Physics-informed neural networks coupled with hybrid phase-field modeling for rock crack propagation
YUE Jiahao1, WANG Guilin1, 2, 3, WANG Runqiu1, HUANG Jianming1, LIAO Mingyong1, LI Baiyi1
(1. School of Civil Engineering, Chongqing University, Chongqing 400045, China; 2. State Key Laboratory of Safety and Resilience of Civil Engineering in Mountain Area, Chongqing 400045, China; 3. National Joint Engineering Research Center
of Geohazards Prevention in the Reservoir Areas, Chongqing 400045, China)
Abstract：To address the limitations of traditional fracture phase-field models—including their challenges in accurately describing tension-shear mixed-mode cracks in rock, their heavy reliance on meshes within mechanical solution frameworks, and their susceptibility to the “curse of dimensionality” in high-dimensional problems—this paper proposes a novel method that integrates Physics-Informed Neural Networks (PINNs) with a mixed-mode phase-field fracture model to simulate the tension-shear failure behavior of rock under loading. First, based on the modified fracture F-criterion, the method decomposes elastic energy into tensile and shear strain energy components through volumetric-deviatoric splitting. By combining this with the critical energy release rates for tension and shear, it derives the governing equations of the mixed-mode phase-field model for tension-shear cracks. Second, the study employs PINNs to construct an adaptive solution framework and optimizes the energy functional using the Deep Ritz Method (DRM), thereby circumventing the optimization conflicts typically associated with traditional residual-based loss functions. Validation through numerical examples—including the single-edge notched tension test, inclined crack propagation test, and parallel crack staggered propagation test—demonstrates that the calculated results align well with laboratory test data, confirming the correctness and effectiveness of the proposed method. Additionally, by varying the phase-field characteristic length, the regulatory mechanism of crack diffusion effects on fracture behavior is elucidated. This study not only provides a new theoretical framework and numerical implementation pathway for addressing rock mixed-mode failure issues but also establishes a foundation for researching real-time predictions of rock mechanical responses under varying initial conditions, including boundary conditions and material properties, by integrating PINNs with high-fidelity datasets.
[1] BRACE W F，BOMBOLAKIS E G. A note on brittle crack growth in compression[J]. Journal of Geophysical Research，1963，68(12)： 3 709–3 713.
[2] GRIFFITH A A，TAYLOR G I V I. The phenomena of rupture and flow in solids[J]. Philosophical Transactions of the Royal Society of London. Series A，Containing Papers of a Mathematical or Physical Character，1997，221(582/593)：163–198.
[3] NIXON W A，WEBER L J. Fatigue-crack growth in fresh-water ice：preliminary results[J]. Annals of Glaciology，1991，15：236–241.
[4] BA?ANT Z P. Size effect in blunt fracture：concrete，rock，metal[J]. Journal of Engineering Mechanics，1984，110(4)：518–535.
[5] 王桂林，王润秋，孙 帆，等. 单轴压缩下溶隙灰岩声发射RA-AF特征及破裂模式研究[J]. 中国公路学报，2022，35(8)：118–128. (WANG Guilin，WANG Runqiu，SUN Fan，et al. Study on RA-AF characteristics and fracture modes of solution-pore limestone under uniaxial compression using acoustic emission[J]. China Journal of Highway and Transport，2022，35(8)：118–128.(in Chinese))
[6] 王桂林，王 力，王润秋，等. 干湿循环后贯通型锯齿状红砂岩节理面剪切本构模型[J]. 岩土力学，2025，46(3)：706–720.(WANG Guilin，WANG Li，WANG Runqiu，et al. Shear constitutive model for fully-persistent jagged red sandstone joint surfaces after wet-dry cycles[J]. Rock and Soil Mechanics，2025，46(3)：706–720.(in Chinese))
[7] 王润秋，王桂林，张 亮，等. 单轴压缩下溶隙灰岩宏细观耦合损伤能量释放率演化规律研究[J]. 岩土工程学报，2025，47(9)：1 903–1 912(WANG Runqiu，WANG Guilin，ZHANG Liang，et al. Study on evolution mechanisms of macro-meso coupled damage energy release rate in solution-pore limestone under uniaxial compression[J]. Chinese Journal of Geotechnical Engineering，2025，47(9)：1903–1 912. (in Chinese))
[8] BELYTSCHKO T，GRACIE R，VENTURA G. A review of extended/generalized finite element methods for material modeling[J]. Modelling and Simulation in Materials Science and Engineering，2009，17(4)：043001.
