1 Introduction

A detailed investigation and analysis about Rayleigh-type wave propagation exist in various types of isotropic media. The propagation of Rayleigh waves in incompressible visco-elastic media under the effect of initial stress has widened praxis in the realm of geophysics, seismology, civil engineering, soil mechanics, earthquake engineering, mining engineering, and many more. There are multifarious reasons such as external pressure, slow process of creep, difference in temperature, and manufacturing process due to which the hard rocks act as incompressible media in nature. Also, a stress is developed in the media due to the above mentioned parameters, and this stress is recognised as the initial stress. It is a well-established fact that the Earth is a layered structure and the heterogeneity and initial stress are its trivial characteristics. Initial stresses occur in structural elements during their manufacture and assembly, in the crust of the Earth under the action of geostatic and geodynamic forces, in composites when they are created, in rock, etc. The initial stress exists in all constituent layers, but some layers experience it significantly and some modestly. The factors, such as overburdened layer, variation in temperature, slow process of creep, and gravitational field, have pronounced influence on the propagation of waves as they are responsible for the evolution of a large proportion of initial stress in a medium. Comprehensive information can be earned from the disquisition of Biot^[1]. A medium which can remould the attitude of the medium due to some physical or mechanical obligations is designated as the pre-stressed medium whose occupancy may increase or decrease the overall rigidity of an elastic structure. A state of initial stress (pre-stress) in a deformable medium induces mechanical properties which depend mainly on the magnitude of the stress and are quite distinct from those associated with the rigidity of the material itself. The Earth is a highly initially stressed medium. Visco-elasticity associated with incompressibility and inhomogeneity under the effect of initial stress conferred the more real situation of the crust of the Earth. Various authors have made their endeavour to work on the Rayleigh-type wave propagation in a discontinuous medium.

Numerous authors have submitted their elaborated works on the study of wave propagation in different types of anisotropic media. Achenbach^[2] rendered a complete discussion of seismic wave propagation in elastic layered media. Stoneley^[3] discussed the Rayleigh wave propagation in a heterogeneous medium. Dutta^[4] investigated the possibility of Rayleigh wave propagation in an incompressible medium lying over a transversely isotropic semi-infinite base. Chattopadhyay^[5] deliberated the propagation of shear wave in the crustal of visco-elastic material. Dey et al.^[6] investigated the impact of initial stress on the reflection and refraction phenomenon at the boundary present between core-mantle of the Earth. Pal and Chattopadhyay^[7] tried to demonstrate the reflection pattern of plane elastic waves in initially stressed homogeneous orthotropic media. Chattopadhyay et al.^[8] discussed the Rayleigh wave propagation under initial stress in cylindrical coordinates. Sharma and Gogna^[9] obtained the solution to a differential equation representing the motion of elastic wave in a dissipative liquid filled visco-elastic porous solid and applied this solution to obtain the nature of waves. Fu and Rogerson^[10] presented the nonlinear instability analysis of incompressible plate under the effect of initial stress. Again, Rogerson and Fu^[11] used the concept of asymptotic analysis to build up the dispersion relation for wave propagating in an initially stressed incompressible elastic plate.

In the last two decades, some of the results have been found by various authors related to seismic wave propagation in different types of anisotropic media under different types of physical situations. Abd-Alla et al.^[12] performed impeccable effect of initial stress, orthotropy, and gravity field on Rayleigh wave propagation in a magneto-elastic half-space. Again, Abd-Alla et al.^[13-14] made their efforts to confer the effect of various sorts of elastic parameters such as initial stress, gravity field, magnetic field, rotations, and relaxation time on the Rayleigh wave propagation. Sharma^[15] perceived the influence of elasticity, pore-fluid viscosity, frequency, and pore characteristics numerically on the Rayleigh wave velocity in dissipative poro-viscoelastic media. Afterwards, Sharma^[16] made an attempt to discuss the propagation of Rayleigh waves in generalised thermo-elastic media with stress free boundaries, and solved the dispersion relation numerically for exact roots. Ahmed and Abo-Dahab^[17] established the frequency equation for Rayleigh and Stoneley waves in a determinant form of orders twelve and eight, respectively, and obtained the phase velocity and attenuation coefficients for the waves. Ogden and Singh^[18] derived an equation for plane wave motion with small amplitude in a rotating and initially stressed transversely isotropic elastic solid for both compressible and incompressible linearly elastic materials. Wang et al.^[19] remarkably studied the stop band properties of elastic waves in three-dimensional piezoelectric phononic crystals with the initial stress taking the mechanical and electrical coupling into account using the plane wave expansion (PWE) method. The elastic wave localization in disordered periodic piezoelectric rods with the initial stress was studied, and the effects of initial stress on the band gap characteristics were investigated using the transfer matrix and Lyapunov exponent method by Wang et al.^[20]. Some neoteric achievements in this domain have been done by numerous authors including Chatterjee et al.^[21-22], Dhua and Chattopadhyay^[23], Kumari et al.^[24-25], and Khurana and Tomar^[26]. Till now, no authors have made their efforts to dissert the Rayleigh-type wave propagation in an initially stressed inhomogeneous incompressible visco-elastic medium situated over the same semi-infinite elastic medium. Chatterjee and Chattopadhyay^[27] studied the propagation, reflection, and transmission of SH-waves in slightly compressible, finitely deformed elastic media.

