On the transient dynamic antiplane contact problem in the presence of dry friction and slip

Abstract

This article models the elastodynamic transient contact between two elastically similar half planes under antiplane loading and in the presence of friction. Contact is maintained along the positive real line under the presence of a certain remote contact pressure. An antiplane shear load is applied, which entails interfacial shear traction that opposes the frictional force entailed by the contact pressure. In order to balance the surface tractions, the surface must be allowed to slip. We derive the closed form solution of the interfacial traction due to a general antiplanar displacement distribution using a variant of the Wiener-Hopf technique. We also find closed-form expressions for the interfacial shear traction due to this remote antiplane load. In combination with the frictional force, this leads to an integral equation the solution to which is the distribution of relative slip. We quantify both this and the magnitude of the interfacial shear tractions under diverse loading, showing that transient loading leads to partial reverse slip of the contact surfaces. We show that the reverse slip tends to vanish over time, and that it is ameliorated if the friction coefficient is reduced.

On the transient planar contact problem in the presence of dry friction and slip

“…Crucially, with such simple form it is possible to perform an explicit integration of eqn.7.7 with eqn.7.8. A similar advantage to collocation methods in dynamic problems was observed by Martin and Rizzo [85] and in [55], whereupon we reach an analytic form for the convolution of the kernel with N ij . By calling…”

“…In principle, the unknown is b x (x, t) -the slip velocity density -, found as b x , K a + K b . The problem therefore can be thought of as a deconvolution problem, and is agreeable to a treatment similar to the antiplanar case given in [55]. However, for numerical regularity purposes, it becomes easier to use the coordinate transformation introduced in section 4.4.…”

“…This is done in the interest of space, since there are infinitely many ways a steady state load of magnitude P 0 (or Q 0 ) may be reached when time is an explicit field variable. More complex time-dependent loading is nevertheless possible (e.g., the ramp loading considered in [55]). In the event the remote loading can be expressed as the product of a time dependent function P = P 0 f (t), Q = Q 0 g(t), then the ensuing interfacial tractions (call them p * (x, t) and q * (x, t)) can simply be obtained as time convolutions of the result we have presented here, as…”

“…To that end, the traditional Cattaneo-Mindlin style approach introduces a 'corrective' interfacial traction generated by the distribution of slip [3], the description of which can be provided in a number of ways, including as a continuous distribution of straight dislocations (see for instance [52,53,54]). In [55], the author discussed how to express the interfacial shear slip of an antiplanar contact subjected to inertial loads as a distribution of Burgers vectors. In so doing, the contact problem was shown to resemble the classical Cattaneo-Mindlin [49,50] problem, but owing to the time dependencies implied by the inherently transient, elastodynamic contact loads, the problem could only be solved numerically to highlight the presence of a regime of transient forward and reverse slip.…”

The elastodynamic bimaterial interface under mode I and mode II loading

“…Our solutions enable the modelling of these crack's propagation as convolution kernels. Finally, the solutions presented here pave the way for the analytic study of the bimaterial contact problem in the presence of friction, which can now be achieved using an approach similar to that discussed in [64,48]. Consequently, this article offers an analytic treatment for a problem of significance both in geophysics and materials science: elastic wave scattering by a bimaterial interface.…”