Towards Scale-Dependent Constitutive Laws for Plasticity and Fracture of Random Heterogeneous Materials

Abstract

The conventional aim of micromechanics is to derive macroscopic effective constitutive laws for a Representative Volume Element (RVE) ΔV on the scales that are infinitely larger than the microscale. This is tantamount to an assumption that the macroscopic field quantities vary on scales that are much larger than the scale of ΔV However, in many problems of mechanics this is not possible — a typical example is provided by stochastic finite elements, where, due to the finite size L of an element (or window) relative to the microscale d, its stiffness matrix may display statistical fluctuations, Fig. 1. Consequently, one has to set up a stochastic stiffness matrix in relation to the actual microstructure, whereby several features have been established in the setting of elastic materials [1, 2]:

• i) effective stiffness tensor, and hence stiffnes matrix, on a scale δ = L/d depends on the type of boundary conditions — essential or natural,

• ii) this effective stiffness tensor is, in general, anisotropic,

• iii) only triangular finite elements are consistent with a micromechanics formulation,

• iv) the global response can be bounded by solving two finite element problems: one based on the displacement approach, and another based on the force approach.