Search

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.