FUsion-based ConstitutivE model (FuCe): Towards model-data augmentation in constitutive modelling

TL;DR

FuCe addresses learning constitutive behaviour from sparse force–displacement data by fusing a known phenomenological hyperelastic model with an ICNN-based correction that enforces physical constraints. The method yields augmented strain energy $\psi(\mathbf{F}) = \psi_{\text{known}}(\mathbf{F}) + \psi_{NN}(\mathbf{F};\boldsymbol{\theta})$ and uses MC dropout for uncertainty quantification. It is validated on isotropic models (Isihara and Arruda–Boyce) and a single-fiber anisotropic case, across six stress states, and demonstrated in three FEM geometries, with strong extrapolation performance from limited noisy data. The work highlights a practical, interpretable, and uncertainty-aware path to robust constitutive modelling in engineering simulations.

Abstract

Constitutive modelling is crucial for engineering design and simulations to accurately describe material behavior. However, traditional phenomenological models often struggle to capture the complexities of real materials under varying stress conditions due to their fixed forms and limited parameters. While recent advances in deep learning have addressed some limitations of classical models, purely data-driven methods tend to require large datasets, lack interpretability, and struggle to generalize beyond their training data. To tackle these issues, we introduce "Fusion-based Constitutive model (FuCe): Towards model-data augmentation in constitutive modelling". This approach combines established phenomenological models with an ICNN architecture, designed to train on the limited and noisy force-displacement data typically available in practical applications. The hybrid model inherently adheres to necessary constitutive conditions. During inference, Monte Carlo dropout is employed to generate Bayesian predictions, providing mean values and confidence intervals that quantify uncertainty. We demonstrate the model's effectiveness by learning two isotropic constitutive models and one anisotropic model with a single fibre direction, across six different stress states. The framework's applicability is also showcased in finite element simulations across three geometries of varying complexities. Our results highlight the framework's superior extrapolation capabilities, even when trained on limited and noisy data, delivering accurate and physically meaningful predictions across all numerical examples.

Figures (7)

• Figure 1: Training Setup: A square plate with a hole in the bottom-left corner is subjected to asymmetric biaxial tension under displacement control. The noisy data collected from full-field displacements and reaction forces is used to train the proposed constitutive model. This setup is the same as thakolkaran2022nn so as to illustrate the advantage of model-physics fusion.
• Figure 2: The overall framework of the FuCe. Note that for training the proposed approach, differentiable physics simulator is needed that allows backpropagation through the numerical solver.
• Figure 3: Comparison of predicted outputs: (a) Strain energy density $\psi(F(\gamma))$ and (b) First Piola–Kirchhoff stress $\bm{P}(\bm{F}(\gamma))$, produced by the proposed constitutive model, against a data-driven model without fusion (NN-Euclid) and a known phenomenological model, relative to the ground truth across six distinct stress states. The Neo-Hookean model is utilized as the known phenomenological model, along with the known fibre direction $\alpha$, while the Isihara model serves as the ground truth.
• Figure 4: Comparison of predicted outputs: (a) Strain energy density $\psi(F(\gamma))$ and (b) First Piola–Kirchhoff stress $\bm{P}(\bm{F}(\gamma))$, produced by the proposed constitutive model, against a data-driven model without fusion (NN-Euclid) and a known phenomenological model, relative to the ground truth across six distinct stress states. The Neo-Hookean model is utilized as the known phenomenological model, along with the known fibre direction $\alpha$, while the Arruda-Boyce model serves as the ground truth.
• Figure 5: Comparison of predicted outputs: (a) Strain energy density $\psi(F(\gamma))$ and (b) First Piola–Kirchhoff stress $\bm{P}(\bm{F}(\gamma))$, produced by the proposed constitutive model, against a data-driven model without fusion (NN-Euclid) and a known phenomenological model, relative to the ground truth across six distinct stress states. The Neo-Hookean model is utilized as the known phenomenological model, along with the known fibre direction $\alpha$, while the anisotropic model with a single fibre family aligned at $\alpha = 45^\circ$ serves as the ground truth.