物理知情神经网络（PINNs）近年来成为求解偏微分方程（PDE）的强大工具，其中深能法（DEM）因其基于能量的表述，在断裂力学中尤为有效。尽管有这些进展，现有的DEM方法仍要求裂缝附近有密集的配址，面临稳定性挑战，且通常将离散和连续断裂模型分开处理。为克服这些限制，我们引入了扩展深度能量方法（XDEM），这是一个统一的深度学习框架，在离散环境中结合位移不连续性和裂纹尖端渐近，同时在连续环境中灵活耦合位移场和相位场。这种整合使得通过均匀分布、相对稀疏的配位点进行准确的断裂预测成为可能。在包括应力强度因子评估、直线和弯曲裂纹扩展以及复杂裂纹起始等基准问题上的验证表明，XDEM在准确性和效率上始终优于标准DEM。通过在单一框架内桥接离散和相场模型，XDEM为将AI应用于断裂力学奠定了坚实基础，并为工程和材料科学中的预测建模开辟了新途径。

 

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简介

材料及结构失效的准确建模长期以来被视为力学中的一项重大挑战，影响了工程和材料科学数十年的研究。断裂是材料失效的主要驱动因素，而对裂纹起始和传播的准确模拟仍是核心问题。经典的断裂计算方法大致分为两类：离散断裂模型和连续损伤模型。离散方法，包括虚拟裂缝闭合技术（VCCT）1，凝聚区法（CZM） 2,3，以及扩展有限元方法（XFEM4/XIGA 5,6）提供计算效率，但需要显式的裂纹表示和断裂条件，因此难以应用于复杂裂纹网络或三维几何问题。连续方法，如相场法7,8,9,10,11以及周动力学12通过允许裂纹在无任何追踪的情况下成核和演化，提供了更灵活的描述。然而，这种灵活性也以显著更高的计算成本为代价，主要因为需要精细的空间离散化以准确解析裂纹特征。

 

需要注意的是，这些建模框架——无论是离散的还是连续的——最终都会导致必须数值求解的主导偏微分方程（PDE）。传统的数值方案，如有限元法（FEM）、有限差分法（FDM）或无网格法，长期以来一直是这些偏微分方程的主要主力。最近，基于机器学习的新型求解器出现了，特别是物理知情神经网络（PINNs） 13,14，已成为求解此类方程的另一种范式。PINN及其变分对应物深能方法（DEM）直接将底层物理定律嵌入神经网络训练目标中，使偏微分方程能够求解而无需显式网格或标记数据。DEM表述特别适用于断裂力学，其中控制方程自然源自变分能量原理15.DEM已成功应用于离散和连续断裂模型，带来了若干代表性进展。在离散模型中，赵等人使用DEM模拟裂纹传播16尽管他们的方法需要沿着裂缝路径的演变路径增加额外的配位点。Chen 等人。17将强形式PINN与渐近断裂解结合起来，模拟疲劳裂纹扩展，但未充分利用变分DEM框架。Chen 等人。17应用PINNs（采用强偏微分方程）结合渐近断裂解，模拟疲劳裂纹扩展。对于连续模型，Goswami等人率先使用DEM进行相场断裂，证明了其在二阶断裂中的有效性15以及四阶表述18.然而，准确模拟剪切主导（模式II）破坏仍具挑战性。郑19提出了一种受有限元素法启发的方法，其中神经网络预测网格节点的位移场和相位场，随后通过形状函数插值构建位移和相位场。然而，该方法需要沿裂纹路径细化网格，类似于有限元素学。基于这一理念，Manav等人。20进行了更系统的研究，将DEM应用于裂纹成核、扩展、弯曲、分支和聚合。与传统的数值求解器如有限元法（FEM）和等几何分析（IGA）相比，DEM的一个主要优势是允许更大的负载增量15，实现了裂纹传播的加速模拟。尽管取得了这些进展，现有的DEM断裂力学方法已独立处理离散和连续模型。由于每种模型都有其优势，这种分离限制了它们的整体效能：离散模型计算效率更高，且适合相对简单的裂纹扩展问题，而连续相场模型则不需要显式断裂标准，更适合复杂的裂纹图案和三维系统。此外，现有的DEM框架通常需要在裂缝尖端附近进行精细的搭配，以准确捕捉高度局部的场15,16,18,20这又要求先验了解裂纹路径，或开发高效的自适应精炼方案18.

