Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress

The problem of Rayleigh waves in an orthotropic elastic medium under the influence of gravity and initial stress was investigated by Abd-Alla [A. M. Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl. Math. Comput. 99 (1999) 61-69], and the secular equation of the wave in the implicit form was derived. However, due to the uncorrect representation of the solution, the secular equation is not right. The main aim of the present paper is to reconsider this problem. We find the secular equation of the wave in explicit form. By considering some special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity of some previous studies, in which only implicit secular equations were derived.     

Radiation condition and uniqueness for the outgoing elastic wave in a half-plane with free boundary

In this Note we deduce an explicit Sommerfeld-type radiation condition which is convenient to prove the uniqueness for the time-harmonic outgoing wave problem in an isotropic elastic half-plane with free boundary condition. The expression is obtained from a rigorous asymptotic analysis of the associated Green's function. The main difficulty is that the free boundary condition allows the propagation of a Rayleigh wave which cannot be neglected in the far field expansion. We also give the existence result for this problem. To cite this article: M. Durán et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

An energy approach to the proof of the existence of Rayleigh waves in an anisotropic elastic half-space

An approach based on investigating the energy functional is applied for the first time to the classical problem of Rayleigh waves in an anisotropic half-space with a free boundary. The main object of the investigation is an ordinary differential operator in a variable characterizing the depth. An investigation of the spectrum by variational methods enables a new proof to be given of the existence of a Rayleigh wave in a linear elastic half-space with arbitrary anisotropy, which does not rest on the Stroh formalism.     

Investigation of normal waves in a porous layer surrounded by elastic half-spaces

The wave field and dispersion equations are found for a porous layer surrounded by two elastic half-spaces. The porous layer is described by the effective model of a medium in which elastic and fluid layers alternate. To investigate the normal waves, all real roots of dispersion equations are determined and their movements as the wave number increases are investigated. As a result, the dispersion curves of all normal waves are constructed and the dependence of normal waves on the parameters of the porous layer and elastic half-spaces is analyzed. Bibliography: 6 titles.     

Solitary wave in a nonlinear elastic structural element of large deflection

General form nonlinear governing equations for the wave traveling in a nonlinear elastic structural element of large deflection are derived in the present research. An asymptotic solution of solitary wave in the elastic element is derived and investigated by means of a modified complete approximate method. Numerical computations for the solution are carried out. Characteristics of the solitary wave are investigated with various system parameters and initial conditions. Shapes and the propagation of the nonlinear elastic wave are also illustrated with figures. Based on the theoretical and numerical analyses of the research, quantitative conclusions are obtained for the wave motion of the elastic structural element.     

On specific features of the Lebedev scheme in simulating elastic wave propagation in anisotropic media

This paper presents the Lebedev scheme on staggered grids for the numerical simulation of wave propagation in anisotropic elastic media. Primary attention is given to the approximation of the elastic wave equation by the Lebedev scheme. Based on the differential approach, it is shown that the Lebedev scheme approximates a system of equations, which differs from the original equation. It is proved that the approximated system has a set of 24 characteristics, six of them coincide with those of the elastic wave equation and the rest ones are “artifacts.” Requiring the artificial solutions to be equal to zero and the true ones to coincide with those of the elastic wave equation, one comes to the classical definition of the approximation of the initial system on a sufficiently smooth solution. The results obtained and the knowledge of the complete set of characteristics are important for constructing reflectionless boundary conditions during approximation of point sources, etc.     

Dispersion of elastic waves in a micropolar metamaterial plate with periodical arranged resonators

The wave propagation in a micropolar elastic metamaterial is investigated in this paper. The elastic metamaterial is composed of the micropolar elastic host material and the periodically arranged local resonators. Compared with the classical elastic metamaterial, the micropolar elastic metamaterial has more material parameters that can be elaborately designed to manipulate the elastic wave propagation. By introducing additional displacement fields, a multi-displacement continuum model of the micropolar elastic metamaterial is presented to characterize the resonance behavior of the resonators and the microstructure effects of the unit cell. According to this continuum model, two independent wave systems exist: one is a longitudinal system and the other is a shear and rotation coupled transversal system. The dispersive curves and band gaps of the longitudinal and transversal systems are numerically discussed and the influences of the resonators are mainly considered.     

Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory

In the present work, the effect of longitudinal magnetic field on wave dispersion characteristics of equivalent continuum structure (ECS) of single-walled carbon nanotubes (SWCNT) embedded in elastic medium is studied. The ECS is modelled as an Euler–Bernoulli beam. The chemical bonds between a SWCNT and the elastic medium are assumed to be formed. The elastic matrix is described by Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation. The governing equations of motion for the ECS of SWCNT under a longitudinal magnetic field are derived by considering the Lorentz magnetic force obtained from Maxwell’s relations within the frame work of nonlocal elasticity theory. The wave propagation analysis is performed using spectral analysis. The results obtained show that the velocity of flexural waves in SWCNTs increases with the increase of longitudinal magnetic field exerted on it in the frequency range; 0–20 THz. The present analysis also shows that the flexural wave dispersion in the ECS of SWCNT obtained by local and nonlocal elasticity theories differ. It is found that the nonlocality reduces the wave velocity irrespective of the presence of the magnetic field and does not influences it in the higher frequency region. Further it is found that the presence of elastic matrix introduces the frequency band gap in flexural wave mode. The band gap in the flexural wave is found to independent of strength of the longitudinal magnetic field.     

