Shear mode

For the shear mode a Coulomb friction envelope is used. The yield function has the form

$$f_2(\mbox{\boldmath$\sigma$},\kappa_2) = \vert\tau\vert+\sigma\tan\phi(\kappa_2)-c(\kappa_2)$$ (24)

According to [6] the variations of friction angle $\phi$ and cohesion $c$ are assumed as

$$c = c_0\exp\left(-\mbox{$\displaystyle\frac{c_0}{G^{II}_f}$}\kappa_2\right)$$ (25)

$$\tan\phi = \tan\phi_0+(\tan\phi_r-\tan\phi_0)\left(\mbox{$\displaystyle\frac{c_0-c}{c_0}$}\right)$$ (26)

where $c_0$ is initial cohesion of joint, $\phi_0$ initial friction angle, $\phi_r$ residual friction angle, and $G^{II}_f$ fracture energy in mode II failure. A non-associated plastic potential $g_2$ is considered as

$$g_2=\vert\tau\vert+\sigma\tan\Phi-c$$ (27)

Figure: Shear behavior of proposed model for different confinement levels in MPa ( $c_0=0.8\ \rm{MPa},\ \tan\phi_0=1.0,\ \tan\phi_r=0.75,{\rm and}\ G_f^{II}=0.05\ {N/mm}$)