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We study elastic shear waves of small but finite amplitude, composed of an anti-plane shear motion and a general in-plane motion. We use a multiple scales expansion to derive an asymptotic system of coupled nonlinear equations describing their propagation in all isotropic incompressible nonlinear elastic solids, generalizing the scalar Zabolotskaya equation of compressible nonlinear elasticity. We show that for a general isotropic incompressible solid, the coupling between anti-plane and in-plane motions cannot be undone and thus conclude that linear polarization is impossible for general nonlinear two-dimensional shear waves. We then use the equations to study the evolution of a nonlinear Gaussian beam in a soft solid: we show that a pure (linearly polarized) shear beam source generates only odd harmonics, but that introducing a slight in-plane noise in the source signal leads to a second harmonic, of the same magnitude as the fifth harmonic, a phenomenon recently observed experimentally. Finally, we present examples of some special shear motions with linear polarization. EP and GS have been partially supported for this work by the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Italian non-profit research institution Istituto Nazionale di Alta Matematica Francesco Severi (INdAM) and the PRIN2017 project “Mathematics of active materials: From mechanobiology to smart devices” funded by the Italian Ministry of Education, Universities and Research (MIUR). We are most grateful to Gianmarco Pinton and David Esp´ındola for sharing the experimental data used to generate Figure 1. peer-reviewed

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.

We study the propagation of linearly polarized transverse waves in a pre-strained incompressible isotropic elastic solid. Both finite and small-but-finite amplitude waves are examined. Irrespective of the magnitude of the wave amplitude, these waves may propagate only if the (unit) normal to the plane spanned by the directions of propagation and polarization is a principal direction of the left Cauchy–Green deformation tensor associated with the pre-strained state. A rigorous asymptotic analysis of the equations governing the propagation of waves of small but finite amplitude reveals that the time scale over which the nonlinear effects become significant depends on the direction along which the wave travels. Moreover, we design theoretically an experimental procedure to determine the Landau constants of the fourth-order weakly nonlinear theory of elasticity.

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.

A material is of coaxial type if the Cauchy stress tensor T and the strain tensor B are coaxial for all deformations. Clearly a hyperelastic material is of coaxial type if and only if it is isotropic. Here we present a weaker definition of materials of coaxial type. Anisotropic materials may be of a coaxial type in a weak sense if for a given specific B we have that TB = BT . We denote these materials B -coaxial. We show that for transverse isotropic materials weak coaxial constitutive equations may be characterized using universal relations. We discuss the impact of B -coaxial materials in the modeling of soft tissues. We conclude that B -coaxial materials are a strong evidence that in real world materials two anisotropic invariants are always necessary to model in a meaningful and correct way single fiber reinforced materials.

Two classes of elastic constrained materials are considered : the Bell materials where the constraint implies an inextensibility in the simple mean, and the Ericksen material where the constraint implies inextensibility in the quadratic mean. We classify, for both materials, all the universal solutions that fall into the family of the generalized plane deformations. One of these families, which is universal also for incompressible materials, allows us to generate a very interesting new universal relation.

We consider nonlinear viscoelastic materials of differential type and for some special models we derive exact solutions of initial boundary value problems. These exact solutions are used to investigate the reasons for the non-existence of global solutions for such equations.

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.

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.