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Physical Review E Statistical Physics Plasmas Fluids and Related Interdisciplinary Topics, 2005

The paper studies the interaction of a longitudinal wave with transverse waves in general isotropic and unconstrained hyperelastic materials, including the possibility of dissipation. The dissipative term chosen is similar to the classical stress tensor describing a Stokesian fluid and is commonly used in nonlinear acoustics. The aim of this research is to derive the corresponding general equations of motion, valid for any possible form of the strain energy function and to investigate the possibility of obtaining some general and exact solutions to these equations by reducing them to a set of ordinary differential equations. Then the reductions can lead to some exact closed-form solutions for special classes of materials (here the examples of the Hadamard, Blatz-Ko, and power-law strain energy densities are considered, as well as fourth-order elasticity). The solutions derived are in a time/space separable form and may be interpreted as generalized oscillatory shearing motions and generalized sinusoidal standing waves. By means of standard methods of dynamical systems theory, some peculiar properties of waves propagating in compressible materials are uncovered, such as for example, the emergence of destabilizing effects. These latter features exist for highly nonlinear strain energy functions such as the relatively simple power-law strain energy, but they cannot exist in the framework of fourth-order elasticity.

2005

The paper studies the interaction of a longitudinal wave with transverse waves in general isotropic and unconstrained hyperelastic materials, including the possibility of dissipation. The dissipative term chosen is similar to the classical stress tensor describing a Stokesian fluid and is commonly used in nonlinear acoustics. The aim of this research is to derive the corresponding general equations of motion, valid for any possible form of the strain energy function and to investigate the possibility of obtaining some general and exact solutions to these equations by reducing them to a set of ordinary differential equations. Then the reductions can lead to some exact closed-form solutions for special classes of materials ͑here the examples of the Hadamard, Blatz-Ko, and power-law strain energy densities are considered, as well as fourth-order elasticity͒. The solutions derived are in a time-space separable form and may be interpreted as generalized oscillatory shearing motions and generalized sinusoidal standing waves. By means of standard methods of dynamical systems theory, some peculiar properties of waves propagating in compressible materials are uncovered, such as for example, the emergence of destabilizing effects. These latter features exist for highly nonlinear strain energy functions such as the relatively simple power-law strain energy, but they cannot exist in the framework of fourth-order elasticity.

Journal of Elasticity, 2005

Journal of Mechanics of Materials and Structures, 2011

In this paper, the general constitutive equation for a transversely isotropic hyperelastic solid in the presence of initial stress is derived, based on the theory of invariants. In the general finite deformation case for a compressible material this requires 18 invariants (17 for an incompressible material). The equations governing infinitesimal motions superimposed on a finite deformation are then used in conjunction with the constitutive law to examine the propagation of both homogeneous plane waves and, with the restriction to two dimensions, Rayleigh surface waves. For this purpose we consider incompressible materials and a restricted set of invariants that is sufficient to capture both the effects of initial stress and transverse isotropy. Moreover, the equations are specialized to the undeformed configuration in order to compare with the classical formulation of Biot. One feature of the general theory is that the speeds of homogeneous plane waves and surface waves depend nonlinearly on the initial stress, in contrast to the situation of the more specialized isotropic and orthotropic theories of Biot. The speeds of (homogeneous plane) shear waves and Rayleigh waves in an incompressible material are obtained and the significant differences from Biot's results for both isotropic and transversely isotropic materials are highlighted with calculations based on a specific form of strain-energy function.

Journal of Engineering Mathematics

The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media. It begins with a definition of stored energy functions of considered hyperelastic models. The equation of elastodynamics as well as the first-order quasilinear hyperbolic system for plane waves are provided. The eigensystem for this system is determined to study three-wave interaction coefficients. The main part of the paper concerns a discussion of these coefficients. Applying the weakly nonlinear asymptotics method, it is shown that in the case of transverse isotropy the inviscid Burgers’ equation describes an evolution of a single quasi-shear wave. The result contradicts the case of isotropy, where the equation with quadratic nonlinearity cannot describe any shear wave propagation. The paper ends with an example of numerical solutions for the obtained evolution equation.

