Sh surface Waves in a Homogeneous Gradient-Elastic Half-Space with Surface Energy

The existence of SH surface waves in a half-space homogeneous material (i.e. anti-plane shear wave motions which decay exponentially with the distance from the free surface) is shown to be possible within the framework of the generalized linear continuum theory of gradient elasticity with surface energy. As is well-known such waves cannot be predicted by the classical theory of linear elasticity for a homogeneous half-space, although there is experimental evidence supporting their existence. Indeed, this is a drawback of the classical theory which is only circumvented by modelling the half-space as a layered structure (Love waves) or as having non-homogeneous material properties. On the contrary, the present study reveals that SH surface waves may exist in a homogeneous half-space if the problem is analyzed by a continuum theory with appropriate microstructure. This theory, which was recently introduced by Vardoulakis and co-workers, assumes a strain-energy density expression containing, besides the classical terms, volume strain-gradient and surface-energy gradient terms. This revised version was published online in August 2006 with corrections to the Cover Date.     

THE EFFECT OF DUAL-PHASE-LAG MODEL ON REFLECTION OF THERMOELASTIC WAVES IN A SOLID HALF SPACE WITH VARIABLE MATERIAL PROPERTIES

The present article represents an analysis of reflection of P-wave and SV-wave on the boundary of an isotropic and homogeneous generalized thermoelastic half-space when the boundary is stress-free as well as isothermal. The modulus of elasticity is taken as a linear function of reference temperature. The basic governing equations are applied under four theories of the generalized thermoelasticity： Lord-Shulman （L-S） theory with one relaxation time, Green-Naghdi （G-N） theory without energy dissipation and Tzou theory with dual-phase-lag （DPL）, as well as the coupled thermoelasticity （CTE） theory. It is shown that there exist three plane waves, namely, a thermal wave, a P-wave and an SV-wave. The reflection from an isothermal stress-free surface is studied to obtain the reflection amplitude ratios of the reflected waves for the incidence of P- and SV-waves. The amplitude ratios variations with the angle of incident are shown graphically. Also the effects of reference temperature of the modulus of elasticity and dual-phase lags on the reflection amplitude ratios are discussed numerically.     

Dispersive Rayleigh-Wave Propagation in Microstructured Solids Characterized by Dipolar Gradient Elasticity

It is well known that the classical theory of elasticity predicts Rayleigh-wave motions, which are not dispersive at any frequency. Of course, at high frequencies, this is a result that contradicts experimental data and also does not agree with results of the discrete particle theory (atomic-lattice approach). To remedy this shortcoming, the Mindlin–Green–Rivlin theory of dipolar gradient elasticity is employed here to analyze waves of the Rayleigh type propagating along the surface of a half-space. The analysis shows that these waves are indeed dispersive at high frequencies, a result that can be useful in applications of high-frequency surface waves, where the wavelength is often on the micron order. Provided that certain relations hold between the various microstructure parameters entering the theory employed here, the dispersion curves of these waves have the same form as that given by previous analyses based on the atomic-lattice theory. In this way, the present analysis gives also means to obtain estimates for microstructure parameters of the gradient theory.     

Boussinesq–Flamant problem in gradient elasticity with surface energy

The strain gradient elasticity theory with surface energy is applied to Boussinesq–Flamant problem. The solution for the vertical displacements at the surface of half space due to the surface normal line load is presented. The theory includes both volumetric and surface energy terms. Boussinesq–Flamant problem in the strain gradient elasticity is solved by means of Fourier transform. The results obtained show that the vertical displacements of half space in the gradient elasticity are some different from that in the classical elasticity and the effects of the strain gradient parameters (material characteristic lengths) on the vertical displacements do exist.     

Effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a heterogeneous half-space

In the present paper we study the effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a semi-infinite heterogeneous half-space, where the heterogeneity is both in rigidity and density. The present study demonstrates that torsional waves can propagate in the layer. The velocities of torsional waves have been calculated numerically as a functions of KH, (where K is the wave number and H is the thickness of the layer) and are presented in a number of graphs. It is also observed that, for a layer over a homogeneous half-space, the velocity of torsional waves does not coincide with that of Love waves in the presence of the rigid boundary whereas it does at the free boundary.     

Anti-plane shear Lamb''s problem treated by gradient elasticity with surface energy

The consideration of higher-order gradient effects in a classical elastodynamic problem is explored in this paper. The problem is the anti-plane shear analogue of the well-known Lamb's problem. It involves the time-harmonic loading of a half-space by a single concentrated anti-plane shear line force applied on the half-space surface. The classical solution of this problem based on standard linear elasticity was first given by J.D. Achenbach and predicts a logarithmically unbounded displacement at the point of application of the load. The latter formulation involves a Helmholtz equation for the out-of-plane displacement subjected to a traction boundary condition. Here, the generalized continuum theory of gradient elasticity with surface energy leads to a fourth-order PDE under traction and double-traction boundary conditions. This theory assumes a form of the strain-energy density containing, in addition to the standard linear-elasticity terms, strain-gradient and surface-energy terms. The present solution, in some contrast with the classical one, predicts bounded displacements everywhere. This may have important implications for more general contact problems and the Boundary-Integral-Equation Method.     

