The method of composite expansions applied to boundary layer problems in symmetric bending of the spherical shells (Q1063461)

The method of composite expansions which was proposed by \textit{W. Z. Chien} [Asymptotic behavior of a thin clamped circular plate under uniform normal pressure at very large deflection. Sci. Rep. Nat. Tsing Hua Uni. 5, No.1, 71-94 (1948)] is extended to investigate two-parameter boundary layer problems. For the problems of symmetric deformations of spherical shells under the action of uniformly distribution loads q, its nonlinear equilibrium equations can be written as follows: \[ \varepsilon^2 (d^2/dx^2) (x\theta) - (1/4)F \theta - k^2F - \varepsilon ^3p\delta = 0, \quad \delta^2 (d^2/dx^2) (xF) + (1/2)\theta^2 + 4k^2\theta = 0, \] where \(\varepsilon\) and \(\delta\) are undetermined parameters. If \(\delta =1\) and \(\varepsilon\) is a small parameter, the above-mentioned problem is called first boundary layer problem; if \(\varepsilon\) is a small parameter, and \(\delta\) 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 \(\varepsilon\), \(\delta\) and p satisfy the following equation: \(\epsilon^3 p\delta = 1- \varepsilon\). In defining this condition by using the extended method of composite expansions, we find the asymptotic solution of the above- mentioned problems with clamped boundary conditions.