Learning Memory and Material Dependent Constitutive Laws

TL;DR

This work addresses learning memory- and microstructure-dependent constitutive laws in a two-scale homogenization setting by proposing a recurrent Fourier neural operator (FNM–RNO) that maps strain histories and microstructure inputs to homogenized stress. The approach is grounded in Kelvin–Voigt viscoelasticity, with a universal approximation theorem proven for the 1D KV cell problem, and Lipschitz continuity established to guarantee stable microstructure dependence. The method is validated on both linear viscoelastic and nonlinear elasto-viscoplastic materials, showing high generalization across piecewise-constant and continuous microstructures and robustness to discretization, while enabling deployment in macroscale simulations without retraining. The results indicate a scalable, data-driven pathway to compute memory- and microstructure-dependent constitutive laws, potentially accelerating multiscale material simulations in engineering applications.

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.

Figures (10)

• Figure 1: Visualization of samples from the two datasets: piecewise-constant material (PC) and high-memory continuous material (HMC); see \ref{['subsec:data_set']}. Each dataset consists of material samples $(E^{(j)}, \nu^{(j)})$, averaged strain trajectory samples $\overline{\epsilon}^{(j)}$, and the averaged stress trajectory samples $\overline{\sigma}^{(j)}$. We visualize the averaged stress response with (solid lines) and without (dotted lines) memory effects.
• Figure 1: The distributions of the relative $L^{\infty}$ error on 2,500 testing samples from the PC dataset (top) and the HMC dataset (bottom). We visualize the errors in the FNM--RNO predictions where FNM--RNOs (i) are trained with or without the penalty term in \ref{['eq:loss_function']}, and (ii) have a varying number of internal variables. We also visualize the distribution of error given by the linear stress response without memory effects, where the response function is obtained using \ref{['eqn:kernelform']} with $K(t)=0$ for all $t\in[0,1]$.
• Figure 2: The distributions of the relative $L^2$ error on 2,500 testing samples from the PC dataset (left) and the HMC dataset (right). We visualize the errors in FNM--RNOs predictions where the trained FNM--RNOs have a varying number of internal variables. We also visualize the distribution of error given by the linear stress response without memory effects, where the response function is obtained using \ref{['eqn:kernelform']} with $K\equiv0$.
• Figure 2: Visualization of FNM--RNO predictions at the sample from the PC testing dataset with the largest relative $L^{\infty}$ testing error (5 internal variables and without penalty in \ref{['fig:l_infity_error_distribution']}). We compare the predictions by FNM--RNOs trained using the loss function in \ref{['eq:loss_function']} without the penalty term and with the penalty term. For the averaged stress and internal variable predictions, we show their full trajectories in $t\in[0,1]$ and enlarged views in $t\in[0, 0.01]$.
• Figure 3: Visualization of testing samples and FNM--RNO predictions (averaged stress $\overline{\sigma}$ and internal variables $\xi$) with the largest and median relative $L^2$ testing error. We consider the trained FNM--RNO with 5 internal variables. We visualize the averaged stress response with (solid lines) and without (dotted lines) memory effects along with the FNM--RNO prediction (dash-dot lines).

Theorems & Definitions (36)

• Remark 2.1
• Definition 2.2: FNM--RNO Architecture
• Remark 2.3
• Definition 2.4: Fourier Neural Mapping (FNM)
• Remark 2.5
• Lemma 3.1: Lemma 1.1 in bhattacharya2023learning
• Remark 3.2
• Proposition 3.3: Theorem 3.6 in bhattacharya2023learning
• Lemma 3.4
• Proof 1