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.

我们研究了三维新胡克弹性圆柱管在轴向压缩下的分叉，其中外表面的运动受到限制。首次将WKB方法应用于三维特征值问题，并分别计算厚壁和薄壁圆柱体的分岔条件。本文针对两种情况（即大的轴向或周向模数）考虑了两种WKB展开式，并分别获得了每种情况下特征值的渐近展开式。发现轴向拉伸相对于圆柱体厚度的变化具有边界层结构。评估轴向拉伸对模数、长度和壁厚的依赖性。临界拉伸发生在有限临界模式数下，并且是管厚度的递减函数，因此厚圆柱体更容易弯曲。此外，轴向和周向模式之间的转变发生在三个点，模态数为$10,20$以及众数为零的点。采用复合矩阵法对解析结果进行验证，数值结果与解析数据结果比较吻合较好。