phase-field fracture modeling

Existing machine learning methods for phase-field fracture modeling (PFM) are predominantly evaluated on oversimplified benchmarks, failing to capture realistic, complex fracture behaviors. Method: We introduce the first standardized, high-complexity benchmark dataset for brittle fracture—comprising 6,000 samples—featuring diverse energy decomposition schemes, heterogeneous boundary conditions, and stochastic initial crack configurations, enabling multiscale and uncertainty-aware evaluation. Using this dataset, we systematically assess three representative architectures—physics-informed neural networks (PINNs), Fourier neural operators (FNOs), and U-Net—and enhance their generalization and robustness via ensemble learning. Contribution/Results: Experiments uncover shared limitations of current models in capturing strongly nonlinear fracture evolution, validating the dataset’s efficacy for rigorous reliability assessment. This work establishes a scalable, multiphysics-coupled testbed for trustworthy, mechanics-driven AI research.

This work addresses the limitation of classical phase-field fracture models, which lack an explicit strength criterion and thus struggle to capture cohesive fracture behavior and control crack initiation strength. The authors introduce the fracture characteristic strain as a local constitutive variable and, for the first time, reformulate its evolution within a fully local constitutive framework. This formulation is solved at each integration point via a plasticity-like return-mapping algorithm, requiring no additional global degrees of freedom and remaining compatible with standard finite element implementations. The approach explicitly decouples the strength surface from the fracture energy, accommodating both nonsmooth and smooth Drucker-Prager–type strength criteria, and includes a consistent tangent operator to ensure computational efficiency. Numerical examples demonstrate mesh and length-scale insensitivity and accurately reproduce complex fracture phenomena such as brittle-to-cohesive transitions and dynamic crack branching.

This study addresses the artificial toughening and deviation from the Griffith-type linear-elastic fracture propagation threshold caused by improper initial crack representation in phase-field fracture modeling. We systematically evaluate multiple initialization strategies through Γ-convergence analysis and path-following methods, rigorously comparing diffuse phase-field solutions against sharp-interface analytical fracture solutions to identify the origins of excessive crack-band width and force–displacement response overshoot. Two novel, robust initialization criteria are proposed: (1) a single-element-wide damage band and (2) an explicit single-element crack slit. Both effectively suppress artificial toughening and enable the phase-field model to accurately reproduce the theoretical fracture initiation threshold. Results demonstrate that initialization methodology exerts a decisive influence on predictive accuracy. This work establishes essential practical guidelines for reliable quantitative fracture analysis using phase-field methods.

To address the prohibitively high computational cost of phase-field fracture simulation in brittle materials—hindering efficient, full-process prediction of crack nucleation, propagation, and branching—this work proposes a deep neural operator surrogate model integrating physical priors with a novel network architecture. Methodologically, it introduces three key innovations: (i) a two-stage DeepONet backbone that decouples learning tasks; (ii) a physics-informed DeepONet explicitly enforcing the phase-field energy functional, drastically reducing data dependency; and (iii) a Kolmogorov–Arnold network replacing conventional MLPs to enhance functional approximation efficiency. Validated on single-edge-notched specimens and one-dimensional bars, the model achieves accuracy comparable to full-order phase-field solvers, with prediction errors highly localized near crack fronts. Computational time is reduced by one to two orders of magnitude, demonstrating exceptional efficiency while preserving strong physical consistency.

Addressing the critical challenge of hydrogen embrittlement to safe hydrogen utilization, this study develops a multiphysics predictive framework for hydrogen-assisted cracking based on phase-field fracture theory. Methodologically, it implements, for the first time in COMSOL, a hydrogen-concentration-dependent phase-field fracture model that couples nonlinear hydrogen diffusion with elastic–elastoplastic mechanical response—unifying brittle, ductile, and transitional fracture regimes while adaptively capturing hydrogen accumulation at crack tips and associated fracture-mode transitions. Validated against benchmark cases—including single-edge-notched plates, boundary-layer models, and a 3D pressure vessel—the framework quantitatively reproduces hydrogen-induced crack-tip softening and crack-path deviation. The open-source implementation provides a scalable, high-fidelity simulation tool for structural integrity assessment of components operating in hydrogen environments.

This work addresses the challenge of generalizing phase-field fracture modeling across arbitrary geometries throughout the entire process—from crack initiation to propagation—by proposing a hybrid integrated finite element–neural network (IFENN) solver. The method uniquely combines DeepOKAN and convolutional neural networks (CNNs) to separately model the initiation and propagation stages, while incorporating artificial boundary conditions to enhance far-field prediction accuracy. Requiring only a single physics-informed training on a reference geometry and leveraging a Gaussian-point sampling strategy, the approach substantially reduces offline computational costs. Numerical experiments demonstrate that the model achieves high accuracy and strong generalization capabilities on both seen and unseen geometries, significantly lowering computational resource demands.

This study addresses the unclear influence of phase-field regularization on the interaction between cracks and elastic waves in dynamic fracture, particularly the lack of systematic understanding regarding behavioral differences under various damage–displacement coupling formulations. By examining the interaction of tensile and compressive waves with phase-field cracks in a one-dimensional bar, the work analyzes the dynamic responses of three fracture models and, for the first time, extends cohesive-fracture phase-field regularization to an elastoplastic dynamic framework, deriving an analytical law for dynamic cohesive cracking. Integrating phase-field modeling, wave analysis, and two-dimensional simulations of dynamic crack branching, the research identifies key parameters governing wave–crack interactions and demonstrates the proposed model’s capability to accurately reproduce sharp-crack dynamic features and predict crack-branching behavior under varying loading intensities.

该研究提出一种无网格多分辨率深度能量方法，通过单个神经网络表示位移和相场，并直接最小化增量能量来解决脆性断裂问题。

This study addresses the challenge of unifying transient heat conduction and crack growth modeling within neural network solvers for thermo-mechanical crack propagation. To this end, an extended deep energy method is proposed, employing dual networks to represent temperature and displacement fields separately. A scalar embedding function implicitly describes sharp cracks, enabling the treatment of field discontinuities without a regularization length parameter. Coupled solutions are achieved via a staggered minimization strategy, while Williams asymptotic expansions enrich the crack-tip displacement field and hierarchical Monte Carlo integration enhances computational accuracy. Experimental results demonstrate that the stress intensity factor error is merely 0.11%, with predicted crack paths exhibiting excellent agreement with reference solutions, thereby validating the accuracy and effectiveness of the proposed approach under complex loading conditions.

This study addresses persistent bottlenecks in simulating plastic instability, including difficulties in experimental correlation, computational non-convergence, and mesh sensitivity. To overcome these challenges, this work proposes a damage-like variational framework that characterizes plastic softening as an internal variable, thereby decoupling hardening and softening behaviors. Furthermore, a field-independent constitutive model is established to bridge the theoretical gap between plastic instability and continuum damage mechanics, incorporating regularization techniques from fracture mechanics. The proposed approach effectively suppresses spurious localization, significantly enhances numerical convergence, and provides a unified description of diverse plastic instability phenomena.