Precise, efficient and flexible modeling of crystallizing elastomers based on physics-augmented neural networks

TL;DR

The paper tackles accurate, efficient modeling of strain-induced crystallization (SIC) in natural rubber by formulating a two-potential generalized standard material framework and implementing it within a physics-augmented neural network (PANN) to learn the isochoric free energy and the dissipation potential. The approach enforces thermodynamic consistency, objectivity, and isotropy, and bounds crystallinity to $\omega \in [0,1]$ via KKT conditions in an incremental variational FE setting. It demonstrates calibration and validation on unfilled and filled NR using stress–deformation–crystallinity data, and shows that PANNs can match or exceed benchmark models (e.g., MAPC) while offering computational efficiency and robust extrapolation, including cases with auto-discovered pseudo-crystallinity variables when crystallinity data are unavailable. The work provides a versatile, thermodynamically sound framework for SIC in elastomers, enabling accurate field predictions in notched specimens and flexible extension to thermo-mechanical coupling or fracture-phase-field analyses. Such a framework facilitates design and analysis of NR components under complex loading, with potential for improved predictive capabilities and computational performance.

Abstract

We propose a precise and efficient physics-augmented neural network (PANN) to model strain-induced crystallization in natural rubber (NR). The approach is based on a two potential framework, similar to the concept of generalized standard materials (GSMs). To describe the material behavior, neural network-based free energy and dissipation potentials are employed. The evolution of crystallinity is derived from the two potentials and resembles a classical GSM-type equation. Two additional Lagrange multipliers together with the corresponding Karush-Kuhn-Tucker conditions are introduced to ensure boundedness of the crystallinity, such that it can be interpreted as a variable of concentration type. The neural network-based potentials ensure all physically desirable properties by construction. Most importantly, objectivity, material symmetry, and thermodynamic consistency are automatically fulfilled. In addition, an alternative derivation of the governing model equations in time-discrete form is presented based on an incremental variational framework, which also serves as the basis for a finite element implementation. We demonstrate the predictive capability of the PANN using three different experimental data sets from literature, considering both stress and crystallinity evolution at material point level as well as the corresponding field distributions in a notched specimen. Moreover, we demonstrate that our model can be flexibly employed for both unfilled and filled NR.

Figures (11)

• Figure 1: Schematic of the proposed PANN model for SIC. The DOC $\prescript{n}{}{\omega}$ of the current increment follows from the compliance of $\prescript{n}{}{\tau}$ and $\prescript{n}{}{\hat{\tau}}$ with the evolution equations \ref{['eq:biot']}--\ref{['KKT1']}.
• Figure 2: Training scheme of the PANN based on the constrained optimization problem defined in Eq. \ref{['eq:training_optimization']}. At each time step, the current DOC is determined by iteratively solving the unconstrained evolution equation \ref{['eq:biot']}, i.e., under the presumption $\mu_0 = \mu_1 = 0$. Consequently, evaluating the stress at any increment of the training sequence requires the evaluation of all preceding time steps. The illustration is based on rosenkranz2024akalina2025a.
• Figure 3: PANNs trained on the experimental stress and crystallinity data for uniaxial tension of unfilled NR by rault2006a: (a) model version with $\overline{I}_1$ and $\overline{I}_2$ as deformation input and (b) only considering $\overline{I}_1$.
• Figure 4: MAPC model for SIC considered as a benchmark rastak2018. The model parameters are taken from the original contribution, calibrated on the exact considered set of experimental data by rault2006a. Note that we use the symbols $\lambda, \omega$ to denote stretch and crystallinity on the continuum scale. In rastak2018, $\overline{\lambda}, \overline{\omega}$ are used for this purpose instead.
• Figure 5: Finite element mesh used for the SENT specimen with close-up of the notch region. The dimensions of the considered notch are $20m m \times 0.2m m$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4