# A Strain Hardening and Softening Constitutive Model for Hard Brittle Rocks
## 1 Introduction
The studies mentioned above consider the peak point as the starting point of the softening process, simplify the pre-peak process to one of linear elasticity, ignore the pre-peak rock hardening process and only study the post-peak strength parameter evolution law [22]. To overcome the abovementioned shortcomings, relevant scholars took the initial yield point of rock as the starting point and established strength parameter evolution models to describe the strain hardening and softening process. [PERSON] et al. [23] proposed a linear CWFS model (cohesion weakening-friction strengthening) based on experimental data derived from granite and implemented this model in FLAC2D numerical software to analyse the \"V\" damage caused by tunnel excavation. [PERSON] [24] and [PERSON] [25] proposed a nonlinear CWFS model based on the results of triaxial compression tests performed on granite under different confining stresses, and their model curves were in good agreement with the test curves. [PERSON] et al. [26] analysed the results of triaxial cyclic loading and unloading of Beishan granite and found that the cohesion gradually decreased as an exponential function with the plastic parameter increasing after damage occurred in the rock, while the internal friction angle increased as a log-normal distribution function until the peak was reached and then gradually decreased. On the basis of [PERSON]'s [26] research, [PERSON] et al. [27] proposed a strain softening constitutive model considering the elastic modulus degradation effect. [PERSON] et al. [28] revealed the relationship between the softening parameter, cohesion and the internal friction angle through triaxial cyclic loading and unloading tests. [PERSON] et al. [29] established a two-dimensional strength parameter model considering the confining pressure and plastic parameters and revealed the relationship between the parameters, cohesion and the internal friction angle. It is worth noting that by solving for cohesion and the internal friction angle based on the Mohr-Coulomb yield criterion [20; 26; 27; 28; 29], the rock was assumed to experience the same plastic deformation when reaching peak stress and residual stress under different confining stresses, and on this basis, the relationship between confining pressure and stress under the same plastic deformation was established, and then the cohesion and internal friction angle evolution was obtained. However, the experimental results show that the plastic deformation of the rock differs under different confining pressures when the stress value of the yield state (such as peak stress and residual stress) remains constant [30]. Therefore, the strength parameter evolution law that is based on the same plastic deformation will produce results inconsistent with the experimental results. In summary, on the one hand, there are few methods to determine the stress value of the same yield state under different confining pressures; on the other hand, the current model cannot simultaneously calculate the cohesion and internal friction angle at the initial yield stress, peak stress and residual stress and reflect the evolution law of the strain hardening and softening process. Therefore, the main aim of this paper is to develop a strain hardening and softening constitutive model that can consider the nonlinear characteristics of rocks under different confining pressures. The constitutive model proposed in this paper differs from other models based on the following three characteristics:  
(i) The normalised hardening factor and softening factor are defined to characterise the yield state of the rock at the initial yield stress point, peak stress point, residual stress point and any yield stress points. (ii) A unified strength parameter evolution model is proposed, and it can reflect the variations in cohesion and the internal friction angle in the subsequent yield process.