Response of Generalized Thermoelastic Half-space with Voids to Mechanical and Thermal sources

The dynamic response of a homogeneous, isotropic, generalized thermoelastic half-space with voids subjected to normal, tangential force and thermal source is investigated. The displacements, stresses, temperature distribution and change in volume fraction field so obtained in the physical domain are computed numerically and illustrated graphically. The numerical results of these quantities for magnesium crystal-like material are illustrated to depict the response of various sources in the Lord–Shulman (L–S) theory and Green–Lindsay (G–L) theory for an insulated boundary and temperature gradient boundary. Some particular cases have been deduced.     

Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity

This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. Two-dimensional configurations in the form of either a half-space (Flamant–Boussinesq type problem) or a full-space (Kelvin type problem) are treated and the concentrated loads are taken as line forces. Our main concern is to determine possible deviations from the predictions of plane-strain/plane-stress classical linear elastostatics when a more refined theory is employed to attack the problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities and discontinuities exist in the classical solutions. The use of the theory of gradient elasticity is intended here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin (Arch. Ration. Mech. Anal. 11:385–414, 1962) and Mindlin (Arch. Ration. Mech. Anal. 16:51–78, 1964) is employed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient) additional to the standard Lamé constants (Georgiadis et al., J. Elast. 74:17–45, 2004). This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. The solution method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity solutions (Flamant–Boussinesq and Kelvin plane-strain solutions). Indeed, continuous and bounded displacements are predicted at the points of application of the loads. Such a behavior of the displacement fields is, of course, more natural than the singular behavior present in the classical solutions.      

Angular Dependence of Rayleigh-Wave Velocity in Prestressed Polycrystalline Media with Monoclinic Texture

This article is concerned with Rayleigh waves propagating along the free surface of a macroscopically homogeneous, prestressed half-space. In the meso-scale, the half-space in question is taken to be a textured polycrystalline aggregate of cubic crystallites, which has the normal to its free surface being a 2-fold axis of monoclinic sample symmetry. Under the theoretical framework of linear elasticity with initial stress, an angular dependence formula, which shows explicitly how the phase velocity of Rayleigh waves depends on the propagation direction, the prestress, and the crystallographic texture, is derived from a constitutive equation motivated by Hartig's law. This velocity formula includes terms which describe the effects of texture on acoustoelastic coefficients, and it is correct to within terms linear in the initial stress and in the anisotropic part of the incremental elasticity tensor. Since its derivation makes no presumption on the origin of the initial stress, this velocity formula is meant to be applicable when the prestress is induced by plastic deformations such as those incurred during the surface enhancement treatment of low plasticity burnishing. The angular dependence formula assumes a simpler form when the texture of the prestressed half-space is orthorhombic. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer

The paper presents the effect of a rigid boundary on the propagation of torsional surface waves in a porous elastic layer over a porous elastic half-space using the mechanics of the medium derived by Cowin and Nunziato (Cowin, S. C. and Nunziato, J. W. Linear elastic materials with voids. Journal of Elasticity, 13(2), 125–147 (1983)). The velocity equation is derived, and the results are discussed. It is observed that there may be two torsional surface wave fronts in the medium whereas three wave fronts of torsional surface waves in the absence of the rigid boundary plane given by Dey et al. (Dey, S., Gupta, S., Gupta, A. K., Kar, S. K., and De, P. K. Propagation of torsional surface waves in an elastic layer with void pores over an elastic half-space with void pores. Tamkang Journal of Science and Engineering, 6(4), 241–249 (2003)). The results also reveal that in the porous layer, the Love wave is also available along with the torsional surface waves. It is remarkable that the phase speed of the Love wave in a porous layer with a rigid surface is different from that in a porous layer with a free surface. The torsional waves are observed to be dispersive in nature, and the velocity decreases as the oscillation frequency increases.     

Construction and Analysis of an Analytical Solution for the Surface Rayleigh Wave within the Framework of the Cosserat Continuum

The problem of propagation of an acoustic surface Rayleigh wave in an infinite half-space is considered within the framework of the asymmetric theory of elasticity (Cosserat medium). It is assumed that material deformation is described not only by the displacement vector but also by an independent rotation vector. A global analytical solution of the problem in displacements is obtained. A comparative analysis of the solution obtained and the corresponding solution for the classical elastic medium is performed. Macroparameters characterizing the difference of the stress-strain state from that predicted by the classical theory of elasticity are introduced.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 116–124, July– August, 2005.     

Reflection of plane waves from an elastic solid half-space under hydrostatic initial stress without energy dissipation

In this paper, the basic governing equations for isotropic and homogeneous generalized thermoelastic half-space under hydrostatic initial stress are formulated in the context of the Green and Naghdi theory of types II and III. These governing equations are solved analytically to obtain the dimensional velocities in an xy-plane. It is shown that there exist three plane waves, namely a thermal wave, a P-wave and an SV-wave. The reflection from an insulated and isothermal stress-free surface is studied to obtain the reflection amplitude ratios of the reflected waves for the incidence of P- and SV-waves. Numerical computations are carried out and comparisons made with the results predicted in the presence and absence of hydrostatic initial stress. Also the effect of the thermoelastic coupling parameter and the thermal condition on amplitude ratios are shown graphically.     

