#### Vol. 12, No. 4, 2017

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Localized bulging of rotating elastic cylinders and tubes

### Juan Wang, Ali Althobaiti and Yibin Fu

Vol. 12 (2017), No. 4, 545–561
##### Abstract

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 $\omega$. For such a solid cylinder, the homogeneous primary deformation is completely determined by the axial stretch ${\lambda }_{z}$, and it is shown that the bifurcation condition is simply given by $d\omega ∕d{\lambda }_{z}=0$ if the resultant axial force $F$ is fixed. For a tube that is shrink-fitted to a rigid circular cylindrical spindle, the azimuthal stretch ${\lambda }_{a}$ on the inner surface of the tube is specified and the deformation is again completely determined by the axial stretch ${\lambda }_{z}$ although the deformation is now inhomogeneous. For this case it is shown that with $F$ fixed the bifurcation condition is also given by $d\omega ∕d{\lambda }_{z}=0$. 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 $\omega$ and $F$ both viewed as functions of ${\lambda }_{a}$ and ${\lambda }_{z}$, the bifurcation condition for localized bulging is that the Jacobian of $\omega$ and $F$ should vanish. Alternatively, the same bifurcation condition can be derived by fixing $\omega$ and setting the Jacobian of $P$ and $F$ to zero. Illustrative numerical results are presented using the Ogden and Gent material models.

##### Keywords
localized bulging, bifurcation, rotating tubes, nonlinear elasticity
##### Milestones
Received: 20 February 2017
Revised: 12 May 2017
Accepted: 19 May 2017
Published: 28 June 2017
##### Authors
 Juan Wang College of Science University of Shanghai for Science and Technology Shanghai 200093 China Ali Althobaiti School of Computing and Mathematics Keele University Staffordshire ST5 5BG United Kingdom Yibin Fu School of Computing and Mathematics Keele University Staffordshire ST5 5BG United Kingdom