Identifying Constitutive Parameters for Complex Hyperelastic Materials using Physics-Informed Neural Networks

TL;DR

This work addresses robustly identifying constitutive parameters of complex hyperelastic materials under large deformation in plane stress by leveraging Physics-Informed Neural Networks (PINNs) trained on multi-modal data, including full-field DIC displacements and loading history. The method integrates PDE residuals, incompressibility, boundary data, and data-driven measurements, learning both neural surrogates of displacement fields and the AB model parameters $\mu$ and $\lambda_m$ from synthetic experiments generated with FEM. Key findings show accurate recovery of AB parameters with errors below 5% in noisy conditions (up to 5% experimental noise) and robust performance for complex geometries where traditional methods fail, with convergence aided by DIC data and domain-integral loading. This framework extends PINN-based modulus identification to complex solids and can be adapted to other hyperelastic laws and rate-dependent models, offering a practical route for material characterization in bio-inspired and metamaterial applications.

Abstract

Identifying constitutive parameters in engineering and biological materials, particularly those with intricate geometries and mechanical behaviors, remains a longstanding challenge. The recent advent of Physics-Informed Neural Networks (PINNs) offers promising solutions, but current frameworks are often limited to basic constitutive laws and encounter practical constraints when combined with experimental data. In this paper, we introduce a robust PINN-based framework designed to identify material parameters for soft materials, specifically those exhibiting complex constitutive behaviors, under large deformation in plane stress conditions. Distinctively, our model emphasizes training PINNs with multi-modal synthetic experimental datasets consisting of full-field deformation and loading history, ensuring algorithm robustness even with noisy data. Our results reveal that the PINNs framework can accurately identify constitutive parameters of the incompressible Arruda-Boyce model for samples with intricate geometries, maintaining an error below 5%, even with an experimental noise level of 5%. We believe our framework provides a robust modulus identification approach for complex solids, especially for those with geometrical and constitutive complexity.

Figures (6)

• Figure 1: (a) The schematics of experimental data collection for PINNs. A sample with internal geometry defects is subjected to displacement BCs. A camera is used to record the DIC dataset and a load cell is used to record the loading history. Then, PINNs incorporate these two experimental datasets during training to identify the constitutive parameters. (b) The architecture of PINNs for samples under the plane stress condition.
• Figure 2: Forward problem of a rectangular sample with a central hole under the plane stress condition: (a) Schematics of the sample geometry and dimensions; (b) Loss as a function of training epochs. The solid line represents the total loss, the dotted line represents the loss of PDE, and the dashed line represents the loss of BCs; (c) Visual outlines of deformed configurations calculated from PINNs (solid lines) and FEM (dashed lines) under five different stretch ratios; (d) Loading as a function of stretch obtained from the PINNs (filled circles) and FEM (solid line); (e) The normalized Cauchy stress contour in the loading direction $\sigma_{11}$ of PINNs and FEM, and their logarithmic relative error. The contour is plotted in the undeformed configuration.
• Figure 3: Constitutive parameter identification for a rectangular sample with a central hole under the plane stress condition with different ground truths. (a) Convergence of $\mu$ over 1M training epochs;(b) Convergence of $\lambda_m$ over 1M training epochs. The dashed line is the predictions from PINNs, and the solid line is the ground truth.
• Figure 4: Constitutive parameter identification for a rectangular sample with four circular inhomogeneities centered at locations: $(-0.4,0.3)$, $(-0.2,-0.1)$, $(0.1,0.2)$, and $(0.3,-0.3)$. (a) Schematics of the sample geometry and dimension; (b) Loss as a function of training epochs. (c) Visual outlines of deformed configurations calculated from PINNs (solid lines) and FEM (dashed lines) under 5 different stretch ratios; (d-e) Constitutive parameter identification of $\mu$ and $\lambda_m$ over 800k training epochs. The dashed line is the predictions from PINNs and the solid line is the ground truth.
• Figure 5: Influence of the speckle density and noise on the predictions of PINNs. (a-i, ii) Speckle density effect. (b-i, ii) Speckle noise level effect.