  If gradient systems depend on a microstructure, we want to derive a
macroscopic gradient structure describing the effective behavior of the
microscopic effects. We introduce a notion of evolutionary Gamma-convergence
that relates the microscopic energy and the microscopic dissipation potential
with their macroscopic limits via Gamma-convergence. This new notion
generalizes the concept of EDP-convergence, which was introduced in
arXiv:1507.06322, and is called "relaxed EDP-convergence". Both notions are
based on De Giorgi's energy-dissipation principle, however the special
structure of the dissipation functional in terms of the primal and dual
dissipation potential is, in general, not preserved under Gamma-convergence. By
investigating the kinetic relation directly and using general forcings we still
derive a unique macroscopic dissipation potential.
  The wiggly-energy model of James et al serves as a prototypical example where
this nontrivial limit passage can be fully analyzed.
