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Theory Manual Version 3.6
 Subsubsection Frictionless Terms Up Subsection 7.1.9: Sliding-Elastic Subsubsection Integration Scheme Frictional Terms

The linearization of the frictional contribution follows from the second term of eq.( as The remaining quantity to be determined in this expression is the linearization ; according to eq.(, must also be evaluated. As depends on via eq.(, we must first propose a temporal discretization scheme for these rate quantities.
In a similar fashion to eq.(, let the material time derivative in the material frame of the covariant basis vectors of be discretized as where denotes the change in from the previous time step. A temporally discretized form of eq.( may now be written as where The directional derivative of may now be evaluated as where and is defined by analogy with eq.( to be where is a skew-symmetric tensor whose dual vector is .
Utilizing the discrete-time equations Eqs.( and ( in eq.(, we find and thus eq.( may be discretized in time, where and In eq.(, is a spatial increment defined to simplify notation and is a projection tensor.
Defining the tensors and vectors the directional derivative may be written fairly compactly as Finally, the tangential stiffness matrix is found to be where The stiffness matrix of the frictional contribution to the virtual work, eq.(, is nonsymmetric. Summing eq.( and eq.( produces the total stiffness matrix for the case of frictional slip, which is also nonsymmetric.
 Subsubsection Frictionless Terms Up Subsection 7.1.9: Sliding-Elastic Subsubsection Integration Scheme