. shows different material properties and/or material orientations and may itself be inhomogeneous at some smaller length scale(s). To determine the macroscopic overall characteristics of heterogeneous media is an essential problem in many engineering applications. From the time and cost view points, performing straightforward experimental measurements on a number of material samples, for various phase properties, volume fractions and loading histories is a hardly feasible task. On the other hand, due to usually enormous difference in length scales, it is impossible, for instance, to generate a finite element mesh that accurately represents the microstructure and also allows the numerical solution of the macroscopic structural component within a reasonable amount of time on today s computational systems. To overcome this problem several homogenization techniques have been created to obtain a suitable constitutive model to be inserted at the macroscopic level, i.e., homogenization is a mechanics based modeling scheme that transforms a body of a heterogeneous material into a constitutively equivalent body of a homogeneous continuum. In principle, the transformation model should be built on the basis of the composite microstructure, along with the relevant physical laws. A set of effective properties is obtained for the equivalent homogeneous continuum. When a composite specimen is under external load, micro stresses and strains are induced throughout the specimen. Ideally, the micro fields should be computed exactly, given the specimen and its fiber/matrix microstructure. Through the concept of homogenization, the composite specimen is regarded as a body of an effective homogeneous material, whose mechanical behavior is described by a definitive constitutive law. Let us consider a stationary thermo-mechanical problem in a heterogeneous medium. Let the position of a typical point of the body be denoted by three coordinates x x x of a Cartesian system of axes and let the periodic cell of the structure, Y, be defined by the inequalities, h i / <y i <h i /, where y i = x i / denotes the local fields, and 0 <, h i. The linear equation of equilibrium in quasi-static and stationary problem is given by ij j + X i = 0 () where the subscripts assume the values,, and, the comma denotes partial differentiation, and the summation convention is applied. X i is the body force. The stress and strain tensors are related to the temperature change T by the constitutive relations ij x = C ijkl x/ kl x x/ T x ij = C ijkl x/ kl x/ () Eq. () is known as the Duhamel Neumann law. The constitutive laws contain the components of the elasticity four-order tensor C ijkl x/ and the components of the thermal-stress second-order tensor ij x/, and ij x/ are the thermal expansion coefficients. These coefficients are functions of the fast variable y i = x i /, because they express the rapid change of properties of the composite or the high level of the heterogeneity.

6 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September 007 THERMO-MECHANICAL PROPERTIES OF THREE-PHASE COMPOSITES 805 The strain tensor is defined by the Cauchy relations kl x = ( ) uk l + u l k () where u k are the components of displacement. Eqs. () and () perform a system for solving the thermo-mechanical problem in which the influence of the temperature is induced by external body forces arising from temperature difference T. To this system we must adjoin appropriate boundary and initial conditions, which will be omitted here for simplicity. All coefficients C ijkl ij and kl are considered to be piecewise-smooth Y -periodic functions of the fast coordinates y i. The objective of the homogenization approach is to obtain a closed system of equations with constant coefficients, equivalent to the given system (). These new coefficients are considered to represent the global properties of the composite. To solve this classical problem a two-scale homogenization technique has been applied. Rodriguez-Ramos et al. [0] and Guinovart-Diaz et al. [], derived analytical expressions for the macroscopic effective properties of two-phase fiber reinforced composite with perfect interphases using the asymptotic homogenization method (AHM), which means that tractions and displacements are continuous across the interface in a periodic structure. For the prediction of effective elastic coefficients of three-phase composites in Afonso et al. [7] by means of closed form of analytical solutions based on a combination of the modified shear-lag model and the method of cells are considered. It does not require detailed knowledge of the microstructure. Guinovart-Diaz et al. [8] derived a recursive asymptotic homogenization scheme to predict the effective elastic properties of multi-phase composite materials. Recently, analytical expressions for effective material coefficients have been derived in Guinovart-Diaz et al. [9] for three-phase, thermo-elastic composites. A simple closed form of effective properties for a three-phase unidirectional transversely isotropic composite is presented. By using homogenization schemes for periodic media, the local problems are solved and effective thermo-elastic properties moduli are determined. These expressions for the effective thermo-elastic constants can be found in Guinovart-Diaz et al. [9], and are not given here for sake of brevity. The present work is closely connected with these formulations. But here the focus is set on the development of an equivalent numerical homogenization tool that also can be used for more complex composites to exceed the limits of the analytical methods. NUMERICAL HOMOGENIZATION OF RVE USING FINITE ELEMENT ANALYSIS Representative Volume Element For many composites, the macrostructure can be seen as a periodic array of repeated unit cells. Also for particulate reinforced composites, a repeated unit cell can still be constructed after assuming a uniform distribution and the same geometry for the reinforcing phase. Therefore, in most micro-mechanical analyses

