Existence of longitudinal and transverse waves in anisotropic thermoelastic media

Four waves propagate in an anisotropic thermoelastic medium. The fastest among them is a quasi-longitudinal wave. The slowest of them is a thermal wave. The remaining two are called quasi-transverse waves. The prefix ‘quasi’ refers to their polarizations being nearly, but not exactly, parallel or perpendicular to the direction of propagation. The polarizations of these four waves are not mutually orthogonal. Hence, unlike anisotropic elastic media, the existence of a longitudinal wave may not imply the existence of a transverse wave, by default. The existence of a purely longitudinal wave in an anisotropic thermoelastic medium is ensured by the stationary characters of three expressions. These expressions involve components of phase direction with elastic (stiffness and coupling) and thermal coefficients of the thermoelastic medium. The existence of a purely transverse wave is ensured by the two equations restricting the choice of thermoelastic (stiffness and coupling) coefficients. The existence of longitudinal and transverse waves along the coordinate axes and in the coordinate planes are discussed for general anisotropy. The discussion is extended to orthotropic materials, and the existence of pure phases is explored along few specific phase directions.     

Existence of longitudinal waves in anisotropic poroelastic solids

In anisotropic fluid-saturated porous solids, four waves can propagate along a general phase direction. However, solid particles in different waves may not vibrate in mutually orthogonal directions. In the propagation of each of these waves, the displacement of pore–fluid particles may not be parallel to that of solid particles. The polarization for a wave is the direction of aggregate displacement of the particles of the two constituents of a porous aggregate. These polarizations, for different waves, are not mutually orthogonal. Out of the four waves in anisotropic poroelastic medium, two are termed as quasi-longitudinal waves. The prefix ‘quasi’ refers to their polarization being nearly, but not exactly, parallel to the direction of propagation. The existence of purely longitudinal waves in an anisotropic poroelastic medium is ensured by the stationary characters of two expressions. These expressions involve the elastic (stiffness and coupling) coefficients of a porous aggregate and the components of phase direction. Necessary and sufficient conditions for the existence of longitudinal waves are discussed for different anisotropic symmetries. Conditions are also discussed for the existence of the apparent longitudinal waves, i.e., the propagation of wave motion with the particle displacement parallel to the ray direction instead of the phase direction. A graphical solution of a numerical example is shown to check the existence of these apparent longitudinal waves for general directions of phase propagation.     

Waves of dilatancy in a granular material with incompressible granules

As a densely packed granular material begins to flow through a constriction, dilatant waves are often observed. These waves initiate at the constriction and propagate outward into the granular material with an increase in the porosity of the material behind the wave front. As a model for dilatant waves, we consider curved waves carrying a jump in the gradient of porosity in the context of the continuum theory for the flow of granular materials presented by Goodman and Cowin. Their speed of propagation is determined and it is shown that such waves carry a jump in the stress acting on planes transverse to their direction of propagation. For the case when a wave of dilatancy propagates into a uniform region at rest, an explicit formula for the amplitude of the wave is derived for waves of general shape and then specialized to give results for plane, cylindrical, and spherical waves. In this case, the wave is purely longitudinal.     

混凝土表面开口裂缝对近表面声波影响

基于混凝土近表面不同波型的声速,分析开口裂缝对声波的影响。有限元仿真结果表明,激励源辐射的初始纵波P经裂缝端点衍射产生纵波P-P与横波P-T,P-T以一定的角度射向混凝土表面经模式转换产生纵波P-T-P;初始横波T在裂缝端点衍射产生纵波T-P与横波T-P;初始瑞利波R在裂缝端点衍射后除了产生纵波R-P与横波R-S外继续沿着固体表面传播(R-R)。在此基础上,基于变型波到达接收阵元的时刻定量检测裂缝的深度,检测结果较单面平测法具有更高的信噪比与检测精度,可辅助单面平测法更精确地定量检测混凝土开口裂缝的深度。     

