Three-dimensional Trefftz computational grains for the micromechanical modeling of heterogeneous media with coated spherical inclusions

Guannan Wang, Leiting Dong, Junbo Wang, Satya N. Atluri

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5 Scopus citations


Three-dimensional computational grains based on the Trefftz method (TCGs) are developed to directly model the micromechanical behavior of heterogeneous materials with coated spherical inclusions. Each TCG is polyhedral in geometry and contains three phases: an inclusion, the surrounded coating (or interphase) and the matrix. By satisfying the 3D Navier's equations exactly, the internal displacement and stress fields within the TCGs are expressed in terms of the Papkovich-Neuber (P-N) solutions, in which spherical harmonics are employed to further express the P-N potentials. Further, the Wachspress coordinates are adopted to represent the polyhedral-surface displacements that are considered as nodal shape functions, in order to enforce the compatibility of deformations between two TCGs. Two techniques are developed to derive the local stiffness matrix of the TCGs: one is directly using the multi-field boundary variational principle (MFBVP) while the other is first applying the collocation technique for the continuity conditions within and among the grains and then employing a primal-field boundary variational principle (PFBVP). The local stress distributions at the interfaces between the 3 phases, as well as the effective homogenized material properties generated by the direct micromechanical simulations using the TCGs, are compared to other available analytical and numerical results in the literature, and good agreement is always obtained. The material and geometrical parameters of the coatings/interphases are varied to test their influence on the homogenized and localized responses of the heterogeneous media. Finally, the periodic boundary conditions are applied to the representative volume elements (RVEs) that contain one or more TCGs to model the heterogeneous materials directly.

Original languageEnglish
Pages (from-to)505-529
Number of pages25
JournalJournal of Mechanics of Materials and Structures
Issue number4
StatePublished - 2018


  • Coated spherical inclusions
  • Collocation technique
  • Heterogeneous materials
  • Papkovich-Neuber solutions
  • Periodic boundary conditions
  • Spherical harmonics
  • Trefftz computational grains
  • Variational principles


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