7 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September H. BERGER ET AL. Figure (a) Schematic diagram of a periodic unit cell; (b) Notation of surfaces. the repeated unit cell is chosen as the RVE for the composites as shown in Figure. In the present work, micro-mechanical analysis method is applied to periodic RVE. The micro-mechanical method provides the effective thermo-mechanical properties of three phase composites from the known properties of their constituents (fiber, matrix and interphase) for different volume fractions using periodic representative volume element (RVE) or a unit cell model. The purpose of present article is to establish unit cells for three-phase composites and to demonstrate the usage of proper boundary conditions to account for the periodic nature of the stress and strain fields in the composite. The finite element method has been extensively used in the literature to analyze a periodic unit cell, to determine the effective properties of fiber reinforced composites. Here the unit cells are discretized and analyzed using the finite element method to predict the effective thermo-mechanical properties of unidirectional periodic coated (interphase) cylindrical fiber composites for different volume fractions.

8 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September 007 THERMO-MECHANICAL PROPERTIES OF THREE-PHASE COMPOSITES 807 This constitutive law can be determined based on the detailed fields in the selected unit cell through an averaging procedure. Specifically, if the exact micro fields ij and ij in the unit cell are known under the applied load, the averaged stresses and strains over the unit cell are given by ij = V ij dv V (4) ij = V ij dv V where V is the volume of the RVE. The averages are then treated as the effective stress and strain fields in the homogenized RVE. The relations between ij and ij determine the effective constitutive law. The constitutive relation presented in Eq. () can also be expressed in matrix form as shown in Eq. (5). C C C C 4 C 5 C 6 T C C C C 4 C 5 C 6 T = C C C C 4 C 5 C 6 T C 4 C 4 C 4 C 44 C 45 C 46 T (5) C 5 C 5 C 5 C 54 C 55 C 56 T C 6 C 6 C 6 C 64 C 65 C 66 T In the preceding, C ijkl is the effective stiffness of the homogenized composite. The number of independent constants in C ijkl is determined by the assumed symmetry. For these particular composites, transversely isotropic symmetry is assumed. For a transversely isotropic material, thermal expansion cannot induce shear, but the expansion in the three directions need not be equal. Consequently the equation has the form C C C T C C T = C T C symm C 44 0 C 66 (6) For instance, the first column of stiffness matrix is obtained when we impose the boundary conditions in such a way that the macroscopic strain is not equal to zero and all other strains and T are zero. Once all the independent constants of stiffness matrix are obtained, coefficient of thermal expansion (CTE) kl can be obtained by solving the constitutive equation with all kl equal to zero and nonzero T (by providing temperature difference to the unit cell as a body force and fix all the faces in all directions). Due to temperature difference there may be stresses developed inside the RVE. From the above averaging relations stresses ij are calculated. Once we know the ij and stiffness matrix C and temperature difference T, we can solve for the effective coefficient of thermal expansion kl.

9 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September H. BERGER ET AL. For transversely isotropic symmetry, the engineering constants are related to components of the stiffness tensor by E a = ( C C + C C C C C ( ) C) / C C C E t = ( C C + C C C C C ) ( C / C ) C a = ( ( ) C C C) / C C C (7) t = ( ) ( ) C C C C / C C C G = G = C 44 G = C 66 Boundary Conditions One of the most important issue in the finite element analysis of periodic a RVE is an appropriate application of the periodic boundary conditions. The displacement field for the periodic structure can be expressed as u i x x x = 0 ij x j + u i x x x (8) In the above, 0 ij is the global (average) strain tensor of the periodic structure and the first term on the right side represents a linear distributed displacement field. The second term on the right side u i x x x is a periodic function from one unit cell to another. It represents a modification to the linear displacement field due to the heterogeneous structure of the composites. Since the periodic array of the repeated unit cells represents a continuous physical body, two continuities must be satisfied at the boundaries of the neighboring unit cells. One is that the displacements must be continuous, i.e., the adjacent unit cells cannot be separated or penetrated into each other at the boundaries after the deformation. The second condition implies that the traction distributions at the opposite parallel boundaries of a unit cell must be the same. In this manner, the individual unit cell can thus be assembled as a physically continuous body followed by Xia et al. [0]. Obviously, the assumption of displacement field in the form of Eq. (8) meets the first of the above requirements. Unfortunately, it cannot be directly applied to the boundaries since the periodic part, u i x x x is generally unknown. For any unit cell, its boundary surfaces must always appear in parallel pairs, the displacements on a pair of parallel opposite boundary surfaces can be written as u k+ i u k i = 0 ij xk+ j + u i (9) = 0 ij xk j + u i (0) where indices k + and k identify the kth pair of two opposite parallel boundary surfaces of a repeated unit cell. Note that u i x x x is the same at the two parallel boundaries (periodicity), therefore, the difference between the above two equations is u k+ i u k i ( = 0 ik x k+ j ) x k j = 0 ij xk j ()