一种由横波激发的特殊非线性声现象

目前对非线性超声的研究多集中在纵波激发的谐波性质以及对材料微观结构变化的实验检测上,横波激发的非线性声波性质少有研究。对横波激发的一维非线性声波方程入手,利用摄动法求解该方程,并改写为一阶偏微分方程,然后利用交错网格的有限差分形式进行数值求解。结果表明:采用横波激发,能产生线性横波和非线性纵波,且纵波的高次谐波内有两个信号,分别以纵波和横波两种速度传播。若采用较长的激发信号,纵波谐波能形成"拍"现象,成为一种奇特的声传播现象。     

Third-order elastic constants and pressure derivatives of the second-order elastic constants of hexagonal boron nitride

The second and third order elastic constants and pressure derivatives of second order elastic constants of hexagonal boron nitride have been obtained using the deformation theory. The strain energy derived using the deformation theory is compared with the strain dependent lattice energy obtained from elastic continuum model approximation to get the expressions for second and third order elastic constants. Higher order elastic constants are a measure of anharmonicity of crystal lattice. The six second-order elastic constants and the ten non-vanishing third order elastic constants and six pressure derivatives of hexagonal boron nitride are obtained in the present work and are compared with available experimental values. The second order elastic constant C ₁₁ which corresponds to the elastic stiffness along the basal plane of the crystal is greater than C ₃₃. Since C ₃₃ being the stiffness tensor component along the c-axis of the crystal, this result is expected from a layer-like material like boron nitride (BN). The third order elastic constants of hexagonal BN are generally one order of magnitude greater than the second-order of elastic constants as expected of a crystalline solid. The pressure derivative dC ₃₃/dp obtained in the present study is greater than dC ₁₁/dp which indicates that the compressibility along c-axis is higher than that along ab-plane of hexagonal BN.     

On finite amplitude waves in heterogeneous elastic solids

The perturbation technique of the ‘stretching of the coordinates’ is used to obtain first and second order perturbation solutions of finite amplitude plane waves which propagate into an elastic half-space whose material property varies in the direction of the propagation. The interaction between nonlinearity and heterogeneity is discussed, and the results are illustrated by means of two examples: the longitudinal waves propagating in an elastic half-space with harmonic heterogeneity; the shear wave in a half-space whose property varies as A[1 + ?(X/L)ⁿ], where A, ? ? 1, L, and n are constants, and X denotes the initial particle position measured normal to the plane boundary.     

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.     

Comparison Between Using Longitudinal and Shear Waves in Ultrasonic Stress Measurement to Investigate the Effect of Post-Weld Heat-Treatment on Welding Residual Stresses

Fusion welding is a joining process widely used in the industry. However, undesired residual stresses are produced once the welding process is completed. Post-weld heat-treatment (PWHT) is extensively employed in order to relieve the welding residual stresses. In this study, effect of PWHT time and temperature on the residual stresses of a ferritic stainless steel is investigated. Residual stress distributions in eight welded specimens were measured by using an ultrasonic method. Ultrasonic stress measurement is a nondestructive method based on acoustoelasticity law, which correlates mechanical stresses with velocity of an ultrasonic wave propagating within the subject material. The ultrasonic wave employed could be longitudinal or shear wave produced by the longitudinal (normal) or transverse (shear) transducers, respectively. Ultrasonic stress measurements based on longitudinal waves use longitudinal critically refracted (L_CR) waves in this direction, while shear wave methods use an ultrasonic birefringence phenomenon. The results show that the effect of PWHT can be successfully inferred by both longitudinal and shear wave methods, but the former is found to be more sensitive to stress variation. Furthermore, the distribution of subsurface residual stresses is found to be more distinguishable when the L_CR method is employed.     

