# A Strain Hardening and Softening Constitutive Model for Hard Brittle Rocks
## 3 Mechanical Behaviour of Brittle Hard Rock
### Strength Parameter Evolution Model
Figure 4a-d show the strength parameter relationship curves with both axial plastic strain and confining pressure for the granite and diabase samples. As shown in Figure 4, the cohesion and internal friction angle evolution laws for granite and diabase are very similar under different confining pressures. On the one hand, with the increase in the axial plastic strain, the cohesion and internal friction angle values first increase and then decrease; on the other hand, with the increase in the confining pressure, the axial plastic strain at the peak and residual stress increase gradually. The reason is that the brittle failure of rock gradually transitions to plastic failure with increasing confining pressure. As far as one confining pressure is concerned, the cohesion and internal friction angle of both hard brittle rocks increase rapidly in the pre-peak hardening stage, decrease gradually in the post-peak strain softening stage and gradually become constant with increasing axial plastic strain. The reason is as follows: when the rock reaches the yield stress, unstable crack propagation occurs. Because the crack surface is rough and does not form a penetrating failure surface, the friction force and biting force between the cracks increase. At this time, the rock has not been damaged but continues to harden with \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{D_{h}/D_{ss}}\)} & \multicolumn{3}{c}{**Granite**} & \multicolumn{3}{c}{**Diabase**} \\ \cline{2-7}  & _c/MPa_ & \(\mathbf{\varphi}^{c}\) & \(\mathbf{R^{2}}\) & _c/MPa_ & \(\mathbf{\varphi}^{c}\) & \(\mathbf{R^{2}}\) \\ \hline \(D_{h}=0\) & 27.29 & 41.44 & 0.94 & 43.91 & 28.54 & 0.97 \\ \(D_{h}=0.2\) & 27.86 & 42.61 & 0.94 & 45.13 & 29.67 & 0.97 \\ \(D_{h}=0.5\) & 28.69 & 44.20 & 0.94 & 46.91 & 31.26 & 0.96 \\ \(D_{h}=0.8\) & 29.50 & 45.65 & 0.94 & 48.61 & 32.73 & 0.95 \\ \(D_{h}=1.0\) & 30.03 & 46.54 & 0.94 & 49.72 & 33.64 & 0.94 \\ \(D_{ss}=0.1\) & 28.00 & 45.70 & 0.95 & 48.24 & 32.29 & 0.95 \\ \(D_{ss}=0.2\) & 25.90 & 44.82 & 0.95 & 46.71 & 30.84 & 0.95 \\ \(D_{ss}=0.4\) & 21.44 & 42.87 & 0.97 & 43.5 & 27.57 & 0.96 \\ \(D_{ss}=0.5\) & 16.56 & 40.65 & 0.98 & 40.03 & 23.68 & 0.98 \\ \(D_{ss}=0.8\) & 11.17 & 38.07 & 0.99 & 36.23 & 18.94 & 0.99 \\ \(D_{ss}=0.9\) & 8.24 & 36.63 & 1.00 & 34.18 & 16.16 & 1.00 \\ \(D_{ss}=1.0\) & 5.12 & 35.05 & 1.00 & 31.99 & 13.00 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The cohesion \(c\) and internal friction angle \(\varphi\) values corresponding to each hardening and softening factor (Beishan granite [26] and Danjiangkou diabase [37]). the propagation of cracks [40]. As the load increases, more cracks form, grow, connect and slip in the rock sample; therefore, the cohesive strength and occlusal force between the cracks gradually decrease and finally tend to become stable [41]. According to the work of reference [42], a smooth and continuous Gaussian function can be adopted to describe the evolution law of the strength parameters of granite and diabase with respect to the axial plastic strain, as follows:  
\[G\Big{(}\varepsilon_{1}^{ps}\Big{)}=\big{(}c_{p}-c_{r}\big{)}\exp\left[-\left( \frac{\varepsilon_{1}^{ps}-\varepsilon_{1,p}^{ps}}{\xi_{c}}\right)^{2}\right]+ c_{r} \tag{19}\]  
\[F\Big{(}\varepsilon_{1}^{ps}\Big{)}=\big{(}\varphi_{p}-\varphi_{r}\big{)}\exp \left[-\left(\frac{\varepsilon_{1}^{ps}-\varepsilon_{1,p}^{ps}}{\xi_{\varphi} }\right)^{2}\right]+\varphi_{r} \tag{20}\]  
Figure 4: Cohesion/internal friction angle–axial plastic strain relationships of two hard brittle rock types under different confining pressures and their best fits. Data of (**a**,**b**) are for Beishan granite [26] and those of (**c**,**d**) are for Danjiangkou diabase [37]. where \(c_{p}\) and \(\varphi_{p}\) are the cohesion and internal friction angle at the peak stress of the rock and \(c_{r}\) and \(\varphi_{r}\) are the cohesion and internal friction angle at the residual stress of the rock, respectively.