10 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September 007 THERMO-MECHANICAL PROPERTIES OF THREE-PHASE COMPOSITES 809 Since xj k are constants for each pair of the parallel boundary surfaces, with specified 0 ij, the right side becomes constant and such equations can be easily applied in the finite element analysis as nodal displacement constraint equations. Eq. () is a special type of displacement boundary conditions. Instead of giving known values of boundary displacements, it specifies the displacement differences between two opposite boundaries. Obviously, it becomes easier to adopt this form in a finite element procedure, instead of applying Eq. (8) directly as the boundary conditions. To apply the constraint equations () for instance in FEM, it is required to produce the same meshing at each two paired boundary surfaces. Then each constraint equation in () contains only two displacement components of the paired nodes. The number of the constraint equations is usually quite large, certain preprocessing program can be used to produce the data depending on the individual FEM code used. FINITE ELEMENT SIMULATIONS All the finite element calculations were done with commercial finite element program ANSYS within the framework of the small displacements theory, and the materials are assumed to behave as linear elastic and isotropic solids. The finite element mesh is created using three dimensional multi-field 0-node tetrahedral elements. To ensure equal mesh configurations on opposite surfaces for applying periodic boundary conditions three surfaces are first meshed with plane elements. Then the plane mesh configurations are copied to the opposite surfaces and the three dimensional mesh is generated based on the pre-meshed surfaces. With these identical nodal configurations on opposite surface the periodic boundary conditions can be applied as constraint equations between the appropriate nodal pairs. For the evaluation of effective coefficients, the boundary conditions have to be applied to the unit cell in such a way that, except one component of macroscopic strain field vector in Eq. (6), all other strain components and T are equal to zero. Then each effective coefficient can be easily determined by multiplying the corresponding row of material matrix with the strain field vector. In the next subsections, implementations of different boundary conditions to estimate all the effective coefficients of composite are explained in detail with the help of Figure (b). Evaluation of C and C For the calculation of C and C, we impose the boundary conditions in such away that the macroscopic strain is not equal to zero and all other strains and T are zero in Eq. (6). This can be achieved by applying the appropriate constraint equations to the different surfaces of the unit cell. For instance, consider a RVE with unit size x K+ j x K j = and = 0 05, then Eq. () reduces as follows ( ) u A+ u A = x A+ x A = 0 05 () ) = 0 since = 0 () ) = 0 since = 0 (4) u B+ u B = ( x B+ x B ( u C+ u C = x C+ x C Here u u and u are the displacements in x x and x directions, respectively.

11 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September H. BERGER ET AL. For the calculation of average stresses and strains and according to Eq. (6), the integral is replaced by a sum over averaged element values multiplied by the respective element volume. Using these averaged values the coefficients C and C can be calculated from the matrix Eq. (6). Due to zero strains and temperature fields except, the first row becomes = C. Then C can be calculated as the ratio of /. Similarly, C can be evaluated as the ratio of / from the second row of matrix Eq. (6). Evaluation of C and C For the calculation of the effective coefficients C and C we have similar conditions like for the calculation of the effective coefficients C and C. But now prescribed displacements in the form of constraint equations in x direction must be applied on parallel surfaces along x direction and prescribe zero displacements in the form of constraint equations on all other parallel surfaces, i.e., x and x directions. For instance, if we consider the applied far filed strain = 0 05 on the unit size RVE, then applied constraint equations are as follows ( ) u A+ u A = x A+ x A = 0 (5) ( ) u B+ u B = x B+ x B = 0 (6) ( ) u C+ u C = x C+ x C = 0 05 (7) Using the average strains and stresses calculated from Eq. (4) and from the third row of stiffness matrix Eq. (6), we get C as the ratio of / and similarly, C can be evaluated as the ratio of / from first row of matrix Eq. (6). Evaluation of C 44 and C 66 To evaluate the effective coefficient C 66 the in-plane shear strain may have a non-zero value in strain-temperature field vector of Eq. (6) only. This can be achieved by applying the appropriate constraint equations to the different surfaces of the unit cell. For illustration, consider the unit size of RVE and = 0 05, then Eq. () reduced as follows u A+ u A = 0 u A+ u A = 0 05 u A+ u A = 0 (8) u B+ u B = 0 05 u B+ u B = 0 u B+ u B = 0 (9) Using the calculated non-zero average strain and stress values and then from the sixth row in the matrix Eq. (6) we get = C 66 and consequently, C 66 can be computed as the ratio of /. For the evaluation of C 44 the out-of-plane shear strain or may have a non-zero value only. Because of transverse isotropic symmetry, we can consider either x x or x x planes as out-of-shear plane. For example, if we consider shear strain = 0 05 with unit size RVE, then the applied displacement constraint