This article is associated with the study of Rayleigh-type wave propagation in an incompressible visco-elastic layer lying over an incompressible visco-elastic half-space under the effect of initial stresses. Dispersion relation is acquired in a determinant form with complex entries of order six. The real part is associated with the phase velocity of Rayleigh wave, and the imaginary part endows the damping factor.

2 Formulation of problem

We consider propagation of Rayleigh-type wave in a non-homogeneous isotropic visco-elastic layer of finite thickness h overlying a non-homogeneous isotropic visco-elastic semi-infinite medium, both under initial stresses and also incompressible media and in welded contact to each other (see Fig. 1). The origin is taken at the interface between the layer (M₁) and the semi-infinite medium (M₂). We consider a motion in the xy-plane, where the x-axis is taken along the interface and the y-axis is vertically downward. The initial stresses P₁ and P₂ are acting along the x-axis on the layer and the semi-infinite medium, respectively.

The propagation of Rayleigh-type wave is in the x-and y-directions. It is speculated that

(1)

The equations of motion^[1] for the Rayleigh-type wave in the absence of body force under the initial stress can be written as

(2)

where is the rotational component, and P_k (k = 1, 2) are incremental stress components. Following Biot^[1], the stress-strain relations are

(3)

The superscripts k=1 and k=2 are appended to indicate the layer medium (M₁) and the semi-infinite medium (M₂), respectively. The above equation of motion satisfied by the stress-strain relation of Hooke's law is

(4)

where δ_ij, , τ_ij^(k), e_ij^(k), e_ii^(k), μ_k, and λ_k are the Kronecker delta, cubical dilatation, stress component, shear and normal strain components, stiffness, and Lame's constant of elastic media, respectively. Also, the incremental strain relations are , , and . For a visco-elastic medium, μ_k → μ_k, λ_k → λ_k, and the stress-strain relation is

(5)

where and λ_k, λ'_k, μ_k, and μ'_k are Lame's constant and stiffness of the materials due to elastic and visco-elastic properties for k = 1, 2, respectively. Since the medium is reputed to be an incompressible material, we have

(6)

The inhomogeneity of the layer medium (M₁) is considered as

(7)

The inhomogeneity of the semi-infinite medium (M₂) is considered as

(8)

Here, μ_k⁽⁰⁾, ρ_k⁽⁰⁾, and P_k⁽⁰⁾ (k = 1, 2) are rigidities, densities, and initial stresses, respectively. ν and ε are inhomogeneity parameters of the layer medium (M₁) and semi-infinite medium (M₂), respectively.

2.1 Solution for visco-elastic layer medium (M₁)

The equations of motion for the layer medium (M₁) are

(9)

With the help of (6), (7), and (9), we have

(10)

(11)

where

(12)

and ϕ₁=ϕ₁(x, y, t) is a differentiable function. Differentiating (10) with respect to y and (11) with respect to x and then adding the possessed equation, we get

(13)

To obtain the solution for the layer medium (M₁), we consider the solution to (13) as

(14)

where α=lc, α is the frequency of oscillation, and l is the wave number. In view of (13), (14) results in

(15)

where

(16)

The solution to (15) becomes

(17)

where

m₁, m₂, m₃, and m₄ are the roots of (17), and p₁, q₁, and n₁ are defined in Appendix A. Using (17) in (14), we obtain

(18)

Substituting (18) into (12), the displacement components are obtained as

(19)

(20)

2.2 Solution for visco-elastic semi-infinite medium (M₂)

The equations of motion for the semi-infinite medium (M₂) are

(21)

With the help of (6), (8), and (21), we obtain

(22)

(23)

where

(24)

and ϕ₂ = ϕ₂(x, y, t) is a differentiable function. Differentiating (22) with respect to y and (23) with respect to x and then adding the resulting equations, we get

(25)

To obtain the solution for the semi-infinite medium (M₂), we consider the solution to (25) as

(26)

where α= lc, α is the frequency of oscillation, and l is the wave number. In view of (25), (26) results in

(27)

where

(28)

The solution to (27) becomes

(29)

where

m₅ and m₆ are roots of (29), and p₂, q₂, and n₂ are defined in Appendix A. Using (29) in (26), we have

(30)

The displacement components are obtained after using (30) into (24) as

(31)

(32)

3 Boundary condition

The continuity of stresses and displacements with the following boundary conditions is as follows^[1].

