The problem of unilateral contact of flexible non-stretchable rope and rigid solid

The flexible nonstretchable rope is one of simplest mechanical systems with the infinite number of degrees of freedom. Contact problems can be posed for such ropes, as well as for the related one-dimensional systems (strings and beams). The unbounded contact problem for heavy flexible nonstretchable rope and smooth rigid solid of a given shape is considered. The rope has one end fixed and the other end free. The rope density may be variable. The presence of a rigid solid slightly deflects the rope from the vertical line. The problem is to find the rope shape and can be reduced to finding the density of the interacting forces between the rope and the solid. This density is the sum of the piecewise-continuous part and concentrated forces. These forces are described by Dirac’s delta-function. The contact conditions are as follows. The density should be non-negative, the distance between the rope and the body should be non-negative, and this distance should be equal to zero at points where the density is positive. There are two approaches to such problems. The first one is variational. Here, the rope shape is in question. The rigorous problem statements and (rather complicated) proofs of uniqueness and existence of solutions are possible in this approach. However, the analytical solutions usually are not considered here. Even if the hypothetical analytical solution is obtained, it is not easy to establish whether it is really the solution (i.e., whether it minimizes some functional or satisfies some variational inequality). The second approach is based on the theory of strength of materials. It is convenient here to consider the density of the interacting forces as the function to be found. It is often possible to construct and to verify the hypothetical analytical solution. However, this approach lacks for the rigorous problem statement. Even the formulation of the theorem of the uniqueness of the solution is impossible without this rigorous statement. The present article uses the second approach with some improvements and refinements. Then, as in the first approach, the rigorous problem statement can be formulated. Further, it is easy to prove the uniqueness of the solution and to substantiate the constructed analytical solution. This construction also proves the existence of the solution. The substantiation of the solution includes proving the non-negativity of the contact forces and contact distances and proves the existence of a root of transcendental equation that yields the length of the contact segment. It is shown that different contact patterns are possible: a contact at the point and a contact along a rope segment. The pattern kind depends on the shape of the rigid solid.