Gravitational effect on surface waves in a homogeneous fibre-reinforced anisotropic general viscoelastic media of higher and fractional order with voids

In this paper, gravitational effects on propagation of surface waves in a homogeneous fibrereinforced anisotropic general viscoelastic media of higher order with voids is investigated. The general surface wave speed is derived to study the effects of gravity on surface waves. Particular cases for Stoneley and Rayleigh waves are discussed. The results obtained in this investigation are more general in the sense that some earlier published results are obtained from our result as special cases. In the absence of voids our results for viscoelastic of order zero are well agreement to fibre-reinforced materials. Also by neglecting the reinforced elastic parameters, the results reduce to well known isotropic medium. Numerical results for particular materials are given and illustrated graphically. The results indicate that the effect of the gravitational, voids and the reinforced elastic parameters on surface waves are very pronounced.


INTRODUCTION
It is of great interest to study the propagation of surface waves in a homogeneous fibre-reinforced anisotropic general viscoelastic media of higher order with voids as it plays an importent role in material fracture and failure.Such problems have attracted much attention and have undergone a certain development (Bullen, 1965;Ewing and Jardetzky, 1957;Rayleigh, 1885;Stoneley, 1924).
Surface waves have been well recognized in the study of earthquake, seismology, geophysics and geodynamics.These waves usually have greater amplitudes as compared with body waves and travel more slowly than body waves.There are many types of surface waves but we only discussed Stoneley and Rayleigh waves.In earthquake the movement is due to the surface waves.
These are also used for detecting cracks and other defects in materials.Lord Rayleigh (1885) was the first to observe such kind of waves in 1885.That is why we called it Rayleigh waves.Sengupta and Nath (2001) investigated surface waves in fibre-reinforced anisotropic elastic media but their decomposition of displacement vector was not correct due to which some errors are found in their investigations (Sarvajit, 2002).
The idea of continuous self-reinforcement at every point of an elastic solid was introduced by Belfield et al. (1983).The superiority of fibre-reinforced composite materials over other structural materials attracted many authors to study different type of problems in this field.Fibre-reinforced composite structures are used due to their low weight and high strength.Two important components namely concrete and steel of a reinforced medium are bound together as a single unit so that there can be no relative displacement between them, that is, they act together as a single anisotropic unit.The artificial structures on the surface of the earth are excited during an earthquake, which give rise to violent vibrations in some cases.Engineers and architects are in search of such reinforced elastic materials for the structures that resist the oscillatory vibration.The propagation of waves depends upon the ground vibration and the physical properties of the structure material.Surface wave propagation in fiber reinforced media was discussed by various authors.
In classical theory of elasticity, the voids is an important generalization.Nunziato and Cowin (1979) and Cowin and Nunziato (1983) discussed the theory in elastic media with voids.Puri and Cowin (1985) studied the effects of voids on plane waves in linear elastic media and it is evident that pure shear waves remain unaffected by the presence of pores.Theory of thermoelastic material with voids is investigated by Lesan (1986).Good amount of literature on surface wave propagation in a generalized thermoelastic material with voids, is available in Singh and Pal (2011) and references therein.Chandrasekharaiah (1987a, b) discussed the effects of voids on propagation of plane and surface waves.Abo-Dahab (2010) investigated the propagation of P waves from stress-free surface elastic half-space with voids.
The effect of gravity on wave propagation in an elastic solid medium was first considered by Bromwich (1898).Later on gravity effects on wave propagation were discussed by various authors (Abd-Alla et al., 2013;Abd-Alla and Ahmed, 2003;De and Sengupta, 1974;Sengupta and Acharya, 1979) Surface waves in fiber-reinforced,general viscoelastic media of higher order under gravity is discussed by kakar et. al. (2013) whereas Pal and Sengupta (1987) studied the gravitational effects in viscoelastic media.Ren et al. (2012) investigated the coupling effects of void shape and void size on the growth of an elliptic void in a fiberreinforced hyper-elastic thin plate.Vishwakarma et al. (2013) discussed the influence of rigid boundary on the love wave propagation in elastic layer with void pores.Tvergaard (2011) studied the elastic-plastic void expansion in near-self-similar shapes.Fonseca et al. (2011) expressed the material voids in elastic solids with anisotropic surface energies.The extensive literature on the topic is now available and we can only mention a few recent interesting investigations in Abo-Dahab and Abd-Alla (2014), Abd-Alla et al. (2011), Abd-Alla andAhmed (2003), Abd-Alla (1999), Abd-Alla and Ahmed (1999), Abd-Alla et al. (2004), Elnaggar and Abd-Alla (1989), Abd-Alla and Ahmed (1996) Abd-Alla et al. (2012) and Abd-Alla et al. (2013).Aim of this paper is to investigate the gravitational effects on propagation of surface waves in fibre-reinforced viscoelastic anisotropic media of higher order with voids.The general surface wave speed is derived to study the effect of gravity and voids on surface waves.Particular cases for Stonely and Rayleigh waves are discussed.The results obtained in this investigation are more general in the sense that some earlier published results are obtained from our result as special cases.Numerical results are given and illustrated graphically.

