A physics-informed neural network for modeling fracture without gradient damage: formulation, application, and assessment

TL;DR

This work introduces a physics-informed neural network framework that models fracture in elastomers under large deformation without gradient damage, avoiding the numerical complexities of gradient-regularized FEM. The PINN uses a mixed formulation to approximate displacement, stress, and undamaged energy, with a damage mechanism controlled by a distortional-tensile energy split and irreversibility; training relies on collocation-based physics losses rather than data. Validation against FEM with gradient damage across multiple defect configurations shows that PINN crack paths are robust to collocation distribution and closely track baseline solutions, with incremental training offering higher accuracy than a single-step approach. The study also assesses hyperparameters, collocation strategies, pre-training, and normalization, providing practical guidance for applying PINNs to broader fracture problems and motivating extensions to other materials and damage models.

Abstract

Accurate computational modeling of damage and fracture remains a central challenge in solid mechanics. The finite element method (FEM) is widely used for numerical modeling of fracture problems; however, classical damage models without gradient regularization yield mesh-dependent and usually inaccurate predictions. The use of gradient damage with FEM improves numerical robustness but introduces significant mathematical and numerical implementation complexities. Physics-informed neural networks (PINNs) can encode the governing partial differential equations, boundary conditions, and constitutive models into the loss functions, offering a new method for fracture modeling. Prior applications of PINNs have been limited to small-strain problems and have incorporated gradient damage formulation without a critical evaluation of its necessity. Since PINNs in their basic form are meshless, this work presents a PINN framework for modeling fracture in elastomers undergoing large deformation without the gradient damage formulation. The PINN implementation here does not require training data and utilizes the collocation method to formulate physics-informed loss functions. We have validated the PINN's predictions for various defect configurations using benchmark solutions obtained from FEM with gradient damage formulation. The crack paths obtained using the PINN are approximately insensitive to the collocation point distribution. This study offers new insights into the feasibility of using PINNs without gradient damage and suggests a simplified and efficient computational modeling strategy for fracture problems. The PINN's performance has been evaluated through systematic variations in key neural network parameters to provide an assessment and guidance for future applications. The results motivate the extension of PINN-based approaches to a broader class of materials and damage models in mechanics.

Figures (20)

• Figure 1: The architecture of the PINN for large deformation fracture modeling. The formulation is based on temporal discretization into $N_{t}$ time steps. The loss function formulation and training process for a time instant $t^{j}$, $j \in [1, N_{t}+1]$ is summarized. Mechanics-related information of boundary conditions ($BCu$, $BCt$), governing equations ($BLM$, $BAM$) and constitutive relations ($S$, UFE) is encoded in the total loss function $\mathcal{L}_{Total}$. $BLM$ and $BAM$ denote the individual loss functions corresponding to the balance of linear momentum and angular momentum, respectively. $BCu$ and $BCt$ are the individual loss functions corresponding to displacement and traction boundary conditions, respectively. $S$ and UFE are the individual loss functions for stress and undamaged free energy density constitutive relations, respectively.
• Figure 1: Asymmetric double-edge notched plate: effect of loss weights. For PINN (one-time step) application, the solution is evaluated at $t=0.45$ s with 100,000 epochs training. For the baseline weight values $\alpha_{C}=\alpha_{BC}=\alpha_{BLM}=1$, $\mathcal{L}_{C}$ is significantly higher than $\mathcal{L}_{BC}$ and $\mathcal{L}_{BLM}$. The loss and $L_{2}$ error evolution with training epochs for (A) increase in only $\alpha_{C}$, (B) simultaneous increase in $\alpha_{C}$ and $\alpha_{BC}$, and (C) simultaneous increase in all three weights. A common observation is that the loss corresponding to the unchanged weight, and the total loss increases upon an increase in the weights. The baseline weight values provide a good balance between loss and $L_{2}$ error minimization for the material parameters used in this work.
• Figure 2: Asymmetric double-edge notched plate: schematic, FEM with gradient damage and FEM without gradient damage crack path predictions.(A) Schematic for an asymmetrically double-edge notched plate specimen subjected to tension under plane strain conditions. The geometry is stretched at a speed of 2. (B) Mesh not aligned with the anticipated crack path. The FEM without gradient damage solution, as expected, is nonphysical, while the FEM with gradient damage predicts the correct crack path. (C) The crack paths from FEM without gradient damage for two different meshes. The solutions differ significantly, highlighting FEM without gradient damage's mesh dependency.
• Figure 2: Asymmetric double-edge notched plate: PINN solution convergence. For PINN (one-time step) application, the solution is evaluated at $t=0.45$ s and $t=0.65$ s with 1,000,000 training epochs (10 times the number of epochs used). The $L_{2}$ errors and losses decrease monotonically and approximately converge with respect to the number of training epochs. The overall width of the crack path decreases with epochs, and the path converges.
• Figure 3: Asymmetric double-edge notched plate: PINN (incremental) crack path prediction. The collocation grid is shown. For the PINN (incremental) application, the time step size used is 0.0025 s, with the first increment being $t=0.42$ s. Training for each time increment is performed for 25,000 epochs. Damage solution from PINN (incremental) at $t=0.54$ s. The crack paths from PINN (incremental) and the baseline FEM with gradient damage show good agreement.