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Tight bounds for three-dimensional nonlinear incompressible elastic composites. (English) Zbl 1213.74058

Summary: Tighter variational bounds, in the whole range of inclusion volume fraction, that is to say, even near percolation, for the effective energy of nonlinear composites, in the special case of 3D two-phase incompressible elastic composites with isotropic constituents are presented. Following the methodology of Talbot, Willis and Ponte Castañeda, a linear comparison material with the same microgeometry as the nonlinear composite is employed. The asymptotic homogenization method (AHM) combined with a finite element analysis (FEM), is used to find the displacement field as well as the effective properties for the comparison material. An elastic composite with periodically distributed spherical inclusions in a cubic array is considered as an example. Various numerical examples are performed. Comparisons with others theories (i.e. variational bounds, self-consistent estimates, etc.) are shown. Coincidence of the AHM-FEM results with the universal bounds of Nemat-Nasser, Yu and Hori serves as a useful check to the numerical calculation.

MSC:

74B20 Nonlinear elasticity
74E30 Composite and mixture properties
74A40 Random materials and composite materials
74S05 Finite element methods applied to problems in solid mechanics
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