Torsional surface waves in heterogeneous anisotropic half-space under initial stress

The present paper is concerned with the propagation of torsional surface waves in a heterogeneous anisotropic half-space under the initial compressive stress. The heterogeneity in the half-space is caused by the linear variation in rigidity, initial compressive stress and density. The solution part of the problem involves the use of Whittaker function. The dispersion equation has been obtained in a closed form, which shows the variation of phase velocity with corresponding wave number. Effects of anisotropy and initial stress have been shown by the means of graphs for different anisotropic materials. It has found that the phase velocity of torsional waves decreases with increment in initial stress and inhomogeneity. Obtained phase velocity of torsional surface wave is found to be less than the shear wave velocity, which agrees with the standard result.     

Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.     

New Solitary Waves in Elastic Materials: Existence and Nonlinear Interaction

Waves mentioned in the title were revealed in composite materials that are described by the microstructural theory of the second order — the theory of two-phase mixtures. For harmonic periodic waves, a mixture is always a dispersive medium. This medium admits existence of other waves — waves with profiles described by functions of mathematical physics (the Chebyshov–Hermite, Whittaker, Mathieu, and Lamé functions). If the initial profile of a plane wave is chosen in the form of the Chebyshev–Hermite or Whittaker function, then the wave may be regarded as an aperiodic solitary wave. The dispersivity of a mixture as a nonlinear frequency dependence of phase velocities transforms for nonperiodic solitary waves into a nonlinear phase-dependence of wave velocities. This and some other properties of such waves permit us to state that these waves fall into a new class of waves in materials, which is intermediate between the classical simple waves and the classical dispersion traveling waves. The existence of these new waves is proved in a computer analysis of phase-velocity-versus-phase plots. One of the main results of the interaction study is proof of the existence of this interaction itself. Some features of the wave interaction — triplets and the concept of synchronization — are commented on     

On the development of plane waves generated by a periodic pressure applied to the flow of a continuously stratified fluid

In the linear formulation, an investigation is made into the development of undamped (in time) plane waves generated by a. harmonically varying pressure applied to the free surface of an initially undisturbed flow of a continuously stratified fluid of finite depth. The cases of a homogeneous fluid and two-layer fluid are considered in [1–3]. Nonstationary waves in a continuously stratified flow generated by a time-independent pressure were investigated in [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 99–104, July–August, 1980.     

On a representation of the solution of the linear elasticity equations

A new representation of the stress tensor in the linear theory of elasticity is proposed. The representation satisfies the equilibrium equations and the compatibility conditions for strains. In this representation, the stress tensor is expressed in terms of a harmonic vector. The second boundary-value problem for an elastic half-space and elastic layer is considered as an example.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 85–91, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Perturbation Formulas for Polarization Ratio and Phase Shift of Rayleigh Waves in Prestressed Anisotropic Media

This paper completes an earlier study (Tanuma and Man, Journal of Elasticity, 85, 21–37, 2006) where we derive a first-order perturbation formula for the phase velocity of Rayleigh waves that propagate along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a suitably-chosen, comparative, unstressed and isotropic state be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, here we derive first-order perturbation formulas for the changes in polarization ratio and phase shift of Rayleigh waves from their respective comparative isotropic value. Examples are given, which show that the perturbation formulas for phase velocity and polarization ratio can serve as a starting point for investigations on the possible advantages of using Rayleigh-wave polarization, as compared with using wave speed, for acoustoelastic measurement of stress.      

Love solutions in the linear inhomogeneous transversely isotropic theory of elasticity

A general Love solution for the inhomogeneous transversely isotropic theory of elasticity with the elastic constants dependent on the coordinate z is proposed. This result may be considered as a generalization of the Love solutions we recently derived for the inhomogeneous isotropic theory of elasticity. The key steps of deriving the Love solution for the classical linear homogeneous transversely isotropic theory of elasticity are described for further use of the derivation procedure, which is then generalized to the inhomogeneous transversely isotropic case. Some particular cases of inhomogeneity traditionally used in the theory of elasticity are also examined. The significance of the derived solutions and their importance for the modeling of functionally graded materials are briefly discussed     

Evolution of three-dimensional gravitationally warped waves during the movement of a pressure zone of variable intensity

Three-dimensional, unestablished, gravitationally warped waves arising due to the motion of a harmonically time-varying pressure zone over a solid, thin plate floating on the surface of a homogeneous liquid of finite depth have been studied in the linear formulation. In the absence of a plate, three-dimensional waves are generated by the movement of a region of periodic perturbations, where established waves have been studied in [1, 2], and unestablished waves have been investigated in [3–5]. The evolution of three-dimensional, gravitationally warped waves formed during the motion of a constant load over a plate has been considered in [6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 54–60, September–October, 1986.