A geometric optics solution for the thin shell equation

Abstract

This paper is chiefly concerned with the application of the asymptotic ‘geometric optics’ methods to linear shell theory. To accomplish this, it is found to be especially helpful to have the equations in a compact form using tensors and tensor products rather than the components of tensors used by most investigators. In the first portion of the paper, it is shown that this notation also simplifies several steps in the derivation of the equations and that a simple ‘static-geometric’ analogy for the nonlinear theory can be established. The final linear equation with a complex dependent variable has for components the equations obtained by Naghdi, which for lines-of-curvature coordinates have the same form as the Novozhilov equations, but with the modified curvature measure due to Koiter and Sanders. A general solution is sought in terms of an exponential asymptotic expansion that is uniformly valid in the large. As in the geometric optics solution of the reduced wave equation, the ‘transport’ equation reduces to an ordinary differential equation along the ‘rays’, i.e., the characteristics of the ‘eikonel’ equation. Unlike the solution of the reduced wave equation, the ‘wave-fronts’ are not orthogonal to the rays, so a nonorthogonal coordinate system turns out to be advantageous. Although difficulties remain, the exciting possibilities of the approach for general shell analysis are indicated by two examples. One is a simple solution for the heretofore difficult problem of a cylinder with a rigid insert in the wall. The second is a simple interpretation for the previously obtained solution for a shell of revolution with ‘rapidly varying’ edge loads.