Low-frequency acoustic characteristics of biological tissues

The laws of propagation of elastic waves of different types in biological tissues in the acoustic frequency range have been theoretically and experimentally investigated. The contributions of the imaginary and real components of the complex modulus of elasticity to the elastic wave velocity are analyzed. It is shown that in soft tissues, low-frequency elastic disturbances are propagated chiefly by shear (transverse) waves. The geometric dispersion of the elastic wave velocity has been investigated in experiments on gel model systems; the results of the measurements are in agreement with the theoretical dispersion curve.     

Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

Nonstationary love waves of the SH-type in an anisotropic elastic medium. A kinematic approach

In this paper, the propagation of Love waves in anisotropic elastic media is studied. These waves are a similar to the transverse surface SH waves in the isotropic case. Necessary conditions for the existence of Love waves of this polarization type near the surface Σ of an anisotropic elastic body are deduced. The algorithm developed here makes it possible to find the direction (s) of transverse surface wave propagation (at every point on the surface Σ). The algorithm employed is illustrated by some special anisotropic cases. The space-time method is used to construct the asymptotics of Love waves for those types of anisotropic media the eikonal equation of which is valid on the surface of an elastic body. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 262–276 Translated by Z. A. Yanson     

Propagation of small sditurbances in magneto-elastics

The possible modes of propagation of small disturbances in a homogeneous, incompressible initially unstressed, electrically conducting elastic medium at rest in the presence of a uniform external magnetic field are studied and it is found that if the medium is non-rigid (analogous to non-viscous fluids) and perfectly conducting, the disturbances propagate with Alfvén velocity. The effects of rigidity and finite conductivity on these transverse Alfvén waves have been investigated in some detail. It is interesting to note that the series representations obtained are strikingly similar to those got by Chadwick in the case of thermo-elastic plane waves.     

Reflected-refracted and head sh waves formed on a thin layer that is in welded contact with two elastic half-spaces

The reflection and refraction of SH waves by an elastic layer sandwiched between two elastic half-spaces are studied by using the numerical simulation on the basis of contour integration in the complex plane of the horizontal component of the slowness vector. The propagation of body, channel, head, and screened body waves are simulated in time and spectral domains. The wave fields associated with the propagation in the layer have strong attenuation, provided that the wave length is smaller than the value of the thickness of the layer. The stationary wave field of such waves is of resonance nature. Moreover, the maximum of the modulus of the spectral function is shifted to higher frequences as the epicentric distance increases. Thus, the attenuation of such waves depends on spectral characteristics of a source-receiver system. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1997, pp. 91–120. Translated by T. N. Surkova.     

Boundary value problems in the theory of thermomicrostretch elastic solids

In this paper the linear theory of thermomicrostretch elastic solids is considered. The basic properties of wave numbers of the longitudinal and transverse plane waves are established.The existence of eigenfrequencies of the interior homogeneous boundary value problem (BVP) of steady oscillations is studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fragments of the theory of nanotransistors: self-switching of a plane transverse hypersonic wave in nonlinearly elastic nanocomposite materials

We present fragments of the theory of nanotransistors concerning general information about transistors, models of nanocomposite materials, effects of nonlinear interaction of waves, and a theoretical analysis of the interaction of cubically nonlinear elastic plane harmonic waves in materials whose nonlinear properties are described by the Murnaghan potential. Using the method of slowly varying amplitudes, we investigate the interaction of two harmonic vertical transverse plane waves. Shortened and evolution equations, and Manley-Rowe relations, are obtained. We analyze analytically and numerically the mechanism of repumping of the energy of a strong pumping wave that propagates at a frequency of ω to a weak signal wave propagating at a frequency of 3ω . The described mechanism of switching of hypersonic waves in a nonlinearly elastic nanomaterial is similar to the mechanism of switching observed in optical and other transistors.     

