Homogenization techniques for thermoviscoelastic solids containing cracks

this paper mathematical techniques are developed for obtaining locally averaged (homogenized) constitutive equations for heterogeneous linear thermoviscoelastic solids. Homogenization principles will be developed for the cases wherein no internal boundaries are present, and also where internal boundaries in the form of sharp cracks are present, thus resulting in damage dependent macroscopic constitutive equations. The microthermomechanics problem will first be formulated, followed by the construction of the locally averaged equations resulting from the homogenization process. It will be shown that homogenized conservation laws and constitutive equations take the same form as do the local equations when locally linear thermoviscoelastic media are considered. However, the resulting homogenized constitutive equations will be nonlinear in the case wherein time dependent damage occurs. In addition, for materials of convolution type at the local scale, the homogenized equations will be shown to contain a term that depends on the time derivative of the strain localization tensor. Example problems will be discussed and the homogenized results will be given for these examples in order to demonstrate the technique. 0 1998 Elsevier Science Ltd. All rights reserved.