18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS COMPUTATIONAL MICROMECHANICS FOR COMPOSITES WITH FINITE BOUNDARIES S. Nomura 1* and T. Pathapalli 1 1 Mechanical and Aerospace Engineering Department, The University of Texas at Arlington, Arlington, TX 76019-0023 USA * Corresponding author (nomura@uta.edu) Keywords : Micromechanics, elliptic inclusions, computer algebra system primarily involves determining a set of continuous Abstract permissible functions that satisfy the boundary A semi-analytical approach is presented to solve conditions and continuity conditions at the matrix- general boundary value problems (BVPs) that arise inclusion interface, expressing the BVP (governing in the analysis of composite materials. As an equation) in terms of the Sturm-Liouville (S-L) illustration, a rectangular shaped medium that system [3], subjecting the S-L problem (eigenvalue contains elliptical inclusions will be considered. problem) to the Galerkin method to obtain an Permissible functions, which satisfy the orthonormal set of eigenfunctions, and finally, homogeneous boundary conditions and the representing the mechanical/physical field as a linear continuity conditions at the matrix-inclusion combination of the evaluated eigenfunctions. In interface, are analytically derived. The order to facilitate the analytical derivation of the eigenfunctions are subsequently derived from the permissible functions in terms of the relevant permissible functions using the Galerkin method. material and geometrical parameters, a computer The mechanical and physical fields (temperature or algebra system [4, 5] has been extensively used. As elastic deformation), with an arbitrarily given source a demonstration example, a steady-state heat term, can be expressed as a linear combination of the conduction problem for a rectangular shaped eigenfunctions. The method is favorably compared medium with elliptical inclusions is presented. The with numerical results obtained from the finite results are favorably compared with those obtained element method. from the finite element method. 1 Introduction 2 Formulations The conventional micromechanics [1], pioneered by The governing differential equations for both Eshelby [2], has been widely used to analyze the elasticity and steady-state heat conduction can be microstructure of elastic solids that contain expressed as inclusions. However, the major limitation of micromechanics is that the medium is assumed to be �[�(�)] + �(�) = 0, (1) infinitely extended (no boundary) and the geometry of the microstructure is strictly limited, thus making where � is a self-adjoint differential operator, �(�) is it inconvenient to accurately represent actual the unknown physical field (temperature or composite materials. Although purely numerical displacement) and �(�) is the source term (heat methods such as the finite element method can be source or body force). For steady state heat employed for the analysis of composites, analytical conduction, � takes the following form: or semi-analytical solutions, if available, are needlessly valuable. This paper presents a new semi- �[�(�)] = �. ��(�) ∇�(�)�, (2) analytical method for heterogeneous materials that where �(�) is the thermal conductivity and for the makes use of both analytical and numerical static elasticity case, � takes the form: techniques to obtain mechanical and physical fields associated with the given BVP. The method
(�[�(�)]) � = �� ���� � �,� �, �, (3) � �� = ��[� � (�)] � (�)��, � (9) � where � ���� is the elastic modulus. For isotropic materials, it can be expressed in terms of the shear � �� = � � � (�) � (�)��. � (10) modulus, � , and the bulk modulus, � , as: � The components of � �� and � �� are obtained using � ���� = �� − 2 � 3 � � �� � �� + ��� �� � �� �, (4) �� + � �� � equations (9) and (10). Therefore, equation (8) can be solved using a standard numerical techniques to where � �� is the Kronecker delta. obtain the eigenvalues, � � , and the corresponding set of unknown coefficients, � �� . Based on the Sturm-Liouville theory, the solution to In accordance with the Sturm-Liouville theory, the equation (1) can be obtained if the eigenfunction, solution to equation (1) can be expressed as a linear � � (�), is available. The eigenfunction is defined as combination of eigenfunctions as �[� � (�)] + � � � � (�) = 0, (5) � �(�) = � � � where � � and � � (�) are the � �� eigenvalue and � � (�), (11) � � eigenfunction, respectively. An intrinsic property of ��� the Sturm-Liouville system is that its differential where � � is the eigenfunction expansion coefficient operator, � , is Hermitian, hence suggesting that all of the source term, �(�) , and can be obtained as the eigenvalues are real and the eigenfunctions are mutually orthogonal as � � = � �(�) � � (�)��. (12) � � � � (�) � � (�)�� = � �� . (6) The most demanding task is to obtain the � � (�) , that satisfies the permissible function, � The solution to equation (5) can be obtained by boundary conditions and the required continuity expressing the eigenfunctions, � � (�) , as a series of � conditions at the interface. For instance, a 2-D body permissible functions as that contains an elliptical inclusion, the permissible function needs to be defined separately for each � � (�, �) is assumed to phase. For the inclusion phase, � � � (�) = �� �� � � (�) , (7) be in the polynomial form as ��� where � � (�) is a permissible function chosen from � � elementary functions such as polynomials to satisfy � ��� (�, �) = � � � �� ��� � ��� � � , (13) � the homogeneous boundary conditions and the ��� ��� continuity conditions across the matrix-inclusion where � represents the order of the polynomial and interface. The quantity, � �� , is the coefficient of the summation ensures that the permissible function, � (�) of the � �� eigenfunction and is determined � � ��� (�, �) , encompasses all the polynomials up to the using the Galerkin method. By substituting equation � ��� (�, �) � �� order. Similarly, for the matrix phase, � (7) into equation (5), multiplying � � (�) on both sides � is assumed to be of the form: and integrating them over the entire domain, equation (5) can be converted to the following � � generalized eigenvalue problem: ��� (�, �) = �(�, �) � �� �� ��� � ��� � � (14) � , � ��� ��� � � + � � � = 0, (8) where �(�, �) is a function that vanishes at the where boundary (for the boundary condition of the first kind). For example, if the boundary is rectangular
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