  Given a set of inelastic material models, a microstructure, a macroscopic
structural geometry, and a set of boundary conditions, one can in principle
always solve the governing equations to determine the system's mechanical
response. However, for large systems this procedure can quickly become
computationally overwhelming, especially in three-dimensions when the
microstructure is locally complex. In such settings multi-scale modeling offers
a route to a more efficient model by holding out the promise of a framework
with fewer degrees of freedom, which at the same time faithfully represents, up
to a certain scale, the behavior of the system.
  In this paper, we present a methodology that produces such models for
inelastic systems upon the basis of a variational scheme. The essence of the
scheme is the construction of a variational statement for the free energy as
well as the dissipation potential for a coarse scale model in terms of the free
energy and dissipation functions of the fine scale model. From the coarse scale
energy and dissipation we can then generate coarse scale material models that
are computationally far more efficient than either directly solving the fine
scale model or by resorting to FE-square type modeling. Moreover, the coarse
scale model preserves the essential mathematical structure of the fine scale
model.
  An essential feature for such schemes is the proper definition of the coarse
scale inelastic variables. By way of concrete examples, we illustrate the
needed steps to generate successful models via application to problems in
classical plasticity, included are comparisons to direct numerical simulations
of the microstructure to illustrate the accuracy of the proposed methodology.
