Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

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.

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Dynamic Stress and Displacement in an Elastic Half-Space with a Cylindrical Cavity

Shock and Vibration ◽

10.1155/2011/904936 ◽

2011 ◽

Vol 18 (6) ◽

pp. 827-838 ◽

Cited By ~ 3

Author(s):

İ. Coşkun ◽

H. Engin ◽

A. Özmutlu

Keyword(s):

Free Surface ◽

Boundary Conditions ◽

Half Space ◽

Cylindrical Cavity ◽

Circular Cross Section ◽

Elastic Half Space ◽

Least Square ◽

Polar Coordinates ◽

Wave Number ◽

Forcing Function

The dynamic response of an elastic half-space with a cylindrical cavity in a circular cross-section is analyzed. The cavity is assumed to be infinitely long, lying parallel to the plane-free surface of the medium at a finite depth and subjected to a uniformly distributed harmonic pressure at the inner surface. The problem considered is one of plain strain, in which it is assumed that the geometry and material properties of the medium and the forcing function are constant along the axis of the cavity. The equations of motion are reduced to two wave equations in polar coordinates with the use of Helmholtz potentials. The method of wave function expansion is used to construct the displacement fields in terms of the potentials. The boundary conditions at the surface of the cavity are satisfied exactly, and they are satisfied approximately at the free surface of the half-space. Thus, the unknown coefficients in the expansions are obtained from the treatment of boundary conditions using a collocation least-square scheme. Numerical results, which are presented in the figures, show that the wave number (i.e., the frequency) and depth of the cavity significantly affect the displacement and stress.

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Shear Horizontal Surface Waves in a Functionally Graded Magneto-Electro-Elastic Half-Space

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.304.204 ◽

2011 ◽

Vol 304 ◽

pp. 204-207

Author(s):

Qian Yang ◽

Yan Ping Kong ◽

Jin Xi Liu

Keyword(s):

Free Surface ◽

Boundary Conditions ◽

Surface Waves ◽

Half Space ◽

Material Properties ◽

Elastic Half Space ◽

Functionally Graded ◽

Horizontal Surface ◽

Propagation Characteristics ◽

Shear Horizontal

We investigate the propagation of shear horizontal (SH) surface waves in a functionally graded magneto-electro-elastic half-space with hexagonal (6mm) symmetry. The material properties vary in the direction perpendicular to the free surface. The surface of the half-space is mechanically free, but subjected to four types of electromagnetic boundary conditions. These boundary conditions are electrically open/magnetically closed, electrically open/magnetically open, electrically closed/magnetically open and electrically closed/magnetically closed. The effects of the electromagnetic boundary conditions and the variation of material properties on the propagation characteristics of SH surface waves are analyzed. The results obtained show that except for the electrically open/magnetically closed condition, three sets of other electromagnetic boundary conditions sustain the propagation of SH surface waves. Different from the case in homogeneous materials, the existing SH surface waves in functionally graded half-spaces are dispersive.

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Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

Journal of Applied Mechanics ◽

10.1115/1.3564708 ◽

1969 ◽

Vol 36 (3) ◽

pp. 505-515 ◽

Cited By ~ 108

Author(s):

D. C. Gakenheimer ◽

J. Miklowitz

Keyword(s):

Exact Solution ◽

Free Surface ◽

Steady State ◽

Wave Front ◽

Half Space ◽

Displacement Field ◽

Elastic Half Space ◽

Point Load ◽

Limit Case ◽

Transient Waves

The propagation of transient waves in a homogeneous, isotropic, linearly elastic half space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are derived for the interior of the half space and for all load speeds. Wave-front expansions are obtained from the exact solution, in addition to results pertaining to the steady-state displacement field. The limit case of zero load speed is considered, yielding new results for Lamb’s point load problem.

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Rayleigh-type surface wave on a rotating orthotropic elastic half-space with impedance boundary conditions

Journal of Vibration and Control ◽

10.1177/1077546320909972 ◽

2020 ◽

Vol 26 (21-22) ◽

pp. 1980-1987

Author(s):

Baljeet Singh ◽

Baljinder Kaur

Keyword(s):

Surface Wave ◽

Boundary Conditions ◽

Rayleigh Wave ◽

Surface Waves ◽

Half Space ◽

Rotation Rate ◽

Secular Equation ◽

Elastic Half Space ◽

Impedance Boundary ◽

Impedance Boundary Conditions

The propagation of Rayleigh type surface waves in a rotating elastic half-space of orthotropic type is studied under impedance boundary conditions. The secular equation is obtained explicitly using traditional methodology. A program in MATLAB software is developed to obtain the numerical values of the nondimensional speed of Rayleigh wave. The speed of Rayleigh wave is illustrated graphically against rotation rate, nondimensional material constants, and impedance boundary parameters.

