Anisotropic Viscoelastic Media Research Articles

Effect of initial stress on harmonic plane homogeneous waves in viscoelastic anisotropic media

Biot's theory is used to study the effect of initial stress on phase velocity and attenuation of plane homogeneous waves in viscoelastic anisotropic medium. Modified Christoffel equations are derived for three-dimensional wave propagation in a general viscoelastic anisotropic medium under initial stress. Effect of initial stress on wave propagation is observed through the deviations in phase velocity and attenuation–amplitude factor for each of the three existing quasi-waves. A particular numerical example shows that the attenuation is more sensitive to the presence of initial stress as compared to the propagation velocities.

Numerical modeling of seismic wavefield in anisotropic viscoelastic porous medium with the pseudo-spectral method

In order to describe the reservoirs more accurately, the characteristics of the earth medium must be considered comprehensively, such as the anisotropy, viscoelasticity and pore, etc. Consequently, an anisotropic viscoelastic porous medium model is presented in this paper, and the elastic wave equation in this medium has been derived. The seismic wavefield numerical modeling was simulated by using the pseudo-spectral method. Synthetic seismograms are used to show the characteristics of the seismic wave in complex medium. The research work is useful to understanding seismic wave propagation in earth medium.

Growth of a mode II crack in an orthotropic plate made of a viscoelastic composite material

The growth of a straight mode II crack in a viscoelastic orthotropic plate is examined. The plate material is modeled by a viscoelastic anisotropic medium. The shear displacement in the fracture process zone is determined as a function of time using the corresponding elastic solution, the Volterra principle, and the method of operator continued functions. The time dependence of the crack length is constructed as integral equations of three phases of stable growth. The solution of these equations gives kinetic curves

Energy flux in viscoelastic anisotropic media

SUMMARY We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P, SV and SH waves. 1 I NTR O DUCTION In this paper, we study the properties of the energy flux of homogeneous and inhomogeneous time-harmonic plane waves propagating in an unbounded viscoelastic anisotropic medium. The complexvalued energy-flux vector F (also called the Poynting or Umov‐ Poynting vector) is used as the quantity which fully characterizes the energy flux and the dissipation of energy of the wave. The real part of the vector F represents the time-averaged energy flux S. Both real, Re F = S, and imaginary, Im F, parts are closely related to the density of the time-averaged dissipated energy W d : W d equals twice the scalar product of −Im F and propagation vector P (real part of the slowness vector, perpendicular to the wave front) or twice the scalar product of Re F and attenuation vector A (imaginary part of the slowness vector, perpendicular to the plane of constant amplitude). The aim of this paper is to present a concise derivation of the expression for the complex-valued energy-flux vector F of homogeneous and inhomogeneous plane waves propagating in unbounded media of unrestricted anisotropy and viscoelasticity, to propose an algorithm for its calculation, and to perform analytical and numerical studies necessary for correct understanding of energy propagation and dissipation of the relevant plane waves. Considerable attention in seismology has been devoted to the energy flux of plane waves propagating in elastic anisotropic and in viscoelastic isotropic media. For elastic anisotropic media, see, for

Plane waves in viscoelastic anisotropic media-I. Theory

SUMMARY Properties of homogeneous and inhomogeneous plane waves propagating in an unbounded viscoelastic anisotropic medium in an arbitrarily specified direction N are studied analytically. The method used for their calculation is based on the so-called mixed specification of the slowness vector. It is quite universal and can be applied to homogeneous and inhomogeneous plane waves propagating in perfectly elastic or viscoelastic, isotropic or anisotropic media. The method leads to the solution of a complex-valued algebraic equation of the sixth degree. Standard methods can be used to solve the algebraic equation. Once the solution has been found, the phase velocities, exponential decays of amplitudes, attenuation angles, polarization vectors, etc., of P, S1 and S2 plane waves, propagating along and against N, can be easily determined. Although the method can be used for an unrestricted anisotropy, a special case of P, SV and SH plane waves, propagating in a plane of symmetry of a monoclinic (orthorhombic, hexagonal) viscoelastic medium is discussed in greater detail. In this plane the waves can be studied as functions of propagation direction N and of the real-valued inhomogeneity parameter D .F or inhomogeneous plane waves, D �= 0, and for homogeneous plane waves, D = 0. The use of the inhomogeneity parameter D offers many advantages in comparison with the conventionally used attenuation angle γ .I nthe N, D domain, any combination of N and D is physically acceptable. This is, however, not the case in the N, γ domain, where certain combinations of N and γ yield non-physical solutions. Another advantage of the use of inhomogeneity parameter D is the simplicity and universality of the algorithms in the N, D domain. Combined effects of attenuation and anisotropy, not known in viscoelastic isotropic media or purely elastic anisotropic media, are studied. It is shown that, in anisotropic viscoelastic media, the slowness vector and the related quantities are not symmetrical with respect to D = 0a s inisotropic viscoelastic media. The phase velocity of an inhomogeneous plane wave may be higher than the phase velocity of the relevant homogeneous plane wave, propagating in the same direction N. Similarly, the modulus of the attenuation vector of an inhomogeneous plane wave may be lower than that for the relevant homogeneous plane wave. The amplitudes of inhomogeneous plane waves in anisotropic viscoelastic media may increase exponentially in the direction of propagation N for certain D. The attenuation angle γ cannot exceed its boundary value, γ ∗ . The boundary attenuation angle γ ∗ is, in general, different from 90 ◦ , and depends both on the direction of propagation N and on the sign of the inhomogeneity parameter D. The polarization of P and SV plane waves is, in general, elliptical, both for homogeneous and inhomogeneous waves. Simple quantitative expressions or estimates for all these effects (and for many others) are presented. The results of the numerical treatment are presented in a companion paper (Paper II, this issue).

