Neural networks meet hyperelasticity: A monotonic approach

TL;DR

The paper tackles the challenge of modeling parametric, rubber-like materials whose mechanical response depends on manufacturing conditions. It introduces a monotonic, hyperelastic PANN framework that enforces incompressibility and invariance while leveraging neural potentials that are convex/monotonic in isochoric invariants, yielding stable extrapolation and robust finite element performance. Across DM, DLP, Ecoflex, and DM-L datasets, the monotonic PANN often outperforms or matches more restrictive or unrestricted variants, especially under extrapolation and multiaxial loading, while enhancing numerical stability in FE simulations. The work demonstrates that monotonicity in invariant-based NN potentials can serve as a general, robust principle for constitutive modeling of highly nonlinear, parametrized materials, with practical implications for 3D-printed digital materials and related soft-matter applications.

Abstract

We apply physics-augmented neural network (PANN) constitutive models to experimental uniaxial tensile data of rubber-like materials whose behavior depends on manufacturing parameters. For this, we conduct experimental investigations on a 3D printed digital material at different mix ratios and consider several datasets from literature, including Ecoflex at different Shore hardness and a photocured 3D printing material at different grayscale values. We introduce a parametrized hyperelastic PANN model which can represent material behavior at different manufacturing parameters. The proposed model fulfills common mechanical conditions of hyperelasticity. In addition, the hyperelastic potential of the proposed model is monotonic in isotropic isochoric strain invariants of the right Cauchy-Green tensor. In incompressible hyperelasticity, this is a relaxed version of the ellipticity (or rank-one convexity) condition. Using this relaxed ellipticity condition, the PANN model has enough flexibility to be applicable to a wide range of materials while having enough structure for a stable extrapolation outside the calibration data. The monotonic PANN yields excellent results for all materials studied and can represent a wide range of largely varying qualitative and quantitative stress behavior. Although calibrated on uniaxial tensile data only, it leads to a stable numerical behavior of 3D finite element simulations. The findings of our work suggest that monotonicity could play a key role in the formulation of very general yet robust and stable constitutive models applicable to materials with highly nonlinear and parametrized behavior.

Figures (9)

• Figure 1: Illustration of the PANN constitutive models. The convex and monotonic PANN is denoted by $\psi_{\mathfrak{cm}}^{\text{PANN}}$, the monotonic PANN is denoted by $\psi_{\mathfrak{m}}^{\text{PANN}}$, and the unrestricted PANN is denoted by $\psi_{\mathfrak{u}}^{\text{PANN}}$.
• Figure 2: (a): Experimental investigations on a 3D printed digital material (DM) using soft Elastico polymer and stiff VeroWhite polymer as base materials. The DM material is manufactured for three different mix ratios. (b): Hyperparameter study for the PANN constitutive model. MSE for different number of nodes and hidden layers (HL). (c): Number of non-zero NN parameters for different number of nodes and hidden layers.
• Figure 3: Influence of monotonicity and convexity on the PANN model performance in uniaxial tension. Points denote the calibration data and lines denote the PANN models. (I): Interpolation of DM for different mix ratios.
• Figure 4: Influence of monotonicity and convexity on the PANN model performance in uniaxial tension and comparison with a Mooney-Rivlin model from literature zhang2024. Points denote the calibration data, squares denote the test data, and lines denote the PANN models. (II): Interpolation of DLP for different grayscale values.
• Figure 5: Influence of monotonicity and convexity on the PANN model performance in uniaxial tension. Points denote the calibration data, squares denote the test data, and lines denote the PANN models. (III): Extrapolation in the strain space for Exoflex with different base resins, (IV): Extrapolation in the parameter space for Ecoflex with different base resins.

Theorems & Definitions (5)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 3.1