Surface waves in an elastic half-space with voids

Summary Surface waves of general type propagating in a plane-faced, homogeneous and isotropic linear elastic halfspace containing a distribution of vacuous pores (voids) are studied. Assuming that the face of the halfspace is free of external loads, it is found that the motion is not necessarily two-dimensional and that all physical variables associated with the waves are derivable from a single scalar function. The phase speed equation reveals that, unlike in the classical problem, the waves are dispersive, in general. If the frequency is very small, the motion is qualitatively similar to that in the classical problem, but the wave speed is modified due to the presence of voids.     

Surface waves in a periodic two-layered elastic half-space

Summary The paper deals with surface waves propagating through a periodic two-layered elastic half-space. The analysis is performed on the basis of a homogenized model with microlocal parameters. The velocity of the surface wave is obtained as a function of geometric and dynamic properties of the subsequent layers. The numerical examples illustrating the variations of the surface wave velocity are presented.     

Reflection of magneto-thermo-elastic waves from a rotating elastic half-space

Models of generalized magneto-thermo-elasticity based on the Lord-Shulman theory (LS) with one relaxation time, the Green-Lindsay theory (GL) with two relaxation times, as well as the classical dynamic coupled theory (CD), are used to study the electro-magneto-thermo-elastic interactions in a semi-infinite perfectly conducting solid. The elastic medium rotates with uniform angular velocity. The initial magnetic field is parallel to the boundary of the half-space. Reflection of magneto-thermo-elastic waves under generalized thermo-elasticity theory is studied. The reflection coefficients are obtained.     

An approximate secular equation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer

In this paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half-space is assumed to be welded. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate secular equation of fourth order in terms of the dimensionless thickness of the layer is derived. It is shown that this approximate secular equation has high accuracy. From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established.     

Surface-breaking crack in an elastic half-space

A semi-infinite crack terminating at the boundary of an elastic half-space is considered. It is assumed that the crack is subjected to a mode-I load applied in a finite region remote from the crack plane, and the boundary of the half-space and the crack surfaces are free of tractions. The problem is formulated in terms of a hypersingular integral equation with respect to the relative crack-face separation defined over the region occupied by the crack. The behaviour of the solution near the corner point where the crack edge intersects the boundary is analysed and results which show the dependence of the stress singularity exponent on the angle of inclination of the crack edge are presented.     

Reflection characteristics of torsional waves in a semi-infinite cylindrical rod connected to an elastic half-space—II

Using the Pochhammer-Chree equation and applying Laplace transformations with respect to time and numerical inverse Laplace transformations, reflection characteristics of torsional waves in a semi-infinite rod connected to an elastic half-space are analyzed under a condition where the incident stress pulse varies proportionally with the distance from the rod axis. Time histories of surface torsional stress and circumferential displacement, and cross-sectional torsional stress and circumferential displacement distributions of the semi-infinite rod at an arbitrary point are shown. When the shear modulus ratio $\text{G}{}_{\mathrm{R}}\text{/G}$ and the velocity ratio $\text{K}$ are small, i.e. The half-space is hard and the rod is soft, time histories of surface stress and displacement obtained from the Pochhammer-Chree equation coincide with those obtained from the elementary equation which is derived from the assumption that each cross-section of the rod undergoes a pure rotation about the rod axis. However, when $\text{G}{}_{\mathrm{r}}\text{/G}$ and $\text{K}$ are not small, the curves obtained from the Pochhamer-Chree equation differ from those obtained from the elementary equation, and the difference in stress increases as the observed point approaches the interface between the rod and the half-space.     

Scattering of elastic waves by an elastic sphere

The problem of scattering of plane compressional wave by an elastic sphere embedded in an isotropic elastic medium of different material properties is solved. Approximate formulas are derived for the displacement field, stress tensor, stess intensity factors, far-field amplitudes and the scattering cross section. It is assumed that the wave length is large compared to the radius of the scatterer. Various elastostatic limits are also presented.     

Scattering of elastic waves by an elastic sphere

The problem of scattering of plane compressional wave by an elastic sphere embedded in an isotropic elastic medium of different material properties is solved. Approximate formulas are derived for the displacement field, stress tensor, stress intensity factors, far-field amplitudes and the scattering cross-section. It is assumed that the wave length is large compared to the radius of the scatterer. Various elastostatic limits are also presented.     

Application of the generalized rays to transient waves in an elastic half-space due to a buried line source

Summary The aim of this investigation is to study by means of the generalized rays the effects of a free surface on the propagation of elastic transient waves emanating from a line source of explosion, which varies with time as the Heaviside step function with rounded shoulders. This is accomplished by the numerical evaluation and analysis of the exact solutions provided by this theory. Graphs are obtained which show displacements and stresses as functions of time at three different receivers in the interior of an infinite medium and at the free surface of a homogeneous half-space. The character of surface responses, for short observation times, is the same as in the infinite medium, with the difference that they are all amplified (surface receiver effect). The late time surface responses are characterized by the presence of a smooth pulse representing the Rayleigh surface wave. The Rayleigh pulse in the non-vanishing surface stress takes its ultimate shape already close to the epicentre, while the Rayleigh pulse in the surface displacements developes fully at the remote distance from the epicentre.With 6 Figures     

Forced flexural vibrations of a Timoshenko column supported by an elastic half-space

The plane stress problem of forced flexural vibrations of an elastic column supported by an elastic half-space and subjected to a sinusoidally varying force at the free end is investigated analytically, by applying the Timoshenko theory to the column. Numerical results obtained from the Timoshenko theory are compared with those from the Bernoulli-Euler theory, and the effects of column slenderness and foundation stiffness on the column response curves and the resonant frequencies of the system are clarified.     

