## 1 Introduction  
As is widely recognized, the process of rock damage and failure under external loading involves various energy-related phenomena, such as energy input, accumulation, conversion, dissipation, and release [1, 2]. Prior to the occurrence of rock damage, a portion of the external work is stored within the rock as elastic energy, while the remaining portion is dissipated in the form of surface energy, thermal energy, sound energy, electromagnetic radiation energy, and other forms [3]. This part of the dissipated energy is collectively referred to as rock dissipated energy [4, 5]. During this stage, the energy conversion and dissipation within the rock are balanced, whereby any excess energy resulting from the conversion of external work into elastic energy is dissipated as a consequence of rock damage. However, when the elastic energy accumulated within the rock reaches its limit, or when the rate of external work conversion into elastic energy surpasses the capacity for elastic energy conversion, the equilibrium between energy conversion and dissipation within the rock is disrupted, leading to rock failure. Throughout the rock failure process,the rock releases energy to its surroundings in various forms, including thermal energy, sound energy, electromagnetic radiation energy, and kinetic energy, among others. In line with the principles of damage mechanics, the process of rock mechanical performance degradation and deformation failure involves the manifestation of dissipated energy and plastic strain as specific indicators of the influence exerted by damage variables on energy and deformation in rocks [6; 7; 8]. The evolution of dissipated energy and plastic strain reflects the physical processes of mechanical property degradation in rock materials [9; 10]. Both dissipated energy and plastic strain are irreversible, meaning their evolution process is irrecoverable. This is consistent with the irreversible nature of the damage variable [11; 12]. Therefore, dissipated energy and plastic strain can also be regarded as derived variables of rock damage [13; 14]. Notably, dissipated energy and plastic strain are closely intertwined, with the occurrence of plastic strain in rocks being invariably accompanied by the dissipation of internal energy, and the dissipation of energy being accompanied by the irreversible deformation of rocks. As such, the investigation of the correlation characteristics between dissipated energy and plastic strain in the process of rock damage assumes paramount importance in comprehending the evolution of rock damage [15; 16; 17]. The study of energy conversion and dissipation mechanisms within rocks during the processes of damage and instability has garnered considerable attention for a long time [18; 19]. Academician [PERSON] [20; 21] conducted systematic research on the internal energy conversion and dissipation mechanisms in rocks and proposed the concept of characterizing rock damage states through energy release mechanisms. [PERSON] [22] introduced internal constraints to the constitutive relationship, damage internal variables, and variables associated with rock damage, and derived the Clausius-Duhem inequality as well as the damage driving force inequality. This paved the way for the exploration of various aspects such as constitutive models, fissure damage, closure of frictional sliding in fissures, hydraulic coupling, and aging damage. [PERSON] [23; 24; 25] explained the coal-rock damage and failure process based on thermodynamic theory, proposed an energy balance differential equation based on the first law of thermodynamics, and established a coal-rock constitutive model based on energy relationships under uniaxial compression. [PERSON] [26] established a constitutive model suitable for cyclic loading and unloading processes that considers energy dissipation and compaction effects, which can more accurately describe the nonlinear strength and deformation characteristics of coal materials.