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.     

Shock compressibility and spallation strength of cubic modification of polycrystalline boron nitride

Methods of shock-wave physics are used to perform experimental investigations of the elastoplastic and strength properties of cubic modification of polycrystalline boron nitride. The samples are prepared by pressing to a pressure of 7–8 GPa at a temperature of 1700–1800 °C and rank just a little below diamond in hardness. The measurements are performed under conditions of the samples being acted upon by plane shock waves with the pressure behind the front of up to 85 GPa and duration of ～10^?6 s. It is demonstrated that a two-wave configuration consisting of an elastic precursor and a plastic compression wave is formed at the amplitude of shock compression of over 60 GPa. The dynamic yield strength, determined by the amplitude of elastic precursor, depends on the structure of samples and varies in the range from 31 to 49 GPa. The dynamic strength, measured under conditions of pulsed tension in expansion waves, is likewise defined by the structure of samples and, in the elastic region of deformation, lies in the range from 0.7 to 1.6 GPa.     

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     

Effects of surface stresses and voids on rayleigh waves in an elastic solid

Rayleigh waves propagating at the plane material boundary of an elastic half-space containing a distribution of voids (vacuous pores) are considered. It is found that the waves are generally dispersive and that the dispersion is caused by both the surface stresses exerted by the boundary and the voids inside. If the body is incompressible, voids have no influence on the motion. In the case of small frequency, the effect of the voids is just to modify the speed of propagation. If the frequency is small and surface stresses are due to a residual surface tension, there exists a critical wavelength at which waves propagate with the speed of pure shear waves and below which the motion is not possible. The critical wave length varies directly with the surface tension and the effect of the voids is to widen the range of wavelengths for which the waves exist.     

New Poisson's integral formulas for thermoelastic half-space and other canonical domains

In this paper the functions of influence of unit point heat source onto displacements and Poisson-type integral formula for a boundary value problem (BVP) in thermoelastic half-space, free of loadings on the boundary plane are presented in closed form. The thermoelastic displacements are generated by heat source applied at the inner point of the half-space and by heat flux, prescribed on its boundary. All these results are formulated in a special theorem. Furthermore, the advantages and usefulness of the obtained results are also discussed. The main difficulties to obtain such kind of results are to derive the functions of influence of a unit concentrated force onto elastic volume dilatation Θ^(k) and Green's functions in heat conduction G. For canonical Cartesian domains, these difficulties were addressed successfully, and the above-mentioned functions were derived and published earlier. Thus, it can be presumed that for the Cartesian domains, this paper will open a great possibility to derive new thermoelastic influence functions and Poisson's integral formulas in closed form. Moreover, the technique proposed here will also work for any orthogonal canonical domain, as soon as the lists of functions G and Θ^(k) are completed.     

Finite amplitude transverse waves in materials with memory

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids with memory. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part depends not only on current but also on previous deformations. The body is first subjected to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that finite amplitude circularly polarized waves may propagate along either n⁺ or n⁻, where n⁺, n⁻ are the normals to the planes of the central circular section of the ellipsoid x · B⁻¹x = 1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.     

Wave Theory for a Converging Spherical Incident Wave in an Infinite-aperture System

Abstract

Weyl's formula describes the decomposition of a monochromatic spherical wave into an angular spectrum of plane waves. In this paper, explicit formulae for the separate contributions of homogeneous and inhomogeneous plane waves are given, and the results used to obtain an exact solution to the half-space boundary-value problem for the focusing of a converging spherical incident wave in an infinite-aperture system.     

Second-order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media

In this paper we propose a perturbation method to investigate the propagation of the ordinary waves in second-order elastic, isotropic, incompressible, and homogeneous materials. This method allows us to determine the first-order terms of the speeds and the amplitudes both of the principal waves and the waves in any propagation direction, when the undisturbed region is subjected to an arbitrary isochoric deformation. Another application of this method is presented when the undisturbed region is subjected to a simple shear. Finally, some numerical results are presented in Mooney-Rivlin materials.     