12 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September 007 THERMO-MECHANICAL PROPERTIES OF THREE-PHASE COMPOSITES 8 equations are as follows u B+ u B = 0 u B+ u B = 0 u B+ u B = 0 05 (0) u C+ u C = 0 u C+ u C = 0 05 u C+ u C = 0 () From the non-zero average strain and stress, C 44 can be computed as the ratio of /. Evaluation of and As we already know all the independent stiffness constants in matrix Eq. (6), now in order to evaluate the effective coefficients of thermal expansion, we impose the boundary conditions in such a way that, all strain fields of strain vector in Eq. (6) are set to be zero and non-zero temperature field T is applied by providing temperature difference to the unit cell as a body force and fix all the faces in all directions as follows here. u A+ = u A = u A+ = u A = u A+ = u A = 0 () u B+ = u B = u B+ = u B = u B+ = u B = 0 () u C+ = u C = u C+ = u C = u C+ = u C = 0 and (4) T = (5) Due to this temperature loading, stresses are induced inside the RVE. From Eq. (6) we can evaluate the average stresses developed in all directions. Once we know the, and with independent stiffness constants and temperature difference T in Eq. (6), we obtain a system of three linear equations from which we can calculate all effective thermal coefficients. RESULTS AND DISCUSSION The effective thermo-mechanical properties of transversely distributed unidirectional cylindrical fiber composites are evaluated for different fiber volume fractions ranging from 0% up to 70% with 0% intervals using FEM and AHM approaches. All AHM results are calculated with formulae reported in Guinovart- Diaz [9]. The material properties presented in Benveniste et al. [] are used in the micro-mechanical unit cell modeling given in the Table. In all these calculations constant interphase volume fraction equal to.07% is considered, i.e., with increase Table Material properties of composite constituents Thermal expansion Constituent Young s modulus (GPa) Poisson s ratio coefficient (0 6 )/K Nickel matrix Tungsten fiber Carbon coating

13 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September H. BERGER ET AL. of fiber volume fraction, the thickness of the interphase is reduced. From these discrete fractions, graphs are interpolated and shown as comparison between AHM and FEM in Figures and. The results show a good agreement between calculations by AHM and FEM. Figure Effective elastic stiffness constants: comparison FEM and AHM.

14 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September 007 Figure THERMO-MECHANICAL PROPERTIES OF THREE-PHASE COMPOSITES 8 Effective coefficients of thermal expansion (CTE): comparison between FEM and AHM. Now for further validation of proposed unit cell modeling using finite element method, we compare the obtained results with Guinovart-Diaz et al. [9], Pagano- Tandon [7] and Periodic Medium Homogenization (PMH) data reported in Table 5 of Lagache et al. [5]. In the PMH method, the asymptotic homogenization method is applied to determine the effective properties of composite with hexagonal distribution of the fibers and the local problems are solved by the finite element method. In these comparisons we use a composite material made of Nicalon fiber (volume fraction = 0.6) and barium magnesium aluminosilicate (BMAS) matrix with an isotropic interphase material (volume fraction = 0.08), which is the result of intermixing phenomena and migration of coupling agents inside the matrix as reported in Lagache et al. [5]. The material properties of the constituents are listed in Table. Here we investigate the effect of the thickness of coating material on the behavior of the Nicalon /BMAS composite system by considering the two different volume fractions (0 and 0.08) of interphase coating material. Table shows that, with less than 8% of interphase volume fraction, the effective moduli E t, G a and G t of the composite decrease abruptly lower than 50% of the composite properties without interphase. Also, we can observe a good agreement between the different approaches. Table Material properties of BMAS/Nicalon composite with interphase Constituent Young s modulus (GPa) Shear modulus (GPa) BMAS matrix Nicalon fiber Interphase material.45.