Modelling of wave phenomena in the Zahorski material based on modified library for ADINA software

This paper presents numerical modelling of wave phenomena in simple elastic structures such as rods and shields made of hyperelastic Zahorski material. The main difference between the Zahorski material, which is an elastic material in the Green sense, and the commonly used Mooney–Rivlin material lies in the non-linear term including the constant C₃. Consequently, qualitative and quantitative differences are observed compared to the Mooney–Rivlin material, for example in the values of effective stresses. The extension to the ADINA software developed by the author, which helps create 2D and 3D libraries, significantly facilitates modelling of the Zahorski material. The modification can be used for comparison of wave phenomena that are observed during the propagation of disturbances in the Mooney–Rivlin and Zahorski materials. It should be emphasised that the Zahorski material behaves much better at high strains during the analysis of incompressible rubber and rubber-like hyperelastic materials and can be used in various fields of science wherever the model of Mooney–Rivlin material is successfully applied. The results of numerical computations for both Mooney–Rivlin and Zahorski materials were presented in a graphical form and compared in order to illustrate the differences.     

An Effective Model of a Fractured Medium

A homogeneous isotropic elastic medium intersected by three systems of fractures on which the jumps of stresses are proportional to displacements is considered. An effective model of this medium is described by equations differing from the respective equations of the elastic medium by additional terms. On the basis of the equations of the effective model, the wave field excited by a point source is established. An investigation of the integral representation of the wave field shows that the velocities of the longitudinal and transversal waves and of the Rayleigh wave are functions of the frequency and the wave numbers. Formulas for the phase and group velocities of these waves are derived. Bibliography: 3 titles.     

Oscillations of an unstretched string

Small oscillations of a linear elastic string that has not been stretched are considered. A nonlinear wave equation that describes the main features of the oscillations is presented. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 36–43, 1993.     

Analysis of Lamb waves with Spectral Finite Elements

For the analysis of the elastic wave propagation at high frequencies, the spectral finite element method (SFEM) is under investigation. The SFEM uses high-order shape functions which are composed of Lagrange polynomials with nodes at the Gauss-Lobatto quadrature points. In this way we obtain a diagonal mass matrix which makes an explicit time scheme more efficient. In a numerical example a computation by Montjoie of an elastic wave propagation within a Reissner-Mindlin (RM) model is presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of elastic bodies connected by a thin soft viscoelastic layer

A dynamic study was performed on a structure consisting of two three-dimensional linearly elastic bodies connected by a thin soft nonlinear Kelvin–Voigt viscoelastic adhesive layer. The adhesive is assumed to be viscoelastic of Kelvin–Voigt generalized type, which makes it possible to deal with a relatively wide range of physical behavior by choosing suitable dissipation potentials. In the static and purely elastic case, convergence results when geometrical and mechanical parameters tend to zero have already been obtained using variational convergence methods. To obtain convergence results in the dynamic case, the main tool, as in the quasistatic case, is a nonlinear version of Trotter?s theory of approximation of semigroups acting on variable Hilbert spaces. The limit problem involves a mechanical constraint imposed along the surface to which the layer shrinks. The meaning of this limit with respect to the relative behavior of the parameters is discussed. The problem applies in particular to wave phenomena in bonded domains.     

A Runge-Kutta/WENO method for solving equations for small-amplitude wave propagation in a saturated elastic porous medium

A high-accuracy Runge-Kutta/WENO method of up to fourth order with respect to time and fifth order with respect to space is developed for the numerical modeling of small-amplitude wave propagation in a steady fluid-saturated elastic porous medium. A system of governing equations is derived from a general thermodynamically consistent model of a compressible fluid flow through a saturated elastic porous medium, which is described by a hyperbolic system of conservation laws with allowance for finite deformations of the medium. The results of numerical solution of one- and two-dimensional wave fields demonstrate the efficiency of the method.     

Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix

This paper reported the result of an investigation into the effect of magnetic field on wave propagation in carbon nanotubes (CNTs) embedded in elastic matrix. Dynamic equations of CNTs under a longitudinal magnetic field are derived by considering the Lorentz magnetic forces. The results obtained show that wave propagation in CNTs embedded in elastic matrix under longitudinal magnetic field appears in critical frequencies at which the velocity of wave propagation drops dramatically. The velocity of wave propagation in CNTs increases with the increase of longitudinal magnetic field exerted on the CNTs in some frequency regions. The critical/cut-off frequency increases with the increase of matrix stiffness, and the influence of matrix on wave velocity is little in some frequency regions. This investigation may give a useful help in applications of nano-oscillators, micro-wave absorbing and nano-electron technology.     

