We investigate axially symmetric localized bulging of an incompressible hyperelastic circular solid cylinder or tube that is rotating about its axis of symmetry with angular velocity ω. For such a solid cylinder, the homogeneous primary deformation is completely determined by the axial stretch λz, and it is shown that the bifurcation condition is simply given by dω/dλz = 0 if the resultant axial force F is fixed. For a tube that is shrink-fitted to a rigid circular cylindrical spindle, the azimuthal stretch λa on the inner surface of the tube is specified and the deformation is again completely determined by the axial stretch λz although the deformation is now inhomogeneous. For this case it is shown that with F fixed the bifurcation condition is also given by dω/dλz = 0. When the spindle is absent (the case of unconstrained rotation), we also allow for the possibility that the tube is additionally subjected to an internal pressure P . It is shown that with P fixed, and ω and F both viewed as functions of λa and λz, the bifurcation condition for localized bulging is that the Jacobian of ω and F should vanish. Alternatively, the same bifurcation condition can be derived by fixing ω and setting the Jacobian of P and F to zero. Illustrative numerical results are presented using the Ogden and Gent material models.

中文翻译：

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旋转弹性圆柱和管的局部膨胀

我们研究了不可压缩的超弹性圆形实心圆柱或管的轴对称局部膨胀，该圆柱或管围绕其对称轴以角速度 ω 旋转。对于这样的实心圆柱体，均匀的初级变形完全由轴向拉伸 λz 决定，如果合轴向力 F 是固定的，那么分岔条件简单地由 dω/dλz = 0 给出。对于收缩配合到刚性圆柱轴的管子，管子内表面上的方位角拉伸 λa 是指定的，变形再次完全由轴向拉伸 λz 决定，尽管现在变形是不均匀的。对于这种情况，表明在 F 固定的情况下，分岔条件也由 dω/dλz = 0 给出。当主轴不存在时（无约束旋转的情况），我们还考虑了管子额外承受内部压力 P 的可能性。结果表明，当 P 固定，并且 ω 和 F 都被视为 λa 和 λz 的函数时，局部膨胀的分岔条件是 ω 和 F 的雅可比行列式应该为零。或者，可以通过固定 ω 并将 P 和 F 的雅可比行列式设置为零来导出相同的分叉条件。使用 Ogden 和 Gent 材料模型提供了说明性的数值结果。通过固定 ω 并将 P 和 F 的雅可比行列式设置为零，可以导出相同的分叉条件。使用 Ogden 和 Gent 材料模型提供了说明性的数值结果。通过固定 ω 并将 P 和 F 的雅可比行列式设置为零，可以导出相同的分叉条件。使用 Ogden 和 Gent 材料模型提供了说明性的数值结果。