15 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September H. BERGER ET AL. Table Comparison between FEM, AHM and Pagano and Tandon [7] and PMH of Lagache et al. [5] Model E a E t G a G t FEM (V = 0) AHM (V = 0) Pagano & Tandon [7] (V = 0) FEM (V = 0 08) AHM (V = 0 08) Pagano & Tandon [7] (V = 0 08) PMH, Lagache et al. [5] (V = 0 08) Table 4 illustrates the comparison between our approach (FEM), AHM, numerical PMH and Self-Consistent Scheme (SCS) double approximations reported in Table of Lagache et al. [5]. The properties of the different media used in the computation are E = 4 GPa, = = 0 4, E = 84 GPa and = 0 and the interphase Young s modulus E is constant and taken in a range between 4 and GPa. The fiber volume fraction is equal to V = 0 5 and the interphase volume content is V = The Young s and shear moduli of the composite increases with increase of coating Young s modulus for the different models. In this case, some matrix material is replaced by a harder material. Also it can be observed that the values for the elastic moduli derived from FEM model are close to those derived from AHM, SCS and PMH models. Table 4 The sensitivity of effective moduli to the elastic modulus of the coating E, which is revealed by different models E Model E a E t G a G t 4 FEM AHM SCS PMH FEM AHM SCS PMH FEM AHM SCS PMH FEM AHM SCS FEM AHM SCS PMH

16 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September 007 THERMO-MECHANICAL PROPERTIES OF THREE-PHASE COMPOSITES 85 CONCLUSIONS A unit cell model is employed to predict the effective thermo-mechanical properties of three-phase coated unidirectional cylindrical fibers using homogenization techniques for different fiber volume fractions. The numerical approach is based on the finite element method. Longitudinal and transversal effective thermo-mechanical coefficients have been calculated with the finite element model and compared with analytical solutions based on the asymptotic homogenization method. The numerical results demonstrate that the developed FEM approach is very accurate and efficient for the analysis of unit cell models of fiber reinforced composites, with the presence of the interphases. The present work has laid down a foundation for further applications of micro-mechanical finite element analysis for problems, such as an investigation of stress field around the fiber in order to understand the onset and the development of inelastic behavior such as plastic deformation and possible damage. Furthermore the proved reliability of the introduced FEM approach opens new possibilities to investigate composites with arbitrary geometrical types of inclusions which cannot be covered by most other homogenization methods. REFERENCES. R. Hill, Theory of Mechanical Properties of Fiber-Strengthened Materials, I Elastic Behavior. J. Mech. Phys. Solids, vol., pp. 99, Z. Hashin, On Elastic Behavior of Fiber-Reinforced Materials of Arbitrary Transverse Phase Geometry. J. Mech. Phys. Solids, vol., pp. 9 4, Z. Hashin, Analysis of Properties of Fiber Composites with Anisotropic Constituents. J. Appl. Mech., vol. 46, pp , L. J. Walpole, A Coated Inclusion in an Elastic Medium. Math. Proc. Comb. Phil. Soc., vol. 8, pp , Y. Mikata and M. Taya, Stress Field in and Around a Coated Short Fiber in an Infinite Matrix Subjected to Uniaxial and Biaxial Loadings. ASME J. Appl. Mech., vol. 5, pp. 9 4, P. S. Theocaris and A. G. Varias, The Influence of the Mesophase on the Transverse and Longitudinal Moduli and the Major Poisson s Ratio in Fibrous Composites. Colloid Polym. Sci., vol. 64, pp , N. J. Pagano and G. P. Tandon, Elastic Response of Multidirectional Coated-Fiber Composites, Compos. Sci. Technol., vol., pp. 7 9, N. J. Pagano and G. P. Tandon, Thermo-Elastic Model for Multi-Directional Coated- Fiber Composites: Traction Formulation. Compos. Sci. Technol., vol. 8, pp , Z. Hashin, Thermo-Elastic Properties of Fiber Composites with Imperfect Interface. Mech. Mater., vol. 8, pp. 48, J. R. Yeh, The Effect of Interphase on the Transverse Properties of Composites. Int. J. Solid Stress., vol. 9, no. 0, pp , 99.. M. Sutcu, A Recursive Concentric Cylinder Model for Composites Containing Coated Fibers. Int. J. Solid Struct., vol. 9, pp. 97, 99.. A. Dasgupta and S. M. Bhandarkar, A Generalized Self-Consistent Mori-Tanaka Scheme for Fiber-Composites with Multiple Interphases. Mech. Mater., vol. 4, pp. 67 8, 99.