Implementation and computational aspects of a 3D elastic wave modelling by PUFEM

This paper presents an enriched finite element model for three dimensional elastic wave problems, in the frequency domain, capable of containing many wavelengths per nodal spacing. This is achieved by applying the plane wave basis decomposition to the three-dimensional (3D) elastic wave equation and expressing the displacement field as a sum of both pressure (P) and shear (S) plane waves. The implementation of this model in 3D presents a number of issues in comparison to its 2D counterpart, especially regarding how S-waves are used in the basis at each node and how to choose the balance between P and S-waves in the approximation space. Various proposed techniques that could be used for the selection of wave directions in 3D are also summarised and used. The developed elements allow us to relax the traditional requirement which consists to consider many nodal points per wavelength, used with low order polynomial based finite elements, and therefore solve elastic wave problems without refining the mesh of the computational domain at each frequency. The effectiveness of the proposed technique is determined by comparing solutions for selected problems with available analytical models or to high resolution numerical results using conventional finite elements, by considering the effect of the mesh size and the number of enriching 3D plane waves. Both balanced and unbalanced choices of plane wave directions in space on structured mesh grids are investigated for assessing the accuracy and conditioning of this 3D PUFEM model for elastic waves.     

Inverse problem for structural acoustic interaction

We consider an inverse problem of determining a source term for a structural acoustic partial differential equation (PDE) model that is comprised of a two- or a three-dimensional interior acoustic wave equation coupled to an elastic plate equation. The coupling takes place across a boundary interface. For this PDE system, we obtain uniqueness and stability estimates for the source term from a single measurement of boundary values of the “structure” (acceleration of the elastic plate). The proof of uniqueness is based on a Carleman estimate (first version) of the wave problem within the chamber. The proof of stability relies on three main points: (i) a more refined Carleman estimate (second version) and its resulting implication, a continuous observability-type estimate; (ii) a compactness/uniqueness argument; (iii) an operator theoretic approach for obtaining the needed regularity in terms of the initial conditions.     

General relations for waves in one-dimensional elastic systems

Common features inherent in waves propagating in one-dimensional elastic systems are pointed out. Local laws of energy and wave momentum transfer when the Lagrangian of an elastic system depends on the generalized coordinates and their derivatives up to the second order inclusive are presented. It is shown that in a reference system moving with the phase velocity, the ratio of the energy flux density to the wave momentum flux density is equal to the phase velocity. It is established that for systems, the behaviour of which is described by linear equations or by nonlinear equations in the unknown function, the ratio of the mean values of the energy flux density to the wave momentum density is equal to the product of the phase and group velocities of the waves.     

On Surface Waves and Deformations in a Pre-stressed Incompressible Elastic Solid

The propagation of infinitesimal surface waves on a half-spaceof incompressible isotropic elastic material subject to a generalpure homogeneous pre-strain is considered. The secular equationfor propagation along a principal axis of the pre-strain isobtained for a general strain-energy function, and conditionswhich ensure stability of the underlying pre-strain are derived.The influence of the pre-stress on the existence of surfacewaves is examined and, in particular, it is found that, undera certain range of hydrostatic pre-stress, a unique wavespeedexists and is bounded above by a limiting speed which correspondsto the shear wave speed in an infinite body. The secular equationis analysed in detail for particular deformations and, for anumber of specific forms of strain-energy function, numericalresults are used to illustrate the dependence of the wave speedon the pre-strain. Particular attention is focused on pre-strainscorresponding to loss of stability, in which case the infinitesimalstrain is time-independent (the wave speed being zero). Thetheory described here encompasses previous work on surface wavesand instabilities in incompressible isotropic elastic materialsand provides a clear delimitation of the range of deformationsfor which surface waves exist.     

Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks

We consider a dynamic linear shallow shell model, subject to nonlinear dissipation active on a portion of its boundary in physical boundary conditions. Our main result is a uniform stabilization theorem which states a uniform decay rate of the resulting solutions. Mathematically, the motion of a shell is described by a system of two coupled partial differential equations, both of hyperbolic type: (i) an elastic wave in the 2-d in-plane displacement, and (ii) a Kirchhoff plate in the scalar normal displacement. These PDEs are defined on a 2-d Riemann manifold. Solution of the uniform stabilization problem for the shell model combines a Riemann geometric approach with microlocal analysis techniques. The former provides an intrinsic, coordinate-free model, as well as a preliminary observability-type inequality. The latter yield sharp trace estimates for the elastic wave—critical for the very solution of the stabilization problem—as well as sharp trace estimates for the Kirchhoff plate—which permit the elimination of geometrical conditions on the controlled portion of the boundary.