Rotational Effect on thermoelastic Stoneley, Love and Rayleigh waves in Fibre-reinforced Anisotropic General Viscoelastic Media of Higher Order

A. M. Abd-Alla, Aftab Khan and S. M. Abo-Dahab

Rotational Effect on thermoelastic Stoneley, Love and Rayleigh waves in Fibre-reinforced Anisotropic General Viscoelastic Media of Higher Order

A. M. Abd-Alla1,2*, Aftab Khan3and S. M. Abo-Dahab1,4

In this paper, we investigated the propagation of the rmoelastic surface waves in fibre-reinforced anisotropic general viscoelastic media of higher order ofnth order, including time rate of strain under the influence of rotation.The general surface wave speed is derived to study the effects of rotation and thermal on surface waves. Particular cases for Stoneley, Love 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. Our results for viscoelastic of order zero are well agreed to fibre-reinforced materials. Comparison was made with the results obtained in the presence and absence of rotation and parameters for fibre-reinforced of the material medium. It is also observed that, surface waves cannot propagate in a fast rotating medium. Numerical results for particular materials are given and illustrated graphically. The results indicate that the effect of rotation on fibrereinforced anisotropic general viscoelastic media are very pronounced.

Fibre-reinforced, viscoelastic, surface waves, rotation, anisotropic,thermoelastic.

1 Introduction

These problems are based on the more realistic elastic model since thermoelastic waves are propagating on the surface of earth, moon and other planets which are rotating about an axis. Schoenberg and Censor (1973) were the first to study the propagation of plane harmonic waves in a rotating elastic medium where it is shown that the elastic medium becomes dispersive and anisotropic due to rotation. Later on, many researchers introduced rotation in different theories of thermoelasticity. Agarwal (1979) studied thermo-elastic plane wave propagation in an infinite non-rotating medium. The normal mode analysis was used to obtain the exact expression for the temperature distribution,the thermal stresses and the displacement components. The purpose of the present work is to show the thermal and rotational effects on the surface waves.Surface waves have been well recognized in the study of earthquake, seismology, geophysics and Geodynamics. A good amount of literaturefor surface waves is available(in Refs. Bullen (1965), Ewing and Jardetzky (1957), Rayleigh (1885), Stoneley (1924)). Acharya andSingupta(1978),Pal and Sengupta(1987)and Sengupta and Nath(2001) and his research collaborators have studied surface waves. These waves usually have greater amplitudes ascompared with body waves and travel more slowly than body waves. There are many types of surface waves but we only discussed Stoneley, Love andRayleigh waves.Earthquakeradiate seismic energy as both body and surface waves. These are also used for detecting cracks and other defects in materials. 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 types 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 i.e. 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(Refs. (Acharya (2009), Samaland and Chattaraj (2011)). 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 material structure. Surface wave propagation in fiber reinforced media was discussed by various authors (Sing (2006), Kakar et al. (2013)). Abd-Alla et al.(2012)investigated the transient coupled thermoelasticity of an annular fin.Reflection of quasi-P and quasi-SV waves at the free and rigid boundaries of a fibrereinforced medium was also discussed by Chattopadhyay et al.(2012). Abd-Alla and Mahmoud(2011) investigated the magneto-thermoelastic problem in rotating non-homogeneous orthotropic hollow cylinder under the hyperbolic , heat conduction model.The extensive literature on the topic is now available and we can only mention a few recent interesting investigations in (Refs. Singh and Singh (2004),Abd-Alla (2013), Singh (2007), Abd-Alla (2011), Abo-Dahab et al. (2016), Alla et al.(2015), Kumar et al. (2016), Said and Othman (2016), Bakora and Tounsi(2015)).The temperature-rate dependent theory of thermoelasticity, which takes into account two relaxation times, was developed by Green and Lindsay (1972).Kumar et al. (2016) investigated the thermomechanical interaction transversely isotropic magnetothermoelastic medium with vacuum and with and without energy dissipation with the combined effects of rotation. Marin (1996) studied the Lagrange identity method in thermoelasticity of bodies with microstructure. Marin (1995) presented the existence and uniqueness in thermoelasticity of micropolar bodies. Marin and Marinescu (1998) investigated the thermoelasticity of initially stressed bodies.Asymptotic equipartition of energies.

Theaim of this paper is to investigate the propagation ofthermoelastic surface wavesin a rotating fibre-reinforced viscoelasticanisotropic media of higher order. The general surface wave speed is derived to study the effectof rotation and thermal on surface waves.The wave velocity equations have been obtained for Stoneley waves, Rayleigh waves and Love waves, and are in well agreement with the corresponding classical result in the absence of viscosity, temperature, rotation as well as homogeneity of the material medium.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. For order zero our results are well agreedto fibre-reinforced materials. It is also observed that the corresponding classical results follow from this analysis, in viscoelastic media of order zero, by neglecting reinforced parameters, rotational and thermal effects. Numerical results are given and illustrated graphically. It is important to note that Love wave remains unaffected by thermal and rotationaleffects.

