By J. A. C. Martinis, M. Raous (eds.)

The publication addresses instability and bifurcation phenomena in frictional touch difficulties. The remedy of this topic has its roots in earlier experiences of instability and bifurcation in elastic, thermoelastic or elastic-plastic our bodies, and in past mathematical, mechanical and computational reports of unilateral difficulties. The salient characteristic of this publication is to place jointly and advance strategies and instruments for balance and bifurcation reviews in mechanics, bearing in mind the inherent non-smoothness and non-associativity (non-symmetry) of unilateral frictional touch legislation. The mechanical foundations, the mathematical conception and the computational algorithms for such experiences are constructed alongside six chapters written by means of the teachers of a CISM path. these options and instruments are illustrated not just with enlightening educational examples but additionally with a few not easy business functions, similar, specifically, to the car industry.

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**Additional info for Friction and Instabilities**

**Example text**

3 Example 3 This example is from (Bjorkman, 1992) and consists of three elastic bars and two flat rigid surfaces, see Figure 5. The elstic surface is placed so that the two upper bars are initially compressed. 52 A. ·. ~:: '· I• 1DD h• 50 0 1• 1 •forcll Figure 5. Example 3: From (Bjorkman, 1992). The geometry. 53 Stability and Critical Points in Large Displacement Frictionless Contact Problems 0.. JOO. Figure 6. Example 3: From (Bjorkman, 1992). Load parameter versus horizontal displacement. The dashed line is the equilibrium path for the same problem without obstacles.

Substituting a solution path into (8), (9) and (10) and taking right hand derivatives with respect to s, using the notations p and ,\ for such derivatives, we conclude that Pnc = 0 and that the following relations hold: u, Kr rN - Nc 9c N'[0 N'0{c1 0 W 0 2 0, f 1 r~cu 1 r0 1+ ,\ r~:: 0 P9 c Pgc 2 0, w W 8 Pgc (11) a>. = 0, (12) where w is a slack variable, and vectors p and 1ft are decomposed into subvectors in an obvious manner; arguments are also omitted in the latter vector. Equations (11) and (12), which we may call the first order system, are to be compared to equation (6) of the non-contact case.

At the contact nodes we have (65) where (66) 29 Instability of Thermoelastic Contact Q and they are given by We denote this subset of non-zero nodal heat sources Q=JVP, (67) where P is the corresponding vector of nodal contact forces normal to the contact interface, f is the coefficient of friction and V = is the local relative sliding velocity. In the axisymmetric geometry, the relative sliding speed varies with radius, being given by V=wr, (68) where w = lw 1 - w2l is the relative angular velocity.