REMARKS ON THE INSTABILITY OF AN INCOMPRESSIBLE AND ISOTROPIC HYPERELASTIC, THICK-WALLED CYLINDRICAL TUBE

The problem of instability of a hyperelastic, Chick-walled cylindrical tube was first studied by Wilkes [1] in 1955. The solution was formul ated within the framework of the theory of small deformations superimp osed on large homogeneous deformations for the general class of incomp ressible, isotropic materials; and results for axially symmetrical buc kling were obtained for the neo-Hookean material. The solution involve s a certain quadratic equation whose characteristic roots depend on th e material response functions. For the neo-Hookean material these roof s always are positive. In fact, here we show for the more general Moon ey-Rivlin material that these roots always are positive, provided the empirical inequalities hold. In a recent study [2] of this problem for a class of internally constrained compressible materials, it is obser ved that these characteristic roots may be real-valued, pure imaginary , or complex-valued. The similarity of the analytical structure of the two problems, however, is most striking; and this similarity leads on e to question possible complex-valued solutions for the incompressible case. Some remarks on this issue will be presented and some new resul ts will be reported, including additional results for both the neo-Hoo kean and Mooney-Rivlin materials.