| Matrix or Vector | Geometry | Shape Functions | Integration Points |
|---|---|---|---|
| Stiffness Matrix | Normal Direction | None | None |
| Sliding Direction | None | None |
| Load Type | Distribution |
|---|---|
| Element Temperature | None - average used for material property evaluation |
| Nodal Temperature | None - average used for material property evaluation |
CONTA178 represents contact and sliding between any two nodes of any types of elements. This node-to-node contact element can handle cases when the contact location is known beforehand.
CONTA178 is applicable to 3-D geometries. It can also be used in 2-D and axisymmetric models by constraining the UZ degrees of freedom. The element is capable of supporting compression in the contact normal direction and Coulomb friction in the tangential direction.
Four different contact algorithms are implemented in this element.
Pure penalty method
Augmented Lagrange method
Pure Lagrange multiplier method
Lagrange multiplier on contact normal penalty on frictional direction
Pure Penalty Method
The Newton-Raphson load vector is:
 | (13–339) |
where:
| Fn = normal contact force |
| Fsy = tangential contact force in y direction |
| Fsz = tangential contact force in z direction |
 | (13–340) |
where:
| Kn = contact normal stiffness (input FKN on R command) |
| un = contact gap size |
 | (13–341) |
where:
| Ks = tangential contact stiffness (input as FKS on R command) |
| uy = contact slip distance in y direction |
μ = coefficient of friction (input as MU on TB command with Lab = FRIC or MP command) |
Augmented Lagrange Method
 | (13–342) |
where:
|
| ε = user-defined compatibility tolerance (input as TOLN on R command) |
The Lagrange multiplier component of force λ is computed locally (for each element) and iteratively.
Pure Lagrange Multiplier Method
The contact forces (i.e., Lagrange multiplier components of forces) become unknown DOFs for each element. The associated Newton-Raphson load vector is:
 | (13–343) |
Lagrange Multiplier on Contact Normal Penalty on Frictional Direction
In this method only the contact normal face is treated as a Lagrange multiplier. The tangential forces are calculated based on penalty method:
 | (13–344) |
The damping capability is only used for modal and transient analyses. Damping is only active in the contact normal direction when contact is closed. The damping force is computed as:
 | (13–345) |
where:
| V = relative velocity between two contact nodes in contact normal direction |
 |
| Cv1 = constant damping coefficient (input as CV1 on R command) |
| Cv2 = linear damping coefficient (input as CV2 on R command) |
If you know that a CONTA178 element will be in sliding status throughout the analysis, and that the relative displacement of the two nodes will be monotonically increasing, the rigid Coulomb friction option (KEYOPT(10) = 7) can be used to avoid convergence problems. This option removes the stiffness in the sliding direction, as shown in Figure 13.37: Force-Deflection Relations for Rigid Coulomb Option. Note that if the relative displacement does not increase monotonically, the convergence characteristics of the rigid Coulomb friction law (KEYOPT(10) = 7) will be worse than for the elastic Coulomb friction law (KEYOPT(10) = 0).
