Qualitative analysis for solutions of a class of nonlinear ordinary differential equations
The boundary value problem (BVP) for a class of nonlinear ordinary differential equations is examined. It can be used to describe the eversion problem for a spherical shell composed of a class of transversely isotropic incompressible Mooney-Rivlin materials. The solutions of the nonlinear equation describing the relation among deformed thickness, initial thickness and material parameters of the spherical shell are obtained. The effects of structure parameter and material parameters on the thickness of the everted spherical shell are discussed by numerical examples.
Asymptotic bifurcation results for the eversion of elastic shells
Abstract. We apply the WKB method to the bifurcation analysis of everted cylindrical and spherical shells composed of Varga material. Both compressible and incompressible cases are considered. The method is degenerate but we obtain explicit bifurcation criteria and compare with previous numerical approximations. Different strategies are compared.

WKB Method with Repeated Roots and Its Application to the Buckling Analysis of an Everted Cylindrical Tube
For a linear ordinary differential equation of variable coefficients in which the highest order derivative $d^ny/dx^n$ is multiplied by a small parameter $\epsilon^n$ say, a WKB solution of the form $y(x)=f(x) \; {\rm exp}\; (\frac{1}{\epsilon}\int^{x} s(x) dx)$ can be sought. To leading order, s(x) satisfies an nth order algebraic equation. It seems that all the existing books on singular perturbation methods have discussed only the case when the roots of this algebraic equation are distinct except at possibly a finite number of points. In this case n independent solutions can readily be obtained and the general solution is a linear combination of these n solutions. When the algebraic equation has repeated roots, it is not immediately clear how to obtain n independent solutions. In this paper we first show, through a simple model problem, how the WKB method should be applied when double roots arise. We then apply the ideas to the WKB analysis of the buckling of an everted circular cylindrical tube. A sim...