Magneto-thermo-elastic plane waves in rotating media

A study is made of the propagation of plane harmonic waves in an infinite conducting thermo-elastic solid permeated by a primary uniform magnetic field when the entire elastic medium is rotating with a uniform angular velocity. A more general dispersion relation is obtained to determine the effects of rotation, relaxation time and the external magnetic field on the phase velocity of the waves. This result indicates that if the primary magnetic field has a transverse component, then the longitudinal and transverse components of the displacement field are linked together. For the case of low frequency (Gc ? 1 where κ is the ratio of the wave frequency to the characteristic frequency), the rotation and the thermal relaxation time are found to have no influence on the phase velocity, and the attenuation factor for both finite and infinite electrical conductivity. In the case of high frequency (κ ? 1), no effect of rotation on the phase speed is observed to the first order of ($\text{1}\text{κ}$), while the relaxation time affects both the phase velocity and the specific energy loss. However, the effects of rotation on the phase speed, the attenuation factor, and the specific energy loss are found only to the second order of ($\text{1}\text{κ}$) in the case of high frequency. Several limiting cases of interest are discussed.     

Plane waves in pyroelectrics with viscous effect

The present paper analyzes the propagation of plane waves in an infinite pyroelectric medium. In order to consider the real situation, a new thermo-electric-elastic model with viscous effect is presented. The theoretical analysis shows that the elastic viscous effect and the Fourier’s law are in the same level in the evolution equations. So, the thermal viscous effect in Cattaneo’s equation belongs to a second-order effect. Numerical calculations are performed for pyroelectric material BaTiO₃ using three models (Kaliski–Lord–Shulman theory, inertial entropy theory and inertial entropy with viscous effect). Results show that the elastic viscous relaxation time τ ₀ plays a large role on the mechanical waves and admits mechanical waves decaying in propagation process. The effects of the thermo-relaxation time τ _s on the attenuation of mechanical waves are also researched and discussed in detail.     

The indentation of a half-space of hexagonal elastic material by a circular punch of arbitrary end-profile

Summary The problem is considered of the indentation by a smooth rigid punch of a half-space composed of linear elastic material of hexagonal symmetry whose plane boundary is parallel to the basal planes. The case is considered in which the area of contact between the punch and the half-space is circular, the end of the punch with is in contact with the half-space having an arbitrary profile. An integral equation is formulated and solved for the boundary value of the normal displacement in the half-space, and an expression is derived for the distribution of pressure under the punch.     

Significance of the deviations of the crack front into the plane perpendicular to the crack propagation direction - II. Crack-front vibration

In this part II, the influence of thermal vibration on the shape of a crack front and on the conditions for crack motion are investigated. We have considered a static crack model in mode I loading in a linear isotropic elastic medium whose front vibrates in the form of a standing wave with displacement ξ perpendicular to both the average crack plane and the crack propagation direction; ξis assumed to be sum of standing sine waves and the crack to be represented by a continuous distribution of vibrating edge dislocations. The displacement and stress fields due to an infinitely long edge dislocation, oscillating in the form of a standing wave, are first given. Then it is shown that, in free vibration conditions, ξ would satisfy the d'Alembert wave equation with a constant phase velocity corresponding to the velocity of a travelling sine wave along an edge dislocation. It is also shown that the sites for the initiation of crack motion are located on the average crack plane at the nodes of the vibrating crack front. The crack opening force G per unit length of the crack front, at the sites of crack propagation initiation, is found to be different from that of the rigid planar crack, when averaged over time, but the magnitude of the difference is relatively small. G decreases with increasing temperature T and is found to be proportional to T ² at very low temperature and to T at higher temperatures. The relation for G leads to view the vibrating crack front as being non straight on average over time. In tungsten, a nearly isotropic material, the theory yields a mean crack front inclination angle of about 1° at absolute zero, 3.2° at room temperature and 10.6° at the melting point.     

