On a functional equation in finite deformation

Abstract

The finite deformation of a rectangular plate into a cylindrical shell gives rise to a functional equation whose general solution gives a variety of shapes into which the plate can be bent. Particular cases include the known forms which are circular, parabolic and one given by rⁿ cosnθ = aⁿ, n = (3–2 σ)⁻¹. The elliptic and hyperbolic forms are found possible only for incompressible plates. An incompressible plate can therefore be bent into a cylinder whose section is any given conic. Five new shapes are obtained in terms of elliptic integrals and three in terms of hyperelliptic integrals. It is found that tractions have to be applied to the curved surfaces in addition to the edge forces and couples in all cases except when the form is circular.