Development of 3D T-Trefftz Voronoi Cell Finite Elements with/without Spherical Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials

In this paper, three-dimensionalT-Trefftz Voronoi Cell Finite Elements (VCFEM-TTs) are developed for micromechanical modeling of heterogeneous materials. Several types of VCFEMs are developed, depending on the types of heterogeneity in each element. Each VCFEM can include alternatively a spherical void, a spherical elastic inclusion, a spherical rigid inclusion, or no voids/inclusions at all.In all of these cases, an inter-element compatible displacement field is assumed at each surface of the polyhedral element, with Barycentric coordinates as nodal shape functions.The T-Trefftz trial displacement fields in each element are expressed in terms of the Papkovich-Neuber solution. Spherical harmonics are used as the Papkovich-Neuber potentials to derive the T-Trefftz trial displacement fields. Characteristic lengthsareused for each element to scale the T-Trefftztrial functions, in order to avoid solving systems of ill-conditioned equations. Two approaches for developing element stiffness matrices are used.The differencesbetween these two approachesare that, the compatibilitybetweenthe independentlyassumed fieldsin the interior of the element with those at the outeras well as the inner-boundary, are enforced alternatively, byLagrange multipliers in multi-fieldboundary variational principles, or by collocation at a finite number of preselected points. These elements are named as VCFEM-TT-BVP and VCFEM-TT-C respectively, following the designations of [Dong and Atluri (2011b, 2012a)].Several three-dimensional computational micromechanics problems are solved using these elements. Computational results demonstrate that both VCFEM-TT-BVP and VCFEM-TT-C can solve three-dimensional problems efficiently and accurately. Especially, these VCFEMTTs can capture the stress concentration around spherical voids/inclusion quite accurately, and the time for computing each element is much less than that for the hybrid-stress version of VCFEM in [Ghosh and Moorthy (2004)]. Therefore, we consider that the 3D Voronoi Cell Finite Elements developed in this study are suit1 Center for Aerospace Research & Education, University of California, Irvine 170 Copyright © 2012 Tech Science Press CMC, vol.29, no.2, pp.169-211, 2012 able for micromechanical modeling of heterogeneous materials. We also point outthat, the process of reducing ellipsoidal coordinates/harmonics to spherical ones in the limiting case cannot work smoothly, which was contrarily presented in an ambiguous way in [Ghosh and Moorthy (2004)]. VCFEMs with ellipsoidal, and arbitrary shaped voids/inclusions will be presented in future studies.