Small longitudinal oscillations of a load on an incompressible, isotropic limited elastic spring

The small amplitude, free longitudinal vibrational motion of a load supported above by an incompressible, isotropic and homogeneous rubberlike spring is studied for a general class of materials having limited extensibility. First, a simple general frequency equation for small motions about a static equilibrium state is derived without specification of the constitutive nature of the spring. The normalized oscillational frequency of the small superimposed motion of the load is then determined for all materials in a broad class of limited elastic materials. This frequency depends on the limiting extensibility constant and the static stretch, but it is independent of any other physical properties of the spring. It is shown that, for the same static stretch, the normalized vibrational frequency of a load on a neo-Hookean support is a lower bound for its normalized vibrational frequency on any limited elastic spring in the general constitutive class. Specific equations for the Gent and Puso limited elastic models are derived, the frequency for the latter being smaller than that for the former under the same static conditions. The effects of their limited extensibility are described both analytically and graphically, and their connection with experiments on certain rubber materials and biological tissue is noted.