Localized bulging of rotating elastic cylinders and tubes

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