Theory Manual Version 3.6
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Subsubsection 7.1.9.7: Stick-Slip Algorithm Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.9: Discretization

#### 7.1.9.8 Linearization

To evaluate the linearization in eq.(7.1.9-6) requires directional derivatives of kinematic quantities, some of which are dependent on the stick-slip status. In an attempt to simplify the presentation, the continuum linearization of only a few kinematic quantities is described below, and the majority of the linearization will be deferred until after the discretization presented in Section 7.1.9.9↓. We note that, as a consequence of the double-loop Uzawa algorithm discussed in Section 7.1.9.6↑, the Lagrange multipliers are updated outside of each Newton step and thus and in the following linearizations. Furthermore, in forthcoming sections, refers to an increment in displacement in the trial solution . All other expressions which employ are slight abuses of notation, used to compactly denote changes in a quantity from the previous time step.
Stick
Parametric coordinates on the primary surface are always invariant since they represent material points where the contact integral is to be evaluated (integration points), and in stick the parametric coordinates of contact on the secondary surface are similarly fixed by definition. Accordingly, directional derivatives of and are given by From the above expressions and Eqs.(7.1.9-2) and (7.1.9-4), we find directional derivatives of and to be where is a skew-symmetric tensor whose dual vector is ; thus for any vector . Given the definitions of Eqs.(7.1.9.2-2) and (7.1.9.5-1), along with the relations of eq.(7.1.9.8-1), it follows that
The linearization operator may be brought inside the contact integral of eq.(7.1.9-5) to find where we recall that from eq.(7.1.9.8-1).
Slip
As in stick, the contact integral over is performed over integration points of prescribed parametric coordinates . However, the point on in contact with has parametric coordinates that change with variations in and , in accordance with eq.(7.1.9.1-1). Thus, directional derivatives of and are given by where we recall that is the spatial location of the intersection point at the current time .
We evaluate in terms of increments in solid displacements by means of our modification [8] of a method proposed by Laursen and Simo [65]. Briefly, recognizing that , the directional derivative of this expression is evaluated and the resulting linear system is inverted to yield where , , and In this expression, are approximate contravariant basis vectors on . In the limit of perfect contact (), and become true mating surfaces and a number of relations emerge, including since (see the discussion following eq.(40) in [8]). In the present formulation, this simplification to perfect contact is not adopted, as it was determined that retaining all terms provides better convergence and stability.
From eq.(7.1.9.5-3) it follows that By eq.(7.1.9.5-2), , and Applying the linearization operator to eq.(7.1.9.3-6) yields where according to eq.(7.1.9.3-3).
To linearize partial time derivatives of positions , we adopt Euler integration and find where represents the time increment and denotes the change in from the previous time step. Since is kept constant when evaluating the partial time derivative, the linearization of this expression reduces to
Finally, linearizing Eqs.(7.1.9-3) and (7.1.9.3-5) produces and
Subsubsection 7.1.9.7: Stick-Slip Algorithm Up Subsection 7.1.9: Sliding-Elastic Subsubsection 7.1.9.9: Discretization