Computational Framework for Multiphysics Problems with Multiple Spatial and Temporal Scales

A systematic approach for analyzing multiple physical processes interacting at multiple spatial and temporal scales is developed. The proposed computational framework is applied to the coupled thermo-viscoelastic composites with microscopically periodic mechanical and thermal properties. A rapidly varying spatial and temporal scales are introduced to capture the effects of spatial and temporal fluctuations induced by spatial heterogeneities at diverse time scales. The initial-boundary value problem on the macroscale is derived by using the double scale asymptotic analysis in space and time. It is shown that an extra history-dependent long-term memory term introduced by the homogenization process in space and time can be obtained by solving a first order initial value problem. This is in contrast to the long-term memory term obtained by the classical spatial homogenization, which requires solutions of the initial-boundary value problem in the unit cell domain. The validity limits of the proposed spatial-temporal homogenized solution are established. Numerical example shows a good agreement between the proposed model and the reference solution obtained by using a finite element mesh with element size comparable to that of material heterogeneity. 1.0 Introduction The primary objective of the manuscript is to develop a systematic approach for analyzing multiple physical processes interacting at multiple spatial and temporal scales. The interacting physical processes may include mechanical, thermal, diffusion, chemical and electromagnetic fields. Most often these phenomena are treated as being uncoupled; hence, few separate analyses of the same system are typically performed for the complete prediction of the response. It is, however, understood that such treatments should be regarded as first-order approximations to the real complex interactions. The coupling of mechanical, thermal, diffusion, chemical and electromagnetic fields (stress/strain, temperature, concentration, current) occurs through diverse phenomena, some of which are depicted in the interaction matrix shown in Table 1. The interaction matrix is “non-symmetric,” with cell (i, j) representing the phenomenon induced by the process corresponding to field i and which influences field j. For instance, cell (1, 2) represents heating due to plastic deformation, while cell (2, 1) corresponds to thermal expansion and thermal stresses. A fully coupled analysis would consider all processes shown in the matrix, while fully uncoupled approach would only consider the diagonal entries. A one-way (or