Plane Waves in Generalized Thermo-microstretch Elastic Solid with Thermal Relaxation Using Finite Element Method

The propagation of plane waves in a thermo-microstretch elastic solid half-space as proposed by Lord?CShulman as well as the classical dynamical coupled theory are discussed. The problem has been solved numerically using a finite element method. Numerical results for the displacement components, force stresses, temperature, couple stresses, and microstress distribution are obtained. The variations of the considered variables through the horizontal distance are given and illustrated graphically. Comparisons are made with the results predicted by the theory of generalized thermoelasticity for different values of the relaxation time.     

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.     

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.     

On wave propagation in laminated composites—II. Propagation normal to the laminates

A continuum model is developed for elastic waves propagating in a direction normal to the laminates of a laminated medium. The developments are analogous to those used in a companion paper for waves traveling along the laminates and indicate that the behavior is “nonlocal” in time as a result of the history-dependence of the current state. A zeroth-order model is deduced which consists of a system of integro-differential equations. The behavior of this system is then analysed and it is found that during an early phase, the motion is confined to a boundary layer and consists of highly damped waves. During a later phase the behavior is found to approach that of a macroscopically homogeneous medium. The behavior during both phases is described by two distinct differential systems. The early phase behavior is then determined for a laminated half-space subjected to a step normal stress.     

The effect of inhomogeneity and anisotropy on Stoneley waves

Summary In this paper we study the dispersion equation of Stoneley waves that are travelling in an inhomogeneous elastic half-space over an anisotropic homogeneous elastic half-space.The phase velocity is calculated as a function of the wave number. The results indicate that the effect of anisotropy on such waves is small and can be neglected, while the effect of inhomogeneity is very pronounced. The results show that Stoneley waves do not exist after some cut-off value of the wave number.     

The propagation of Love waves in a fluid-saturated porous anisotropic layer

Summary The paper concerns the propagation of Love waves in a transverse-isotropic fluid-saturated porous layer overlaying an elastic nonhomogeneous half-space. Using the Biot's theory for the porous layer and the theory of elasticity for the lower medium, the dispersion equation has been derived. This complex transcendental equation that relates frequency, phase velocity, anisotropy factor of the layer and the inhomogeneity character of the half-space has been solved with the aid of successive approximation method. The effect of anisotropy and heterogeneity has been shown graphically.With 2 Figures     

The interfacial waves launched by an explosive line-source in a submerged elastic half-space

The propagation of waves of small amplitude at the interface between an elastic half-space and a fluid half-space is investigated. The effects of an explosive line-source embedded in the elastic medium are studied, consideration being given to both dilatational and distortional cases, and a similar investigation is made for a compressive line-source in the fluid. The exact solutions for the displacement field are obtained. It is shown that for fluids of relatively low density, the interfacial waves are of two kinds. The relative efficiencies of the three sources for the launching of these waves are shown explicitly.     

Broadband Poly(vinylidene fluoride-trifluoroethylene) Ultrasound Focusing Transducers for Determining Elastic Constants of Coating Materials

This paper develops broadband Poly(vinylidene fluoride-trifluoroethylene) [P(VDF-TrFE)] focusing transducers and its surface wave measurement system for determining the elastic constants of coating layers. The measurement is based on a defocussing measurement method and V(f,z) analysis of the focusing transducers. Few P(VDF-TrFE) focusing transducers are successfully fabricated in laboratory and the wave measurements can cover a wide frequency range of 4–120 MHz with great accuracy. Brass and glass substrates electroplated with nickel coating layer with thickness ranging from 15 to 60 μm are tested in this work. Dispersion curves sensitivity analysis is carried out to decide the best approach for inversely determination of coating elastic properties. A searching method based on the downhill simplex algorithm and numerical calculation on waves in a layered half-space model is used for inversely determining the elastic constants of coating layer. The results show good agreement with reported data. Measurement accuracy and potential applications for other types of nondestructive evaluation of the focusing transducers and measurement system are addressed.