Transverse and longitudinal ultrasonic attenuation measurements of the superconducting energy gap in high-purity rhenium

The attenuation of transverse and longitudinal waves propagating along thec axis in high-purity rhenium has been measured in the normal and superconducting states. In the superconducting state the attenuation was compared with the simple BCS theory using as adjustable parameters the gap parameter and the residual attenuation atT _c (for transverse waves). Using transverse-wave attenuation a gap parameter of 3.50±0.10 was found, while the longitudinal attenuation data yielded a gap parameter of 2.90±0.10. At present, the only apparent explanation for this difference is that the integrals for transverse-and longitudinal wave energy loss in an ultrasonic wave contain different azimuthal angle dependencies and thus average an anisotropic energy gap differently.Research sponsored by the Air Force Office of Scientific Research under AFOSR Grant No. 71-2079.     

Nonlinear response of electron-phonon interaction inn-type nondegenerate piezoelectric semiconductors

The nonlinear response of the electron-phonon interaction in nondegenerate piezoelectric semiconductors such asn-type InSb in the presence of a dc magnetic fieldB directed along the propagation of acoustic waves has been studied by using a quantum mechanical treatment. The effect of electron scattering in solids has been taken into consideration, so the electron relaxation time cannot be neglected. The nonlinear nature of the energy band is corrected for, using the Heisenberg equation of motion. It is found that the nonlinear response is proportional to a nonlinear longitudinal conductivity tensor _zzz when acoustic waves propagate parallel to the [111] direction for both piezoelectric and deformation-potential couplings. Numerical calculations for _zzz of n-type InSb at low temperatures are presented. Results show that the nonlinear response decreases rapidly with the sound frequency, and decreases slowly with the dc magnetic field and temperature. Therefore, the electron relaxation time and the nonparabolicity of energy bands play important roles for the electron-phonon interaction due to the piezoelectric and deformation-potential couplings in the microwave region.     

Second harmonic generation of shear waves in crystals

Nonlinear self-interaction of shear waves in electro-elastic crystals is investigated based on the rotationally invariant state function. Theoretical analyses are conducted for cubic, hexagonal, and trigonal crystals. The calculations show that nonlinear self-interaction of shear waves has some characteristics distinctly different from that of longitudinal waves. First, the process of self-interaction to generate its own second harmonic wave is permitted only in some special wave propagation directions for a shear wave. Second, the geometrical nonlinearity originated from finite strain does not contribute to the second harmonic generation (SHG) of shear waves. Therefore, unlike the case of longitudinal wave, the second-order elastic constants do not involve in the nonlinear parameter of the second harmonic generation of shear waves. Third, unlike the nonlinearity parameter of the longitudinal waves, the nonlinear parameter of the shear wave exhibits strong anisotropy, which is directly related to the symmetry of the crystal. In the calculations, the electromechanical coupling nonlinearity is considered for the 6 mm and 3 m symmetry crystals. Complement to the SHG of longitudinal waves already in use, the SHG of shear waves provides more measurements for the determination of third-order elastic constants of solids. The method is applied to a Z-cut lithium niobate (LiNbO/sub 3/) crystal, and its third-order elastic constant c/sub 444/ is determined.     

Dispersion analysis of numerical approximations to plane wave motions in an isotropic elastic solid

 It is well known that isotropic, nondispersive continuous hyperbolic problems become dispersive and anisotropic upon discretization. The purpose of this paper is to conduct a dispersion analysis of the nondissipative numerical approximations to plane wave motions in isotropic elastic solids. The discrete formulations considered are: an explicit, second-order accurate finite difference scheme, a consistent mass matrix formulation with linear quadrilateral elements and the corresponding lumped mass matrix formulation. Dispersion relation is derived for each of these formulations. In the context of the finite difference scheme, expressions for group velocity for both the shear and longitudinal waves are derived and the effect of using meshes of unequal size in x and y directions is studied. Results from numerical experiments confirming the predictions of analysis are also presented. Received 22 October 1999     

Stress dependency on ultrasonic wave propagation velocity

The velocity of an ultrasonic wave propagating in the uniformly deformed isotropic solid was analysed by the Eulerian viewpoint. The pseudo elastic coefficient (PEC) was used to solve the equation of motion of the elastic wave under finite deformation. The infinitesimal displacement gradients are connected to the stress increments by thePEC. Using thePEC and the partial differential equation of motion, the velocity of ultrasonic wave was quantitatively related to applied stress, moreover, the stress dependence on longitudinal and transverse wave velocities propagating in the direction parallel or perpendicular to the uniaxial tensile direction could be cleared. Consequently, the Murnaghan's third order elastic constants can be calculated by precisely measuring the uniaxial tensile stress and ultrasonic wave velocity.     

