Bounds and estimates for the overall plastic behaviour of composites with strain gradient effects

The macroscopic plastic response is estimated for a composite with each phase satisfying a strain gradient constitutive description. In a J(2) deformation theory of strain gradient plasticity, the: strain energy function is assumed to depend on both the von Mises strain invariant epsilon(e) and on a curvature invariant chi(e). For a general coupling between epsilon(e) and chi(e), a nonlinear variational principle is formulated generalizing that of Ponte Castaneda (1992 J. Mech. Phys. Solids 40, 1757). The minimum principle is used to derive bounds and estimates for the overall plastic response of statistically homogeneous and isotropic strain gradient composites. An essential ingredient of the strategy is the use of previous results of the authors (1994 J. Mech. Phys. Solids 42, 1851) for a linear comparison composite. For a particular coupling between epsilon(e) and chi(e), Suggested earlier by Fleck & Hutchinson (1993), the Hashin-Shtrikman upper bounds and lower estimates are calculated for a power-law hardening solid (including the rigid-perfectly plastic limit). The results demonstrate a size effect.