The macroscopic thermomechanical behaviour of heterogeneous media may depend strongly on their microstructure. For example, the macroscopic behavior of polycrystals depends on the size and orientation of underlying single crystals. Another situation is the complex rheological behavior of vesicle suspensions. In such cases, the constitutive behavior of the microstructure needs to be considered when modelling the bulk material response.
In this talk, I will describe a homogenization method to connect the microscopic and macroscopic scales based on extensive physical quantities assuming that both scales can be modeled using continuum mechanics. The method extends to the continuum-on-continuum setting the celebrated approach of Irving & Kirkwood, which forms the basis for upscaling atomistic variables of classical statistical mechanics, such as position, momentum and interatomic forces to continuum variables, such as stress and heat flux. An application of this extended method will be explored within the context of finite element-based homogenization of solids in quasi-static conditions. Finally, I will discuss the microscale inertial or dynamical effects on the macroscale behavior for a 1-dimensional elastic-layered medium.