Material responses at micro- and macro-scales

Material properties at micro- and macro-scales used in micromechanics and continuum mechanics are random and deterministic and so are the corresponding material responses. We view material properties and responses in continuum mechanics as approximations of those in micromechanics. Our premise is that solutions of problems using material properties at various scales must agree in some sense, e.g., continuum mechanics solutions should match on average micromechanics solutions. Continuum solutions with this property are said to be consistent. Theoretical arguments and numerical examples are presented to demonstrate that the continuum solutions may or may not be consistent and may miss essential features of material response depending on the problem and quantity of interest. The examples include beams with random stiffness and one- and two-dimensional specimens with random conductivity. (C) 2015 Elsevier B.V. All rights reserved.