Propagation of plane waves in microstretch fluid

The propagation of plane harmonic waves are studied in a microstretch fluid medium. It is found that five basic waves can propagate at distinct speeds in an infinite linear homogeneous isotropic microstretch fluid. Out of these five waves, one is a longitudinal micro-rotational wave, two are coupled longitudinal waves and remaining two are coupled transverse waves. The longitudinal micro-rotational wave travels independently and is not influenced by the microstretching of the medium, while the coupled longitudinal waves arise due to the presence of microstretching and coupled transverse waves arise due to the presence of micro-rotation in the medium. Speed of propagation of all the waves are found to be complex valued and dispersive at low frequency, but almost non-dispersive at high frequency. Due to complex valued speeds of propagation, all the waves are attenuating but differently. Coupled sets of longitudinal waves reduce to a longitudinal wave of micropolar fluid in the absence of microstretching. Reflection phenomena of a set of coupled longitudinal waves incident obliquely at the free surface of a microstretch fluid half-space has been investigated. Closed formulae for the reflection coefficients are presented and computed numerically for a particular medium. The real and imaginary parts of the complex speeds of all the waves and their corresponding attenuation coefficients have also been studied numerically and depicted graphically against frequency parameter.     

微极各向同性弹性板中的Rayleigh-Lamb波

研究了无应力作用条件下，均匀、各向同性、圆柱形微极结构弹性板中波的传播．导出了对称和斜对称模式下波传播的特征方程．对短波这一极端情况，无应力圆板中对称和斜对称模态波的特征方程退化为Pmyle曲表面波频率方程．并得到薄板的计算结果．给出了位移和微转动分量，并绘制了相应图形．给出了若干特殊情况的研究结果及对称和斜对称模态特征方程的图示．     

Love waves of SH type in an inhomogeneous transversely isotropic elastic medium

The asymptotics of high-frequency Love waves, which are analogous to transverse surface SH waves, is considered for a special type of anisotropy (transverse isotropy) of elastic media. The wave field is represented as a sum of the space-time (ST) caustic expansion and two additional ST ray series for faster (relative to the transverse surface wave) body waves, decaying exponentially with depth. Near the surface, the coefficients of the ST caustic and ray series, as well as the eikonals of waves, are determined in the form of expansions in a small parameter, which characterizes the proximity of the caustic of the ray field to the surface. With regard for the specific structure of the elasticity tensor of a transversely isotropic medium, the surface is treated as a plane. Interrelations between the parameters of elasticity, which are consistent with the conditions of the positivity of the elastic deformation energy and provide for the origination of the surface waves considered, are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 243–262.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Reflection of plane waves in generalized thermoelasticity of type III with nonlocal effect

The generalized thermoelasticity theory based upon the Green and Naghdi model III of thermoelasticity as well as the Eringen's nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves, which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear-type wave, which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear-type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients and the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted.     

Reflection of plane waves from the boundary of a pre-stressed compressible elastic half-space

The effect of pre-stress on the propagation and reflection ofplane waves in an incompressible isotropic elastic half-spacehas been examined recently by the authors (Ogden & Sotiropoulos,1997). In the present paper the corresponding analysis for compressiblematerials is detailed. In the two-dimensional context consideredfor incompressible materials the (homogeneous) plane waves werenecessarily shear waves. By contrast, in the compressible contextpure shear waves can propagate only in specific directions inthe considered principal plane and, in a general direction,a quasi-shear wave may be accompanied by a quasi-longitudinalwave, as is the case in the anisotropic linear theory. The dependenceof the (in-plane) slowness section on the pre-stress (and finitedeformation) and on the choice of constitutive law is elucidated.This information is used to determine the reflection coefficientsfor reflection of either a (quasi-) shear wave or a (quasi-)longitudinal wave from the boundary of the half-space and tocharacterize the different cases which arise depending on thegeometry of the slowness section. The theoretical results are illustrated by numerical calculationsfor the range of possible types of behaviour with referenceforms of strain-energy function and different states of finitedeformation and to the question of stability of the half-space.     

Nonlinear modulation of SH waves in an incompressible hyperelastic plate

Propagation of nonlinear shear horizontal (SH) waves in a homogeneous, isotropic and incompressible elastic plate of uniform thickness is considered. The constituent material of the plate is assumed to be generalized neo-Hookean. By employing a perturbation method and balancing the weak nonlinearity and dispersion in the analysis, it is shown that the nonlinear modulation of waves is governed asymptotically by a nonlinear Schr?dinger (NLS) equation. Then the effect of nonlinearity on the propagation characteristics of asymptotic waves is discussed on the basis of this equation. It is found that, irrespective of the plate thickness, the wave number and the mode number, when the plate material is softening in shear then the nonlinear plane periodic waves are unstable under infinitesimal perturbations and therefore the bright (envelope) solitary SH waves will exist and propagate in such a plate. But if the plate material is hardening in shear in this case nonlinear plane periodic waves are stable and only the dark solitary SH waves may exist.     

