Necking, beading, and bulging in soft elastic cylinders

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 subcritically 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 √3
2 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.