# A Strain Hardening and Softening Constitutive Model for Hard Brittle Rocks
## 1 Introduction  
With the advancement of technology, human society's demand for deep underground rock engineering construction has increased [1; 2]. Many deep underground structures are built in hard brittle rock masses such as granite and diabase [3; 4]. The fracture or even failure of hard brittle rock is the most important challenge in engineering construction. Hard brittle rocks often exhibit obvious nonlinear mechanical properties such as strain hardening/softening and plastic dilatation under high in situ stress [5]. Therefore, the in-depth study of the nonlinear mechanical behaviours of hard brittle rocks and the establishment of a hardening and softening constitutive model is of great significance for engineering design and engineering safety assessment [6]. The phenomenon in which a significant increase or decrease in stress accompanies increasing strain after a rock has yielded under compression conditions is known as strain hardening or strain softening of rocks. In view of this nonlinear mechanical behaviour, many constitutive models have been proposed. Based on the traditional strength yield criterion, the damage constitutive models have been proposed [7]. These models can describe the nonlinear behaviour of the rock under loading but do not consider the true changes in the subsequent yield surface of the rock at the strain hardening and strain softening stages [8]. In recent years, some scholars have used plastic parameters as independent variables to characterise the strain hardening/softening behaviour of rocks and represent the evolution of subsequent yield surfaces and have proposed elastic-plastic strain hardening and softening constitutive models [9; 10]. Because the Mohr-Coulomb yield criterion can satisfactorily describe the shear failure mechanism, it has been widely adopted as the yield surface equation. The following two strength parameters are needed to determine the evolution law of the yield surface: cohesion and the internal friction angle [11]. Yielding occurs once the shear stress of the rock reaches a stress threshold. Based on the Mohr-Coulomb yield criterion, various strength parameter evolution models are proposed. According to the strength parameters at peak stress and residual stress, the strength parameters decrease linearly with the increase in the plastic parameter [12; 13]. Based on the degradation index concept proposed by [PERSON] and [PERSON] [14], the cohesion and internal friction angle strength degradation index was proposed to describe the process of linear reduction of the post-peak strength parameters [15], and [PERSON] et al. [16] proposed three strain softening models by adopting the drop modulus variation. [PERSON] and [PERSON] [17] and [PERSON] and [PERSON] [18] obtained a finite strain solution for circular underground tunnels in a Mohr-Coulomb elastoplastic coupling strain softening rock mass, assuming that cohesion and the internal friction angle decrease linearly with increasing plastic parameters. The application of the linear reduction model of post-peak strength parameters is relatively simple, but the actual softening process of the post-peak stage is ignored. To accurately describe the post-peak mechanical behaviour of rock, [PERSON] et al. [19] used multiple piecewise linear functions to approximate the nonlinear softening behaviour after the peak stress was reached. [PERSON] et al. [20] proposed that the decreasing law of the internal friction angle conforms to a quadratic function and that cohesion decreases linearly after the peak stress is reached. Based on the triaxial test data of sandstone, mudstone and argillaceous sandstone, [PERSON] et al. [21] proposed a strength parameter evolution model of constant friction and cohesion weakening, assuming the internal friction angle remains constant during the post-peak softening process.