| Matrix or Vector | Shape Functions | Integration Points |
|---|---|---|
| Stiffness Matrix | None (nodes may be coincident) | None |
| Load Type | Distribution |
|---|---|
| Element Temperature | None - average used for material property evaluation |
| Nodal Temperature | None - average used for material property evaluation |
CONTAC12 may have one of three conditions if the elastic Coulomb friction option (KEYOPT(1) = 0) is used: closed and stuck, closed and sliding, or open. The following matrices are derived assuming that θ is input as 0.0.
Closed and stuck. This occurs if:
(1–21)
where:
μ = coefficient of friction (input as MU on TB command with Lab= FRIC or MP command)Fn = normal force across gap Fs = sliding force parallel to gap The normal force is:
(1–22)
where:
kn = normal stiffness (input as KN on R command un,I = displacement of node I in normal direction un,J = displacement of node J in normal direction 
d = distance between nodes The sliding force is:
(1–23)
where:
ks = sticking stiffness (input as KS on R command) us,I = displacement of node I in sliding direction us,J = displacement of node J in sliding direction uo = distance that nodes I and J have slid with respect to each other The resulting element stiffness matrix (in element coordinates) is:
(1–24)
and the Newton-Raphson load vector (in element coordinates) is:
(1–25)
Closed and sliding. This occurs if:
(1–26)
In this case, the element stiffness matrix (in element coordinates) is:
(1–27)
and the Newton-Raphson load vector is the same as in Equation 1–25. If the unsymmetric option is chosen (NROPT,UNSYM), then the stiffness matrix includes the coupling between the normal and sliding directions; which for STAT = 2 is:
(1–28)
Open - When there is no contact between nodes I and J. There is no stiffness matrix or load vector.
Figure 1.2: Force-Deflection Relations for Standard Case shows the force-deflection relationships for this element. It may be seen in these figures that the element is nonlinear and therefore needs to be solved iteratively. Further, since energy lost in the slider cannot be recovered, the load needs to be applied gradually.
The element is normally oriented based on θ (input as THETA on R command). If KEYOPT(2) = 1, however, θ is not used. Rather, the first iteration has θ equal to zero, and all subsequent iterations have the orientation of the element based on the displacements of the previous iteration. In no case does the element use its nodal coordinates.
If the user knows that a gap element will be in sliding status for the life of the problem, and that the relative displacement of the two nodes will be monotonically increasing, the rigid Coulomb friction option (KEYOPT(1) = 1) can be used to avoid convergence problems. This option removes the stiffness in the sliding direction, as shown in Figure 1.3: Force-Deflection Relations for Rigid Coulomb Option. It should be noted that if the relative displacement does not increase monotonically, the convergence characteristics of KEYOPT(1) = 1 will be worse than for KEYOPT(1) = 0.
