The method of composite expansions applied to boundary layer problems in symmetric bending of the spherical shells

Abstract

In this paper, the method of composite expansions which was proposed by W. Z. Chien (1948)^[5] is extended to investigate two parameter boundary layer problems.

For the problems of symmetric deformations of the spherical shells under the action of uniformly distribution load q, its nonlinear equilibrium equations can be written as follows:

$$\begin{gathered} \varepsilon ^2 \frac{{d^2 }}{{dx^2 }}(x\theta ) - \frac{1}{4}F\theta - k^2 F - \varepsilon ^3 p\delta = 0 \hfill \\ \delta ^2 \frac{{d^2 }}{{dx^2 }}(xF) + \frac{1}{2}\theta ^2 + 4k^2 \theta = 0 \hfill \\ \end{gathered} $$

where ɛ and δ are undetermined parameters. If δ=1 and ɛ is a small parameter, the above-mentioned problem is called first boundary layer problem; if ɛ is a small parameter, and δ is a small parameter, too, the above-mentioned problem is called second boundary layer problem.

For the above-mentioned problems, however, we assume that the constants ɛ, δ andp satisfy the following equation:

$$\varepsilon ^3 p\delta = 1 - \varepsilon $$

In defining this condition by using the extended method of composite expansions, we find the asymptotic solution of the above-mentioned problems with the clamped boundary conditions.