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 outer- as 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 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.