Thermodynamic Framework of Multiscale Homogenization Schemes for Dissipative Materials

Abstract

The prediction of failure processes in composite, heterogeneous materials require multiscale analysis to account for the complex mechanisms and features taking place. Between the different multiscale schemes the more commonly used are those based on homogenization procedures, due to their versatility. In this work a thermodynamically consistent homogenisation based multiscale approach is formulated for modelling thermo-plastic materials. The proposal is valid for arbitrary multiscale procedures, including local or nonlocal methods, and continuum or discontinuum methods in either scale. The necessary and sufficient conditions for fulfilling the thermodynamic consistency are defined. It is demonstrated that the Hill-Mandel variational criterion for homogenization scheme is a necessary, but not a sufficient condition when dissipative material responses are involved at any scale. On this point, the additional condition that needs to be fulfilled is established. The general case of temperature-dependent, higher order elastoplasticity is considered as theoretical framework to account for the material dissipation at micro and macro scales of observation. Additionally, it is shown that the thermodynamic consistency enforces the homogenization of the nonlocal terms of the micro scale’s free energy density; however, this does not necessarily lead to nonlocal effects on the macro scale. Finally, the particular cases of local isothermal elastoplasticity and continuum damage are considered for the purpose of the proposed approach for multiscale homogenizations.