Receding Contact in Plane Viscoelasticity

These last two observations were made for Elasticity by Dundurs (1975). 

To show this is left as an exercise. The essential point is contained in Problem 1.2.2, which tells us that the Stieltjes product (1.2.29), in the non-aging case, is commutative, so that k(t) can be moved across the viscoelastic functions. It must be positive in the sense that [k du] 0 for all positive functions u(t), if the inequalities are to remain unaffected. 

It can be shown, by means of two partial integrations, that (2.9.11) holds. 

More detailed results are possible if the bodies in contact are isotropic and such that plane strain, or stress, conditions apply and body forces are zero. Let us suppose, as in the last section, that outside of the contact region, only surface tractions (as opposed to displacements) are prescribed on the boundary and further that any line integral of the stresses around a closed contour T, covering any portion of either or both bodies, is zero  

The vector rij is the outward normal to the contour at a given point. This integral is proportional to the vector sum of the forces acting on 7. It will be zero if no net forces are acting across the contours. If the bodies are simply connected, 

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