 

为解决这些限制，我们提出了扩展深能量方法（XDEM）（见图）。1），一个统一的人工智能框架，连接了断裂力学的离散与连续表述。在离散区间，XDEM引入了裂纹函数来表示位移不连续，并引入了扩展函数以捕捉近端渐近场。在连续相场领域中，XDEM通过灵活的神经架构解耦位移和损伤演化，通过惩罚或历史场策略强制执行不可逆性。这些发展共同消除了针对特定问题的搭配优化需求，同时大幅提升了准确性、鲁棒性和计算效率。具体来说，XDEM建立了一个统一的基于DEM的框架，适用于离散和连续断裂模型。引入的裂纹和扩展函数使得通过稀疏且均匀分布的配点，准确识别应力强度因子。XDEM的多功能性通过基准研究得以展示，这些研究涉及应力强度因子评估、直线、弯曲和分支裂纹的扩展以及裂纹起始。

 

图1：扩展深能法（XDEM）的示意图：XDEM由离散和连续模型组成。

图1：扩展深能法（XDEM）的示意图：XDEM由离散和连续模型组成。

全尺寸图像

连续表述用虚线表示。

 

选举结果

本节介绍了所提出的扩展深能方法（XDEM）在不同载荷和边界条件下断裂演化建模中的表现。XDEM有两种形式：离散（XDEM-D）和连续（XDEM-C）。正文中讨论的结果涉及XDEM-D，而XDEM-C的结果则见补充说明2.2。结果分为两部分：（i）应力强度因子（SIF）的预测，用于量化尖端附近的应力场;（ii）不同断裂模式（模式I–III）下裂纹路径传播的模拟。

 

应力强度因子

应力强度因子是线性弹性断裂力学中的一个基本量，用以表征裂缝尖端附近应力场的强度。我们首先评估XDEM准确预测SIF的能力。考虑三种代表性情况：模式I（开启）、模式II（滑动）和模式III（撕裂），以及混合模式加载条件。由于I–III模式的各个裂纹可视为混合模断裂的特例，我们这里重点介绍混合模态构型，而纯模态的详细结果见补充说明2.1。

 

混合模问题涉及一块长度为2b = 2、高度为2h = 2的矩形板，包含一个长度为2a = 1、倾斜的中心裂纹β（见图。材料参数在平面应变条件下为E = 1000 MPa，泊松比 ν = 0.3。均匀拉伸应力σ0= 100 MPa作用于顶边界，而左右边在x方向（u）受限x= 0）以及y方向的底部边界（u）y= 0）。XDEM表述中使用的位移场定义见补充说明2.1.4。图2b比较了XDEM预测的模式I和模式II应力强度因子（K）I以及K二）采用有限元法（FEM）参考解，∈15°、30°、45°、60°、75°的裂纹倾角β。XDEM在所有方向上表现出极佳的一致性，准确捕捉了应力强度因子的两个分量。SIFs通过交互积分法计算，详见补充说明3.8。

 

图2：混合模式裂纹问题。

图2：混合模式裂纹问题。

Full size image

a geometry, material properties, and boundary conditions; b comparison of XDEM-predicted SIFs (KI and KII) with FEM reference solutions for a = 0.5 and σ0 = 100 MPa at different crack inclination angles β. c–f Displacement and stress contours predicted by XDEM for different crack angles β in the mixed-mode crack problem.

 

These results demonstrate that XDEM reliably captures near-tip stress fields-a key prerequisite for accurate crack propagation modeling, which is explored further in Supplementary Note 2.2. Training parameters are consistent with the mode-I case, using 100 × 100 uniformly distributed collocation points. The displacement and stress contours for various crack angles (Fig. 2c–f) clearly exhibit the expected discontinuities across the crack surface, underscoring the effectiveness of the crack function in representing displacement jumps within the neural architecture.

 

To examine the sensitivity of XDEM to collocation density, Table 1 summarizes its accuracy and computational efficiency across different point distributions. The results indicate that, for most crack angles, a moderate resolution of 30 × 30 points suffices to achieve high accuracy. At very low point densities, however, integration inaccuracies during training can occasionally lead to spurious stress fields or non-physical fracture patterns (see Supplementary Note 3.1). Such effects can be mitigated by employing early stopping criteria or by modestly increasing the collocation resolution.

 

Table 1 Accuracy and efficiency of XDEM for the mixed-mode crack problem

Full size table

Finally, to further illustrate the generality of XDEM, Fig. 3 presents results for intersecting cracks. XDEM accurately reproduces displacement discontinuities along multiple crack interfaces in close agreement with the reference solution16. These results highlight the robustness of XDEM in representing complex crack topologies without requiring adaptive meshing or problem-specific refinement.

 

Fig. 3: Displacement contours predicted by XDEM and FEM (reference solution) for intersecting cracks.

