Scale-dependent bounds on effective elastoplastic response of random composites

We consider the mechanical response of random, heterogeneous materials, where each phase is elastic-plastic with an associated flow rule, and the microstructure's statistics is homogeneous and ergodic. Under proportional monotonic loading, the effective (in the macroscopic sense, or overall) elastoplastic response is shown to be bounded from above and below by those obtained, respectively, from displacement and traction boundary conditions applied to finite size domains (square shaped windows). A scale dependent hierarchy of these bounds is obtained by extending the methods used earlier for the elastic moduli estimation: the larger the scale relative to the heterogeneity, the closer are the bounds. A fiber reinforced metal matrix composite is employed to illustrate the theoretical results. Its constitutive response and plastic strain field are investigated by computational micromechanics for different window sizes under both types of boundary conditions; it is found here that the displacement conditions result in denser and more uniformly distributed slip band patterns, while the traction conditions lead to more localized fields. We also investigate a mixed boundary condition, under which the mechanical response of composite is found to fall between those under displacement and traction controlled boundary conditions.