Concurrent enforcement of polyconvexity and true-stress-true-strain monotonicity in incompressible isotropic hyperelasticity: application to neural network constitutive models

TL;DR

This paper establishes a theoretical link between polyconvexity and stress-strain monotonicity in hyperelastic materials, and applies these principles to design and evaluate physics-augmented neural network models for material behavior.

Contribution

It demonstrates that polyconvexity ensures true-stress-true-strain monotonicity and informs neural network architecture for hyperelasticity modeling, with experimental calibration and analysis.

Findings

Polyconvexity implies stress-strain monotonicity for many hyperelastic models.

Different neural network parametrizations exhibit varying approximation and extrapolation capabilities.

Neural network models show significant differences in predictive power outside calibration data.

Abstract

The design of physics-augmented neural networks (PANNs) for the purposes of constitutive modeling has received considerable attention as of late for a variety of material behaviors. Here, we revisit the classical framework of isotropic incompressible hyperelasticity in light of recent advances in the study of constitutive inequalities. We show that polyconvexity implies true-stress-true-strain monotonicity for a large class of incompressible strain-energy functions. The resulting elastic law obeys the physically reasonable Legendre-Hadamard (or ellipticity) condition as well as the notion of increasing stress with increasing strain. These results then inform the architecture of four distinct PANNs which are subsequently calibrated to three different sets of experimental data each. We show that different PANN parametrizations - satisfying the same constitutive constraints a priori - have varying approximation power for the description of material behavior. Moreover, even when distinct parametrizations perform comparatively well within the calibration regime, they show pronounced differences in extrapolation. This observation motivates a critical discussion about the predictive power of PANNs which also has implications for the modeling of more complex material behavior by virtue of neural networks.