Fig. 3: Displacement contours predicted by XDEM and FEM (reference solution) for intersecting cracks.

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a Geometry, boundary conditions, and crack configuration. b–f Displacement contours for intersecting cracks with β = 15, 30, 45, 60, and 75, respectively. For each angle, the u°°°°°x and uy fields predicted by XDEM are compared with the FEM reference solution, and the corresponding error maps are computed with FEM taken as the reference solution. Source data are provided as a Source Data file.

 

Crack propagation

The preceding section established that XDEM can accurately predict SIFs, forming the basis for modeling dynamic crack evolution. Here, we further validate XDEM’s performance in simulating crack propagation under mixed-mode loading, focusing on three representative scenarios: straight crack growth, crack kinking, and crack initiation. Detailed results for the additional examples are provided in Supplementary Note 2.2.

 

We first examine the classical Bittencourt problem21, a widely used benchmark in computational fracture mechanics. Experimental observations for this setup are available in ref. 22. As shown in Fig. 4a, this configuration is particularly interesting because the crack trajectory varies depending on the initial crack position, leading the crack to deflect toward different holes. The displacement field formulation used in XDEM for this problem is given in Supplementary Note 2.2.2.

 

Fig. 4: XDEM results for the Bittencourt problem.

Fig. 4: XDEM results for the Bittencourt problem.

Full size image

a Geometry of the Bittencourt setup. b Predicted crack propagation paths by XDEM compared against experimental observations22 for two scenarios: without a hole (Experiment 1) and with a hole (Experiment 2). c Displacement and stress contours for Experiment 1 (without hole). d Displacement and stress contours for Experiment 2 (with hole).

 

In the XDEM simulations, crack propagation initiates at a normalized displacement \bar{u}\approx 0.4, consistent with previous experimental and numerical findings23. The predicted crack paths (Fig. 4b) show excellent agreement with the experimental observations22, accurately reproducing the deflection behavior observed in both scenarios-with and without a hole. The corresponding displacement and stress contours (Fig. 4c, d) clearly illustrate XDEM’s ability to capture displacement discontinuities and stress concentration near the crack tip during propagation.

 

To further assess the robustness of XDEM, we next consider crack propagation in materials with stiffness inclusions, where the primary challenge arises from the interaction between materials of differing elastic properties. The benchmark setup follows24,25 and is illustrated in Fig. 5a. The displacement field used in XDEM is detailed in Supplementary Note 2.2.3. Figure 5b compares the predicted crack paths for soft and hard inclusions in linear elastic materials with corresponding reference solutions24,25. XDEM reproduces the reference trajectories with high fidelity for both cases, confirming its robustness across strong material heterogeneities. The load-displacement response predicted by XDEM (Fig. 5c) exhibits excellent agreement with established results, while the displacement and stress contours (Fig. 5d, e) demonstrate accurate resolution of both the stress concentration near the crack tip and the stress discontinuity across the inclusion interface.

 

Fig. 5: Performance of XDEM on crack inclusion problems.

Fig. 5: Performance of XDEM on crack inclusion problems.

Full size image

a schematic of the inclusion setup with two scenarios. Case 1: linear elasticity with soft inclusion, where the inclusion is a softer material with E1 = 210, 000N mm−2 and E2 = 21, 000N mm−2. Case 2: linear elasticity with hard inclusion, where the inclusion is stiffer, E1 = 21, 000N mm−2 and E2 = 210, 000N mm−2. The Poisson’s ratio and fracture energy are ν = 0.3 and Gc = 2.7N m−1 for all materials. b Comparison of crack propagation paths between XDEM and the reference solutions24,25. c Load-displacement curve predicted by XDEM. d Displacement and stress contour at \bar{u}=0.015 for the soft inclusion case. e Displacement and stress contour at \bar{u}=0.0302 for the hard inclusion case.

 

