Architecture and training constraints for deep neural network constitutive models of elasto-plastic metamaterials

Nonlinear elastic metamaterials (NLEMs) leverage complex subwavelength geometries to achieve superior energy dissipation and redistribution. The resulting geometric scales of interest are often unfeasible for direct numerical simulation at full resolution, especially for NLEMs undergoing complex deformation behaviors such as history- and rate-dependent plasticity, contact, and buckling. Therefore, low-order effective medium models based on mass-spring lattices have been proposed for the simulation of nonlinear wave propagation in NLEMs Wallen et al., In Press, https://doi.org/10.1016/j.jmps.2025.106276. However, the empirical constitutive relations of the effective discrete-element unit cells require significant effort to obtain. In this work, using the results of fine-scale, large-deformation finite-element simulations, deep neural networks are trained to represent the relationships between loading and elasto-plastic deformation of NLEM unit cells. The trained neural networks are then incorporated into the differential-algebraic equations of motion for simulation of wave propagation. The present study considers various steps in the implementation of the machine-learning model, such as sampling of training data, constraints on network architecture to ensure physical consistency, and application of automatic differentiation, to maximize the accuracy of the approach.