Learning Memory and Material Dependent Constitutive Laws

TL;DR

This paper introduces a neural operator framework for learning complex, memory- and microstructure-dependent constitutive laws in heterogeneous materials, enabling accurate macroscale simulations without microstructure resolution.

Contribution

It develops a novel neural operator-based approach for data-driven learning of microstructure-dependent constitutive laws, supported by theoretical foundations and universal approximation guarantees.

Findings

Accurately learns viscoelastic and elasto-viscoplastic models from data.

Successfully applies learned models in macroscale simulations across different microstructures.

Provides theoretical analysis and universal approximation proof for the proposed framework.

Abstract

We propose and study a neural operator framework for learning memory- and material microstructure-dependent constitutive laws for heterogeneous materials. We work in the two-scale setting where homogenization theory provides a systematic approach to deriving macroscale constitutive laws, obviating the need to resolve complex microstructure repeatedly. However, the unit cell problems defining these constitutive models are typically not amenable to explicit evaluation. It is therefore of interest to learn constitutive models from data generated by the unit cell problem. Our proposed framework models homogenized constitutive laws with both memory- and microstructure-dependence through the use of Markovian recurrent and Fourier neural operators. The homogenization problem for Kelvin-Voigt viscoelastic materials is studied to provide firm theoretical foundations for our model. The theoretical properties of the cell problem in this Kelvin-Voigt setting motivate the proposed learning framework; and are also used to prove a universal approximation theorem for the learned macroscale constitutive model. Numerical experiments show that the proposed learning framework accurately learns memory- and microstructure-dependent viscoelastic and elasto-viscoplastic constitutive models, beyond the setting of the theory. Furthermore, we show that the learned constitutive models can be successfully deployed in macroscale simulation of material deformation for different microstructures without retraining.