# Elastoplastic Constitutive Model for Energy Dissipation and Crack Evolution in Rocks
## 1 Introduction  
The failure process of rocks subjected to load can be characterized in terms of stress, strain, energy, and crack evolution [1, 2]. Stress and strain, which can be directly measured through experimental methods, are commonly employed to develop elastoplastic constitutive models that delineate the mechanical properties of rock materials [3, 4]. However, it is essential to recognize that the failure of rock materials is fundamentally an energy-driven instability phenomenon, serving as a comprehensive representation of crack evolution [5, 6]. Thus, there is an urgent need to establish a rock elastoplastic constitutive model that accurately captures energy dissipation and crack propagation to enhance our ability to describe and predict the actual failure processes of rocks. Rocks do not behave as linear elastic materials. Following plastic yielding, their stress-strain curves demonstrate considerable hardening and softening behaviors, while volumetric strains exhibit distinct dilatancy characteristics [7, 8]. (1) To quantitatively describe the hardening and softening behaviors of rock materials, researchers model the material as an elastoplastic element with uniform internal strain, reflecting the overall deformation effects of rocks under load. They developed corresponding hardening-softeningmodels aimed at accurately predicting the stress-strain curves of rock materials [9; 10]. Currently, these models are deeply integrated with advanced computational capabilities, providing reliable computational tools for rock engineering applications [11; 12]. For instance, the strain softening model implemented in FLAC\({}^{\text{3D}}\) 6.0 allows users to define various mechanical parameters through piecewise linear functions of equivalent plastic strain, thus transforming the challenge of simulating rock hardening and softening into the determination of evolving model parameters. This significantly simplifies the application of these models in engineering numerical simulations [13; 14]. (2) To quantitatively characterize the dilatancy characteristics of rock materials, the dilatancy angle is utilized to quantify the nonlinear volumetric strain of rocks. This concept was first introduced by [PERSON] [15], representing the ratio of plastic volumetric strain to shear strain, which can be derived from flow rules and the corresponding plastic potential function. Subsequently, [PERSON] [16; 17], building on the non-associated flow rule, derived a calculation formula for the dilatancy angle, stipulating that it should lie between zero and the internal friction angle. They highlighted that the associated flow rule would result in no energy dissipation for plastic strain. This formula has since been extensively applied in the stability analysis of rock engineering. For example, [PERSON] [18; 19; 20] systematically examined the significance of the dilatancy angle in predicting rock failure and displacement near excavation boundaries, proposing improved methodologies for the dilatancy angle model, which have been effectively validated through laboratory tests, numerical simulations, and field practices. The process of rock failure under load fundamentally involves the input, accumulation, dissipation, release, and transformation of energy, along with the dynamics of crack closure, initiation, propagation, and coalescence [21; 22; 23; 24], specifically the following: (1) The failure process of rock materials is invariably associated with energy dissipation, which drives the redistribution of internal stresses and nonlinear strains, subsequently influencing the macroscopic mechanical properties of the rock and resulting in mechanical behaviors such as hardening and softening [25; 26]. Excluding the thermal energy generated by environmental temperature fluctuations, the energy input from external work is partially accumulated as elastic strain energy and partially dissipated as dissipative energy.