Study on the mesoscopic fracture behaviors of granite based on the elastoplastic phase-field model
LI Mingyao1，2，LI Shaojin1，PENG Lei1，DING Yufei1，ZUO Jianping1，2
(1. School of Mechanics and Civil Engineering，China University of Mining and Technology(Beijing)，Beijing 100083，China；
2. State Key Laboratory for Fine Exploration and Intelligent Development of Coal Resources，China University of
Mining and Technology(Beijing)，Beijing 100083，China)
Abstract：The mesoscopic heterogeneity of granite has an important influence on its damage and failure behavior. In this paper，an elastoplastic phase-field model is established based on the thermodynamic principle as well as the balance equations of forces and energy，and the non-associative plastic constitutive relationship suitable for rock-like materials within the framework of the classical phase field model is introduced. By comparing with the analytical solution and experimental data，the accuracy and reliability of the elastoplastic phase-field model are verified. Furthermore，a heterogeneous numerical model reflecting the real mesostructure of granite is established by digital image processing. The triaxial compression test of granite is numerically simulated，and the macroscopic mechanical behavior and the crack propagation mechanism of rock is analyzed at the mesoscale. The results show that compared with the experimental data and the traditional elastic-brittle phase-field methods，the elastoplastic phase-field model based on the real mesostructure of granite is capable of well capturing its macroscopic nonlinear mechanical behavior. The initiation and propagation of internal cracks in granite and the distribution of local stress field are affected by the mechanical properties，geometric shape and distribution of mineral particles. The research method of this paper provides a simple and effective way to study the multi-scale damage and failure of rock in the future，and it has important engineering practical significance for evaluating the mechanical properties of surrounding rock in underground engineering.
李明耀1，2，李绍金1，彭 磊1，丁宇飞1，左建平1，2. 基于相场法的花岗岩弹塑性损伤模型及其细观力学行为研究[J]. 岩石力学与工程学报, 2024, 43(3): 611-622.
LI Mingyao1，2，LI Shaojin1，PENG Lei1，DING Yufei1，ZUO Jianping1，2. Study on the mesoscopic fracture behaviors of granite based on the elastoplastic phase-field model. , 2024, 43(3): 611-622.
[1] 杨圣奇，田文岭，董晋鹏. 高温后两种晶粒花岗岩破坏力学特性试验研究[J]. 岩土工程学报，2021，43(2)：281–289.(YANG Shengqi，TIAN Wenling，DONG Jinpeng. Experimental study on failure mechanical properties of granite with two grain sizes after thermal treatment[J]. Chinese Journal of Geotechnical Engineering，2021，43(2)：281–289.(in Chinese))
[2] 田文岭，杨圣奇，黄彦华，等. 花岗岩高温高压损伤破裂细观机制模拟研究[J]. 岩石力学与工程学报，2022，41(9)：1 810–1 819. (TIAN Wenling，YANG Shengqi，HUANG Yanhua，et al. Meso-fracture mechanism of granite specimens under high temperature and confining pressure by numerical simulation[J]. Chinese Journal of Rock Mechanics and Engineering，2022，41(9)：1 810–1 819.(in Chinese))
[3] 张世瑞，邱士利，李邵军，等. 北山花岗岩细观非均质性对单轴压缩力学特性影响的FDEM数值研究[J]. 岩石力学与工程学报，2022，41(增1)：2 658–2 672.(ZHANG Shirui，QIU Shili，LI Shaojun，et al. Study on mechanical properties of Beishan granite meso-heterogeneity under uniaxial compression with FDEM[J]. Chinese Journal of Rock Mechanics and Engineering，2022，41(Supp.1)：2 658–2 672.(in Chinese))
[4] 李 宁，张 平，陈蕴生. 裂隙岩体试验研究进展与思考[C]// 中国岩石力学与工程学会第七次学术大会论文集. 西安：[s. n.]，2002：63–69.(LI Ning，ZHANG Ping，CHEN Yunsheng. The progress and thought of the experimental study for the cracked rock mass[C]// Symposiums on the 7th Academic Meeting of China Society of Rock Mechanics and Engineering. Xi?an：[s. n.]，2002：63–69.(in Chinese))
[5] 张艳博，吴文瑞，姚旭龙，等. 单轴压缩下花岗岩声发射、红外特征及损伤演化试验研究[J]. 岩土力学，2020，41(增1)：139–146. (ZHANG Yanbo，WU Wenrui，YAO Xulong，et al. Acoustic emission，infrared characteristics and damage evolution of granite under uniaxial compression[J]. Rock and Soil Mechanics，2020，41(Supp.1)：139–146.(in Chinese))
[6] HUNSWECK M J，SHEN Y，LEW A J. A finite element approach to the simulation of hydraulic fractures with lag[J]. International Journal for Numerical and Analytical Methods in Geomechanics，2013，37(9)：993–1 015.
