Creep, recovery and vibration of an incompressible viscoelastic material of the rate type: Simple tension case

We consider three-dimensional nonlinear viscoelastic models that account for both stress relaxation and creep/recovery phenomena. These models are based on different frame indifferent time derivatives: the Oldroyd (or upper-convected) derivative, the Jaumann (or co-rotational) derivative and the Cotter-Rivlin (or lower-convected) derivative. Under a simple tension creep process, these constitutive equations predict the same stress relaxation but lead to different situations. The models based on the Oldroyd and the lower-convected derivative require restrictions on the values of the material parameters as well as on the traction/compression stress. The model based on the Jaumann derivatives does not require any restriction. All the constitutive models examined are used to study the finite amplitude, horizontal oscillatory motion of a mass attached to a rate-type viscoelastic string. In this way we generalize a classical results by Beatty and Zhou (1991}.