Plane waves in viscoelastic anisotropic media-II. Numerical examples

SUMMARY Properties of homogeneous and inhomogeneous plane waves propagating in unbounded viscoelastic anisotropic media in an arbitrarily specified direction are studied numerically. The analytical expressions and algebraic equations of the sixth degree derived in the companion paper (Paper I, this issue) are used for this purpose. Studied characteristics are phase velocity, amplitude decay along the propagation direction, attenuation, attenuation angle and polarization vector. Simple analytical expressions are used to describe the behaviour of SH plane waves propagating in a plane of symmetry of a viscoelastic anisotropic medium. Characteristics of P and S plane waves are determined by solving a complex-valued algebraic equation of the sixth degree. Combined effects of attenuation and anisotropy are illustrated on several models of viscoelastic anisotropic media. The explanation of the so-called forbidden directions known from the directional specification of the slowness vector is given. Non-physical results related to forbidden directions do not arise in the algorithm used.

Factorization of integro-differential equations of the acoustics of dispersive viscoelastic anisotropic media and the tensor integral operators of wave packet velocities

Factorization of tensor integro-differential wave equations of the acoustics of dispersive viscoelastic anisotropic media is performed for the one-dimensional case. The resulting first-order partial differential equations include integral tensor functionals of the sound velocities of polarized plane wave pulses. The velocity operators \(\hat V\) are expressed in a coordinate-free compact form. They are determined by the kernels of the integral representations and provide a general description of the kinematics and dynamics of wave packets for arbitrary propagation directions in anisotropic viscoelastic media.

Acoustic waves generated by a line source in a viscoelastic anisotropic medium

The acoustic signature provided by laser-generated ultrasound in polymer matrix composites is difficult to interpret on account of the dispersive nature of these materials. In order to simulate such wave forms, a double numerical inverse transform is used to calculate the surface displacement generated by a line impact. This calculation method is preferred due to the frequency dependence connected with the viscoelastic rheology. The focusing effects that anisotropy gives rise to are represented. In addition, the spreading of the signals issued from absorption is correlated with the anisotropic changes of the slowness versus frequency. The predicted displacements are in good agreement with the result of experiments performed with a composite plate.

Reflection and refraction of antiplane shear waves at a plane boundary between viscoelastic anisotropic media

We consider two monoclinic viscoelastic media in contact, with the incidence and refraction planes coincident with the respective planes of symmetry. Then, an incident homogeneous antiplane shear wave generates pure reflected and refracted antiplane waves, whose slowness and Umov–Poynting vectors lie in the planes of symmetry. The simplicity of the problem permits a detailed investigation of the phenomena caused by the combined anisotropic––anelastic properties of the media and waves. A general approach and the analysis of a numerical example provide a complete picture of the physics. In general, the reflected and refracted waves are inhomogeneous, i.e. equiphase planes do not coincide with equiamplitude planes. The reflected wave is homogeneous only when the incidence medium is transversely isotropic, i.e. its symmetry axis is perpendicular to the interface. If the refraction medium is elastic, the refracted wave is inhomogeneous of the elastic type, i.e. the attenuation vector is perpendicular to the Umov–Poynting vector (energy direction). The angle between the attenuation and the real slowness vectors may exceed 90 degrees, but the angle between the attenuation and the Umov–Poynting vector is always less than 90 degrees. If the incidence medium is elastic, the attenuation of the refracted wave is perpendicular to the interface. As in the anisotropic elastic case, energy flow parallel to the interface is the criterion for obtaining a critical angle. As in the isotropic viscoelastic case, critical angles exist only in rare instances. Indeed, they do not exist if one of the media is elastic. The existence of Brewster angles (related to a zero reflection coefficient) is also severely restricted by anelasticity. To balance the energy flux at the boundary, it is necessary to consider the interference flux between the incident and reflected waves (this flux vanishes in the elastic case). For the particular example, the refracted flux is always greater than zero and there is transmission for all the incident angles. This phenomenon is related to the absence of critical angles. For a transversely isotropic incidence medium, attenuations, quality factors and phase and energy velocities of the incident and reflected waves coincide for all the incidence angles. It is important to point out that the relevant physical phenomena are related to the energy flow direction (Umov–Poynting vector) rather than to the propagation direction (real slowness vector). For instance, the characteristics of the elastic type inhomogeneous waves, the existence of critical angles, and the fact that the amplitudes of the reflected and refracted waves decay in the direction of energy flow despite the fact that they grow in the direction of phase propagation.

