T-Trefftz Voronoi Cell Finite Elements with Elastic/Rigid Inclusions or Voids for Micromechanical Analysis of Composite and Porous Materials

In this paper, we develop T-Trefftz Voronoi Cell Finite Elements (VCFEM-TTs) for micromechanical modeling of composite and porous materials. In addition to a homogenous matrix in each polygon-shaped element, three types of arbitrarily-shaped heterogeneities are considered in each element: an elastic inclusion, a rigid inclusion, or a void. In all of these three cases, an inter-element compatible displacement field is assumed along the element outer-boundary, and interior displacement fields in the matrix as well as in the inclusion are independently assumed as T-Trefftz trial functions. Characteristic lengths are used for each element to scale the T-Trefftz trial functions, in order to avoid solving systems of ill-conditioned equations. Two approaches for developing element stiffness matrices are used. The differences between these two approaches are that, the compatibility between the independently assumed fields at the outeras well as the innerboundary, are enforced alternatively, by Lagrange multipliers in multi-field boundary variational principles, or by collocation at a finite number of preselected points. Following a previous paper of the authors, these elements are denoted as VCFEMTT-BVP and VCFEM-TT-C respectively. Several two dimensional problems are solved using these elements, and the results are compared to analytical solutions and that of VCFEM-HS-PCE developed by [Ghosh and Mallett (1994); Ghosh, Lee and Moorthy (1995)]. Computational results demonstrate that VCFEM-TTs developed in this study are much more efficient than VCFEM-HS-PCE developed by Ghosh, et al., because domain integrations are avoided in VCFEM-TTs. In addition, the accuracy of stress fields computed by VCFEM-HS-PCE by [Ghosh and Mallett (1994); Ghosh, Lee and Moorthy (1995)] seem to be very poor as compared to analytical solutions, because the polynomial Airy stress function is highly incomplete for problems in a doubly-connected domain (as in this case, when a inclusion or a void is present in the element). However, the results of VCFEM-TTs 1 Department of Civil & Environmental Engineering, University of California, Irvine, [email protected] 2 Center for Aerospace Research & Education, University of California, Irvine 184 Copyright © 2012 Tech Science Press CMES, vol.83, no.2, pp.183-219, 2012 developed in the present paper are very accurate, because, the compete T-Trefftz trial functions derived from positive and negative power complex potentials are able to model the singular nature of these stress concentration problems. Finally, out of these two methods, VCFEM-TT-C is very simple, efficient, and does not suffer from LBB conditions. Because it is almost impossible to satisfy LBB conditions a priori, we consider VCFEM-TT-C to be very useful for ground-breaking studies in micromechanical modeling of composite and porous materials.