# A Strain Hardening and Softening Constitutive Model for Hard Brittle Rocks
## 3 Mechanical Behaviour of Brittle Hard Rock
### Strength Parameter Evolution Model  
After the rock has reached the initial yield stress, plastic yielding begins to occur. Considering the axial strain \(\varepsilon_{18}\) at the initial yield stress as the zero point of the plastic parameter, the strength parameter and plastic parameter values under the same yield state are determined by the normalised hardening and softening factors based on the Mohr-Coulomb yield criterion. Then, a cohesion and internal friction angle evolution model under different confining pressures with respect to the axial plastic strain \(\varepsilon_{1}^{ps}\) is established. The modelling process is as follows:  
(1) Establish the corresponding relationship between the hardening and softening factors and the axial plastic strain under different confining pressures. Since the hardening and softening factors are defined by the ratio of the stress difference, it is appropriate to consider more points near the peak and residual stress where the stress changes less so that the points are more uniformly distributed on the stress curve. In this way, the evolution law of the strength parameters and axial plastic strain can be discussed better. Considering that less plastic deformation occurs in the strain hardening process than in the strain softening process, five hardening factors \(D_{h}\) = 0, 0.2, 0.5, 0.8, 1.0 and eight softening factors \(D_{ss}\) = 0, 0.1, 0.2, 0.4, 0.6, 0.8, 0.9, 1.0 are selected and substituted into Equations (10) and (11), respectively, to obtain the twelve axial stresses under each confining pressure. The initial yield stress is taken as the starting point of the axial stress, and the relationship curve between the axial stress and axial plastic strain is established. The linear interpolation method is used for 12 axial stress values, and the data regarding the relationship between each hardening and softening factor and axial plastic strain under different confining pressures are approximately obtained. Beishan granite [26] data are shown as an example in Figure 3.  
(2) Solve the cohesion \(c\) and internal friction angle \(\varphi\) corresponding to each hardening and softening factor under different confining pressures. Based on the Mohr-Coulomb yield criterion, the axial stress and confining pressure of rock under different hardening and softening factors are substituted into the following equation as the maximum principal stress \(\sigma_{1}\) and the maximum principal stress \(\sigma_{3}\), respectively. \[\sigma_{1}=a\sigma_{3}+b \tag{16}\]  
where \(a\) is the slope and \(b\) is the intercept. Figure 3: Relationship between hardening factors and softening factors and axial plastic strain under different confining pressures. The coefficients \(a\) and \(b\) obtained by linear fitting can be substituted into Equations (17) and (18), as follows:  
\[\varphi=\arcsin\biggl{(}\frac{a-1}{a+1}\biggr{)} \tag{17}\]  
\[c=\frac{b}{2\sqrt{a}} \tag{18}\]  
The cohesion \(c\) and internal friction angle \(\varphi\) of the rock under each hardening and softening factor can be calculated, as given in Table 2.  
As Table 2 shows, the linear fitting results for axial stress and confining pressure under the same hardening and softening factors are good, and the fitting correlation coefficients are greater than 0.94, which conforms to the shear failure law of rock described by the Mohr-Coulomb yield criterion. It also shows that the hardening factor and softening factor can determine the stress values under the same yield state and different confining pressures. (3) Strength parameter evolution model considering axial plastic strain and confining pressure. After the above two steps, the one-to-one relationship between the strength parameters and the axial plastic strain for the same hardening and softening factors under different confining pressures can be obtained, so the relationship between the cohesion \(c\) and internal friction angle \(\varphi\) of two kinds of hard brittle rocks and the axial plastic strain considering the confining pressure can be captured.