# Elastoplastic Constitutive Model for Energy Dissipation and Crack Evolution in Rocks
## 1 Introduction
When the accumulated elastic strain energy surpasses its storage capacity, it converts into dissipated energy and is released externally, culminating in rock failure [27; 28]. In this context, [PERSON] [29; 30] examined the energy transformation laws during rock failure from a thermodynamic perspective, indicating that these laws more accurately reflect the essence of rock failure and asserting that dissipated energy provides a more comprehensive description of the progressive failure process of rock materials. Presently, dissipated energy is computed by integrating stress-strain curves and subtracting the elastic strain energy, thus becoming a crucial parameter for characterizing the transition from deformation to instability and failure in rocks. This methodology has facilitated the development of constitutive models of rock based on energy conservation principles and has enhanced their application in numerical simulations [31; 32; 33]. (2) The failure process of rock materials is also characterized by crack evolution, which results in stress concentration and uneven deformation within the material, thereby impacting the overall mechanical properties of the rock and causing the volumetric strain to exhibit nonlinear characteristics, such as dilatancy [34; 35; 36]. For an extended period, the internal failure and overall deformation attributed to crack evolution have been focal points of research, primarily due to the random distribution of cracks and the difficulties associated with predicting their evolution [37; 38; 39]. To tackle this issue, researchers have introduced the concept of crack volumetric strain as a means to quantitatively characterize crack evolution. This value is determined by calculating the volumetric strain and subtracting the elastic volumetric strain [40; 41]. This approach circumvents the complexities associated with random crack distribution by concentratingon the total volumetric change resulting from crack progression, effectively sidestepping the analytical challenges presented by randomness. In recent years, crack volumetric strain has been extensively utilized in experimental research and theoretical analysis. It not only elucidates the mechanical mechanisms underlying rock deformation and failure but also effectively delineates the rules governing crack strain evolution and wing crack length through the establishment of corresponding crack propagation models [42; 43; 44]. In summary, despite significant advancements in the prediction of hardening-softening behavior and dilatancy characteristics in elastoplastic constitutive models for rocks, there has been limited research on the representation of energy dissipation and crack evolution within these frameworks. Consequently, this paper firstly analyzes the elastoplastic characterization of dissipated energy and crack volumetric strain in rocks subjected to conventional triaxial compression, elucidating their specific representation in a theoretical elastoplastic constitutive model. Subsequently, a three-dimensional finite difference format is derived to implement the numerical algorithm of the theoretical model in FLAC\({}^{\text{3D}}\). Finally, using argillaceous siltstone as a case study, the model is validated, demonstrating its capability to accurately predict energy dissipation and crack evolution. The findings of this research provide a theoretical basis for the elastic-plastic modeling of energy dissipation and crack evolution in rocks, as well as important insights for the development and validation of such models.