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We consider the nonlinear boundary value problem of specifying the displacement of the lateral surface of a cylindrical body subject to zero normal stresses on the top and the bottom and sliding conditions (i.e. no tangential components of the surface traction) at the lateral surface. We restrict our analysis to study the existence of axisymmetric deformations assuming that the material of the body is homogeneous, isotropic and hyperelastic. We study the linearization of the nonlinear equations about a trivial solution and show that smooth solutions of the linear problem must be separable. We classify the nontrivial axisymmetric solutions of the linearized problem in two types that we call buckling and barrelling like solutions. We characterize the eigenvalues for both solutions types as well as those displacements of the lateral surface at which the complementing condition for the linearized equations fails to be satisfied. For a class of Blatz–Ko type materials we give a complete characterization of the existence, multiplicity and disposition of the corresponding eigenvalues. We show, for such material, that the eigenvalues of buckling and barrelling types are simple, and that they form monotone sequences (decreasing for the former and increasing for the latter) both of which converge to a value at which the complementing condition fails. Moreover, it is shown that the cylinder looses stability first to buckling rather than to barrelling. |
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