Polyconvex anisotropic hyperelasticity with neural networks

TL;DR

This paper introduces two neural network-based hyperelastic models that ensure polyconvexity and material stability for anisotropic finite deformations, demonstrating high accuracy and generalization on complex metamaterial data.

Contribution

The paper presents novel neural network models that guarantee polyconvexity and anisotropy, improving stability and predictive capability in finite deformation modeling.

Findings

The deformation gradient-based model predicts material behavior accurately.

Invariant-based model shows limitations in certain deformation modes.

Both models perform well with transversely isotropic data.

Abstract

In the present work, two machine learning based constitutive models for finite deformations are proposed. Using input convex neural networks, the models are hyperelastic, anisotropic and fulfill the polyconvexity condition, which implies ellipticity and thus ensures material stability. The first constitutive model is based on a set of polyconvex, anisotropic and objective invariants. The second approach is formulated in terms of the deformation gradient, its cofactor and determinant, uses group symmetrization to fulfill the material symmetry condition, and data augmentation to fulfill objectivity approximately. The extension of the dataset for the data augmentation approach is based on mechanical considerations and does not require additional experimental or simulation data. The models are calibrated with highly challenging simulation data of cubic lattice metamaterials, including finite deformations and lattice instabilities. A moderate amount of calibration data is used, based on deformations which are commonly applied in experimental investigations. While the invariant-based model shows drawbacks for several deformation modes, the model based on the deformation gradient alone is able to reproduce and predict the effective material behavior very well and exhibits excellent generalization capabilities. In addition, the models are calibrated with transversely isotropic data, generated with an analytical polyconvex potential. For this case, both models show excellent results, demonstrating the straightforward applicability of the polyconvex neural network constitutive models to other symmetry groups.