An asymmetric wave field in a transversely isotropic elastic medium

A transversely isotropic homogeneous elastic medium excited by a point force perpendicular to the anisotropic axis is considered. The wave field in this medium is constructed and investigated. The front sets of the SV and SH waves are in contact with one another at a point. The front sets in the vicinity of this point are investigated additionally. If we consider the SH wave (or the SV wave) separately, then a false plane front set arises in this region. In considering the SH and SV waves in combination, this false front set disappears. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 163–174.     

On the first-order speeds in any directions of acceleration waves in prestressed second-order isotropic,compressible, and homogeneous materials

In this paper, using the perturbation method we proposed in [A. Marasco, A. Romano, On the ordinary waves in second-order elastic, isotropic, compressible, and homogeneous materials, Math. Comput. Modelling 49 (7–8) (2009) 1504–1518], the first-order terms of the speeds and the amplitude of the principal waves and of the waves in any propagation direction are determined in second-order elastic, isotropic, compressible, and homogeneous materials. Moreover, for the general waves we determine the relations among the second-order constitutive constants which ensure that the waves are longitudinal or transverse.     

Longitudinal and Transverse Waves in Finite Elastic Strain. Hadamard and Green Materials

This work is concerned with the propagation of purely longitudinaland purely transverse waves in homogeneously deformed isotropicelastic materials. Two types of compressible material are alsodiscussed. A Hadamard material, so called by John in the hyperelasticcase, is one in which longitudinal waves may propagate in everydirection when the material is homogeneously deformed. A secondmaterial, called a "Green material" is introduced. In it twotransverse waves can propagate in every direction when the materialis homogeneously deformed. It is seen that a Mooney materialis the only isotropic incompressible elastic material in whichtwo transverse waves can propagate in every direction when itis homogeneously deformed, while the pressure stays constantthroughout the material. The propagation of finite amplitudewaves in these materials is discussed. Finally, it is shownthat the only motions which can be maintained in all homogeneouscompressible elastic Hadamard materials under the action ofsurface forces alone, are necessarily homogeneous and accelerationless.     

Reflection of plane waves from the stress-free isothermal and insulated boundaries of a nonlocal thermoelastic solid

The generalized thermoelasticity theory based upon the Green and Naghdi model II of thermoelasticity as well as the Eringen’s nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves which are dispersive in nature and associated with attenuation. In addition to the coupled waves, there also exists one independent vertically shear type wave which is dispersive but without any attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear type wave is found to to be associated with a critical frequency, while the coupled longitudinal waves may have critical frequencies under constraints. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on the phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients as well as the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted.     

Time-dependent waves of rayleigh type near the surface of an inhomogeneous elastic body

The well studied, high-frequency Rayleigh waves polarized in a plane normal to a cross section of the surface of an inhomogeneous elastic body with phase speed close to the speed of transverse waves are generalized to the case of the time-dependent equations of elasticity. For the wave field uniform asymptotics are obtained in the form of space-time ray expansions of two types: with a real eikonal (for transverse waves diffracted at the surface) and with a complex eikonal (for a longitudinal wave damped away from the surface).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 168–183, 1986.     

Vibrations of the surface of an elastic double-layered half-space with a periodic system of cracks

The two-dimensional problem of the normal incidence of a plane transverse wave from the far field on to the free surface of an elastic double-layered half-space, comprising a homogeneous layer attached to a semi-infinite base of a different elastic material, is considered. At the boundary between the two media there is a system of plane cracks, arranged periodically along the separation line, which models the fracture zone at the interface between dense solid rock and soft sedimentary rock. The effect of the fractures on the transmission of a transverse seismic wave generated by a deep-focus earthquake, and of the type of vibrations of the free surface of the ground that result, is studied. It is difficult to predict whether the seismic wave is strengthened or weakened by the fracture zone. The effect of the system of cracks on vibrations of the free surface largely depends on the physical and geometrical parameters and, primarily, on the vibration frequencies.