Intersonic shear crack growth along weak planes

Classical dynamic fracture theories predict the Rayleigh surface wave speed (c _R ) to be the limiting speed of propagation for mode-I cracks in constitutively homogeneous, isotropic, linear elastic materials subjected to remote loading. For mode-II cracks, propagating along prescribed straight line paths, the same theories, while excluding the possibility of crack growth in the speed regime between c _R and the shear wave speed, c _s , do not exclude intersonic (c _s <υ<c _l ) crack tip speeds. In the present study, we provide the first experimental evidence of intersonic crack growth in such constitutively homogeneous and isotropic solids, ever recorded in a laboratory setting. Intersonic shear dominated crack growth, featuring shear shock waves, was observed along weak planes in a brittle polyester resin under far-field asymmetric loading. The shear cracks initially propagate at speeds just above c _s and subsequently accelerate rapidly to the longitudinal wave speed (c _l ) of the solid. At longer times, when steady state conditions are attained, they propagate at speeds slightly higher than √2–c _s . The experimental results compare well with existing asymptotic theories of intersonic crack growth, and the significance of the preferred speed of √2–c _s is discussed. Received: 13 September 1999 / Reviewed and occerted: 19 November 1999     

Investigation of the kinetics of cracks in polymethyl methacrylate by dynamic photoelasticity

Methods have been developed for experimental investigation of the kinetics of fast cracks in plates of optically active model material making it possible to load by pulsed pressure varying the amplitude and length of the pulse and also to use the photoelastic method for recording dynamic processes with use of a superfast motion picture camera. The relationship of the dynamic stress intensity factor to crack growth rate and plate thickness was obtained taking into consideration the influence of longitudinal and transverse elastic waves. A clear relationship between stress intensity factor and v_cr was established. It was shown that with a fast crack growth rate (v_cr < 270 m/sec) the uniqueness between stress intensity factor and v_cr is disturbed by the action of reflected waves. The character of action on the stress intensity factor and crack growth rate of longitudinal and transverse waves reflected from the boundaries of the plate and also the influence of inertial effects caused by stress waves, high crack growth rates, and a change in the stressed and strained state with an increase in plate thickness were determined. It was determined that in polymethyl methacrylate in incidence on a crack at an angle of 90° a longitudinal wave increases the stress intensity factor and v_cr (with v_cr<625 m/sec) while a transverse wave decreases these characteristics.Translated from Problemy Prochnosti, No. 9, pp. 40–46, September, 1991.     

Elastic constants of MoSi2 and WSi2 single crystals

Single crystals of MoSi₂ and WSi₂ with a body-centred-tetragonal C 1 1_b structure were fabricated using a floating-zone method. The elastic wave velocity was measured for samples with various orientations using a simple pulse echo method at room temperature, and six elastic stiffness constantsc _ij were calculated. The stiffness constants were a little higher for WSi₂ than for MoSi₂.c ₁₁ andc ₃₃ of these compounds were approximately equal toc ₁₁ of tungsten and molybdenum, respectively, althoughc _ij (i j) was a little higher for these compounds than for molybdenum and tungsten. Young's modulus 1/s ₁₁ was the highest in the <0 0 1> direction, and the lowest in the <1 0 0> direction. The shear modulus 1/s ₆₆ was high on the {0 0 1} plane and independent of shear direction. It was generally low on the close-packed {1 1 0} plane and largely dependent on shear direction. The elastic constants for the polycrystalline materials were estimated fromc _ij ands _ij . Poisson's ratiov was 0.15 for MoSi₂ and for WSi₂, and these values were much lower than for ordinary metals and alloys. The Debye temperature _D was estimated using the elastic-wave velocity of the polycrystalline materials via the elastic constants such as Young's modulus and shear modulus: it was 759 K for MoSi₂ and 625 K for WSi₂.     