[7] 郭历伦，陈忠富，罗景润，等. 扩展有限元方法及应用综述[J]. 力学季刊，2011，32(4)：612–625.(GUO Lilun，CHEN Zhongfu，LUO Jingrun，et al. A review of the extended finite element method and its applications[J]. Chinese Quarterly of Mechanics，2011，32(4)：612–625.(in Chinese))
[8] LI K，HUANG L H，HUANG X C. Propagation simulation and dilatancy analysis of rock joint using displacement discontinuity method[J]. Journal of Central South University，2014，21(3)：1 184–1 189.
[9] ARMERO F，KIM J. Three-dimensional finite elements with embedded strong discontinuities to model material failure in the infinitesimal range[J]. International Journal for Numerical Methods in Engineering，2012，91(12)：1 291–1 330.
[10] VIJAYAKUMAR S，YACOUB T E，CURRAN J H. A node-centric indirect boundary element method：three-dimensional displacement discontinuities[J]. Computers and Structures，2000，74(6)：687–703.
[11] YAN X Q. A special crack tip displacement discontinuity element[J]. Mechanics Research Communications，2004，31(6)：651–659.
[12] 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.
[13] MOLNAR G，GRAVOUIL A. 2D and 3D Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture[J]. Finite Elements in Analysis and Design，2017，130：27–38.
[14] ZHOU S，RABCZUK T，ZHUANG X. Phase field modeling of quasi-static and dynamic crack propagation：COMSOL implementation and case studies[J]. Advances in Engineering Software，2018，122：31–49.
[15] 吴建营. 固体结构损伤破坏统一相场理论、算法和应用[J]. 力学学报，2021，53(2)：301–329.(WU Jianying. On the unified phase-field theory for damage and failure in solids and structures：Theoretical and numerical aspects[J]. Chinese Journal of Theoretical and Applied Mechanics，2021，53(2)：301–329.(in Chinese))
[16] 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.
[17] 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.
[18] BOURDIN B. Numerical implementation of the variational formulation for quasi-static brittle fracture[J]. Interfaces and Free Boundaries，2007，9(3)：411–430.
[19] 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.
[20] 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(45-48)：2 765–2 778.
[21] CHU D，LI X，LIU Z，et al. A unified phase field damage model for modeling the brittle-ductile dynamic failure mode transition in metals[J]. Engineering Fracture Mechanics，2019，212：197–209.
[22] ZHOU S，ZHANG C，XU Y，et al. A hybrid phase field method for modeling thermal fractures in brittle rocks：fracture diversity from a modified driving force[J]. International Journal of Fracture，2022，238(2)：185–201.
[23] 李鹏飞，朱其志，顾水涛，等. 岩石类材料裂隙形成和扩展的相场方法模拟[J]. 工程力学，2018，35(3)：41–48.(LI Pengfei，ZHU Qizhi，GU Shuitao，et al. A phase field method to simulate crack nucleation and crack propagation in rock-like materials[J]. Engineering Mechanics，2018，35(3)：41–48.(in Chinese))
[24] ZHOU S，ZHUANG X，ZHU H，et al. Phase field modelling of crack propagation，branching and coalescence in rocks[J]. Theoretical and Applied Fracture Mechanics，2018，96：174–192.
[25] 王苏生，杨圣奇，田文岭，等. 预制裂隙岩石裂纹扩展相场模拟方法研究[J]. 岩石力学与工程学报，2023，42(12)：365–377.(WANG Susheng，YANG Shengqi，TIAN Wenlin，et al. Research on phase-field simulation method of crack propagation of rock with pre-existing fissures[J]. Chinese Journal of Rock Mechanics and Engineering，2023，42(12)：365–377.(in Chinese))
[26] WANG M，YU Z，JIN Y，et al. Modeling of damage and cracking in heterogeneous rock-like materials by phase-field method[J]. Mechanics Research Communications，2021，114：103612.
