# A Strain Hardening and Softening Constitutive Model for Hard Brittle Rocks
## 3 Mechanical Behaviour of Brittle Hard Rock
### Strength Parameter Evolution Model
Compared with the Gaussian function before correction, the modified model can compensate for the disadvantage that the strength parameter evolution process in the pre-peak hardening stage cannot be accurately described, as shown in \[G\Big{(}\varepsilon_{1}^{ps}\Big{)}=\big{(}c_{p}-c_{r}\big{)}\exp\left[-\left( \frac{\varepsilon_{1}^{ps}-\varepsilon_{1,p}^{ps}}{\xi_{c}}\right)^{2}\right]+ c_{r} \tag{19}\]0 The fitting correlation coefficients of the strength parameter evolution model are mostly distributed in the range from 0.96 to 1.00 (Table 3), which illustrates that the established strength parameter evolution model is correct and reasonable. As Table 3 shows, the peak parameter \(\varepsilon_{1,p}^{ps}\) and the fitting parameters \(\xi_{c}\) and \(\xi_{\varphi}\) gradually increase with increasing confining pressure. The reason is that the rock has more obvious plastic characteristics under high confining pressures, and more plastic deformation occurs after reaching the peak and residual stresses, so the peak parameter \(\varepsilon_{1,p}^{ps}\) becomes larger and the stress-strain curve appears wider, and the fitting parameters \(\xi_{c}\) and \(\xi_{\varphi}\), which control the width of the model, also increase gradually with increasing confining pressure. The confining pressure dependence of \(\varepsilon_{1,p}^{ps}\), \(\xi_{c}\) and \(\xi_{\varphi}\) can be observed in \[G\Big{(}\varepsilon_{1}^{ps}\Big{)}=\big{(}c_{p}-c_{r}\big{)}\exp\left[-\left( \frac{\varepsilon_{1}^{ps}-\varepsilon_{1,p}^{ps}}{\xi_{c}}\right)^{2}\right]+ c_{r} \tag{19}\]1 The variations in these parameters with confining pressures can be described by a unified empirical function, as follows:  
\[G\Big{(}\varepsilon_{1}^{ps}\Big{)}=\big{(}c_{p}-c_{r}\big{)}\exp\left[-\left( \frac{\varepsilon_{1}^{ps}-\varepsilon_{1,p}^{ps}}{\xi_{c}}\right)^{2}\right]+ c_{r} \tag{19}\]2  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Rock** \\ **Type** \\ \end{tabular}  } & \multirow{2}{*}{\(\sigma_{3}\)/MPa} & \multirow{2}{*}{\(\varepsilon_{1,p}^{ps}\)/\%} & \multicolumn{5}{c}{\(c\)} & \multicolumn{5}{c}{\(c\)} & \multicolumn{5}{c}{\(\varphi\)} \\ \cline{3-13}  & & & \(\xi_{c}\) & & \(c_{s}\)/MPa & \(c_{p}\)/MPa & \(c_{r}\)/MPa & \(\mathbf{R^{2}}\) & \(\xi_{\varphi}\) & \(\varphi_{s}\)/\(c^{0}\) & \(\varphi_{p}\)/\(c^{0}\) & \(\varphi_{p}\)/\(c^{0}\) & \(\mathbf{R^{2}}\) \\ \hline \multirow{4}{*}{Granite} & 0.2 & 0.003 & 0.223 & 27.29 & 30.03 & 5.12 & 0.98 & 0.200 & 41.44 & 46.54 & 35.05 & 0.91 \\  & 1 & 0.057 & 0.282 & 27.29 & 30.03 & 5.12 & 0.99 & 0.282 & 41.44 & 46.54 & 35.05 & 0.98 \\  & 3.5 & 0.078 & 0.467 & 27.29 & 30.03 & 5.12 & 0.99 & 0.462 & 41.44 & 46.54 & 35.05 & 0.97 \\  & 10 & 0.120 & 0.379 & 27.29 & 30.03 & 5.12 & 0.99 & 0.364 & 41.44 & 46.54 & 35.05 & 0.98 \\  & 20 & 0.149 & 0.525 & 27.29 & 30.03 & 5.12 & 1.00 & 0.517 & 41.44 & 46.54 & 35.05 & 0.99 \\  & 40 & 0.239 & 0.672 & 27.29 & 30.03 & 5.12 & 0.99 & 0.663 & 41.44 & 46.54 & 35.05 & 0.98 \\ \hline \multirow{4}{*}{Diabase} & 5 & 0.031 & 0.154 & 43.91 & 49.72 & 32.00 & 0.96 & 0.166 & 28.54 & 33.64 & 13.00 & 0.98 \\  & 10 & 0.060 & 0.212 & 43.91 & 49.72 & 32.00 & 0.99 & 0.214 & 28.54 & 33.64 & 13.00 & 0.98 \\ \cline{1-1}  & 20 & 0.085 & 0.  
where \(P\) represents one of the parameters \(\varepsilon_{1,p}^{ps}\), \(\xi_{c}\) or \(\xi_{\varphi}\). \(P_{0}\), \(A_{1}\), \(A_{2}\), \(t_{1}\), and \(t_{2}\) are fitting coefficients, and all values are positive. In addition, \(P_{0}\) represents the range of the confining pressure function; \(A_{1}\) and \(A_{2}\) jointly control the increase in the amplitude of the confining pressure function; and \(t_{1}\) and \(t_{2}\) jointly control the growth rate of the confining pressure function.