FORMULATION OF THE PROBLEM
The constitutive relation of an anisotropic and elastic solid is expressed by the generalized Hooke's law, which can be written as: where, ij  are the Cartesian components of the stress and ij ε is the strain tensor which is related with the displacement vector, u i ; ijkl C are the components of a fourth-order tensor called the elasticities of the medium.The Einstein convention for repeated indices is used.
In the absence of body forces, the field equations in the presence of voids may be taken as follows: In these equations,  is the so-called volume fraction field.
It is assumed that the waves travel in the positive direction of the x 1 -axis and at any instant, all particles have equal displacements in any direction parallel to Ox 3 .In view of those assumptions, the propagation of waves will be independent of x 3 .Therefore all derivatives with respect to x 3 will be zero.
The general equation for a fibre-reinforced linearly elastic anisotropic media with respect to a direction 1 2 3 ( , , ) a a a a  is as follows (Sengupta and Nath, 2001): An Einstein summation convention for repeated indices over "k" is used and comma followed by an index denotes the derivative with respect to coordinate.(1, 0, 0) a  , the components of stress becomes as follows: By choosing the fibre direction as (1, 0, 0) a  ; also by taking all derivatives w.r.t.

3
x zero.The Equation (4) of motion takes the following form: From Equation (2), we have: Similarly, we can get similar relations in where

SOLUTION OF THE PROBLEM
To solve the coupled thermoelastic equations, we make the assumptions: Thus coupled equations (6a, b and c)) becomes: where Above set of equation can be written as where From above set of equations, for non-trival solution, we have: )( , , ) 0

Let D m 
Auxiliary equation becomes: E,F,G and H must be positive for real positive roots (m).
In the absence of gravity the above equation is cubic and if there are no voids then the above equation is quadratic in m and it is easy to solve.Let mi (i=1,2,3,4) be four positive real roots, then solution by normal mode method has the following form: Hence we obtain the expressions of the displacement components, volume fraction field and stresses as follows Similar expressions can be obtained for second mediun and present them with super script dashes as follows: Also it is found that: In order to determine the secular equations, we have the following boundary conditions.

BOUNDARY CONDITIONS
1.The displacement components and volume fraction field between the mediums are continuous, that is, where h is a constant.Boundary conditions implies the following equations:

Stoneley waves
Equation ( 14) is the secular equation for Stonely waves in a fibre reinforced viscoelastic media of higher order.For k = 0, results are similar to Abd-Alla (2003).If rotational, voids and fiber-reinforced parameters are ignored, then for k = 0, the results are same as Stoneley (1924).

