Localized necking and bulging of finitely deformed residually stressed solid cylinder

Abstract

In this paper, we investigate the influence of residual stress on the stability of a solid circular cylinder subject to axial extension. The nonlinear theory of elasticity is used to derive the equations governing the linearized incremental deformations superimposed on a known finitely deformed configuration. Specialized to the neo-Hookean model with residual stress, the bifurcation analysis results in an exact bifurcation condition for zero and periodic modes, based on the Stroh formalism. An expansion technique is adopted to treat singularities in the governing equations. We investigate three loading scenarios, specifically, gradual increase in axial stretch with constant residual stress and gradual increase in residual stress with fixed axial length or fixed axial force. In each case, the zero mode, corresponding to a localized bifurcation, occurs first. An explicit bifurcation condition for the zero mode is derived, which accounts for the effect of geometric dimensions, the residual stress, and the axial elongation on the stability of the solid cylinder. The critical values of the residual stress and the axial force for localized necking or bulging to occur are identified. In particular, we show that a reduction in residual stress delays the onset of localized necking. At constant values of pre-stretch, with $1.5<\lambda <2.10692$ , localized necking occurs at a critical residual stress, whereas localized bulging occurs when $\lambda >2.10692$ . We use the Maxwell equal-area rule to characterize the “two-phase” deformation consisting of necked and bulged regions. It is shown that the Maxwell values of stretch identify the radii of the two regions, which are connected by a transition zone that translates along the cylinder. At the completion of “two-phase” deformation, the stretch is again uniform.