Influence of Compressibility and Hardening on Cavitation

This paper examines the effect of material compressibility and material hardening on the phenomenon of internal rupture. Specifically, a bifurcation problem for a solid circular cylinder composed of a particular class of compressible elastic materials is studied. The surface of the cylinder is subjected to a radial stretch $\lambda \left(>1\right)\text{,}$ and the cylinder is in a state of plane strain. One solution to this problem, for all values of λ, is that of a pure homogeneous stretching in which the cylinder expands radially. However, a second solution bifurcates from this homogeneous solution at a critical value of $\lambda \left(\mathrm{say}\text{,}{\lambda }_{\mathrm{cr}}\right)$ at which the homogeneous solution becomes unstable. For $\lambda >{\lambda }_{\mathrm{cr}}\text{,}$ a circular cylindrical cavity forms at the axis of the cylinder. We find that ${\lambda }_{\mathrm{cr}}\text{.}$ decreases as the material becomes softer and also as it becomes less compressible. The corresponding value of radial stress ${\tau }_{\mathrm{cr}}\text{.}$ also decreases as the material becomes softer but increases as it becomes less compressible.