*3.1. Contact Conditions*

Figure 1 shows a typical contact problem. Ω<sup>1</sup> and Ω<sup>2</sup> are two objects, whose boundaries are Γ<sup>1</sup> and Γ2, respectively. *Su* and *S<sup>σ</sup>* are the given displacement and stress boundaries, respectively. *P* and *p* are the external loads. *l* and *l* are a contact point pair and *n*<sup>1</sup> and *t*<sup>1</sup> are the unit normal and tangential vectors of the contact interface, respectively. When the two objects are in contact, the contact system should satisfy the contact displacement and contact force conditions in addition to the boundary and initial conditions. This paper assumes that there is no initial gap between the two objects.

**Figure 1.** Contact sketch of two objects.

(1) Contact Displacement Conditions

For two squeezed objects, it is generally considered that they will not embed into each other, so the contact point pair *l* and *l* should satisfy the displacement condition of no embedding in the normal direction:

$$
\mu\_1^\mathrm{T} (\mathbf{U}\_{l'}^{t+\Delta t} - \mathbf{U}\_{l}^{t+\Delta t}) = 0 \tag{5}
$$

where *Ut*+Δ*<sup>t</sup> <sup>l</sup>* and *<sup>U</sup>t*+Δ*<sup>t</sup> <sup>l</sup>* are the displacement vectors of *l* and *l*, respectively, at time *t* + Δ*t*. The direction of *n*<sup>1</sup> is from *l* to *l* .

For a contact point pair without relative sliding, the displacement condition of no relative sliding in the tangential direction within time *t*–*t* + Δ*t* should be satisfied:

$$\mathbf{t}\_1^T(\mathbf{U}\_{l'}^{t+\Delta t} - \mathbf{U}\_{l}^{t+\Delta t}) = \mathbf{t}\_1^T(\mathbf{U}\_{l'}^t - \mathbf{U}\_{l}^t) \tag{6}$$

#### (2) Contact Force Conditions

For two objects in contact, the interaction forces between the contact point pair should satisfy Newton's third law; that is, the following contact force conditions should be satisfied:

$$\mathbf{N}\_{l'}^{t} = -\mathbf{N}\_{l'}^{t} \ \mathbf{T}\_{l'}^{t} = -\mathbf{T}\_{l}^{t} \tag{7}$$

where *N<sup>t</sup> <sup>l</sup>* and *<sup>N</sup><sup>t</sup> <sup>l</sup>* are the normal contact forces of *l* and *l*, respectively. *T<sup>t</sup> <sup>l</sup>* and *<sup>T</sup><sup>t</sup> <sup>l</sup>* are the tangential contact forces of *l* and *l*, respectively.