The upper surface of layer medium (M₁) is stress free, i.e.,

(33)

The stresses and displacements are continuous at the interface of layer medium (M₁) and semi-infinite medium (M₂), i.e.,

(34)

where

4 Dispersion relation

Using the boundary conditions (33) and (34) in (19), (20), (31), and (32), the following relations are obtained:

(35)

(36)

(37)

(38)

(39)

(40)

Eliminate the arbitrary constants c₁, c₂, c₃, c₄, c₅, and c₆. From the relations (35)-(40), we can obtain the following determinant:

(41)

where

5 Particular cases

Case Ⅰ When the layer and the semi-infinite medium are free from the initial stress, i.e., P₁⁽⁰⁾ = 0 and P₂⁽⁰⁾ = 0, (41) reduces to

(42)

where

a'₁, a'₂, b'₁, b'₂, c'₁, c'₂, d'₁, d'₂, e'₁, e'₂, p'₁, p'₂, q'₁, q'₂, n'₁, and n'₂ are defined in Appendix A.

Case Ⅱ When the layer and the semi-infinite medium are isotropic, i.e., μ'₁⁽⁰⁾= 0, μ'₂⁽⁰⁾ = 0, ε= 0, ν = 0, P₁⁽⁰⁾ = 0, and P₂⁽⁰⁾ = 0, (41) reduces to

(43)

where

with p''₁, p''₂, q''₁, and q''₂ defined in Appendix A.

Case Ⅲ In the absence of inhomogeneity parameters, i.e., ν → 0 and ε → 0, (41) reduces to

(44)

where

with R₁, R₂, Q₁, and Q₂ defined in Appendix A.

6 Numerical discussion

Numerical computation and graphical demonstration are obtained for the derived dispersion relation of Rayleigh-type wave propagation in a non-homogeneous isotropic visco-elastic layer lying over a non-homogeneous isotropic visco-elastic semi-infinite medium both under initial stress and also incompressible medium. The influence of the non-dimensional visco-elasticity , non-dimensional stresses , and inhomogeneity parameters νh and εh on the phase and damping velocities of Rayleigh-type wave is analysed, and their pictorial delimitation is accomplished through figures.

The relevant parameters for the visco-elastic layer medium (M₁) are (Gubbins^[28] and Caloi^[29])

and those for the visco-elastic semi-infinite medium (M₂) are (Gubbins^[28] and Caloi^[29])

Unless otherwise stated, the values of the parameters used for the purpose of numerical computation are provided in Table 1.

The real part of (41) provides the phase velocity (c/β₁), and the imaginary part of this equation is for the damping velocity. Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting, or preventing its oscillation. Figures 2-11 show the variation of the dimensionless phase and damping velocities against the dimensionless wave number, whereas Fig. 12 specifies the variation of visco-elasticity with initial stresses.

6.1 Effect of inhomogeneity parameters on phase and damping velocities of Rayleigh-type wave

Figures 2-5 manifest the effect of inhomogeneity parameter on the phase and damping velocities of Rayleigh-type wave propagating in an inhomogeneous incompressible visco-elastic layer lying over an inhomogeneous incompressible visco-elastic semi-infinite medium both in presence and absence of initial stresses. In particular, Figs. 2(a) and 2(b) show the variation of phase and damping velocities of Rayleigh-type wave, respectively, for different values of the inhomogeneity parameter of the layer (M₁) in presence of initial stress. It is clear that the phase velocity and damping velocity of Rayleigh-type wave are encouraged with the increase in the inhomogeneity parameter of layer (M₁). Figures 3(a) and 3(b) exhibit the impact of inhomogeneity parameter on the phase and damping velocities of the Rayleigh-type wave, respectively, in absence of initial stress in the layer (M₁). It is evident that from the figures, as the inhomogeneity parameter of the layer (M₁) increases, the phase velocity as well as the damping velocity of Rayleigh-type wave increases both in presence and absence of initial stress in the layer. Further, Figs. 4(a), 4(b), 5(a), and 5(b) demonstrate the variation of phase and damping velocities of Rayleigh-type wave propagating in the considered model for different values of inhomogeneity parameter associated with the semi-infinite medium (M₂) both in presence and absence of initial stress in the medium. It is clear from Figs. 4 and 5 that the phase velocity as well as the damping velocity decreases as inhomogeneity increases in either of the medium.

