Study on Elastoplastic Damage and Crack Propagation Mechanisms in Rock Based on the Phase Field Method
Abstract
:1. Introduction
2. Formulation and Theoretical Derivation of the Elastoplastic Phase Field Fracture Model
2.1. Crack Surface Energy
2.2. Elastic Strain Energy
2.3. Plastic Damage Strain Energy
2.4. Derivation of Phase Field Governing Equations and Equilibrium Equations
3. Numerical Implementation of Phase Field Fracture Model Based on Finite Elements
3.1. Finite Element Discretization
3.2. Numerical Implementation Process
4. Numerical Examples
4.1. Single-Edge Notched Tensile Test
4.2. Uniaxial Compression Test on Rock with a Prefabricated Crack
5. Discussion
- (1)
- Advantages and Contributions: The model’s primary strength lies in its ability to overcome the limitations of purely elastic phase field approaches. Traditional models often neglect plastic deformation mechanisms such as shear band localization and residual strength, leading to overly brittle predictions. By integrating plastic strain energy and non-associated flow rules, this work quantifies the reduction in crack propagation resistance caused by plastic slip, enabling a seamless transition from elastic loading to plastic damage accumulation and unstable fracture. Numerical validation through single-edge notch tension and uniaxial compression tests confirms the model’s accuracy, with load–displacement curves exhibiting less than 5% error in elastic stiffness and a 1.2% deviation in peak load compared to experimental benchmarks. These results highlight its potential for predicting residual strength in underground engineering structures, where stress redistribution and long-term stability are paramount.
- (2)
- However, the model’s reliance on small-deformation theory and isotropic hardening assumptions introduces constraints. Scenarios involving large rotations, such as shear band widening or rockbursts, may require extensions to finite-deformation frameworks to account for geometric nonlinearity. Additionally, the omission of crack interaction mechanisms—such as coalescence or branching—limits its applicability to multi-fracture systems. Furthermore, while key parameters (e.g., critical energy release rates, regularization length scale, and residual strength coefficient) are calibrated through benchmark tests, their sensitivity to crack propagation behavior and energy dissipation remains to be systematically quantified. A comprehensive parametric sensitivity analysis would refine calibration protocols for heterogeneous rock masses, though such an exploration is deferred to future work to maintain focus on the foundational coupling framework proposed here.
- (3)
- Comparison with Existing Studies: When compared to existing studies, the proposed model distinguishes itself through its explicit integration of plasticity and fracture mechanics. Prior hydromechanical phase field models, such as those by Wang et al. [15] and Liu et al. [16], omitted plastic energy dissipation, while cohesive zone approaches employed by Liu et al. [14] relied on predefined crack paths. In contrast, this work eliminates such constraints, enabling autonomous crack initiation and propagation through a unified energy functional. This theoretical innovation not only enhances computational flexibility but also provides a more holistic representation of rock behavior under compressive-shear loading, as evidenced by its alignment with classical experimental observations.
- (4)
- Future Directions: Future research should focus on expanding the model’s scope to address multifield coupling and dynamic loading scenarios. Incorporating thermo-hydro-mechanical (THM) interactions would improve predictions for saturated or thermally altered rock masses, where pore pressure and temperature gradients significantly influence fracture behavior. Extending the framework to dynamic regimes, such as blasting or seismic loading, could unravel crack propagation mechanisms under rapid stress changes. Preliminary studies on cyclic loading and crack branching already show promise, with results correlating well with experimental acoustic emission data. Furthermore, integrating machine learning techniques could streamline parameter inversion processes, particularly for heterogeneous formations where empirical calibration is impractical. These advancements would transform the model into a versatile tool for both academic research and industrial applications, ultimately advancing the safety and efficiency of rock engineering projects.
6. Conclusions and Prospects
- (1)
- This study develops an elastoplastic phase field fracture model, incorporating plastic deformation mechanisms and overcoming the limitations of conventional elastic models under compressive-shear conditions. By integrating plastic strain energy and non-associated flow rules, the framework couples plasticity with phase field damage evolution, explicitly quantifying how plastic slip and shear band formation regulate crack driving forces and propagation paths. The model successfully simulates complete energy evolution processes—from elastic loading to plastic damage accumulation and unstable fracture—providing a novel theoretical tool for analyzing mesoscale rock failure mechanisms under compressive-shear stresses.
- (2)
- The numerical implementation leverages finite element discretization and nonlinear solvers, validating the model’s capability to simulate complex crack propagation. The coupled interpolation of displacement and phase fields, staggered algorithms, and Newton–Raphson iterations address elastoplastic coupling nonlinearities. Benchmark tests—including single-edge notched tension and prefabricated-crack compression—demonstrate the accurate capture of crack initiation dynamics, wing crack propagation, and through-going failure. Specifically, in the elastic deformation stage, the load–displacement curve exhibits a slope error < 5% and a correlation coefficient R2 > 0.99. At the peak load, the relative error between this model and data from the literature is only 1.2%. These quantitative results further validate the reliability of this model.
- (3)
- The findings offer practical engineering relevance for hydraulic fracturing and underground energy storage applications. By quantifying plastic deformation’s impact on crack driving forces, the model enables reliable residual strength predictions, reducing dependency on costly physical tests for the long-term stability assessments of underground structures. Additionally, full-process energy evolution simulations support parameter inversion in real-time monitoring systems, enhancing safety and cost-efficiency in rock engineering projects.
- (4)
- Future research should focus on multifield coupling extensions and parameter optimization under dynamic loads. Integrating thermal, hydraulic, and mechanical (THM) couplings into phase field frameworks will advance the modeling of complex geological conditions. Developing data-driven parameter inversion methods and incorporating non-local plasticity theory to refine shear band width simulations are critical directions for advancing rock fracture mechanics research.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Symbol of Material Parameter | Name of Material Parameter | Values | Unit |
---|---|---|---|
elastic modulus | |||
Poisson’s ratio | |||
angle of internal friction | |||
cohesion | |||
critical energy release rate for tensile fracture | |||
critical energy release rate for tension–shear fracture | |||
critical energy release rate for compression–shear fracture | |||
critical energy release rate for plastic damage | |||
plastic bulk modulus | |||
plastic shear modulus | |||
material parameter | |||
hardening parameter | |||
scale parameter |
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Zhang, J.; Qin, G.; Wang, B. Study on Elastoplastic Damage and Crack Propagation Mechanisms in Rock Based on the Phase Field Method. Appl. Sci. 2025, 15, 6206. https://doi.org/10.3390/app15116206
Zhang J, Qin G, Wang B. Study on Elastoplastic Damage and Crack Propagation Mechanisms in Rock Based on the Phase Field Method. Applied Sciences. 2025; 15(11):6206. https://doi.org/10.3390/app15116206
Chicago/Turabian StyleZhang, Jie, Guang Qin, and Bin Wang. 2025. "Study on Elastoplastic Damage and Crack Propagation Mechanisms in Rock Based on the Phase Field Method" Applied Sciences 15, no. 11: 6206. https://doi.org/10.3390/app15116206
APA StyleZhang, J., Qin, G., & Wang, B. (2025). Study on Elastoplastic Damage and Crack Propagation Mechanisms in Rock Based on the Phase Field Method. Applied Sciences, 15(11), 6206. https://doi.org/10.3390/app15116206