## Abstract

In this paper, as an extension to the authors's work in [Dong and Atluri (2011a,b, 2012a,b,c)], three-dimensional Trefftz Voronoi Cells (TVCs) with ellipsoidal voids/inclusions are developed for micromechanical modeling of heterogeneous materials. Several types of TVCs are developed, depending on the types of heterogeneity in each Voronoi Cell(VC). Each TVC can include alternatively an ellipsoidal void, an ellipsoidal elastic inclusion, or an ellipsoidal rigid inclusion. In all of these cases, an inter-VC compatible displacement field is assumed at each surface of the polyhedral VC, with Barycentric coordinates as nodal shape functions. The Trefftz trial displacement fields in each VC are expressed in terms of the Papkovich-Neuber solution. Ellipsoidal harmonics are used as the Papkovich-Neuber potentials to derive the Trefftz trial displacement fields. Characteristic lengths are used for each VC to scale the Trefftz trial functions, in order to avoid solving systems of ill-conditioned equations. Two approaches for developing VC stiffness matrices are used. The differences between these two approaches are that, the compatibility between the independently assumed fields in the interior of the VC with those at the outer- as well as the inner-boundary, are enforced alternatively, by Lagrange multipliers in multi-field boundary variational principles, or by collocation at a finite number of preselected points. These VCs are named as TVC-BVP and TVC-C respectively. Several three-dimensional computational micromechanics problems are solved using these TVCs. Computational results demonstrate that both TVC-BVP and TVC-C can efficiently predict the overall properties of composite/porous materials. They can also accurately capture the stress concentration around ellipsoidal voids/inclusions, which can be used in future to study the damage of materials, in combination of tools of modeling micro-crack initiation and propagation. Therefore, we consider that the 3D TVCs developed in this study are very suitable for ground-breaking micromechanical study of heterogeneous materials.

Original language | English |
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Pages (from-to) | 39-81 |

Number of pages | 43 |

Journal | Computers, Materials and Continua |

Volume | 30 |

Issue number | 1 |

State | Published - 2012 |

## Keywords

- Barycentric coordinates
- Collocation
- Completeness
- Efficiency
- Ellipsoidal harmonics
- Inclusion
- LBB conditions
- Matrix
- Papkovich-Neuber solution
- Trefftz Voronoi Cell
- Variational principle
- Void