TH 18 INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS
COMPUTATIONAL MICROMECHANICS FOR COMPOSITES WITH FINITE BOUNDARIES
S. Nomura1* and T. Pathapalli1 1Mechanical and Aerospace Engineering Department, The University of Texas at Arlington, Arlington, TX 76019-0023 USA * Corresponding author ([email protected])
Keywords: Micromechanics, elliptic inclusions, computer algebra system
Abstract primarily involves determining a set of continuous 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 makes use of both analytical and numerical where ( ) is the thermal conductivity and for the techniques to obtain mechanical and physical fields static elasticity case, takes the form: 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:
2 The components of and are obtained using = − + + , (4) 3 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 equation (1) can be obtained if the eigenfunction, In accordance with the Sturm-Liouville theory, the