Variational linear comparison bounds for nonlinear composites with anisotropic phases. I. General results

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

1 LMS - Laboratoire de mécanique des solides

This work is concerned with the development of bounds for nonlinear composites with anisotropic phases by means of an appropriate generalization of the ‘linear comparison’ variational method, introduced by Ponte Castañeda for composites with isotropic phases. The bounds can be expressed in terms of a convex (concave) optimization problem, requiring the computation of certain ‘error’ functions that, in turn, depend on the solution of a non-concave/non-convex optimization problem. A simple formula is derived for the overall stress–strain relation of the composite associated with the bound, and special, simpler forms are provided for power-law materials, as well as for ideally plastic materials, where the computation of the error functions simplifies dramatically. As will be seen in part II of this work in the specific context of composites with crystalline phases (e.g. polycrystals), the new bounds have the capability of improving on earlier bounds, such as the ones proposed by deBotton and Ponte Castañeda for these specific material systems.

Mots clés: nonlinear homogenization; variational methods; anisotropy

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

Martin I Idiart, Pedro Ponte Castañeda. Variational linear comparison bounds for nonlinear composites with anisotropic phases. I. General results. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007, 463, pp.907-924. ⟨10.1098/rspa.2006.1797⟩. ⟨hal-00311889⟩