Waves in Nonlocal Elastic Solid with Voids

In this paper, the governing relations and equations are derived for nonlocal elastic solid with voids. The propagation of time harmonic plane waves is investigated in an infinite nonlocal elastic solid material with voids. It has been found that three basic waves consisting of two sets of coupled longitudinal waves and one independent transverse wave may travel with distinct speeds. The sets of coupled waves are found to be dispersive, attenuating and influenced by the presence of voids and nonlocality parameters in the medium. The transverse wave is dispersive but non-attenuating, influenced by the nonlocality and independent of void parameters. Furthermore, the transverse wave is found to face critical frequency, while the coupled waves may face critical frequencies conditionally. Beyond each critical frequency, the respective wave is no more a propagating wave. Reflection phenomenon of an incident coupled longitudinal waves from stress-free boundary surface of a nonlocal elastic solid half-space with voids has also been studied. Using appropriate boundary conditions, the formulae for various reflection coefficients and their respective energy ratios are presented. For a particular model, the effects of non-locality and dissipation parameter (\(\tau \)) have been depicted on phase speeds and attenuation coefficients of propagating waves. The effect of nonlocality on reflection coefficients has also been observed and shown graphically.     

Wave propagation at the boundary surface of elastic and initially stressed viscothermoelastic diffusion with voids media

The present investigation is concerned with the wave propagation at the boundary surface of elastic half-space and initially stressed viscothermoelastic diffusion with voids half-space. The longitudinal and transverse waves are incident obliquely at the plane interface between uniform elastic half-space and initially stressed viscothermoelastic diffusion with voids half-space. It is found that the amplitude ratios of various reflected and transmitted waves are functions of angle of incidence, frequency of incident wave and are influenced by the initial stress, diffusion, voids, elastic and viscoelastic properties of media. The expressions of amplitude ratios and energy ratios are obtained in closed form and computed numerically for a specific model. The variations of energy ratios with angle of incidence are shown graphically. The conservation of energy at the interface is verified.     

Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids

Wave propagation in a porous elastic medium saturated by two immiscible fluids is investigated. It is shown that there exist three dilatational waves and one transverse wave propagating with different velocities. It is found that the velocities of all the three longitudinal waves are influenced by the capillary pressure, while the velocity of transverse wave does not at all. The problem of reflection and refraction phenomena due to longitudinal and transverse wave incident obliquely at a plane interface between uniform elastic solid half-space and porous elastic half-space saturated by two immiscible fluids has been analyzed. The amplitude ratios of various reflected and refracted waves are found to be continuous functions of the angle of incidence. Expression of energy ratios of various reflected and refracted waves are derived in closed form. The amplitude ratios and energy ratios have been computed numerically for a particular model and the results obtained are depicted graphically. It is verified that during transmission there is no dissipation of energy at the interface. Some particular cases have also been reduced from the present formulation.     

Reflection and refraction of attenuated waves at boundary of elastic solid and porous solid saturated with two immiscible viscous fluids

The propagation of elastic waves is studied in a porous solid saturated with two immiscible viscous fluids.The propagation of three longitudinal waves is represented through three scalar potential functions.The lone transverse wave is presented by a vector potential function.The displacements of particles in different phases of the aggregate are defined in terms of these potential functions.It is shown that there exist three longitudinal waves and one transverse wave.The phenomena of reflection and refraction due to longitudinal and transverse waves at a plane interface between an elastic solid half-space and a porous solid half-space saturated with two immiscible viscous fluids are investigated.For the presence of viscosity in pore-fluids,the waves refracted to the porous medium attenuate in the direction normal to the interface.The ratios of the amplitudes of the reflected and refracted waves to that of the incident wave are calculated as a nonsingular system of linear algebraic equations.These amplitude ratios are used to further calculate the shares of different scattered waves in the energy of the incident wave.The modulus of the amplitude and the energy ratios with the angle of incidence are computed for a particular numerical model.The conservation of the energy across the interface is verified.The effects of variations in non-wet saturation of pores and frequencies on the energy partition are depicted graphically and discussed.     

