Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading

Abstract

As a direct extension of the asymptotic spatial homogenization method we develop a temporal homogenization scheme for a class of homogeneous solids with an intrinsic time scale significantly longer than a period of prescribed loading. Two rate-dependent material models, the Maxwell viscoelastic model and the power-law viscoplastic model, are studied as an illustrative examples. Double scale asymptotic analysis in time domain is utilized to obtain a sequence of initial-boundary value problems with various orders of temporal scaling parameter. It is shown that various order initial-boundary value problems can be further decomposed into: (i) the global initial-boundary value problem with smooth loading for the entire loading history, and (ii) the local initial-boundary value problem with the remaining (oscillatory) portion of loading for a single load period. Large time increments can be used for integrating the global problem due to smooth loading, whereas the integration of the local initial-boundary value problem requires a significantly smaller time step, but only locally in a single load period. The present temporal homogenization approach has been found to be in good agreement with a closed-form analytical solution for one-dimensional case and with a numerical solution in multidimensional case obtained by using a sufficiently small time step required to resolve the load oscillations.