Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architectures

TL;DR

This work tackles the high computational cost of phase-field fracture simulations by developing operator-learning approaches that map input boundary conditions and notch configurations to fracture fields. It introduces a data-driven two-step DeepONet, a physics-informed two-step DeepONet, and a DeepOKAN variant (KAN trunk with an MLP branch) to predict displacement $\mathbf{u}$ and phase field $\alpha$ for 1D and 2D brittle fracture problems, including nucleation, propagation, kinking, and branching. The data-driven method achieves high accuracy with as few as 45 FE-derived samples, while the physics-informed variant reduces data needs to 10 samples but requires energy minimization in the trunk; DeepOKAN provides competitive performance with smaller networks but demands careful hyperparameter tuning. Collectively, these approaches enable fast, physics-consistent, and scalable fracture predictions, with potential to extend to 3D and real-time screening tasks in engineering design.

Abstract

Phase-field modeling reformulates fracture problems as energy minimization problems and enables a comprehensive characterization of the fracture process, including crack nucleation, propagation, merging, and branching, without relying on ad-hoc assumptions. However, the numerical solution of phase-field fracture problems is characterized by a high computational cost. To address this challenge, in this paper, we employ a deep neural operator (DeepONet) consisting of a branch network and a trunk network to solve brittle fracture problems. We explore three distinct approaches that vary in their trunk network configurations. In the first approach, we demonstrate the effectiveness of a two-step DeepONet, which results in a simplification of the learning task. In the second approach, we employ a physics-informed DeepONet, whereby the mathematical expression of the energy is integrated into the trunk network's loss to enforce physical consistency. The integration of physics also results in a substantially smaller data size needed for training. In the third approach, we replace the neural network in the trunk with a Kolmogorov-Arnold Network and train it without the physics loss. Using these methods, we model crack nucleation in a one-dimensional homogeneous bar under prescribed end displacements, as well as crack propagation and branching in single edge-notched specimens with varying notch lengths subjected to tensile and shear loading. We show that the networks predict the solution fields accurately, and the error in the predicted fields is localized near the crack.

Figures (22)

• Figure 1: (a) A solid body containing a crack-like notch and a sharp crack, and (b) the plot of the phase field $\alpha$ regularizing them.
• Figure 2: Schematic representation of the DeepONet structure with three different types of trunk networks. The branch network processes the boundary condition or the initial crack size, while the trunk network processes the spatial coordinate $\bm{x}$ to predict the phase field $\alpha(\bm{x})$ and the displacement field $\bm{u}(\bm{x})$.
• Figure 3: Schematic of the two-step DeepONet for predicting the phase field $\alpha$ and the components $u$ and $v$ of the displacement field $\bm{u}$. During the training, first, the trunk network parameters, along with matrices ${A}_{u}$, ${A}_{v}$, and ${A}_{\alpha}$, are optimized, followed by the QR factorization of ${\Phi}^*$, and finally the training of the branch network.
• Figure 4: Case 1: One-dimensional homogeneous bar: geometry and boundary conditions.
• Figure 5: Case 1: The evolution of the total loss (losses associated with both $u$ and $\alpha$) for the trunk network during training eventually reaching $10^{-6}$. Similarly, the loss function for the branch network, responsible for predicting both $u$ and $\alpha$, also converges to $10^{-6}$.