Small-amplitude superimposed horizontal motion of a load supported symmetrically by rubber springs

Abstract

The undamped small-amplitude superimposed horizontal motion of a load supported symmetrically between identical isotropic hyperelastic springs, each subjected to an initial finite uniaxial static stretch, is reviewed in general terms. The small superimposed motion is discussed for the classical incompressible Mooney–Rivlin, James–Guth, and Gent models, as well as two limit classes of compressible Blatz–Ko material models, $f=0$ and $f=1$ . The small-amplitude vibrational frequency is presented for each model, and the effects of limited extensibility are demonstrated for the James–Guth and Gent materials. Unstable equilibrium states are exhibited for the Blatz–Ko compressible foamed rubber material with $f=0$ , while the others exhibit infinitesimally stable motion for an essentially arbitrary initial static stretch. It is shown that unstable equilibrium states exist for the general compressible Blatz–Ko model with $0<f<1$ , and these states are characterized graphically in terms of the static stretch and the material Poisson ratio. The article concludes with a discussion of the Blatz–Ko and Gent–Thomas foamed rubber materials and their relation to the classical molecular-based uni-constant theory of elasticity.