Asymptotic analysis of the nonlinear equations for an infinite, rubber-like slab under an equilibrated vertical line load

Abstract

An infinite slab of incompressible Rivlin-Saunders material of constant thickness 2H is subject to an equilibrated, radially varying, vertical body force, comprising a concentrated, downward line load and a smooth, upward, exponentially distributed load with a characteristic decay length R. The deformation is axisymmetric and described by three stretches and a shear strain (or, equivalently, four strains) and a rotation which satisfy three relatively simple compatibility conditions. Force equilibrium is satisfied identically by the introduction of three stress functions. The incompressibility constraint is used to eliminate the normal stretch. With the introduction of stress-strain relations, the field equations are reduced to a set of seven, first-order, quasilinear partial differential equations. The loads, the radial distance, and the unknowns are scaled by the small parameter ɛ=H/R. As ɛ→0, 11 possible sets of field equations are found, including linear plate theory, von Kármán plate theory, Föppl membrane theory, large-strain membrane theory, and Wu's large-stretch (asymptotic) membrane theory. Notably absent as limiting cases are thick plate theories.