Finite plane deformation of thine elastic sheets reinforced with inextensible cords

Abstract

A general theory is formulated, in tensor notation, for the finite elastic deformation of curvilinearly aeolotropic materials which are subject to constraints, the aeolotropy and the system of constraints being related to different curvilinear frames of reference in the undeformed body. The results obtained are utilized in considering the general two-dimensional deformation of a thin, plane, uniform sheet of elastic material, reinforced by means of a continuous layer of thin, flexible, inextensible cords lying in the plane midway between its major surfaces. These cords coincide with either or both of the families of curves which constitute a general curvilinear coordinate system in the plane in which they lie. The theory is formulated initially for materials which exhibit a type of aeolotropy which is general, except for the restriction required for the structure of the plate to be symmetrical about the reinforcing layer. When there are two sets of cords, the governing differential equations are hyperbolic, with the characteristic curves coinciding with the paths followed by the cords, and the solution of these equations, subject to various types of boundary condition, is investigated. When the cords lie initially in two sets of parallel straight lines, the stress resultants and displacements can be expressed in terms of real arbitrary functions. This type of solution is employed to examine the deformation of an infinite plane sector, with a system of stresses and displacements prescribed along its edges. The general theory for a sheet reinforced with a single set of cords is also developed briefly. The linear equations for classically small deformations are derived both for this case, and for a sheet containing two sets of cords, and when the cords lie initially in parallel straight lines these equations are solved in terms of arbitrary functions.