On a Method for Multiscale Solid Mechanics

Abstract

Given a fine-scale physical theory characterized by an evolutionary system of equations and a set of quantities, defined from the variables of the fine theory, that serve as a coarse representation of the fine scale phenomena, a systematic method for developing an evolutionary set of equations for the coarse quantities is presented. This set may be considered as governing equations for a coarse theory corresponding to the fine one since the solutions of the coarse theory define a small class of solutions to the fine theory. The method for generating the coarse theory is technically straightforward. The conditions required for the procedure to succeed are understood qualitatively, but a rigorous result for the intended applications, is not known - hence the method is formal. The method is illustrated on a few model problems.