# Coupled Elastic-Plastic Damage Modeling of Rock Based on Irreversible Thermodynamics
###### Abstract  
Shale is a common rock in oil and gas extraction, and the study of its nonlinear mechanical behavior is crucial for the development of engineering techniques such as hydraulic fracturing. This paper establishes a new coupled elastic-plastic damage model based on the second law of thermodynamics, the strain equivalence principle, the non-associated flow rule, and the Drucker-Prager yield criterion. This model is used to describe the mechanical behavior of shale before and after peak strength and has been implemented in ABAQUS via UMAT for numerical computation. The model comprehensively considers the quasi-brittle and anisotropic characteristics of shale, as well as the strength degradation caused by damage during both the elastic and plastic phases. A damage yield function has been established as a criterion for damage occurrence, and the constitutive integration algorithm has been derived using a regression mapping algorithm. Compared with experimental data from La Biche shale in Canada, the theoretical model accurately simulated the stress-strain curves and volumetric-axial strain curves of shale under confining pressures of 5 MPa, 25 MPa, and 50 MPa. When compared with experimental data from shale in Western Huber and Eastern Chonging, China, the model precisely fitted the stress-strain curves of shale at pressures of 30 MPa, 50 MPa, and 70 MPa, and at bedding angles of \(0^{\circ}\), \(22.5^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\). This proves that the model can effectively predict the failure behavior of shale under different confining pressures and bedding angles. Additionally, a sensitivity analysis has been performed on parameters such as the plastic hardening rate \(b\), damage evolution rate \(B_{\rm{\omega}}\), weighting factor \(r\), and damage softening parameter \(a\). This research is expected to provide theoretical support for the efficient extraction technologies of shale oil and gas. s 2024 14 2  
However, the efficient and safe extraction of oil and gas from shale still poses challenges. Unlike common granite or sandstone, shale is a typical sedimentary rock. Its naturally formed bedding planes during sedimentation give it typical heterogeneity and anisotropy at different scales, significantly affecting the shale's strength characteristics, deformation behavior, and failure modes [12; 13; 14; 15; 16]. Additionally, the inherent plastic deformation and damage behavior of shale also impact the extraction of shale oil and gas [17; 18; 19]. Therefore, correctly understanding the stress-strain behavior of shale and establishing appropriate constitutive models to explain, describe, and predict the plastic and damage laws of shale is crucial for the development of shale oil and gas extraction technologies such as hydraulic fracturing, the stability of wellbore, and core issues such as energy production. Quasi-brittleness and anisotropy are two characteristics of shale. On one hand, shale exhibits quasi-brittleness [20; 21; 22]. Under compressive stress, quasi-brittle geomaterials exhibit inelastic deformation, including degradation of elastic stiffness, strain hardening or softening, and confinement dependency. The stress-strain curve of shale initially shows linear elastic deformation, followed by nonlinear deformation due to the accumulation of minor plastic strains caused by microcrack initiation, crack propagation, and coalescence [17; 23; 24; 25]. After reaching peak stress, shale enters a strain-softening phase where crack accumulation is significant, and irreversible strain markedly increases. The failure behavior in quasi-brittle rocks includes plastic effects; at the mesoscale, the failure process in shale can be understood as two physical phenomena: one is damage behavior caused by microcrack propagation, and the other is macroscopic irreversible strain caused by frictional sliding along crack surfaces, i.e., elastoplastic mechanical behavior [26; 27; 28; 29; 30; 31].