Title: Towards Trustworthy Geometric Deep Learning for Elastoplasticity
Description: Recent advances in machine learning have unlocked new potential for innovation in engineering science. Neural networks are used as universal function approximators that harness high-dimensional data with excellent learning capacity. While this is an opportunity to accelerate computational mechanics research, application in constitutive modeling is not trivial. Machine learning material response predictions without enforcing physical constraints may lack interpretability and could be detrimental to high-risk engineering applications. This dissertation presents a meta-modeling framework for automating the discovery of elastoplasticity models across material scales with emphasis on establishing interpretable and, hence, trustworthy machine learning modeling tools. Our objective is to introduce a workflow that leverages computational mechanics domain expertise to enforce / post hoc validate physical properties of the data-driven constitutive laws. Firstly, we introduce a deep learning framework designed to train and validate neural networks to predict the hyperelastic response of materials.

We adopt the Sobolev training method and adapt it for mechanics modeling to gain control over the higher-order derivatives of the learned functions. We generate machine learning models that are thermodynamically consistent, interpretable, and demonstrate enhanced learning capacity. The Sobolev training framework is shown through numerical experiments on different material data sets (e.g. β-HMX crystal, polycrystals, soil) to generate hyperelastic energy functionals that predict the elastic energy, stress, and stiffness measures more accurately than the classical training methods that minimize L2 norms. To model path-dependent phenomena, we depart from the common approach to lump the elastic and plastic response into one black-box neural network prediction. We decompose the elastoplastic behavior into its interpretable theoretical components by training separately a stored elastic energy function, a yield surface, and a plastic flow that evolve based on a set of deep neural network predictions.

We interpret the yield function as a level set and control its evolutionas the neural network approximated solutions of a Hamilton-Jacobi equation that governs the hardening/softening mechanism. Our framework may recover any classical literature yield functions and hardening rules as well as discover new mechanisms that are either unbeknownst or difficult to express with mathematical expressions. Through numerical experiments on a 3D FFT-generated polycrystal material response database, we demonstrate that our novel approach provides more robust and accurate forward predictions of cyclic stress paths than black-box deep neural network models. We demonstrate the framework's capacity to readily extend to more complex plasticity phenomena, such as pressure sensitivity, rate-dependence, and anisotropy. Finally, we integrate geometric deep learning and Sobolev training to generate constitutive models for the homogenized responses of anisotropic microstructures (e.g. polycrystals, granular materials).

Commonly used hand-crafted homogenized microstructural descriptors (e.g. porosity or the averaged orientation of constitutes) may not adequately capture the topological structures of a material. This is overcome by introducing weighted graphs as new high-dimensional descriptors that represent topological information, such as the connectivity of anisotropic grains in an assemble. Through graph convolutional deep neural networks and graph embedding techniques, our neural networks extract low-dimensional features from the weighted graphs and, subsequently, learn the influence of these low-dimensional features on the resultant stored elastic energy functionals and plasticity models.