Infinite-contrast periodic composites with strongly nonlinear behavior: Effective-medium theory versus full-field simulations

M. I. IDIART; F. WILLOT; Y. P. PELLEGRINI; P. PONTE CASTAÑEDA

INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES

Elsevier

Año: 2009 vol. 46 p. 3365 - 3382

0020-7683

This paper presents a combined numerical-theoretical study of the macroscopic behavior and local field distributions in a special class of two-dimensional periodic composites with viscoplastic phases. The emphasis is on strongly nonlinear materials containing pores or rigid inclusions. Full-field numerical simulations are carried out using a Fast-Fourier Transform algorithm [H. Moulinec, P. Suquet, C. R. Acad. Sci. Paris II 318, 1417 (1994)], while the theoretical results are obtained by means of the `second-order´ nonlinear homogenization method [P. Ponte Casta~neda, J. Mech. Phys. Solids 50, 737 (2002)]. The effect of nonlinearity and inclusion concentration is investigated in the context of power-law (with strain-rate sensitivity $m$) behavior for the matrix phase under in-plane shear loadings. Overall, the `second-order´ estimates are found to be in good agreement with the numerical simulations, with the best agreement for the rigidly reinforced materials. For the porous systems, as the nonlinearity increases ($m$ decreases), the strain field is found to localize along shear bands passing through the voids (the strain fluctuations becoming unbounded) and the effective stress exhibits a singular behavior in the dilute limit. More specifically, for small porosities and fixed nonlinearity $m>0$, the effective stress decreases linearly with increasing porosity. However, for ideally plastic behavior ($m = 0$), the dependence on porosity becomes non-analytic. On the other hand, for rigidly-reinforced composites, the strain field adopts a tile pattern with bounded strain fluctuations, and no singular behavior is observed (to leading order) in the dilute limit.