On the existence of surface waves in an elastic half-space with impedance boundary conditions (original) (raw)
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Reflection of Elastic Waves from Plane Surface of a Half-space with Impedance Boundary Conditions
Geosciences Research, 2017
In the present work, a problem on the reflection of elastic waves at a plane surface of an elastic half-space is considered. The elastic half-space is assumed homogeneous and isotropic. The plane surface of half-space is subjected to impedance boundary conditions, where normal and tangential tractions are proportional to normal and tangential displacement times frequency, respectively. The reflection coefficients of reflected P and SV are obtained in closed form for incidence of P or SV waves. These reflection coefficients depend on the angle of incidence, impedance parameters and other material parameters. The material parameters of Diabase (dark-colored igneous rock), Limestone (sedimentary rock) and Gneiss (high grade metamorphic rock) are chosen to compute the reflection coefficients for certain ranges of angle of incidence and impedance parameters. The effect of impedance boundary on the reflection phenomena is shown graphically for three different rock materials (Diabase, Limestone and Gneiss).
Rayleigh-type waves on a coated elastic half-space with a clamped surface
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Elastodynamics of a half-space coated by a thin soft layer with a clamped upper face is considered. The focus is on the analysis of localized waves that do not exist on a clamped homogeneous half-space. Non-traditional effective boundary conditions along the substrate surface incorporating the effect of the coating are derived using a long-wave high-frequency procedure. The derived conditions are implemented within the framework of the earlier developed specialized formulation for surface waves, resulting in a perturbation of the shortened equation of surface motion in the form of an integral or pseudo-differential operator. Non-uniform asymptotic formula for the speeds of the sought for Rayleigh-type waves, failing near zero frequency and the thickness resonances of a layer with both clamped faces, follow from the aforementioned perturbed equation. Asymptotic results are compared with the numerical solutions of the full dispersion relation for a clamped coated half-space. A similar...
The Effect of Pure Shear on the Reflection of Plane Waves at the Boundary of an Elastic Half-Space
Iranian Journal of Mathematical Sciences and Informatics, 2007
This paper is concerned with the effect of pure shear on the reflection from a plane boundary of infinitesimal plane waves propagating in a half-space of incompressible isotropic elastic material. For a special class of constitutive laws it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave (with angle of reflection equal to the angle of incidence) although its amplitude may vanish at certain discrete angles but is independent of the state of deformation. Reflected wave amplitude is exactly equal to one in this case. For a second class of constitutive laws similar behavior is found for certain combinations of angle of incidence, material properties and de- formations, but additional possibilities also arise. In particular, there may be two reflected waves instead of one reflected wave and a surface wave. Here surface wave amplitude depends upon the pure shear and the reflected wave amplitude ...
2018
The wave phenomena on the contact of two isotropic elastic half-spaces with canonical surface protrusions is investigated. The junction of two half-spaces is modeled as a three-layer waveguide consisting of two homogeneous halfspaces and embedded, periodically inhomogeneous inner layer. The conditions of wave propagation of allowed frequencies are obtained in periodically inhomogeneous layered structure. The problem of wave formation in transversely periodic cells of the composite waveguide is solved. As a particular case, the propagation of high-frequency (shortwave) wave signal along the composite waveguide is numerically investigated. The variety of generated waves through the thickness of composite waveguide are given depending on the relative linear dimensions of the layers and physico-mechanical characteristics of materials of the composite waveguide. The bands of allowed (or forbidden) frequencies are defined for these forms.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
The dynamic response of a homogeneous half-space, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the ‘local’ problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of non-local elastic phenomena on the Rayleigh wave speed is investigated.
Sh surface Waves in a Homogeneous Gradient-Elastic Half-Space with Surface Energy
1997
The existence of SH surface waves in a half-space of 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 halfspace 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.
Rayleigh waves with impedance boundary conditions in anisotropic solids
Wave Motion, 2014
The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x 3 = 0. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.
Surface Shear Waves in a Half-Plane with Depth-Variant Structure
Journal of Optimization Theory and Applications
We consider the propagation of surface shear waves in a half-plane, whose shear modulus µ(y) and density ρ(y) depend continuously on the depth coordinate y. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency ω and wave number k as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum ω(k) for the corresponding surface wave. We establish the criteria for non-existence of surface waves and for the existence of N (k) surface wave solutions, with N (k) → ∞ as k → ∞. The most intriguing result is a possibility of the existence of infinite number of solutions, N (k) = ∞, for any given k. These three options are conditioned by the properties of µ(y) and ρ(y).
The author considers reflection and transmission of plane harmonic waves at the interface of two incompressible isotropic elastic half-spaces having the same material properties but different strain-energy functions. It is shown that an incident SV plane wave generates a single reflected wave and an interfacial wave, while a transmitted wave and an interfacial wave are generated in the second half-space. Some illustrative graphs are presented.