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On Higher Order Effective Boundary Conditions for a Coated Elastic Half-Space

Advanced Structured Materials - Problems of Nonlinear Mechanics and Physics of Materials ◽

10.1007/978-3-319-92234-8_25 ◽

2018 ◽

pp. 449-462 ◽

Cited By ~ 1

Author(s):

Julius Kaplunov ◽

Danila Prikazchikov ◽

Leyla Sultanova

Keyword(s):

Boundary Conditions ◽

Half Space ◽

Elastic Half Space ◽

Higher Order ◽

Effective Boundary ◽

Effective Boundary Conditions

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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 ◽

10.1098/rsta.2019.0111 ◽

2019 ◽

Vol 377 (2156) ◽

pp. 20190111 ◽

Cited By ~ 2

Author(s):

J. Kaplunov ◽

D. Prikazchikov ◽

L. Sultanova

Keyword(s):

Half Space ◽

Numerical Solutions ◽

Substrate Surface ◽

Numerical Data ◽

Elastic Half Space ◽

Asymptotic Results ◽

Homogeneous Half Space ◽

Upper Face ◽

Localized Waves ◽

Perturbed Equation

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 similarity with Love-type waves proves to be useful for interpreting numerical data. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

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MATHEMATICAL ANALYSIS OF ELASTIC SURFACE WAVES IN TOPOGRAPHIC WAVEGUIDES

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202599000373 ◽

1999 ◽

Vol 09 (05) ◽

pp. 755-798 ◽

Cited By ~ 22

Author(s):

A. S. BONNET-BEN DHIA ◽

J. DUTERTE ◽

P. JOLY

Keyword(s):

Free Surface ◽

Half Space ◽

Guided Waves ◽

Transverse Energy ◽

Eigenvalue Problems ◽

Low Frequency ◽

Elastic Half Space ◽

High Frequencies ◽

Guided Mode ◽

Adjoint Operators

We present here a theoretical study of the guided waves in an isotropic homogeneous elastic half-space whose free surface has been deformed. The deformation is supposed to be invariant in the propagation direction and localized in the transverse ones. We show that finding guided waves amounts to solving a family of 2-D eigenvalue problems set in the cross-section of the propagation medium. Then using the min-max principle for non-compact self-adjoint operators, we prove the existence of guided waves for some particular geometries of the free surface. These waves have a smaller speed than that of the Rayleigh wave in the perfect half-space and a finite transverse energy. Moreover, we prove that the existence results are valid for arbitrary high frequencies in the presence of singularities of the free boundary. Finally, we prove that no guided mode can exist at low frequency, except maybe the fundamental one.

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Impedance of Strip-Traveling Waves on an Elastic Half Space: Asymptotic Solution

Journal of Applied Mechanics ◽

10.1115/1.3423302 ◽

1974 ◽

Vol 41 (2) ◽

pp. 412-416

Author(s):

S. H. Crandall ◽

A. K. Nigam

Keyword(s):

Asymptotic Solution ◽

Rayleigh Wave ◽

Half Space ◽

Traveling Wave ◽

Load Distribution ◽

Wave Speed ◽

Elastic Half Space ◽

Surface Load ◽

Shear Wave Speed ◽

Rayleigh Wave Speed

The dynamic normal-load distribution across a strip that is required to maintain a plane progressive wave along its length is studied for the case where the strip is of infinite length and lies on the surface of a homogeneous isotropic elastic half space. This configuration is proposed as a preliminary idealized model for analyzing the dynamic interaction between soils and flexible foundations. The surface load distribution across the strip and the motion of the strip are related by a pair of dual integral equations. An asymptotic solution is obtained for the limiting case of small wavelength. The nature of this solution depends importantly on the propagation velocity of the strip-traveling wave in comparison with the Rayleigh wave speed, the shear wave speed and the dilatational wave speed. When the strip-traveling wave propagates faster than the Rayleigh wave speed, a pattern of trailing Rayleigh waves is shed from the strip. The limiting amplitude of the trailing waves is provided by the asymptotic solution.

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