*3.2. Contact States*

The above contact conditions only apply when two objects are in contact, and they do not exist at the same time. There are generally three types of contact states: bond contact, separation, and sliding contact. The static contact state is also considered in this paper.

#### (1) Bond Contact State

In the bond contact state, the contact interface has good bond properties and has no relative sliding, so Equations (5)–(7) constitute the contact conditions. In addition, the normal and tangential contact forces should not exceed the ultimate tensile load and shear load, respectively:

$$\|\mathbf{N}\_{l}^{t}\| \leq D^{t}f\_{\mathbb{C}}A\_{1} \tag{8}$$

$$\|\|T\_I^t\|\| \le D^t(\mu\_\ast \|\mathbf{N}\_I^t\| + f\_\mathbf{c} A\_1) \tag{9}$$

where *f*<sup>c</sup> is the cohesive force of the contact interface, *A*<sup>1</sup> is the control area of *l*, and *μ*<sup>s</sup> is the static friction coefficient of the contact interface.

#### (2) Separation State

In the separation state, the bond properties of the contact interface are destroyed, and there is no contact between the two objects. Therefore, the contact interface is not restricted by the contact conditions, giving

$$\|\mathbf{N}\_I^t\| = \|T\_I^t\| = 0 \tag{10}$$

#### (3) Sliding Contact State

In the sliding contact state, the bond properties of the contact interface are destroyed, and there is relative sliding. This state is a sliding friction state, so Equations (5) and (7) constitute the contact conditions. At this point, the tangential contact force should be calculated according to the dynamic friction law:

$$\|\|T\_l^t\|\| = D^t \mu\_\mathbf{d} \|\| \mathbf{N}\_l^t\|\|\tag{11}$$

where *μ*<sup>d</sup> is the dynamic friction coefficient of the contact interface.

#### (4) Static Contact State

In the static contact state, the bond properties of the contact interface are destroyed, but there is no relative sliding. This state is a static friction state, so Equations (5)–(7) constitute the contact conditions. At this point, the tangential contact force should not exceed the ultimate shear load:

$$\|\|T\_l^t\|\| \le D^t \mu\_s \|\|\mathbf{N}\_l^t\|\|\tag{12}$$

The above contact conditions are the relationships between the contact displacement and contact force that the contact interface should satisfy when the contact state is known. In the numerical calculation process, the contact state is often not known in advance, so it is necessary to assume and judge it in each calculation step.

#### **4. Dynamic Contact Analysis Method for Rock Mass and Fault**

In the seismic loading process of a system, the contact states between the rock mass and the fault can directly affect the seismic safety of the tunnel structure. Based on the point-to-point contact type in the traditional dynamic contact force method, the pointto-surface contact type is also considered in this paper, thus improving the method for studying the nonlinear dynamic large sliding problem between a rock mass and a fault.
