# Contact problems modeling

I am a mathematician.I want to understand the physical setting of my problem.

Physical setting. A large variety of situations involving contact phenomena can be cast in the following general physical setting. Let $$\Omega$$ be a domain in $$\mathbb{R}^{d}$$ representing the reference configuration of a deformable body which, as a result of actions of body forces and boundary tractions, may come in contact with a rigid or a deformable foundation.

The surface of the body $$\Gamma=\partial \Omega$$ is assumed to be composed of three parts $$\Gamma_{1}, \Gamma_{2}$$, and $$\Gamma_{3}$$, relatively closed with mutually disjoint relatively open interiors. Let $$\nu$$ be the unit outward normal vector on $$\Gamma$$. In this chapter we assume that the boundary $$\Gamma$$ is piecewise smooth so that $$\nu$$ exists everywhere except at corner points and relations involving $$\boldsymbol{\nu}$$ are understood to be valid where $$\boldsymbol{\nu}$$ exists. On $$\Gamma_{1}$$ the body is held fixed (clamped), on $$\Gamma_{2}$$ known tractions act and $$\Gamma_{3}$$ is the potential contact surface. At each time instant $$\Gamma_{3}$$ is divided into two parts: one part where the body and the foundation are in contact, and the other part where they are separated. The boundary of the contact part is a free boundary, determined by the solution of the problem. For the sake of generality, we assume that in the reference configuration there exists a gap, denoted by $$g=g(\boldsymbol{x})$$, between $$\Gamma_{3}$$ and the foundation, which is measured along the outer normal $$\nu$$. The setting is depicted in Figure $$1.1$$. We are interested in mathematical models that describe the evolution of the mechanical state of the body during the time interval $$[0, T], T>0$$. To this end, we denote by $$\boldsymbol{u}=\boldsymbol{u}(\boldsymbol{x}, t), \boldsymbol{\sigma}=\boldsymbol{\sigma}(\boldsymbol{x}, t)$$, and $$\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}(\boldsymbol{u})$$ the displacement vector, the stress tensor, and the linearized strain tensor, respectively. The state of the system is completely determined by $$(\boldsymbol{u}, \boldsymbol{\sigma})$$,

the same problem but without gap

My questions are:

1- they say that the body is clamped on $$\Gamma_{1}$$ so how the body can move?

2- the forces of tractions , we imagine that there is a person who tracts this body from the side $$\Gamma_{2}$$ (picture2)?

3-Why the arrow are oriented like that ($$f_0$$ is is the density of applied forces, such as gravity orelectromagnetic forces)?

4- the body moves from right to left?

2. You can assume there is a "person who tracts" if you want, but $$f_2$$ could be an external pressure load for example.
3. There is no definition in the question of $$f_0$$ so it could be anything.