Variational linear comparison bounds for nonlinear composites with anisotropic phases. II. Crystalline materials

Martin Idiart (1, 2) , Pedro Ponte Castañeda (1, 2)

1 LMS - Laboratoire de mécanique des solides
2 University of Pennsylvania

In part I of this work, bounds were derived for the effective potentials of nonlinear composites with anisotropic constituents, making use of an appropriate generalization of the linear comparison variational method. In this second part, the special case of nonlinear composites with crystalline constituents is considered. First, it is shown that, for this special but very important class of materials, the ‘variational’ bounds of part I are at least as good as an earlier version of the bounds due to deBotton & Ponte Castañeda. Then, the relative merits of these two types of bounds are studied in the context of a model, two-dimensional, porous composite with a power-law crystalline matrix phase, under anti-plane loading conditions. The results show that, indeed, the variational bounds of part I improve, in general, on the earlier bounds, with the former becoming progressively sharper than the latter as the number of slip systems of the crystalline matrix phase increases. In particular, it is shown that, unlike the bounds of deBotton & Ponte Castañeda, the variational bounds of part I are able to recover the variational bound for composites with an isotropic matrix phase, as the number of slip systems, all having the same flow stress, tends to infinity.

Mots clés: nonlinear homogenization; variational methods; crystalline materials

Domaines: Mécanique des solides [physics.class-ph]; Mécanique des solides [physics.class-ph]

Martin Idiart, Pedro Ponte Castañeda. Variational linear comparison bounds for nonlinear composites with anisotropic phases. II. Crystalline materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007, 463, pp.925-943. ⟨10.1098/rspa.2006.1804⟩. ⟨hal-00311890⟩