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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.

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.

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 analytic response for the Cauchy extra stress in large amplitude oscillatory shear (LAOS) is computed from a constitutive model for isotropic incompressible materials, including viscoelastic contributions, and relaxation time. Three cases of frame invariant derivatives are considered: lower, upper, and Jaumann. In the first two cases, the shear stress at steady-state includes the first and third harmonics, and the difference of normal stresses includes the zeroth, second, and fourth harmonics. In the Jaumann case, the stress components are obtained in integral form and are approximated with a Fourier series. The behavior of the coefficients is studied parametrically, as a function of relaxation time and constitutive parameters. Further, the shear stress and the difference of normal stresses are studied as functions of shear strain and shear rate, and are visualized by means of the elastic and viscous Lissajous–Bowditch (LB) plots. Sample results in the Pipkin plane are reported, and the influence of the constitutive parameters in each case is discussed.

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).

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.

As tandard method based on the use of differential invariants of a Lie group, G ,e nables us to reduce any ordinary differential equation invariant under the action of G .W e show that this method is applicable to vector fields more general than those associated with Lie symmetries. We characterize all such vector fields and study their relationship with nonlocal symmetries and λsymmetries (Govinder K S and Leach P G L 1995 J. Phys. A: Math. Gen. 28 5349–59, Muriel C and Romero L 2001 IMA J. Appl. Math. 66 111–25).

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.

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.

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.

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.

The group theoretical explanation is given of the quasisolution method for partial differential equations recently introduced by Rubel. The examples show that the group approach is simpler from the computational point of view.

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.

The correspondence between a contact symmetry of a second-order PDE E and a point symmetry of the equivalent first order system S is used to determine a class of solutions for E which may not always be invariant under contact transformation. We name these solutions pseudo-invariant because they are determined from a point symmetry of S via a reduction method. Invariant solutions under the related contact transformation are included in the family of pseudo-invariant solutions.

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 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 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.

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.

New universal relations for nonlinear isotropic materials with internal isotropic constraints are found. It is also shown that two families of nonhomogeneous deformations, with constant principal invariants, are universal solutions for all materials in the above-mentioned class.

The characterization of potential systems derived from first order conservation laws and of the related potential symmetries is carried out for the Fokker-Planck equation \( {u_t} = {u_{{xx}}} + {(a(x)u)_x} \). The potential symmetries for the natural potential system are classified. Via a generalization of the classical reduction method, we obtain classes of exact solutions which contain invariant solutions as a particular case.

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.

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.

This paper is devoted to a discussion of the reduction methods for evolution equations based on invariant surface conditions related to functional separation of variables. The relationship of these methods with nonclassical and weak point symmetries is stressed. Applications to diffusion equations with an inhomogeneous reaction term or with saturating dissipation are provided.

We consider the strain energy recently proposed, on a phenomenological basis by Alan Gent to take into account limiting chain extensibility in rubber-like materials. We show that this model gives the simplest rational approximation for the reduced tensile force associated with uniaxial extension that satisfies the usual basic assumptions of continuum mechanics. Then by examination of the classical Treloar data on uniaxial extension of rubber, we explain why the Gent model cannot give good predictions for small and moderate strains. We propose some modifications and find a particular one which is able with a minimum number of phenomenological coefficients to give a very good fit to uniaxial data over the full range of deformations.