*2.2. Calculation Formula for the Vibration Deterioration Coefficient*

Li et al. [23] conducted rock shear tests under different shear velocities, normal stresses, and undulating angles, using artificial concrete simulation samples. It was found that as the shear deformation velocity increased, the shear strength of the rock joint decreased, and the decreasing rate also decreased gradually. The attenuation laws of the joint strength presented negative exponential changing characteristics, so the influence coefficient of the relative velocity could be quantitatively described using a negative exponential model:

$$
\gamma^t = \alpha \left( \left| \Delta \dot{U}^t \right| + \beta \right)^{-\lambda} \tag{2}
$$

where *<sup>α</sup>*, *<sup>β</sup>*, and *<sup>λ</sup>* are undetermined coefficients and <sup>Δ</sup> . *Ut* is the relative shear velocity in mm/s.

Experimental research [19] indicates that as the number of shear cycles increases, the shear strength of the structural plane decreases, and decreases rapidly in the initial stage. After several shear cycles, the strength value remains unchanged, which can also be described by negative exponential attenuation laws. Therefore, the influence coefficient of the vibration wear can be expressed as

$$\eta^t = P\_0 + (1 - P\_0) \exp(-\xi \frac{t}{t\_{\rm W}}) \tag{3}$$

where *ξ* is an undetermined coefficient, *t*<sup>w</sup> is the whole duration of the earthquake, and *P*<sup>0</sup> is a convergence constant.

Combining Equations (1)–(3), the calculation formula for the vibration deterioration coefficient of the contact interface can be obtained:

$$D^t = \left[ a \left( \left| \Delta \dot{U} \right|^t + \beta \right)^{-\lambda} \right] \left[ P\_0 + (1 - P\_0) \exp(-\xi \frac{t}{t\_{\rm W}}) \right] \tag{4}$$

The undetermined coefficients *α*, *β*, *λ*, *P*0, and *ξ* in Equation (4) should be determined according to the shear tests of rock materials in practical engineering. The relative shear velocity <sup>Δ</sup> . *Ut* of the contact interface is obtained by the time-step integration of finite elements, so that the vibration deterioration coefficient *D<sup>t</sup>* can be solved explicitly in each time step for seismic loading.

#### **3. Contact Conditions and Contact States**
