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The Mullins effect in the oscillatory motion of a load under gravity and attached to a stress-softening, neo-Hookean rubber string is investigated. Equations for the small amplitude vertical oscillations of the load superimposed on the finite static stretch of both the virgin and stress-softened cords, the latter subjected to varying degrees of preconditioning, are derived. The vibrational frequency of the small motion exhibits behavior similar to that observed in experiments by others on postmortem, human aortic tissue for which no stress-softening is reported. Standard numerical methods are applied to study the finite amplitude motion of the load in the stress-softened case. The resultant motions and their various physical aspects under free-fall and general initial conditions are described in several examples. Oscillations that engage all three phases of motion consisting of the suspension, the free-flight, and the retraction of the load in its general vertical motion are illustrated. Effects due to the degree of stress-softening are discussed; and the motion response for two values of the model softening parameter is compared in several examples. All results are illustrated graphically and numerous tabulated numerical results are provided. In a separate study the effects of material nonlinearity in a virgin rubberlike material on the ballooning motion of a hyperelastic string are investigated. The material nonlinearity is characterized by the neo-Hookean material model and the hyperelastic string is fixed at one boundary, where the other boundary is rotated at a constant angular speed. The governing equations of motion include the influence of air drag. Numerical solution of nonlinear steady state equation reveals the dependence of fixed eyelet tension on the length of the hyperelastic string and air drag coefficient. Linear stability of the steady sate motion is obtained by Galerkin’s method for a small perturbation about the steady state configuration. This study shows that the neo-Hookean material may generalize the linearly elastic string and inextensible string for a specific range of values of speed and material constant. Finally, the theoretical results are verified experimentally. The effect of stress-softening on the ballooning motion is investigated and it is found that the effect gives rise to some improvement in the stability under some conditions.