Rayleigh waves
Rayleigh wave is a special case of the above general surface wave.In this case we consider a model where the medium,

       
It is assumed that gravitational field produces a hydrostatic initial stress.It produced by a slow process of creep where the shearing stresses tend to small or vanish after a long period of time.Equilibroim conditions of initial stress are: Thus above set of equations reduces to: ) .
Equation ( 15) is the secular equation for Rayleigh wave for the medium M 1 .For k = 0 and by ignoring the voids and gravitational effects our results are same as that of Sengupta and Nath (2001).If one ignores the fibrereinforced parameters also then results are same as Rayleigh (1885).

NUMERICAL SIMULATION AND DISCUSSION
The following values of elastic constants are considered The numerical technique outlined above was used to obtain secular equation, surface waves velocity and attenuation coefficients under the effects of rotation in two models with voids.
For the sake of brevity some computational results are being presented here.The variations are shown in Figures 1 and 2, respectively.
Figure 1a to i show the variation of the magnitude of the frequency equation  , Stoneley wave velocity ) Re(  and attenuation coefficient ) Im( with respect to the frequency  for different values of order , k gravity field g and phase velocity c .The magnitude of the frequency equation increases with increasing of frequency, while it decreases with increasing of order and gravity field and when effect of phase velocity it increases with increasing of phase velocity, as well, Stoneley wave velocity decreases with increasing of frequency, while it increases with increasing of order and gravity field and when effect of phase velocity, it decreases with increasing of phase velocity and the attenuation coefficient increases with increasing of frequency, except when effect of phase velocity it decreases with increasing of frequency, while it increases with increasing of order , as well it decreases with increasing of gravity field and phase velocity.c .The magnitude of the frequency equation increases with increasing of frequency, while it decreases with increasing of order and gravity field and when effect of phase velocity it increases with increasing of phase velocity, as well, Stoneley wave velocity decreases with increasing of frequency, while it increases with increasing of order and gravity field and when effect of phase velocity, it decreases with increasing of phase velocity and the attenuation coefficient increases with increasing of frequency and when effect of phase velocity it increases and decreases gradually with increasing of frequency, while it decreases with increasing of phase velocity. Finally, one can see that there is a similarity between  the graphs of two waves types (that is, Stoneley and Rayleigh) in the behavior but there are differences between the values and part of their behavior.

CONCLUSION
Due to the complicated nature of the governing equations of the fibre-reinforced anisotropic general viscoelastic media of higher order with voids, the work done in this field is unfortunately limited in number.The method used in this study provides a quite successful in dealing with such problems.This method gives exact solutions in the fibre-reinforced anisotropic elastic media without any assumed restrictions on the actual physical quantities that appear in the governing equations of the problem considered.Important phenomena are observed in all these computations: 1.It was found that the solutions obtained in the context of the fibre-reinforced anisotropic general viscoelastic media of higher integer and fractional order with voids, however, exhibit the behavior of speeds of wave propagation.
2. By comparing Figures 1 and 2, it is found that the wave velocity has the same behavior in both media.But with the passage of gravity field, numerical values of wave velocity in the viscoelastic media are large in comparison due to the viscoelastic fiber-reinforced.
3. Special cases are considered as Stoneley and Rayleigh waves only.4. The results presented in this paper should prove useful for researchers in material science, designers of new materials.5. Study of the phenomenon of rgravity field is also used to improve the conditions of oil extractions.Finally, if the rotation is neglected, the relevant results obtained are deduced to the results obtained by Sengupta and Nath (2001).
parameters of higher order, s , defined as:

iu
are the displacement vectors components.By choosing the fibre direction as adjacent to vacuum.It is free from surface traction.So the stress boundary condition in this case may be expressed as: Figures 2a to i show the variation of the magnitude of the frequency equation  , Stoneley wave velocity ) Re(  and attenuation coefficient ) Im( with respect to the frequency  for different values of order , k gravity field g and phase velocity