Temporal Homogenization of Viscoelastic and Viscoplastic Solids Subjected to Locally Periodic Loading

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 initialboundary 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. 1.0 Introduction Mathematical homogenization method has been widely used for solving initial-boundary value problems with oscillatory coefficients. The validity of the asymptotic homogenization depends on the existence of distinct multiple length scales in the physical processes so that a small positive scaling parameter quantifying the ratio between the scales can be identified. In general, multiple length scales may exist in both space and time domains, although most of the recent research efforts have been focussing on the spatial homogenization (see, for instance, [12][14]). In contrast to the spatial scale separation, which is typically induced by spatial heterogeneities, the multiple temporal scales can be attributed to at least three sources (or their combinations): • the interaction of multiple physical processes Different physical processes, such as mechanical, thermal, diffusion, and chemical reaction, may evolve along different time frames. Interaction between multiple physical processes requires consideration of relevant time frames within a single reference time coordinate. An example problem fall-