SH-waves at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space

Summary A problem of reflection and transmission of a plane SH-wave incident at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space is investigated. Rayleigh's method of approximation is applied to derive the reflection and transmission coefficients for the first and second order approximation of the corrugation. The expressions for reflection and transmission coefficients for the first order approximation of the corrugation are obtained in closed form for a special type of interface, and the energy partition relation is derived. It is found that these coefficients are proportional to the amplitude of corrugation and are functions of elastic properties of the half-spaces and also of the angle of incidence. Numerical examples illustrating the effects of the sandiness, the anisotropy, the corrugation of the interface, the frequency, and the angle of the incidence on the coefficients are presented.     

Reflection of plane waves at the free surface of a fibre-reinforced elastic half-space

The propagation of plane waves in fibre-reinforced, anisotropic, elastic media is discussed. The expressions for the phase velocity of quasi-P (qP) and quasi-SV (qSV) waves propagating in a plane containing the reinforcement direction are obtained as functions of the angle between the propagation and reinforcement directions. Closed form expressions for the amplitude ratios for qP and qSV waves reflected at the free surface of a fibre-reinforced, anisotropic, homogeneous, elastic half-space are obtained. These expressions are used to study the variation of amplitude ratios with angle of incidence. It is found that reinforcement has a significant effect on the amplitude ratios and critical angle     

Second order effects in an elastic half-space acted upon by a non-uniform shear load

Summary A closed form solution to the second order elasticity problem, when an isotropic compressible elastic half-space undergoes a deformation owing to a non-uniformly distributed shear load, is presented. The method of integral transform is employed to determine the solutions. An example is discussed in detail to illustrate the second order effects. Numerical calculations for the second order elastic material for thez-direction displacement and the stresst _rz are carried out. It is found that the second order effect is to reduce thez-direction displacement and to dereaset _rz inside the circle but to increase its value outside the circle.     

Surface waves in a periodic two-layered elastic half-space with the boundary normal to the layering

The paper deals with the problem of surface waves in a vertically layered elastic half-space. The transmitting medium is composed of periodically repeated two-constituent laminae, and its boundary is assumed to be normal to the layering. The problem is solved using an approximate model proposed by Wo?niak (Int J Eng Sci 25:483–499, 1987) and Matysiak and Wo?niak (Int J Eng Sci 25:549–559, 1987), which is referred to as the homogenized model with microlocal parameters. The velocity of the surface wave is obtained as a function of geometric and dynamic properties of the laminae. The variation of the surface wave velocity with respect to composite constituents is illustrated by numerical examples.     

Transient behavior of a few dies on an elastic half-space

Summary An asymptotic approach to dynamic interaction between a few distant dies and an elastic half-space is proposed. The transient motion of the dies under low-frequency vertical load is under consideration. The explicit expression for the fundamental singular solution of Lamb's problem is used to derive the boundary integral equation of contact. Then this equation is asymptotically simplified and solved numerically in combination with equations of motion of the dies.Equations obtained in the asymptotic limit describe both the die-medium dynamic interaction and the interaction between dies through the elastic medium. These equations take into account the energy dissipation phenomenon associated with energy transfer deep into the medium by outgoing elastic waves, of so called geometrical damping.Equations proposed are asymptotically correct within the corresponding range of parameters, as such improving the state-of-the-art.     

Disturbance of SH-type waves due to moving stress discontinuity in an anisotropic soil layer overlying an inhomogeneous elastic half-space

The disturbance and propagation of SH-type waves in an anisotropic soil layer overlying an inhomogeneous elastic half-space by a moving stress discontinuity is considered. Stress discontinuity moves with non-uniform velocity and is impulsive in nature. The displacements are obtained in exact form by the method due to Cagniard modified by de Hoop. The numerical result is calculated for special cases and the natures are depicted graphically.     

Resonance scattering of elastic waves by an elastic inclusion in an elastic matrix. Numerical calculations

The scattering of an incident plane ultrasonic (longitudinal) wave by an elastic spherical inhomogeneity contained within an elastic matrix is studied. The emphasis is on the computation and analysis of basic multiple cases that result when different material behaviours are present in the matrix and in the inclusion. These calculations are useful in underwater acoustic applications. The behaviors are dominated by the soft or rigid backgrounds of the resonance scattering theory (RST). The first three multiple coefficients appearing in the expansions for the total elastodynamic fields developed around the inhomogeneity during the scattering process have been calculated in suitable frequency bands in all the cases considered. The examination of modulus, the real parts, and the imaginary parts of these (complex) coefficients under the RST approach allows the quantitative assessment of the conditions under which monopole or dipole resonances will occur and their relative magnitudes. The decomposition of the multiple coefficients into their resonance and background portions shows that it is the upward frequency shift of the background curves that controls the dominance of either radial (monopole) or translation (dipole) oscillations of the inclusion. This has an effect on the dispersion curves of the composite, which develop optical as well as acoustical branches. The real and imaginary parts of the multiple coefficients are respectively proportional to the attenuation and the effective wave speed in this simple inhomogeneous composite.