Infinitesimal Stability of the Equilibrium States of an Incompressible, Isotropic Elastic Tube Under Pressure

Abstract

The infinitesimal stability of the equilibrium states of an arbitrary incompressible, isotropic and homogeneous elastic cylindrical shell in a pure radial expansion under a constant inflation pressure is studied for both thick- and thin-walled shells. The classical criterion of infinitesimal stability yields a general stability theorem relating the frequency and pressure response and reveals that points at which the pressure is stationary define the domain of unstable or neutrally stable states. All results are expressed in terms of a general shear response function, and specific results are provided for the Mooney-Rivlin, Gent and Ogden models, the second having limited extensibility, the last including experimental data. Every static state of a Mooney-Rivlin tube is stable so long as the pressure is less than an asymptotic limit that increases with the thickness. Otherwise, only the Ogden model exhibits static states of instability for all long cylindrical tubes of thickness less than a transitional value above which all static states are infinitesimally stable. A long cylindrical cavity in all three unbounded models is stable for all pressures. All results are illustrated graphically.