What question did this study set out to answer?

This work aims to enhance simulations of nonlinear elastic metamaterials using deep neural networks for accurate elasto-plastic deformation modeling.

May 14, 2026

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

Key Points

This work aims to enhance simulations of nonlinear elastic metamaterials using deep neural networks for accurate elasto-plastic deformation modeling.
Trained deep neural networks on data from large-deformation finite-element simulations of nonlinear elastic metamaterials.
Incorporated neural networks into differential-algebraic equations for wave propagation simulations.
Considered architecture constraints and automatic differentiation for improved model accuracy.
Demonstrated neural networks effectively represent loading and deformation relationships in metamaterial unit cells.
Enhanced accuracy of wave propagation simulations when using machine-learning models compared to traditional methods.

Abstract

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.

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