# Coupled Elastic-Plastic Damage Modeling of Rock Based on Irreversible Thermodynamics
###### Abstract
These behaviors have an intrinsic coupling during the failure process of shale, where the accumulation of frictional sliding leads to an increase in microcracks, which in turn enhances frictional sliding, meaning damage evolution and plastic flow are intrinsically coupled [32; 33; 34; 35; 36; 37; 38]. Therefore, the mechanical behavior of shale should be described using a damage model coupled with plasticity theory. On the other hand, shale exhibits anisotropy. Many experimental studies have shown that the mechanical behavior of rocks has strong anisotropy [39; 40; 41; 42; 43; 44; 45; 46; 47; 48]. Anisotropic constitutive models are inherently complex; to simplify theoretical calculations, previous scholars mostly considered rock materials as isotropic [49; 50; 51; 52; 53; 54; 55]. However, the strength and deformation characteristics of shale vary significantly under different loading directions, and ignoring anisotropy would cause models to inaccurately simulate and predict the bear  
Figure 1: Map of basins with assessed shale oil and shale gas formations [5]. ing capacity, stability, and stress-strain laws of shale in actual conditions. Therefore, the anisotropy of shale must be fully considered. The failure mechanisms in rock mechanics are multifaceted, including but not limited to microcrack slippage, micropore fragmentation, elastic failure of mineral particles, crystal dislocation movement, crack growth speed and direction, and clustering effects between cracks or pores [56; 57; 58; 59; 60; 61; 62]. These mechanisms are both the root causes of rock damage and reasons for energy dissipation. Previous studies have proposed two types of quasi-brittle rock elastoplastic damage coupling models. One is based on micromechanics approaches [63; 64; 65; 66], which consider the behavior of cracks in detail but lead to complex formulations that are often difficult to compute or apply practically. The other is based on continuous damage models, usually grounded in irreversible thermodynamics, analyzing rock damage mechanisms from the perspective of energy dissipation [67; 68; 69; 70; 71]. The resulting constitutive equations comply with the fundamental laws of thermodynamics [51; 72; 73; 74; 75; 76] and effectively characterize the irreversible energy dissipation processes and the thermodynamic state of the material system, grounded in deep thermodynamic principles and rigorous theoretical bases [77; 78; 79; 80]. Thus, determining the quantitative relationship between energy dissipation and damage variables is key to establishing a thermodynamic constitutive model. Previous research has helped understand the mechanical properties of shale under various conditions, essential for optimizing drilling technology, ensuring wellbore stability, and enhancing the efficiency and safety of shale oil and gas extraction. However, current research on shale constitutive models still faces some limitations. (1) Few studies on shale constitutive models within the framework of thermodynamics, leading to models that do not adequately express the irreversible energy dissipation processes in shale materials. (2) Existing models often only consider parts of shale's quasi-brittleness, anisotropy, and thermodynamic principles, lacking a comprehensive theoretical model. (3) The mechanical behavior of shale should be described using damage models coupled with plasticity theory. (4) Some elastoplastic damage coupling models for shale are too complex for calculation or practical application. (5) The understanding of constitutive model parameters is still insufficient, requiring further discussion and summary of model parameter sensitivity. This paper integrates elastoplastic models and theories related to irreversible thermodynamics to construct a shale elastoplastic damage coupling constitutive model. An anisotropy parameter related to loading direction is introduced to reflect the direction dependency and anisotropy of shale's mechanical behavior. Through UMAT subroutine programming, it has been applied within the ABAQUS 2022.