It is proved that elliptically polarized finite-amplitude inhomogeneous plane waves may not propagate in an elastic material subject to the constraint of incompressibility. The waves considered are harmonic in time and exponentially attenuated in a direction distinct from the direction of propagation. The result holds whether the material is stress-free or homogeneously deformed.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019

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.

CISM Courses and Lectures, 2007

We study incremental wave propagation for what is seemingly the simplest boundary value problem, namely that constitued by the plane interface of a semi-infinite solid. With a view to model loaded elastomers and soft tissues, we focus on incompressible solids, subjected to large homogeneous static deformations. The resulting strain-induced anisotropy complicates matters for the incremental boundary value problem, but we transpose and take advantage of powerful techniques and results from the linear anisotropic elastodynamics theory. In particular we cover several situations where fully explicit secular equations can be derived, including Rayleigh and Stoneley waves in principal directions, and Rayleigh waves polarized in a principal plane or propagating in any direction in a principal plane. We also discuss the merits of polynomial secular equations with respect to more robust, but less transparent, exact secular equations.

Journal of Geophysical Research, 1973

The mathematical framework for describing plane waves in elastic and linear anelastic media is presented. Theoretical results suggest that the nature of plane waves in anelastic materials is distinctly different from the nature of plane waves in elastic materials. In elastic media. the only type of inhomogeneous plane wave (P or S) that can propagate is one for which planes of constant phase are perpendicular to planes of constant amplitude. However, in anelastic media this is the only type of inhomogeneous wave that cannot propagate. For an inhomogeneous P or $ plane wave the particle motion is elliptical, the velocity is less than that of a corresponding homogeneous wave, the maximum attenuation is greater than that of a corresponding homogeneous wave, and the direction of maximum energy flow is not the direction of phase propagation. Expressions for the energy flux, energy densities, dissipated energy, stored energy, and Q-X are derived from an explicit energy conservation relation, valid for an arbitrary steady state viscoelastic radiation field. Each energy expression is valid for homogeneous or inhomogeneous P or S plane waves in elastic or linear anelastic media. A theory of plane wave propagation in materials cannot propagate in anelastic maanelastic materials is of interest in the broad terials, and vice versa. This basic property field of solid mechanics. Additional fields of implies that the results of the reflection probapplication are (1) earthquake engineering, for lem and the Rayleigh-type surface wave probpredicting the dynamic response of soils and lem for anelastic media are substantially difman-made structures, (2) solid earth geo-ferent from the corresponding results for elastic physics, for improving estimates of the earth's media. internal composition, (3)electrical engineering, Buchen [1971] published an excellent acfor analyses of delay lines, and (4) applied count of the physical properties and energy mechanics and physics, for dynamic analyses of associated with plane waves in homogeneous materials. viscoelastic materials. The basic purpose of this The general theory of linear viscoelasticity paper parallels that of Buchen's paper. Howdescribes the behavior of both elastic and linear ever, the derivations presented here differ from anelastic materials. Lockerr [1962], Cooper Buchen's in that they are based on an explicit and Reiss [1966], and Cooper [1967] applied energy conservation relation, valid for an the g•neral theory to the problem of an incident arbitrary steady state viscoelastic radiation homogeneous plane wave on a plane boundary field. In addition, each of the expressions is between viscoelastic materials. They found written in a form valid for both P and S waves that, in general, both the reflected and the (either homogeneous or inhomogeneous). The transmitted waves were inhomogeneous for all general expression derived for Q-• differs from angles of incidence. Borcherdt [1971, 1972, that derived by Buchen. also manuscript in preparation, 1973] extended This paper provides the basic mathematical Loekett's results to include the problem of an framework necessary for concurrent papers by incident inhomogeneous wave and a Rayleigh-Borcherdt [1972, also manuscript in preparatype surface wave on a viscoelastic half-space. tion, 1973]. Botchertit [1971] and Buchen [1971] independently showed that the kind of inhomo-EqvA?toN OF MOTtON FOR V•SCOEnAS?•C geneous wave that can propagate in elastic CONTINUUMS A linear viscoelastic continuum is defined

International Journal of Non-Linear Mechanics, 2009

Weakly nonlinear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically nonlinear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is four-fold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.