6.2 Effect of visco-elasticity on phase and damping velocities of Rayleigh-type wave

Figures 6-9 are carved out to investigate the effect of visco-elasticity on the propagation of Rayleigh-type wave in the reckoned model. Figures 6 and 7 display that the phase velocity decreases whereas the damping velocity increases as the visco-elasticity of the layer (M₁) increases. Figures 8 and 9 portray the effect of visco-elasticity of the semi-infinite medium (M₂) on the phase and damping velocities of the Rayleigh-type wave. It is marked that the phase velocity increases whereas the damping velocity decreases with the increasing visco-elasticity in the semi-infinite medium (M₂). An overview of Figs. 6-9 concludes that the increasing visco-elasticity associated with the layer (M₁) and semi-infinite medium (M₂) always have antagonistic effects on the phase and damping velocities of the Rayleigh-type wave.

6.3 Effect of initial stresses on phase and damping velocities of Rayleigh-type wave

Figures 10 and 11 reflect the effect of initial stresses (ζ₁) and (ζ₂) on the phase and damping velocities of Rayleigh-type wave, respectively. It may be observed from the figures that the phase velocity as well as the damping velocity increases as the initial stress associated with the layer medium (M₁) and semi-infinite medium (M₂) increases. Also, it may be concluded that the initial stress present in either of the media has the most significant impact on the phase velocity and the damping velocity of the Rayleigh-type wave as compared with the presence of other parameters under consideration.

6.4 Effect of initial stress on visco-elasticity of medium

It is reported through Figs. 12(a) and 12(b) that visco-elasticity in the layer (M₁) and semi-infinite medium (M₂) decreases with the increase in the initial stress acting in the layer (M₁) and semi-infinite medium (M₂). It may be due to the fact that as the initial stress in the visco-elastic medium grows, the compaction of the medium increases, leading to the decrease in the inter-particle distance of the medium as compared with its previous state. As a result, the internal friction (dissipation factor), which resists the motion of the particle during wave propagation, decreases gradually. Therefore, the increment of initial stress has an adverse influence on the visco-elasticity of the medium. Moreover, with the increase of initial stress, the deformation (or distortion) in structure of tiny element inside the medium may occur in such a way that the size of the particle inside the medium may vary at different locations. For this reason, the initial stress also favors the inhomogeneity inside the medium.

7 Conclusions

In this paper, an analytical approach is used to investigate the propagation of Rayleigh-type wave in a non-homogeneous visco-elastic layer lying over non-homogeneous visco-elastic semi-infinite media in presence of initial stresses and incompressibility. The closed form of dispersion equation is carried out to demonstrate graphically the influence of inhomogeneity, visco-elasticity, and initial stress parameters prevailing in the layer and semi-infinite medium on the phase velocity as well as the damping velocity of the Rayleigh-type wave. The following outcomes can be encapsulated through this study.

(ⅰ) It is observed that the phase and damping velocities of Rayleigh-type wave decrease with the wave number.

(ⅱ) It is elucidated that the increase in the inhomogeneity parameters associated with the layer encourages the phase velocity as well as the damping velocity, and that associated with the semi-infinite medium significantly discourages the phase velocity as well as the damping velocity of the Rayleigh-type wave.

(ⅲ) As long as the visco-elasticity prevails in the layer medium, the phase velocity of the Rayleigh-type wave decreases, whereas the damping velocity increases.

(ⅳ) As the initial stresses prevail in the layer, the semi-infinite medium phase velocity and the damping velocity of the Rayleigh-type wave are favoured significantly.

(ⅴ) The internal friction (dissipation factor), which resists the motion of the particle during wave propagation, decreases gradually with the increase in the initial stress acting in the respective medium. Further, the initial stress also favors inhomogeneity inside the medium.

The study of Rayleigh-type wave in a non-homogeneous, visco-elastic, incompressible medium has a great application in geophysical prospecting and in understanding the cause and valuation of damages due to earthquake. Hence, the present study may be useful in predicting the nature of Rayleigh wave in non-homogeneous geo-media.

Acknowledgements

Ms. P. SINGH conveys her sincere thanks to Indian Institute of Technology (Indian School of Mines), Dhanbad, India for providing Junior Research Fellowship.

Appendix A