|March 15, 2017
J. Eng. Mater. Technol
, (2017); doi: 10.1115/1.4036243
In this paper the necessary and sufficient conditions for fulfilling the thermodynamic consistency of computational homogenisation schemes in the framework of hierarchical multiscale theories are defined. The proposal is valid for arbitrary homogenisation based multiscale procedures, including continuum and discontinuum methods in either scale. It is demonstrated that the well-known Hill-Mandel variational criterion for homogenisation scheme is a necessary, but not a sufficient condition for the micro-macro thermodynamic consistency when dissipative material responses are involved at any scale. In this sense, the additional condition to be fulfilled regarding multiscale thermodynamic consistency 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. It is shown that the thermodynamic consistency enforces the homogenisation of the non-local terms of the finer scale's free energy density, however this does not lead to non-local gradient effects on the coarse scale. Then, the particular cases of local isothermal elastoplasticity and continuum damage are considered for the purpose of the proposed thermodynamically consistent approach for multiscale homogenisations.