Discussion

In this work, we introduced the Extended Deep Energy Method (XDEM): a unified, physics-informed AI framework for fracture mechanics that seamlessly integrates discrete and continuous damage representations within a single variational formulation. By embedding explicit crack discontinuities and near-tip enrichments in the discrete setting, and coupling displacement and phase fields in the continuous setting, XDEM effectively overcomes several long-standing limitations of conventional DEM frameworks. In particular, XDEM eliminates the need for refined collocation near crack tips, maintains high accuracy with uniformly distributed sampling points, and enhances numerical stability across a wide spectrum of fracture scenarios. Comprehensive validation across canonical benchmarks-including stress intensity factor prediction, straight and kinked crack growth, crack initiation, inclusion-induced fracture, and three-dimensional crack propagation-demonstrates that XDEM achieves accuracy and efficiency on par with, and in many cases surpassing, traditional numerical approaches and standard DEM formulations. These results highlight XDEM’s ability to accurately capture both localized crack-tip fields and complex crack trajectories, thereby bridging the methodological gap between discrete and phase-field models. Beyond addressing key challenges in fracture simulation, XDEM establishes a scalable foundation for AI-driven computational mechanics. Its unified structure, variational grounding, and compatibility with transfer learning techniques position it as a powerful and data-efficient framework for large-scale simulations of complex materials and structures. Future extensions of XDEM to dynamic fracture, multiphysics coupling (e.g., thermo-mechanical or fluid-driven cracking), and neural operator-based generalization could further broaden its applicability and impact. In summary, XDEM provides a robust, accurate, and versatile framework that advances the state of the art in fracture mechanics while exemplifying how physics-informed AI can transform the predictive modeling of complex failure phenomena in engineering and materials science.

 

Method

Extended deep energy method framework

Extended Deep Energy Method (XDEM) unifies both discrete and continuous formulations of the Deep Energy Method (DEM) within a single framework, as illustrated schematically in Fig. 1. The discrete formulation captures sharp displacement discontinuities across cracks, while the continuous formulation models diffusive damage evolution through a phase-field representation. In the following, we introduce the discrete crack formulation and the continuous damage formulation, respectively.

 

Discrete crack formulation

The loss functional of the discrete XDEM is defined as:

 

{{{\bf{u}}}}^{n+1} =\arg {{\min }_{{{\bf{u}}}}}{{\Pi }},\\ {{\Pi }} ={U}_{{{\rm{e}}}}-{W}_{{{\rm{ext}}}},\\ {U}_{{{\rm{e}}}} =\int_{\Omega }\frac{1}{2}\,{{\boldsymbol{\varepsilon }}}({{\bf{x}}};{{{\mathbf{\theta }}}}_{{{\rm{u}}}}):{{\bf{C}}}:{{\boldsymbol{\varepsilon }}}({{\bf{x}}};{{{\mathbf{\theta }}}}_{{{\rm{u}}}})\,dV,\\ {W}_{{{\rm{ext}}}} =\int_{\Omega }{{\bf{f}}}\cdot {{\bf{u}}}({{\bf{x}}};{{{\mathbf{\theta }}}}_{{{\rm{u}}}})\,dV+\int_{{\Gamma }^{{{\rm{t}}}}}\bar{{{\bf{t}}}}\cdot {{\bf{u}}}({{\bf{x}}};{{{\mathbf{\theta }}}}_{{{\rm{u}}}})\,dS,\\ \,{\mbox{s.t.}}\, {u}_{i}({{\bf{x}}};{{{\mathbf{\theta }}}}_{{{\rm{u}}}}) ={\bar{u}}_{i}({{\bf{x}}},{t}^{n+1}),\,{{\bf{x}}}\in {\Gamma }^{{{\rm{u}}}};\quad {u}_{i}^{+}\not\equiv {u}_{i}^{-},\,{{\bf{x}}}\in {\Gamma }^{{{\rm{c}}}}. (1)

Here, σ, C, ε, and u represent the stress tensor, stiffness tensor, strain tensor, and displacement vector, respectively. Ω denotes the computational domain, while Γu, Γt, and Γc对应狄利克雷边界、诺依曼边界和裂纹边界。外部载荷贡献f、\（\bar{{\bf{t}}}\）和\（\bar{{{\bf{u}}}}\）分别代表体力、规定牵引力和施加的位移。裂纹表面施加无牵引条件Γc，且混合（罗宾）边界不在此考虑。在此表述中，所提出的裂纹函数自动强制执行跨Γ位移不连续性c而扩展函数则增强了近端奇异场的表示，显著提高了准确性和收敛性。

 

XDEM在离散裂纹模型中面临的关键挑战是准确表示裂纹表面的位移不连续点以及裂纹尖端附近高度局部的场。为此，我们采纳了XFEM中Heaviside阶梯函数的思想4以及渐近裂纹尖端解，转化为深能法。具体来说，阶梯函数用于捕捉位移不连续点，而裂纹尖端浓缩函数则用于表示裂纹尖端附近的奇异行为。以下内容中，我们将描述XDEM如何引入这两个组成部分，即裂纹函数和扩展函数。

 

裂纹函数

XDEM中位移不连续的处理通过引入裂纹函数实现，该函数基于Zhao等人提出的嵌入函数概念。16.