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     

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.     

Reflection of plane waves at the boundary of a pre-stressed elastic half-space subject to biaxial stretch and simple shear

This paper is concerned with the effect of finite pure homogeneous biaxial stretch together with simple shear deformation on the reflection from a plane boundary of elastic waves propagating in a half-space of incompressible isotropic elastic material. This generalizes the previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of constitutive law, 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 for every angle of incidence. The amplitude of the surface wave may vanish at certain discrete angles depending on the state of stress, biaxial stretch and simple shear deformation and then specialized to recover results obtained previously. The amplitude of the reflected wave is independent of the pre-stress but does depend upon the magnitude of deformation under consideration. The dependence of the reflected and surface wave behavior on the angle of incidence, amount of shear strain, biaxial stretch and the state of stress is illustrated graphically.     

Reflection of piezothermoelastic waves from the charge and stress free boundary of a transversely isotropic half space

The present paper concentrates on the study of propagation and reflection characteristics of waves from the stress free, thermally insulated/isothermal boundary of a piezothermoelastic half space. The non-classical (generalized) theories of linear piezo-thermoelasticity have been employed to investigate the problem. In the two-dimensional model of the transversely isotropic piezothermoelastic medium, there are three types of plane waves quasi-longitudinal (QL), quasi-transverse (QT) and thermal wave (T-mode), whose velocities depend on the angle of incidence and frequency. These waves are dispersive in character and are also affected by piezoelectric as well as pyroelectric properties of the materials. The low and high frequency approximations for the speeds of propagation and the attenuation coefficients of these waves have been obtained. The quasi-longitudinal (QL), quasi-transverse (QT) and thermal wave (T-mode) incident cases at the stress free, thermally insulated or isothermal open circuit boundary of a transversely isotropic piezothermoelastic half space are considered to discuss the reflection characteristics of various waves. The amplitude ratios of reflected waves to that of incident one in each case have been obtained. The special cases of normal and grazing incidence are also derived and discussed. Finally, the numerical computations of reflection coefficients are carried out for cadmium Selenide (CdSe) material by using Gauss elimination procedure. In addition the phase velocities and attenuation coefficients are also computed along various directions of wave propagation. The obtained results in each case are presented graphically.     

Transient reflection of TM-polarized plane waves from a Lorentz-medium half-space

The time-domain reflection coefficient for a TM-polarized plane wave obliquely incident on a Lorentz-medium half-space is determined analytically by inversion of the frequency-domain reflection coefficient. The resulting expression contains only the convolution of simple functions. This allows the temporal behavior of the reflection coefficient to be predicted, and the relationship between the material parameters and the oscillation of the response to be easily identified. The time-domain expression is validated numerically through comparison with the inverse fast Fourier transform of the frequency-domain reflection coefficient.     

Transient reflection of TE-polarized plane waves from a Lorentz-medium half-space

The time-domain reflection coefficient for a plane wave obliquely incident on a Lorentz-medium half-space is determined analytically by inversion of the frequency-domain reflection coefficient. The resulting expression contains only simple functions and a single convolution of these functions. Owing to its simplicity, this form of the reflection coefficient provides insight into its temporal behavior, specifically how the relationship between the damping coefficient and the oscillation frequency determines the shape of the response. The simple form of the reflection coefficient is validated numerically through comparison with the inverse fast Fourier transform of the frequency-domain reflection coefficient.     

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.     

Plane waves in a thermally conducting viscous liquid

The aim of this paper is to investigate plane waves in a thermally conducting viscous liquid half-space with thermal relaxation times. There exist three basic waves, namely; thermal wave, longitudinal wave and transverse wave in a thermally conducting viscous liquid half-space. Reflection of plane waves from the free surface of a thermally conducting viscous liquid half-space is studied. The results are obtained in terms of amplitude ratios and are compared with those without viscosity and thermal disturbances.     

Transient elastic waves in a half-space: comparison of the DBIE-method with the method of generalized ray

Summary The two-dimensional problem of wave propagation in an elastic half-space is studied by the DBIEM (Direct Boundary Integral Equation Method) combined with the finite difference procedure applied to the time variable. The present hybrid formulation employs the fundamental solution depending neither on the frequency nor on the time variable. Time records of surface responses of the half-space are computed and compared with those obtained by the numerical evaluation of exact analytical solutions to this problem.     

Reflection of PlaneWaves from Electro-magneto-thermoelastic Half-space with a Dual-Phase-Lag Model

The aim of this paper is to study the reflection of plane harmonic waves from a semi-infinite elastic solid under the effect of magnetic field in a vacuum. The expressions for the reflection coefficients, which are the relations of the amplitudes of the reflected waves to the amplitude of the incident waves, are obtained. Similarly, the reflection coefficient ratio variations with the angle of incident under different conditions are shown graphically. Comparisons are made with the results predicted by the dual-phase-lag model and Lord-Shulman theory in the presence and absence of magnetic field.     

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     

Influence of the squirt flow on reflection and refraction of elastic waves at a fluid/fluid-saturated poroelastic solid interface

In this paper, the reflection and refraction problem of a plane wave moving in perfect fluid incident on a viscous fluid-saturated poroelastic solid is investigated. Attention is focus on the effect of squirt flow on the reflection and refraction coefficients predicted by BISQ model. The variations of these coefficients with the characteristic squirt flow length have been shown graphically. The results are also compared with those based on Biot theory without considering the squirt flow effects. The results indicate that the effect of the squirt flow is noticeable in the reflection and refraction phenomenon.     

