An analytical model of the motion of a conformable sheet over a general convex surface in the presence of frictional coupling

Friction is important across a wide range of applications. In particular, in health care, friction is thought to be the cause of some pressure ulcers in largely immobile patients, and abrasion due to friction contributes to the deterioration of skin health in incontinence pad users, especially in the presence of liquid. Some of these frictional forces are due to stress in materials wrapped around curved anatomical surfaces, which are often complicated shapes. The little work to date that has considered friction arising by this mechanism has assumed very simplified geometries (prisms, or even cylinders), which have enabled coefficients of friction to be extracted from laboratory tests on arms, but which are certainly not applicable to, for example, the diaper region. This work describes the development of a much more general mathematical model for friction between a draped, stressed sheet and the substrate, relating geometry, material mechanical properties and stress for essentially any convex surface. A general, wide, class of frictional interfaces is described (which includes those which obey Amontons’ law), and the model is presented in differential form for a generic member of this class. Finally, an analytical solution is developed for convex, instantaneously rigid substrates isomorphic to the plane draped with a low-density sheet exhibiting no Poisson contraction, a fair approximation to some anatomical situations. The solution is explicitly calculated for a general prism and a general cone, producing expressions consistent with previous published models and with limited new experimental data.