Towards Unified AI-Driven Fracture Mechanics: The Extended Deep Energy Method (XDEM)

TL;DR

The paper introduces the Extended Deep Energy Method (XDEM), a unified AI framework that simultaneously handles discrete crack representations and continuous phase-field damage within a single variational formulation. By embedding explicit crack discontinuities through a crack function and enriching near-tip fields with an extended function in the discrete setting, alongside decoupled displacement and phase-field evolution in the continuous setting (with irreversibility enforced), XDEM achieves accurate fracture predictions using relatively uniform and sparse collocation. Validation spans stress intensity factors for mode I/II/III and mixed-mode cracks, straight and kinked propagation, crack inclusion, and crack initiation, consistently showing improved accuracy and efficiency over standard DEM and competitive performance against FEM. The framework leverages transfer learning (LoRA) to accelerate load-step training, extends to 3D problems, and discusses practical considerations such as zero-energy modes, loss landscapes, and network architectures (RBF for φ, KAN for u), offering a scalable AI-driven approach for predictive fracture mechanics and materials engineering applications.

Abstract

Physics-Informed Neural Networks (PINNs) have recently emerged as powerful tools for solving partial differential equations (PDEs), with the Deep Energy Method (DEM) proving especially effective in fracture mechanics due to its energy-based formulation. Despite these advances, existing DEM approaches require dense collocation near cracks, face stability challenges, and typically treat discrete and continuous fracture models separately. To overcome these limitations, we introduce the Extended Deep Energy Method (XDEM), a unified deep learning framework that incorporates both displacement discontinuities and crack-tip asymptotics in the discrete setting, while flexibly coupling displacement and phase fields in the continuous setting. This integration enables accurate fracture predictions using uniformly distributed, relatively sparse collocation points. Validation across benchmark problems including stress intensity factor evaluation, straight and kinked crack growth, and complex crack initiation demonstrates that XDEM consistently outperforms standard DEM in accuracy and efficiency. By bridging discrete and phase-field models within a single framework, XDEM establishes a robust foundation for applying AI to fracture mechanics and opens new avenues for predictive modeling in engineering and materials science.

Figures (36)

• Figure 1: Schematic illustration of the Extended Deep Energy Method (XDEM), comprising unified discrete and continuous formulations. The continuous model is indicated by dashed lines.
• Figure 1: Schematic illustration of the Extended Deep Energy Method (XDEM), which consists of discrete and continuous models. The continuous formulation is indicated by dashed lines.
• Figure 2: Mixed-mode crack problem: (a) geometry, material properties, and boundary conditions; (b) comparison of XDEM-predicted stress intensity factors ($K_{I}$ and $K_{II}$) with FEM reference solutions for $a=0.5$ and $\sigma_{0}=100,\text{MPa}$ at different crack inclination angles $\beta$; (c–f) displacement and stress contours predicted by XDEM for different crack angles $\beta$ in the mixed-mode configuration.
• Figure 2: Mixed-mode crack problem: (a) geometry, material properties, and boundary conditions; (b) comparison of XDEM-predicted SIFs ($K_{I}$ and $K_{II}$) with FEM reference solutions for $a=0.5$ and $\sigma_{0}=100\,\text{MPa}$ at different crack inclination angles $\beta$. (c-f) Displacement and stress contours predicted by XDEM for different crack angles $\beta$ in the mixed-mode crack problem.
• Figure 3: Displacement and stress contours predicted by XDEM for intersecting cracks, demonstrating accurate capture of displacement discontinuities and stress concentration at crack tips.