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A simple model for history dependent nonlinear viscoelasticity is considered. The determining equation governing shear motions is derived and investigated in the quasistatic approximation and under the traveling waves ansatz. Traveling waves are possible only if an inequality involving the constitutive parameters is satisfied. This fact is in contrast to what happens in viscoelasticity of the Kelvin–Voigt type. On the other hand, in the quasi-static approximation (classical creep and recovery experiments) the behavior of the history dependent model is similar to analogous rate dependent models.

We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin–Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier–Stokes fluid in an orthogonal rheometer).

A classical problem in the framework of nonlinear elasticity theory is the characterization of the materials that may sustain a pure state of anti-plane shear in the absence of body forces. This problem has been solved by Knowles and by Hill in the framework of isotropic and incompressible elasticity in the seventies. Here we provide a simpler and shorter proof of these classical results. Moreover, we extend these results to nonlinear elastodynamics and we provide some new special solutions.

The problem of universal motions of a uniform and isotropic simple material subject to a generic isotropic internal constraint is studied. Complete results are achieved for motions with space-dependent strain invariants. As an application, free oscillations of an elastic Bell-constrained spherical shell are investigated.

The authors determine the characterization of all the universal generalized plane deformations for homogeneous isotropic elastic materials with internal constraint.

This paper is devoted to a systematic study of local universal relations in continuum mechanics. We show that it is possible to determine the complete set of independent universal relations whose characterization is obtained by linear universal rules. A historical review of the literature on the topic and various significant examples are given.

We show that finite-amplitude shearing motions superimposed on an 'unsteady' simple extension are admissible for incompressible isotropic viscoelastic materials of differential kind. The amplitude of these motions is determined by solving a non-linear non-autonomous differential equation for which we show that limit cycles are possible.

We investigate some possible nonlinear generalizations of the Kelvin—Voigt viscoelastic models. We use the usual idealization of creep and recovery experiments to discuss the mechanical significance of some constitutive requirements and we show that in the case of a shear-rate dependent viscosity localization of the solution is possible under a simple constitutive characterization.

We show that finite amplitude shearing motions superimposed on an unsteady simple extension are admissible in any incompressible isotropic elastic material. We show that the determining equations for these shearing motions admit a general reduction to a system of ordinary differential equations (ODEs) in the remarkable case of generalized circularly polarized transverse waves. When these waves are standing and the underlying unsteady simple extension is composed of a harmonic perturbation of a static stretch it is possible to reduce the determining ODEs to linear or non-linear Mathieu equations. We use this property for a detailed study of the phenomenon of parametric resonance in non-linear elastodynamics.

We prove that the controllable states for the models of Elastic Dielectrics and Magnetoelastic Solids proposed by Eringen and Maugin can be derived from the catalog of controllable states of the Singh and Pipkin theory of Elastic Dielectrics.

An analysis of the propagation of acceleration waves is performed for a solid-gas flow in adiabatic or isothermal cases. The flow considered is defined by linear constitutive laws in the condition of phase separation.