Small amplitude, free longitudinal vibrations of a load on a finitely deformed stress-softening spring with limiting extensibility

Abstract.

A constitutive theory for a general class of incompressible, isotropic stress-softening, limited elastic rubberlike materials is introduced. The model is applied to study the small amplitude, free longitudinal vibrational frequency of a load about a suspended static equilibrium stretch of a finitely deformed, stress-softening spring with limiting extensibility. A number of physical results, including bounds on the frequency, are reported. It is proved, for example, that the normalized vibrational frequency for the ideally elastic neo-Hookean oscillator is a lower bound for the normalized frequency of every incompressible, isotropic stress-softening, limited elastic oscillator within the general class. All results are illustrated for the special limited elastic Gent and the purely elastic Demiray biomaterial models, both with stress-softening characterized by a Zúñiga–Beatty front factor damage function. The results for both stress-softening models are compared with experimental data for several gum rubbers and thoracic aortic tissue provided by others; and, overall, it is found that the stress-softening, limited elastic Gent model best characterizes the data.