Sobolev training of thermodynamic-informed neural networks for smoothed elasto-plasticity models with level set hardening

We introduce a deep learning framework designed to train smoothed elastoplasticity models with interpretable components, such as a smoothed stored elastic energy function, a yield surface, and a plastic flow that are evolved based on a set of deep neural network predictions. By recasting the yield function as an evolving level set, we introduce a machine learning approach to predict the solutions of the Hamilton-Jacobi equation that governs the hardening mechanism. This machine learning hardening law may recover classical hardening models and discover new mechanisms that are otherwise very difficult to anticipate and hand-craft. This treatment enables us to use supervised machine learning to generate models that are thermodynamically consistent, interpretable, but also exhibit excellent learning capacity. Using a 3D FFT solver to create a polycrystal database, numerical experiments are conducted and the implementations of each component of the models are individually verified. Our numerical experiments reveal that this new approach provides more robust and accurate forward predictions of cyclic stress paths than these obtained from black-box deep neural network models such as a recurrent GRU neural network, a 1D convolutional neural network, and a multi-step feedforward model.