17 Downloaded By: [Otto-von-guericke-universitaet] At: 5:5 0 September H. BERGER ET AL.. Y. Benveniste, G. J. Dvorak, and T. Chen, Stress Fields in Composites with Coated Inclusions. Mech. Mater., vol. 7, pp. 05 7, Y. C. Chu and S. I. Rokhlin, Determination of Fiber Matrix Interphase Moduli from Experimental Moduli of Composite with Multi-Layered Fibers. Mech. Mater., vol., pp. 9 5, M. Lagache, A. Agbossou, and J. Pastor, Role of Interphase on Elastic Behavior of Composite Materials: Theoretical and Experimental Analysis. J. Comp. Mater., Vol. 8 no., pp. 4 57, C. S. Chouchaoui and M. L. Benzeggagh, The Effect of Interphase on the Elastic Behavior of a Glass/Epoxy Bundle. Compos. Sci. Technol., vol. 57, pp. 67 6, A. Agbossou and J. Pastor, Thermal Stresses and Thermal Expansion Coefficients of n- Layered Fiber-Reinforced Composites, Compos. Sci. Technol., vol. 57, pp , Y. J. Liu, N. Xu, and J. F. Luo, Modeling of Interphases in Fiber-Reinforced Composites under Transverse Loading Using the Boundary Element Method. J. Appl. Mech., vol. 67, pp. 4 49, Z. Hashin, Thin Interphase/Imperfect Interface in Elasticity with Application to Coated Fiber Composites. J. Mech. Phys. Solids, vol. 50, pp , R. Rodriguez Ramos, F. J. Sabina, R. Guinovart Diaz, and J. Bravo Castillero, Closed- Form Expressions for the Effective Coefficients of Fiber-Reinforced Composite with Transversely Isotropic Constituents. I: Elastic and Square Symmetry. Mech. Mater., vol., pp. 5, 00.. R. Guinovart-Diaz, J. Bravo-Castillero, R. Rodriguez-Ramos, and F. J. Sabina, Closed- Form Expressions for the Effective Coefficients of a Fiber-Reinforced Composite with Transversely Isotropic Constituents-I: Elastic and Hexagonal Symmetry. J. Mech. Phys. Solids, vol. 49, pp , 00.. D. Golanski, K. Terada, and N. Kikuchi, Macro and Micro Scale Modeling of Thermal Residual Stresses in Metal Matrix Composite Surface Layers by the Homogenization Method. Comput. Mech., vol. 9, no., pp. 88 0, K. Terada and N. Kikuchi, Global-Local Constitutive Modeling of Composite Materials by the Homogenization Method, Mater. Sci. Res. Int., vol., no., pp. 7 80, H. Berger, S. Kari, U. Gabbert, R. Rodriguez-Ramos, R. Guinovart, J. A. Otero, and J. Bravo-Castillero, An Analytical and Numerical Approach for Calculating Effective Material Coefficients of Piezoelectric Fiber Composites, Int. J. Solid Struct., vol. 4, pp , S. Kari, H. Berger, R. Rodriguez-Ramos, and U. Gabbert, Computational Evaluation of Effective Material Properties of Composites Reinforced by Randomly Distributed Spherical Particles. Composite Struct., vol. 77, pp., Z. Xia, C. Zhou, Q. Yong, and X. Wang, On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites. Int. J. Solid. Struct., vol. 4, pp , J. C. Afonso and G. Ranalli, Elastic Properties of Three-Phase Composites: Analytical Model Based on the Modified Shear-Lag Model and the Method of Cells. Compo. Sci. & Tech., vol. 65, pp , R. Guinovart-Diaz, R. Rodriguez-Ramos, J. Bravo-Castillero, F. J. Sabina, J. A. Otero and G. A. Maugin, A Recursive Asymptotic Homogenization Scheme for Multi-Phase Fibrous Elastic Composites. Mech. Mater., vol. 7, pp. 9, 005.

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