Transverse wave at a plane interface between isotropic elastic and unsaturated porous elastic solid half-spaces

Based on the poroelasticity theory, this article investigates the reflection and transmission characteristics of an incident plane transverse wave at a plane interface between an isotropic elastic half-space and an unsaturated poroelastic solid half-space. For this purpose, the effect of the saturation degree and frequency on the properties of the four bulk waves in unsaturated porous medium, i.e., three longitudinal waves and one transverse wave, are discussed at first. Two general cases of mode conversion are considered: (i) The initial transverse wave is incident from an unsaturated poroelastic half-space to the interface, and (ii) the initial transverse wave is incident from an elastic solid half-space to the interface. The expressions for the partition of energy at the interface during transmission and reflection process of waves are presented in explicit forms. At last, numerical computations are performed for these two cases and the results obtained are depicted, respectively. The variation of the amplitude ratios and energy ratios with the saturation degree and incident angle is illustrated in detail. It is also verified that, at the interface, the sum of energy ratios is approximately equal to unity as expected.     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

流体饱和标准线性粘弹性多孔介质中的平面波

研究了流体饱和不可压标准线性粘弹性多孔介质中平面波的传播和反射问题.在固相骨架小变形的假定下,得到了粘弹性多孔介质中波动方程的一般解,讨论了弥散关系和波的衰减特性.结果表明:在流体饱和不可压粘弹性多孔介质中,仅存在一个耦合纵波和一个耦合横波,纵波和横波的波速、衰减率等取决于孔隙流体与固相骨架间的相互作用以及固相骨架本身的粘性.同时,研究了半空间自由边界上入射波(纵波、横波)的反射问题,得到了非均匀反射波的波速、反射系数、衰减率等的表达式及其相关的数值结果.     

Seismic Reflection from an Interface Between an Elastic Solid and a Fractured Porous Medium with Partial Saturation

The theory of Tuncay and Corapcioglu (Transp Porous Media 23:237–258, 1996a) has been employed to investigate the possibility of plane wave propagation in a fractured porous medium containing two immiscible fluids. Solid phase of the porous medium is assumed to be linearly elastic, isotropic and the fractures are assumed to be distributed isotropically throughout the medium. It has been shown that there can exist four compressional waves and one rotational wave. The phase speeds of these waves are found to be affected by the presence of fractures, in general. Of the four compressional waves, one arises due to the presence of fractures in the medium and the remaining three are those encountered by Tuncay and Corapcioglu (J Appl Mech 64:313–319, 1997). Reflection and transmission phenomena at a plane interface between a uniform elastic half-space and a fractured porous half-space containing two immiscible fluids, are analyzed due to incidence of plane longitudinal/transverse wave from uniform elastic half-space. Variation of modulus of amplitude and energy ratios with the angle of incidence are computed numerically by taking the elastic half-space as granite and the fractured porous half-space as sandstone material containing non-viscous wetting and non-wetting fluid phases. The results obtained in case of porous half-space with fractures, are compared graphically with those in case of porous half-space without fractures. It is found that the presence of fractures in the porous half-space do affect the reflection/transmission of waves, which is responsible for raising the reflection and lowering the transmission coefficients.     

Influence of magnetic field on wave propagation at liquid-microstretch solid interface

The reflection and refraction of a longitudinal wave at an interface between a perfectly conducting nonviscous liquid half-space and a perfectly conducting microstretch elastic solid half-space are studied. The appropriate solutions to the governing equations are obtained in both the half-spaces satisfying the required boundary conditions at the interface to obtain a system of five non-homogeneous equations in the amplitude ratios of various reflected and transmitted waves. The system is solved by a Fortran program of the Gauss elimination method for a particular example of an interface between water and aluminum-epoxy composite. Numerical values of the amplitude ratios are computed for a certain range of the incidence angle both in the presence and absence of an impressed transverse magnetic field. The effects of the presence of the transverse magnetic field on the amplitude ratios of various reflected and transmitted waves are shown graphically.     

Relection of elastic waves at the elastically supported boundary of a couple stress elastic half-space

The relection elastic waves at the elastically supported boundary of a couple stress elastic half-space are studied in this paper. Different from the classical elastic solid, there are three kinds of elastic waves in the couple stress elastic solid, and two of them are dispersive. The boundary conditions of a couple stress elastic half-space include the couple stress vector and the rotation vector which disappear in the classical elastic solids. These boundary conditions are used to obtain a linear algebraic equation set, from which the amplitude ratios of relection waves to the incident wave can be determined. Then, the relection coeficients in terms of energy lux ratios are calculated numerically, and the normal energy lux conservation is used to validate the numerical results. Based on these numerical results,the inluences of the boundary parameters, which relect the mechanical behavior of elastic support, on the relection energy partition are discussed. Both the incident longitudinal wave(the P wave) and incident transverse wave(the SV wave) are considered.     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p² with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x₂ of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.     

The Peculiarities of Linear Wave Propagation in Double Porous Media

The features of propagation of longitudinal and transverse waves (LW and TW) in fractured porous medium (FPM) saturated with liquid are investigated by methods of multiphase mechanics. The mathematical model of FPM accounting for inequality of velocities and pressures of liquid in pores and fractures, liquid mass exchange and nonstationary interaction forces is developed. Processes of monochromatic wave propagation are studied. The dispersion relation is obtained and the effect of model parameters on wave propagation is analysed. It is established that one transverse and three longitudinal waves propagate in FPM saturated with liquid. The fastest LW is a deformational wave and the two others are filtrational. Filtrational waves attenuate much stronger than deformational and transverse waves. Distinction of velocities and pressures in liquid in various pore systems provides an explanation for the existence of the two filtrational waves in porous medium with two different characteristic sizes of pores.     

