Microstructural disorder, mesoscale finite elements and macroscopic response

Consideration of boundary-value problems in mechanics of materials with dis ordered microstructures leads to the introduction of an intermediate scale, a mesoscale, which specifies the resolution of a finite-element mesh relat ive to the microscale. The effective elastic mesoscale response is bounded by the Dirichlet and Neumann boundary-value problems. The two estimates, se parately, provide inputs to two finite-element schemes-based on minimum pot ential and complementary energy principles, respectively-for bounding the g lobal response. While in the classical case of a homogeneous material, thes e bounds are convergent with the finite elements becoming infinitesimal, th e presence of a disordered non-periodic microstructure prevents such a conv ergence and leads to a possibility of an optimal mesoscale. The method is d emonstrated through an example of torsion of a bar having a percolating two -phase microstructure of over 100 000 grains. By passing to an ensemble set ting, we arrive at a hierarchy of two random continuum fields! which provid es inputs to a stochastic finite-element method.