Wave dispersion and attenuation in fiber composites

 This work deals with the dispersion and attenuation of elastic plane waves propagating in a single layer fiber reinforced composite, in a direction which is perpendicular to the fibers. An iterative effective medium model, based on single scattering considerations, for the quantitative estimation of wave dispersion and attenuation is proposed. The single scattering problem is solved numerically by means of a 2-D boundary element methodology. Numerical results concerning the plane velocity and the attenuation coefficient of longitudinal or transverse SH, SV waves propagating in two types of fiber reinforced composite materials, are presented. The obtained results are compared to those taken either experimentaly or numericaly by other investigators.     

Effective spring stiffness for a planar periodic array of collinear cracks at an interface between two dissimilar isotropic materials

Explicit analytical expressions are obtained for the longitudinal and transverse effective spring stiffnesses of a planar periodic array of collinear cracks at the interface between two dissimilar isotropic materials; they are shown to be identical in a general case of elastic dissimilarity (the well-known open interface crack model is employed for the solution). Since the interfacial spring stiffness can be experimentally determined from ultrasound reflection and transmission analysis, the proposed expressions can be useful in estimating the percentage of disbond area between two dissimilar materials, which is directly related to the residual strength of the interface. The effects of elastic dissimilarity, crack density and crack interaction on the effective spring stiffness are clearly represented in the solution. It is shown that in general the crack interaction weakly depends on material dissimilarity and, for most practical cases, the crack interaction is nearly the same as that for crack arrays between identical solids. This allows approximate factorization of the effective spring stiffness for an array of cracks between dissimilar materials in terms of an elastic dissimilarity factor and two factors obtained for cracks in a homogeneous material: the effective spring stiffness for non-interacting (independent) cracks and the crack interaction factor. In order to avoid the effect of the crack surface interpenetration zones on the effective spring stiffness, the range of the tensile to transverse load ratios is obtained under the assumption of small-scale contact conditions. Since real cracks are often slightly open (due to prior loading history and plastic deformation), it is demonstrated that for ultrasound applications the results obtained are valid for most practical cases of small interfacial cracks as long as the mid-crack opening normalized by the crack length is at least in the order of 10⁻⁵.     

Effective anisotropic elastic constants of bimaterial interphases: comparison between experimental and analytical techniques

The effective elastic constants of a bimaterial composite were experimentally measured with the goal of validating the numerical predications of these constants made by homogenization theory. Secondly, solutions predicted by homogenization theory were compared to predictions made with more standard composite theories. Composite specimens consisting of titanium and epoxy were developed to mimic a porous titanium/tissue interphase. Tensile and shear tests (ASTM D3983) measured the stiffness along the porous coating/epoxy interphase (E _L), across the interphase (E _T) and in shear (G _LT). No significant differences in moduli were found between the experimental measurements and predictions made with homogenization theory, nor between the experimental measurements and Hashin-Shtrikman estimates. Homogenization theory predicted results usually within 20% of Hashin-Shtrikman estimates, but typically more than 50% different from what is predicted by the rule of mixtures. However, homogenization theory allows calculation of anisotropic stiffness estimates and local strains, neither of which is possible using Hashin-Shtrikman estimates. With this experimental validation, the accuracy of homogenization theory for use in implant/tissue interface mechanics applications is confirmed. Since the composite interphase is anisotropic and more compliant in the transverse direction, with stiffness an order of magnitude lower across the interphase, local mechanics, tissue ingrowth and remodeling may be strongly directional dependent.Department of Surgery, Orthopaedic Research LaboratoriesDepartment of Biologic and Materials Sciences, School of Dentistry