  This paper concerns the study of history dependent phenomena in heterogeneous
materials in a two-scale setting where the material is specified at a fine
microscopic scale of heterogeneities that is much smaller than the coarse
macroscopic scale of application. We specifically study a polycrystalline
medium where each grain is governed by crystal plasticity while the solid is
subjected to macroscopic dynamic loads. The theory of homogenization allows us
to solve the macroscale problem directly with a constitutive relation that is
defined implicitly by the solution of the microscale problem. However, the
homogenization leads to a highly complex history dependence at the macroscale,
one that can be quite different from that at the microscale. In this paper, we
examine the use of machine-learning, and especially deep neural networks, to
harness data generated by repeatedly solving the finer scale model to: (i) gain
insights into the history dependence and the macroscopic internal variables
that govern the overall response; and (ii) to create a computationally
efficient surrogate of its solution operator, that can directly be used at the
coarser scale with no further modeling. We do so by introducing a recurrent
neural operator (RNO), and show that: (i) the architecture and the learned
internal variables can provide insight into the physics of the macroscopic
problem; and (ii) that the RNO can provide multiscale, specifically FE2,
accuracy at a cost comparable to a conventional empirical constitutive
relation.
