Prev Subsubsection 188.8.131.52: Facet-To-Facet Sliding Up Subsection 7.1.8: Alternative Formulations Section 7.2: Biphasic Contact Next
The second alternative differs more significantly from the method described above. It also begins with the definition of a single contact integral over the slave surface. But a different derivation is followed to obtain the linearization of this contact integral. The main reason for this difference is a subtly alternative definition for the gap function. In this method, it is defined as follows. where, is the normal of the slave surface (opposed to the master normal as used in the derivation above). In this case, the point is no longer the closest point projection of onto the master surface, but instead is the normal projection along . The linearization of equation (184.108.40.206-2) now becomes, where, are the tangent vectors to the master surface at . Note that since is normal to the slave surface, equation (220.127.116.11-3) does not reduce to equation (7.1.2-7).
In one assumes frictionless contact, the contact traction can be written as follows, where, are the tangent vectors to evaluated at . Using (18.104.22.168-4) we can rewrite the contact integral as follows. where we assumed that the integration domain can be mapped to a 2D parametric domain, .
The linearization of (22.214.171.124-5) now proceeds in the usual fashion. Omitting the details, it can be shown that the linearization of the contact integral results in, where, and .
The discretization of the contact integral and its linearization now proceeds as usual. We will not derive the details, but it is important to point out that the resulting stiffness matrix for this particular contact formulation is not symmetric. Although this method has shown to give good results, especially in large compression problems, it was desirable to derive a symmetric version as well. Because of this, a slightly different formulation was also developed that does reduce to a symmetric stiffness matrix although this symmetric version did not seem to perform as well as the non-symmetric one.