[27] 李明耀，李绍金，左建平，等. 基于相场法的岩石三点弯曲细观破坏数值模拟[J]. 科学技术与工程，2022，22(30)：13 441–13 449.(LI Mingyao，LI Shaojin，ZUO Jianping，et al. Numerical Simulation of the Three-point Bending Failure Behavior of Basalt Based on the Phase Field Method[J]. Science Technology and Engineering，2022，22(30)：13 441–13 449.(in Chinese))
[28] LI M，ZUO J，HU D，et al. Experimental and numerical investigation of microstructure effect on the mechanical behavior and failure process of brittle rocks[J]. Computers and Geotechnics，2020，125：103639.
[29] MOLNAR G，GRAVOUIL A，SEGHIR R，et al. An open-source Abaqus implementation of the phase-field method to study the effect of plasticity on the instantaneous fracture toughness in dynamic crack propagation[J]. Computer Methods in Applied Mechanics and Engineering，2020，365：113004.
[30] LI C，FANG J，WU C，et al. Phase field fracture in elasto-plastic solids：Incorporating phenomenological failure criteria for ductile materials[J]. Computer Methods in Applied Mechanics and Engineering，2022，391：114580.
[31] CHOO J，SUN W. Coupled phase-field and plasticity modeling of geological materials：From brittle fracture to ductile flow[J]. Computer Methods in Applied Mechanics And Engineering，2018，330：1–32.
[32] WANG M，YU Z，SHEN W，et al. Numerical study of time-dependent deformation and cracking in brittle rocks with phase-field method and application to slope instability analysis[J]. International Journal of Rock Mechanics and Mining Sciences，2022，155：105144.
[33] 刘世藩，王 伟，曹亚军，等. 基于相场方法的岩石弹塑性损伤模型研究及其数值实现[J]. 岩石力学与工程学报，2023，42(4)：896–905.(LIU Shifan，WANG Wei，CAO Yajun，et al. Study on the elastoplastic damage model for rock based on the phase-field method and its numerical implementation[J]. Chinese Journal of Rock Mechanics and Engineering，2023，42(4)：896–905.(in Chinese))
[34] 朱泽奇，肖培伟，盛 谦，等. 基于数字图像处理的非均质岩石材料破坏过程模拟[J]. 岩土力学，2011，32(12)：3 780–3 786.(ZHU Zeqi，XIAO Peiwei，SHENG Qian，et al. Numerical simulation of fracture propagation of heterogeneous rock material based on digital image processing[J]. Rock and Soil Mechanics，2011，32(12)：3 780–3 786.(in Chinese))
[35] 李明耀，彭 磊，左建平，等. 基于真实细观结构的FFT数值方法对岩石材料非线性力学行为的研究[J]. 矿业科学学报，2022，7(4)：456–466.(LI Mingyao，PENG Lei，ZUO Jianping，et al. Study of the nonlinear mechanical behavior of rock materials using the FFT numerical method based on the actual mesostructure[J]. Journal of Mining Science and Technology，2022，7(4)：456–466.(in Chinese))
[36] MIEHE C，KIENLE D，ALDAKHEEL F，et al. Phase field modeling of fracture in porous plasticity：A variational gradient-extended Eulerian framework for the macroscopic analysis of ductile failure[J]. Computer Methods in Applied Mechanics and Engineering，2016，312：3–50.
[37] MIEHE C，TEICHTMEISTER S，ALDAKHEEL F. Phase-field modelling of ductile fracture：a variational gradient-extended plasticity-damage theory and its micromorphic regularization[J]. Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences，2016，374：20150170.
[38] YU Z，SHAO J F，VU M N，et al. Numerical study of thermo-hydro-mechanical responses of in situ heating test with phase-field model[J]. International Journal of Rock Mechanics and Mining Sciences，2021，138：104542.
[39] JIANG T，SHAO J F. Micromechanical analysis of the nonlinear behavior of porous geomaterials based on the fast Fourier transform[J]. Computers and Geotechnics，2012，46：69–74.
[40] 李明耀，彭 磊，左建平，等. 基于DIP-FFT数值方法的花岗岩多尺度力学特性研究[J]. 岩石力学与工程学报，2022，41(11)：2 254–2 267.(LI Mingyao PENG Lei，ZUO Jianping，et al. Study on multi-scale mechanical properties of granite based on DIP-FFT numerical method[J]. Chinese Journal of Rock Mechanics and Engineering，2022，41(11)：2 254–2 267.(in Chinese))
2024, Vol. 43 