Propagation and loss in layered viscoelastic composites

The acoustic response of a source in an anisotropic viscoelastic medium has not only a directionally dependent speed of propagation, but also a directionally dependent rate of attenuation. The long wavelength properties of a layered composite comprised of two alternating constituent layers, both of which are isotropic, but with one elastic and the other viscoelastic (modeled as a ‘‘standard solid’’), are investigated. Such a structure behaves as a homogeneous transversely isotropic viscoelastic medium. The plane-wave phase slowness and quality factor as a function of direction are shown for three cases. The first two are (a) when the slower medium is lossy and (b) when the faster medium is lossy. In these cases, the curves for loss versus angle are highly directional, but the actual patterns depend dramatically upon the choice of (a) or (b). The third case examines the behavior when the velocities of the constituents are approximately the same, but the loss is such that which constituent is fast and which is slow is dependent on frequency.

Energy balance and fundamental relations in anisotropic-viscoelastic media

The first part of this work analyses the energy balance equation for inhomogeneous time-harmonic waves propagating in a linear anisotropic-viscoelastic medium whose constitutive equation is described by a general time-dependent relaxation matrix of 21 independent components. This matrix includes most linear anisotropic-viscoelastic rheologies and the generalized Hooke's law. The balance of energy allows the identification of the potential and loss energy densities, which are related to the real and imaginary parts of the complex stiffness matrix. The second part establishes some fundamental relations valid for inhomogeneous viscoelastic plane waves. The scalar product between the complex wavenumber and the complex power flow vector is a real quantity proportional to the time-average kinetic energy density. As in the anisotropic-elastic case, it is confirmed that the phase velocity is the projection of the energy velocity vector onto the propagation direction. A similar equation is obtained by replacing the energy velocity with a velocity related to the dissipated energy. Finally, as in isotropic-viscoelastic media, the time-average energy density can be obtained from the projection of the average power flow vector onto the propagation direction, and the time-average dissipated energy density from the projection of the average power flow vector onto the attenuation direction.

The interaction of non-linear waves in a slightly anisotropic viscoelastic medium

Motions in the form of plane slightly non-linear quasi-transverse waves in a viscoelastic medium with small anisotropy are investigated. The problem of the interaction between two shock waves moving in opposite directions is considered.

The problem of a rigid punch sliding on an elastic or a viscoelastic layer

The problem of a rigid punch moving over a general anisotropic viscoelastic medium is considered both for the case where the medium is a finite layer and where it is an infinite half-space. Formal equations are derived, using Fourier Transform techniques, for the most general case where no assumptions are made concerning the shape of the punch. However, attention is then focussed on the case where the punch is uniformly infinite in one direction. Both the general inertial problem and the non-inertial approximation are considered for the half-plane problem. Previously known results are shown to emerge from the general formalism with relative ease. The general viscoelastic problem is shown to be formally identical to the isotropic problem treated in [1] though algebraically far more complex. In the non-inertial approximation, however, the extra complexity is not great, and in particular, if all the viscoelastic functions are assumed to have similar time behaviour, the resulting equations are essentially no more complex than the isotropic equations solved in [2]. For this case, the relationship between the hysteretic friction coefficients along the two axes of anisotropy on an orthotropic medium is quantified. Displacementtraction relationships are given for an isotropic layer, including inertial effects and some numerical results are given for the thick elastic layer problem.

Viscoelastic model and acoustic wave propagation in anisotropic media

The model of the standard linear solid is extended to the viscoelastic anisotropic medium by defining a relaxation‐time tensor. The velocity and attenuation of sound waves are calculated as a function of frequency from the components of this tensor and low‐ and high‐frequency elastic constants. The anisotropic behavior of relaxation curves can be described in this formalism.Subject Classification: 35.50.

Stress Distribution in Anisotropic Viscoelastic Hollow Sphere

The stress and deformation analysis of an anisotropic linear viscoelastic hollow sphere under a symmetrical loading on the boundaries is considered. In order to reduce some of the mathematical difficulties, assumptions of small strain with quasistatic stress-strain conditions are consistently made. The solution is found for a system of non-homogeneous anisotropic viscoelastic media, and subsequently reduced to that for a homogeneous medium. Numerical results for both short-time and long-time behavior are presented for a number of simple model media in the anisotropic viscoelastic case and compared with the result for the elastic or viscoelastic medium in the isotropic case.