Mechanism of shear attenuation near the interface under combined compression and shear impact loading

We investigate the compressional/shear coupling plastic wave propagation characteristics analytically for ideal elastic–plastic materials in both stress and particle velocity spaces, focusing on the shear wave attenuation near the interface occurring in pressure–shear plate impact experiments. The results show that the shear attenuation is strongly associated with the wave propagation characteristics of the coupling waves. In the stress space, as the shear stress increases, an adjustment of the stress components is observed and the final stress state along the wave path is a combined pure shear- and hydrostatic pressure-state. In the particle velocity space, the wave structures with different loading and maximal transverse particle velocity are obtained. The maximal transverse particle velocity varies with the longitudinal velocity and forms a boundary line. Once the loading transverse velocity exceeds this line, a transverse particle velocity discontinuity occurs at the impact interface. If the bonding strength is sufficiently high, there will be a shear band in the target in the extreme vicinity of the interface.     

Propagation and Evolution of Wave Fronts in Two-Phase Porous Media

An investigation is made into the propagation and evolution of wave fronts in a porous medium which is intended to contain two phases: the porous solid, referred to as the skeleton, and the fluid within the interconnected pores formed by the skeleton. In particular, the microscopic density of each real material is assumed to be unchangeable, while the macroscopic density of each phase may change, associated with the volume fractions. A two-phase porous medium model is concisely introduced based on the work by de Boer. Propagation conditions and amplitude evolution of the discontinuity waves are presented by use of the idea of surfaces of discontinuity, where the wave front is treated as a surface of discontinuity. It is demonstrated that the saturation condition entails certain restrictions between the amplitudes of the longitudinal waves in the solid and fluid phases. Two propagation velocities are attained upon examining the existence of the discontinuity waves. It is found that a completely coupled longitudinal wave and a pure transverse wave are realizable in the two-phase porous medium. The discontinuity strength of the pore-pressure may be determined by the amplitude of the coupled longitudinal wave. In the case of homogeneous weak discontinuities, explicit evolution equations of the amplitudes for two types of discontinuity waves are derived.     

Wave propagation in thermally conducting linear fibre-reinforced composite materials

The propagation of plane waves in a fibre-reinforced, anisotropic, generalized thermoelastic media is discussed. The governing equations in x–y plane are solved to obtain a cubic equation in phase velocity. Three coupled waves, namely quasi-P, quasi-SV and quasi-thermal waves are shown to exist. The propagation of Rayleigh waves in stress free thermally insulated and transversely isotropic fibre-reinforced thermoelastic solid half-space is also investigated. The frequency equation is obtained for these waves. The velocities of the plane waves are shown graphically with the angle of propagation. The numerical results are also compared to those without thermal disturbances and anisotropy parameters.     

Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space

The problem of reflection and transmission due to longitudinal and transverse waves incident obliquely at a plane interface between uniform elastic solid half-space and fractional order thermoelastic solid half-space has been studied. It is found that the amplitude ratios of various reflected and refracted waves are functions of angle of incidence and frequency of incident wave and are influenced by the fractional order thermoelastic properties of media. The expressions of amplitude ratios and energy ratios have been computed numerically for a particular model. The variation of amplitude and energy ratios with angle of incidence is shown graphically. The conservation of energy at the interface is verified.     

Effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a heterogeneous half-space

In the present paper we study the effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a semi-infinite heterogeneous half-space, where the heterogeneity is both in rigidity and density. The present study demonstrates that torsional waves can propagate in the layer. The velocities of torsional waves have been calculated numerically as a functions of KH, (where K is the wave number and H is the thickness of the layer) and are presented in a number of graphs. It is also observed that, for a layer over a homogeneous half-space, the velocity of torsional waves does not coincide with that of Love waves in the presence of the rigid boundary whereas it does at the free boundary.     

Elastic Anisotropic Material with Purely Longitudinal and Transverse Waves

The simplest form of the matrix of elasticity moduli of an anisotropic material conducting purely longitudinal and transverse waves with an arbitrary direction of the wave normal is obtained. A generic solution of equations in displacements is represented in terms of three functions satisfying independent wave equations. In the case of planar deformation, this solution yields a complex representation coinciding with the Kolosov–Muskhelishvili formulas for an isotropic material. The formulas in the present work also determine an anisotropic material with Young's modulus identical for all directions, as in an isotropic medium.