Peeling, Slipping and Cracking—Some One-Dimensional Free-Boundary Problems in Mechanics

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Abstract
We examine several problems involving a one-dimensional continuum with a free boundary. Apart from their intrinsic interest, they illuminate the related but far more complex problems which arise in fracture mechanics, in frictional sliding such as may occur in shallow earthquakes, and in the testing of adhesives. First we treat peeling of adhesive tape from a surface. The tape is regarded as a flexible string, part of which adheres to the surface and part of which is separated from it. The free boundary is the point (or points) at which separation takes place and the problem is to find the trajectory of this point (or points) together with the motion of the separated section of the string. We treat both infinitesimal and finite deformations. We also treat the related problem of the sliding of a segment of a stretched string pressed against a rough plane under the influence of transverse loads. Adhesive forces may be either present or absent at the endpoints of the moving segment, which form the free boundaries. Next the lengthwise splitting of a beam is considered. The free boundary is the point of separation of the beam into two sections. Two problems wherein the beam is split by the insertion of a wedge are solved. Then we discuss the stability of a beam split further by a machine which pulls the ends (two halves of one end) apart quasi-statically. Depending upon the stiffness of the machine, the crack may extend quasi-statically or run catastrophically. Finally we treat the peeling of an elastics from a plane surface to which it adheres. This analysis applies to the peeling of a still adhesive tape or to the splitting of a beam with large deformation.

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