WKB analysis of the bifurcation for a three-dimensional neo-Hookean cylindrical tube under restricted compression

Abstract

We study the bifurcation of a three-dimensional neo-Hookean elastic cylindrical tube under axial compression, where the movement of the outer surface is restricted. For the first time, the WKB method is applied to the three-dimensional eigenvalue problem and the bifurcation conditions are calculated for thick and thin-walled cylinders, separately. In this paper, two WKB expansions are considered for the two cases (i.e. for large axial or circumferential mode number) and the asymptotic expansion of the eigenvalue is obtained for each case separately. It is found that the changes of the axial stretch relative to the thickness of the cylinder have a boundary layer structure. The dependency of the axial stretch on the mode numbers, length and the wall thickness is assessed. The critical stretches occur for the finite critical mode numbers and are an decreasing function of the thickness of the tube, so thick cylinders are easier to buckle. Also, transitions between axial and circumferential modes occur in three points for mode numbers of 10,20 and at a point for mode number of zero. Using the compound matrix method to verify the analytical results, the comparison of outcomes of the numerical and analytical data shows a good agreement.