#### Tangent contact stiffness

To find the tangent contact stiffness matrix, please look at Figure 132, part a). At the beginning of a concrete time increment, characterized by time , the slave node at position corresponds to the projection vector on the master side. At the end of the time increment, characterized by time both have moved to positions and , respectively. The differential displacement between slave and master surface changed during this increment by the vector satisfying:

 (228)

Here, satisfies

 (229)

Since (the dependency of on and is dropped to simplify the notation)

 (230)

 (231)

 (232)

 (233)

this also amounts to

 (234)

Notice that the local coordinates take the values of time (the superscript m denotes iteration m within the increment). The differential tangential displacement now amounts to:

 (235)

where

 (236)

Derivation w.r.t. satisfies (straightforward differentiation):

 (237)

 (238)

and

 (239)

The derivative of w.r.t. can be obtained from the derivative of w.r.t. by keeping and fixed (notice that the derivative is taken at , consequently, all derivatives of values at time disappear):

 (240)

Physically, the tangential contact equations are as follows (written at the location of slave node p):

• an additive decomposition of the differential tangential displacement in stick and slip :

 (241)

• a stick law ( , where is the stick slope and the representative slave area for the slave node at stake) defining the tangential force exerted by the slave side on the master side at the location of slave node p:

 (242)

• a Coulomb slip limit:

 (243)

• a slip evolution equation:

 (244)

First, a difference form of the additive decomposition of the differential tangential displacement is derived. Starting from

 (245)

one obtains after taking the time derivative:

 (246)

Substituting the slip evolution equation leads to:

 (247)

and after multiplying with :

 (248)

Writing this equation at , using finite differences (backwards Euler), one gets:

 (249)

where and . The parameter is assumed to be independent of time.

Now, the radial return algorithm will be described to solve the governing equations. Assume that the solution at time is known, i.e. and are known. Using the stick law the tangential forc can be calculated. Now we would like to know these variables at time , given the total differential tangential displacement . At first we construct a trial tangential force which is the force which arises at time assuming that no slip takes place from till . This assumption is equivalent to . Therefore, the trial tangential force satisfies (cf. the stick law):

 (250)

Now, this can also be written as:

 (251)

or

 (252)

Using Equation (249) this is equivalent to:

 (253)

or

 (254)

From the last equation one obtains

 (255)

and, since the terms in brackets in Equation (254) are both positive:

 (256)

The only equation which is left to be satisfied is the Coulomb slip limit. Two possibilities arise:

1. .

In that case the Coulomb slip limit is satisfied and we have found the solution:

 (257)

and

 (258)

No extra slip occurred from to .

2. .

In that case we project the solution back onto the slip surface and require . Using Equation (256) this leads to the following expression for the increase of the consistency parameter :

 (259)

which can be used to update (by using the slip evolution equation):

 (260)

The tangential force can be written as:

 (261)

Now since

 (262)

and

 (263)

where and are vectors, one obtains for the derivative of the tangential force:

 (264)

where

 (265)

One finally arrives at (using Equation (258)

All quantities on the right hand side are known now (cf. Equation (213) and Equation (237)).

In CalculiX, for node-to-face contact, Equation (228) is reformulated and simplified. It is reformulated in the sense that is assumed to be the projection of and is written as (cf. Figure 132, part b))

 (267)

Part a) and part b) of the figure are really equivalent, they just represent the same facts from a different point of view. In part a) the projection on the master surface is performed at time , and the differential displacement is calculated at time , in part b) the projection is done at time and the differential displacement is calculated at time . Now, the actual position can be written as the sum of the undeformed position and the deformation, i.e. and leading to:

 (268)

Since the undeformed position is no function of time it drops out:

 (269)

or:

 (270) (271)

Now, the last two terms are dropped, i.e. it is assumed that the differential deformation at time between positions and is neglegible compared to the differential motion from to . Then the expression for simplifies to:

 (272)

and the only quantity to be stored is the difference in deformation between and at the actual time and at the time of the beginning of the increment.