# A Strain Hardening and Softening Constitutive Model for Hard Brittle Rocks
## 3 Mechanical Behaviour of Brittle Hard Rock
### Strength Parameter Evolution Model
\(\varepsilon_{1,p}^{ps}\) is the axial plastic strain at the peak stress of the rock under each confining pressure. \(\xi_{c}\) and \(\xi_{\varphi}\) are the cohesion and internal friction angle fitting parameters, which control the width of the model curve. Although the Gaussian function can adequately describe the evolution process of the strength parameters in the strain softening and residual stages, there is still a considerable difference in the description of the evolution process of the strength parameters in the hardening stage. As shown in Figure 5, the difference between the value calculated by the Gaussian function and the experimental value decreases continuously with increasing axial plastic strain at the pre-peak hardening stage, and the difference value is zero at the peak stress. Therefore, [PERSON]'s sigmoid mathematical function is introduced to properly modify the Gaussian function to eliminate the difference in the Gaussian function at the strain hardening stage. After the Gaussian function was modified, a unified strength parameter evolution model of cohesion/internal friction angle-axial plastic strain for two hard brittle rocks under different confining pressures can be expressed as follows:  
\[c\Big{(}\varepsilon_{1}^{ps}\Big{)}=\big{(}c_{p}-c_{r}\big{)}\exp\left[-\left( \frac{\varepsilon_{1}^{ps}-\varepsilon_{1,p}^{ps}}{\xi_{c}}\right)^{2}\right] -\frac{2\Delta X_{c}}{1+\exp\left(\left|\Delta X_{c}\right|\cdot\varepsilon_{1 }^{ps}/\varepsilon_{1,p}^{ps}\right)^{0.9}}+c_{r} \tag{21}\]  
\[\varphi\Big{(}\varepsilon_{1}^{ps}\Big{)}=\big{(}\varphi_{p}-\varphi_{r} \big{)}\exp\left[-\left(\frac{\varepsilon_{1}^{ps}-\varepsilon_{1,p}^{ps}}{ \xi_{p}}\right)^{2}\right]-\frac{2\Delta X_{\varphi}}{1+\exp\left(\left|\Delta X _{\varphi}\right|\varepsilon_{1}^{ps}/\varepsilon_{1,p}^{ps}\right)^{0.9}}+ \varphi_{r} \tag{22}\]  
where:  
\[\Delta X_{c}=G(0)-c_{s} \tag{23}\]  
Figure 5: The fitting results of the Gaussian function before and after modification (the internal friction angle of Beishan granite [26] under 40 MPa). \[\Delta X_{\varphi}=F(0)-\varphi_{s} \tag{24}\]  
where \(c_{s}\) and \(\varphi_{s}\) are the cohesion and friction angle at the initial yield stress, respectively. \(G(0)\) and \(F(0)\) represent the initial cohesion and internal friction angle values calculated by Gaussian Equations (19) and (20). \(\Delta X_{c}\) and \(\Delta X_{\varphi}\) are temporary parameters, which represent the differences between the initial values calculated by Equations (19) and (20) and the experimental values. The modified strength parameter evolution model has the following two advantages: (i) It has fewer fitting parameters and a unified function form. The fitting parameters \(\xi_{c}\) and \(\xi_{\varphi}\) are obtained by Equations (19) and (20), respectively, and the temporary parameters \(\Delta X_{c}\) and \(\Delta X_{\varphi}\) can be calculated by entering the fitting parameters \(\xi_{c}\) and \(\xi_{\varphi}\) into Equations (23) and (24), respectively. The peak parameter \(\varepsilon_{1,p}^{ps}\) can be obtained from the axial plastic strain at the peak stress. \(c_{s}\) and \(\varphi_{s}\) can be obtained by fitting the initial yield stress at each confining pressure. \(c_{p}\) and \(\varphi_{p}\) can be obtained by fitting the peak stress at each confining pressure. \(c_{r}\) and \(\varphi_{r}\) can be obtained by fitting the residual stress at each confining pressure. All fit coefficients for the cohesion/internal friction angle-axial plastic strain curves of granite and diabase are shown in Table 3.  
(ii) The model can smoothly simulate the nonlinear cohesion and internal friction angle processes in the pre-peak hardening stage and the post-peak softening stage, as shown in Figure 4a-d.