[9] SUKUMAR N，MOËS N，MORAN B，et al. Extended finite element method for three-dimensional crack modelling[J]. International Journal for Numerical Methods in Engineering，2000，48(11)：1 549–1 570.
[10] CUNDALL P A，STRACK O D L. A discrete numerical model for granular assemblies[J]. Géotechnique，1979，29(1)：47–65.
[11] POTYONDY D O，CUNDALL P A. A bonded-particle model for rock[J]. International Journal of Rock Mechanics and Mining Sciences，2004，41(8)：1 329–1 364.
[12] LISJAK A，GRASSELLI G. A review of discrete modeling techniques for fracturing processes in discontinuous rock masses[J]. Journal of Rock Mechanics and Geotechnical Engineering，2014，6(4)：301–314.
[13] BARENBLATT G I. The mathematical theory of equilibrium cracks in brittle fracture[J]. Advances in Applied Mechanics，1962，7(1)：55–129.
[14] ZHANG Z，PAULINO G H. Cohesive zone modeling of dynamic failure in homogeneous and functionally graded materials[J]. International Journal of Plasticity，2005，21(6)：1 195–1 254.
[15] MIEHE C，SCHÄNZEL L M，ULMER H. Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids[J]. Computer Methods in Applied Mechanics and Engineering，2015，294：449–485.
[16] BOURDIN B，FRANCFORT G A，MARIGO J J. Numerical experiments in revisited brittle fracture[J]. Journal of the Mechanics and Physics of Solids，2000，48(4)：797–826.
[17] FRANCFORT G A，MARIGO J J. Revisiting brittle fracture as an energy minimization problem[J]. Journal of the Mechanics and Physics of Solids，1998，46(8)：1 319–1 342.
[18] RAISSI M，PERDIKARIS P，KARNIADAKIS G E. Physics-informed neural networks：A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics，2019，378：686–707.
[19] KARNIADAKIS G E，KEVREKIDIS I G，LU L，et al. Physics- informed machine learning[J]. Nature Reviews Physics，2021，3(6)：422–440.
[20] WEINAN E，BING Y. The deep Ritz method：a deep learning-based numerical algorithm for solving variational problems[J]. Communications in Mathematics and Statistics，2018，6(1)：1–12.
[21] GOSWAMI S，ANITESCU C，CHAKRABORTY S，et al. Transfer learning enhanced physics informed neural network for phase-field modeling of fracture[J]. Theoretical and Applied Fracture Mechanics，2020，106：102447.
[22] GOSWAMI S，YIN M，YU Y，et al. A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials[J]. Computer Methods in Applied Mechanics and Engineering，2022，391：114587.
[23] MANAV M，MOLINARO R，MISHRA S，et al. Phase-field modeling of fracture with physics-informed deep learning[J]. Computer Methods in Applied Mechanics and Engineering，2024，429：117104.
[24] SHEN B，STEPHANSSON O. Modification of the G-criterion for crack propagation subjected to compression[J]. Engineering Fracture Mechanics，1994，47(2)：177–189.
[25] GRIFFITH A，GILMAN J J. The phenomena of rupture and flow in solids[J]. Transactions of the ASM，1968，61：855–906.
[26] AMOR H，MARIGO J J，MAURINI C. Regularized formulation of the variational brittle fracture with unilateral contact：Numerical experiments[J]. Journal of the Mechanics and Physics of Solids，2009，57(8)：1 209–1 229.
[27] MIEHE C，HOFACKER M，WELSCHINGER F. A phase field model for rate-independent crack propagation：Robust algorithmic implementation based on operator splits[J]. Computer Methods in Applied Mechanics and Engineering，2010，199(45)：2 765–2 778.
[28] ZHANG X，SLOAN S W，VIGNES C，et al. A modification of the phase-field model for mixed mode crack propagation in rock-like materials[J]. Computer Methods in Applied Mechanics and Engineering，2017，322：123–136.
[29] MIEHE C，WELSCHINGER F，HOFACKER M. Thermodynamically consistent phase-field models of fracture：Variational principles and multi-field FE implementations[J]. International Journal for Numerical Methods in Engineering，2010，83(10)：1 273–1 311.
[30] AMARI S I. Natural gradient works efficiently in learning[J]. Neural Computation，1998，10(2)：251–257.
[31] WONG L N Y, EINSTEIN H H. Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression[J]. International Journal of Rock Mechanics and Mining Sciences，2009，46(2)：239–249.
[32] BOBET A，EINSTEIN H H. Numerical modeling of fracture coalescence in a model rock material[J]. International Journal of Fracture，1998，92(3)：221–252.
2025, Vol. 44 
