Theory Manual Version 3.4
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Section 7.1: Sliding Interfaces Up Section 7.1: Sliding Interfaces Subsection 7.1.2: Weak Form of Two Body Contact

### 7.1.1 Contact Kinematics

For the most part the notation of this section follows [49], with a few simplifications here and there since the implementation in FEBio is currently for quasi-static, frictionless, two body contact problem.
The volume occupied by body in the reference configuration is denoted by where . The boundary of body is denoted by and is divided into three regions , where is the boundary where tractions are applied, the boundary where the solution is prescribed and the part of the boundary that will be in contact with the other body. It is assumed that .
The deformation of body is defined by . The boundary of the deformed body , that is the boundary of is denoted by where is the boundary in the current configuration where the tractions are applied and similar definitions for and . See the figure below for a graphical illustration of the defined regions.

The two-body contact problem.
Points in body 1 are denoted by in the reference configuration and in the current configuration. For body 2 these points are denoted by and . To define contact, the location where the two bodies are in contact with each other must be established. If body 1 is the slave body and body 2 is the master body, then for a given point on the slave reference contact surface there is a point on the master contact surface that is in some sense closest to point . This closest point is defined in a closest point projection sense: With the definition of established the gap function can be defined, which is a measure for the distance between and , where is the local surface normal of surface evaluated at . Note that when has penetrated body 2, so that the constraint condition to be satisfied at all time is .
Section 7.1: Sliding Interfaces Up Section 7.1: Sliding Interfaces Subsection 7.1.2: Weak Form of Two Body Contact