Reflection of quasi-P and quasi-SV waves at the free and rigid boundaries of a fibre-reinforced medium

The propagation of plane waves in fibre-reinforced media is discussed. The expressions of phase velocities of quasi-P (qP) and quasi-SV (qSV) waves propagating in plane symmetry are obtained in terms of propagation vectors. We have established a relation from which the displacement vector can be obtained in terms of the propagation vector. Expressions for the reflection coefficients of qP and qSV waves are obtained. Numerical results of reflection coefficients are obtained and presented graphically. The partition of energy between qP and qSV waves reflected on free and rigid boundaries due to incident qP and qSV waves are also obtained and presented graphically.     

A crack within a half-space of orthotropic elastic material

Summary The dislocation layer method combined with a technique of images is used to study a mode III loaded Griffith-type elastic strip crack situated parallel to the free-surface of a semi-infinite orthotropic crystal. The density function of the proposed distribution of dislocations is shown to satisfy a complex singular integral equation. Its closed-form solution provides a compact expression for an appropriate combination of the resulting stress field components. Some representative numerical results are presented in tabular form and discussed.     

Reflection of Magneto-Thermoviscoelastic Waves Under Generalized Thermoviscoelasticity

Considering each of the thermal fields, viscoelastic and electromagnetic fields contribute to the total deformation of a body and interact with each other. Reflection of magneto-thermoelastic waves under generalized thermoviscoelastic theories is employed to study the reflection of plane harmonic waves from a semi-infinite elastic solid in a vacuum. The expressions for the reflection coefficients, which are the ratios of the amplitudes of the reflected waves to the amplitudes of the incident waves, are obtained, and the reflection coefficient ratios variation with the angle of incidence under different conditions are shown graphically when acrylic plastic materials are considered.     

Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media

The paper analyzes the reflection and refraction of a plane wave incidence obliquely at the interface between piezoelectric and piezomagnetic media. The materials are assumed to be transversely isotropic. Numerical calculations are performed for BaTiO₃/CoFe₂O₄ material combination. Four cases, incidence of the coupled quasi-pressure (QP) and quasi-shear vertical (QSV) wave from BaTiO₃ or CoFe₂O₄ media, are discussed. The reflection and transmission coefficients and energy coefficients varying with the incident angle are examined. Calculated results are verified by considering the energy conservation. Results show that the reflected and transmitted wave fields in the sagittal plane consist of six kinds of waves, i.e. the coupled QP and QSV waves, evanescent electroacoustic (EA) and magnetic potential (MP) waves in the piezoelectric medium (BaTiO₃), evanescent magnetoacoustic (MA) and electric potential (EP) waves in the piezomagnetic medium (CoFe₂O₄), among which the EA, MA, MP and EP waves propagate along the interface. The most amount of the incident energy goes with the waves that are the same type as the incident wave, while the energy arising from wave mode conversion occupies a less part of the incident energy. The electric energy in BaTiO₃ is higher than the magnetic energy in CoFe₂O₄; they both attain their maximum values at/before the critical angle. Critical angles have little effect on evanescent waves except when the total reflection takes place. These results would provide useful complementary information for magnetoelectric composite materials.     

Rayleigh-Lamb waves in magneto-thermoelastic homogeneous isotropic plate

The propagation of magnetic-thermoelastic plane wave in an initially unstressed, homogeneous isotropic, conducting plate under uniform static magnetic field has been investigated. The generalized theory of thermoelasticity is employed, by assuming electrical behaviour as quasi-static and the mechanical behaviour as dynamic, to study the problem. The secular equations for both symmetric and skew-symmetric waves have been obtained. The magneto-elastic shear horizontal (SH) mode of wave propagation gets decoupled from rest of the motion and it is not influenced by thermal variations and thermal relaxation times. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation, because a finite thickness plate in such a situation behaves like a semi-infinite medium. Thin plate results are also deduced at the end. Dispersion curves are represented graphically for various modes of wave propagation in different theories of thermoelasticity. The amplitudes of displacement, perturbed magnetic field and temperature change are also obtained analytically and computed numerically. The result in case of elastokinetics, magneto-elasticity and coupled magneto-elasticity has also been deduced as special cases at appropriate stages of this work.     

Reflection and Refraction of Micropolar Magneto-thermoviscoelastic Waves at the Interface between Two Micropolar Viscoelastic Media

Using micropolar generalized thermoviscoelastic theories, problems of reflection and refraction of magneto-thermoeviscoelastic waves at the interface between two viscoelastic media are studied when a uniform magnetic field permeates the media. Coefficient ratios of reflection and refraction are obtained using continuous boundary conditions. Some special cases are considered, i.e., the absence of micropolar and viscous effects. By numerical calculations, variations of the amplitude ratios of reflection and refraction coefficients with the angle of incidence are shown graphically for incident rotational and dilatational waves at the interface between two media (one medium is aluminium-epoxy micropolar iscoelastic material, and the other is magnesium crystal micropolar viscoelastic material). Comparing the generalized thermoelastic theories developed by Lord and Shulman (LS) and by Green and Lindsay (GL) in this paper to conventional dynamics (CD) theory the effects of a magnetic field and viscosity are shown numerically in this paper.