Necking, beading, and bulging in soft elastic cylinders

Abstract

Due to surface tension, a beading instability takes place in a long enough fluid column that results in the breakup of the column and the formation of smaller packets with the same overall volume but a smaller surface area. Similarly, a soft elastic cylinder under axial stretching can develop an instability if the surface tension is large enough. This instability occurs when the axial force reaches a maximum with fixed surface tension or the surface tension reaches a maximum with fixed axial force. However, unlike the situation in fluids where the instability develops with a finite wavelength, for a hyperelastic solid cylinder that is subjected to the combined action of surface tension and axial stretching, a linear bifurcation analysis predicts that the critical wavelength is infinite. We show, both theoretically and numerically, that a localized solution can bifurcate sub-critically from the uniform solution, but the character of the resulting bifurcation depends on the loading path. For fixed axial stretch and variable surface tension, the localized solution corresponds to a bulge or a depression, beading or necking, depending on whether the axial stretch is greater than a certain threshold value that is dependent on the material model and is equal to 23 when the material is neo-Hookean. At this single threshold value, localized solutions cease to exist and the bifurcation becomes exceptionally supercritical. For either fixed surface tension and variable axial force, or fixed axial force and variable surface tension, the localized solution corresponds to a depression or a bulge, respectively. We explain why the bifurcation diagrams in previous numerical and experimental studies look as if the bifurcation were supercritical although it was not meant to. Our analysis shows that beading in fluids and solids are fundamentally different. Fluid beading resulting from the Plateau–Rayleigh instability follows a supercritical linear instability whereas solid beading in general is a subcritical localized instability akin to phase transition.