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In the framework of the theory of isotropic incompressible nonlinear elasticity we derive an asymptotic system of equations using a multiple scales expansion and considering waves of finite but small amplitude composed by an anti-plane shear superposed to a general plane motion. The system of equations generalizes the classical Zabolotskaya equation. Moreover, we show that the hyperbolic system, we derive, has a mathematical structure similar to the systems determining the propagation of transverse waves in nonlinear elasticity.

In this paper we consider the structure of the symmetry group of some important mechanical theories (nonlinear elasticity and fluids of grade n ). We discuss why the invariance with respect to some well-known transformations must be used with care and we explain why some of these universal transformations are useless to obtain invariant solutions of physical significance.

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 provide a systematic and complete analysis of the overdetermined problem that one obtains while considering the balance equations of unconstrained isotropic nonlinear Cauchy elastic bodies undergoing anti-plane shear deformations.

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 problem of bending of a pretwisted bar, cantilevered and loaded by transverse forces at the end, is dealt within the framework of the linear three-dimensional theory of elasticity. The stress components are assumed representable as a suitable series in the pretwist parameter k , whose coefficients can be determined by the solution of plane problems. This is possible if a suitable dependence on the axial coordinate is assigned. The boundary value problems which characterize the coefficients up to the second power are explicitly determined, and the solutions are calculated for an elliptical section. In this way we determine the approximate solution up to terms of order k 2 , to account for a moderate rate of pretwist. From the results of the mathematical analysis, some indications can be deduced regarding the effect of pretwist and a comparison can be made with the classical case of a straight bar.

We define an algorithm to characterize all weak symmetry groups of a partial differential equation. New invariant solutions to the Fokker-Planck equation are found.

In this paper, following the lines of a Noether's theorem, is established the complete set of generators of conservation laws related to geometric symmetries for linear theory of isotropic micropolar elastostatic.

We present the results of a systematical investigation of invariance properties of a semilinear hyperbolic equation u xt = ƒ(u) , under a one-parameter Lie group of transformations, for arbitrary ƒ(u). The infinitesimals, resulting generators of the Lie algebra, and ordinary differential equations determining invariant solutions, are determined.

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.