Time-incremental schemes for nonlinear viscoelasticity minimize, in each step of size $\tau$, the elastic energy plus $\frac{1}{2\tau}D(\nabla u_{\mathrm{old}},\nabla u)^2$ for a dissipation distance $D$. Frame indifference forces $D(F,\cdot)$ to vanish on the entire orbit $\mathrm{SO}(n)F$. The dissipation is therefore never convex, and genuine distances such as the additive choice $D(F,G)=|F^\top F-G^\top G|$ are not even rank-one convex. Since this term dominates for small $\tau$, one would expect microstructure in every step, as in other settings with nonconvex dissipation such as single-crystal plasticity.
So, where is this microstructure? We linearize the incremental problem by $\Gamma$-convergence as $\tau\to0$, around an arbitrary deformed configuration $y$ that need not be an equilibrium. The limit is convex: the first variation $\int_\Omega \partial_F W(\nabla y):\nabla z\,dx$ of the elastic energy plus a quadratic dissipation $\frac12\int_\Omega \mathbb{A}(\nabla y)\nabla z:\nabla z\,dx$ that acts only on linearized strains. No microstructure appears: small increments see only the quadratic expansion of $D^2$ near the orbit, and geometric rigidity prevents mixing of different rotations.
Joint work with Martin Jesenko and Martin Kružík.