Development of 3D T-Trefftz voronoi cell finite elements with/without spherical voids &/or elastic/rigid inclusions for micromechanical modeling of heterogeneous materials

L. Dong, S. N. Atluri

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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 outer- as 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 suitable 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.

Original languageEnglish
Pages (from-to)169-211
Number of pages43
JournalComputers, Materials and Continua
Issue number2
StatePublished - 2012


  • Barycentric coordinates
  • Collocation
  • Completeness
  • Efficiency
  • Inclusion
  • LBB conditions
  • Matrix
  • Papkovich-Neuber solution
  • Spherical harmonics
  • T-Trefftz
  • Variational principle
  • Void


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