2 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

If a body is rotating about an axis with a constant angular velocity ?then the equation of motioncan be written as follows((Abd-Alla et al. (2013)).

The equations (5) of motiontake the following form

From equation (2.3), we have

Thus the above set of equations (8-11) becomes (For convenience dashes are omitted)

3 Solution of the problem

Now our main objective to solve the eqs. (10), (12), (13) and (14).We seek the solution of(10), (12), (13), (14) in the following forms

Thusthe coupled equations (12-14)become

Substituting from Equation (15) into Equation (10), we obtain the following solution,

The above set of equations (17--19) can be written as

From aboveset of equations (21), we have

The auxiliary equation (23) becomes

A, B andC must be positive for real positive roots (m). If there is no thermal effect, then the above equation is quadratic inmand it is easy to solve. But in the case of thermoelastic, it is cubic.A, BandC must positively impose a necessary and sufficient condition upon the frequency of rotation of the medium. Through which a surface wave cannot propagate in a fast rotating medium. If there is no thermal effect then

This means that in a fast rotating medium, surface wave cannot propagate.

Hence we obtain the expressions of the displacement components, temperature distribution function and stresses as follows

Also it is found that

Similar expressions can be obtained for second medium and present them with dashes as follows

Also it is found that

In order to determine the secular equations, we have the following boundary conditions.

4 Boundary conditions

1) The displacement components between the mediums are continuous, i.e.

3) Thermal boundary conditions [19] gives

Boundary conditions imply the following equations

From the above equations containingEandF, we have

From equation (45), we get the velocity of surface waves in common boundary between two viscoelastic, fiber-reinforced solid media of Voigt type, where the viscosity is of general nth order involving time rate of change of strain.

5 Particular cases

5.1 Stoneley waves

It is the generalized form of Rayleigh waves in which we assume that the waves are propagated along the common boundary of two semi - infinite media M1 and M2.Therefore, equation (45) determines the wave velocity equation for Stoneley waves in the case of general viscoelastic, fibre- reinforced solid medal of nth order involving time rate of strain. Clearly from the equation (45), it follows that wave velocity of the Stoneley waves depends upon the parameters for fibre-reinforced of the material medium and the viscosity. Since the wave velocity equation (45) for Stoneley waves under the present circumstances does not contain ω explicitly, such types of waves are not dispersive like the classical one. In case of absence of parameters for fibre-reinforced and isotropic viscoelastic medium of 1st order involving time rate of change of strain is taken.Equation (45) is the secular equation for Stonely waves in a fibre reinforced viscoelastic media of orders. Fork=0, results are similar to Abd-Alla, et al. (2013)and Lotfy(2012). If rotational, thermal and fiber-reinforced parameters are ignored, then fork=0, the results are same as Stoneley (1924).

Then equation (45) reduces to,

Figure 1: Variation of the magnitude of the frequency equation coefficient for Stoneley waves velocity with respect to ?with variation of c,ωand k

Equation (46) gives the wave velocity equation of Stoneley waves in a viscoelastic medium of Voigt type where the viscosity is of Ist order involving time rate of change of strain which is completely in agreement with classical results given by Sengupta and Nath [9]. Further equation (46), of course, is in complete agreement with the corresponding classical result, when the effect of rotation, viscosity and parameters of fibre-reinforcement are ignored.

5.2 Love waves

To investigate the rotational effects on Love waves in a fibre reinforced viscoelastic media of higher order, we replace mediumM1by an infinitely extended horizontal plate of finite thickness d and bounded by two horizontal plane surfacesx2=0andx2=d. MediumMis semi infinite as in the general case.

The boundary conditions of Love wave are as follows

The displacement componentu3andτ12between the mediums is continuous, i.e.

This implies

For non trivial solution implies

This gives the wave velocity of Love waves propagating in a fiber-reinforced viscoelastic medium of orders. For k=0, the results are exactly same as in [23]. It is interesting to note that rotation and thermal does not affect the velocity of Love waves.

Figure 2: Variation of the magnitude of the frequency for Love waves with respect to with the variation of c,ωand k

5.3 Rayleigh waves

Rayleigh wave is a special case of the above general surface wave. In this case we consider a model where the mediumM1is replaced by vacuum. Since the boundary is adjacent to vacuum. It is free from surface traction. So the stress boundary condition in this case may be expressed as

Thus the above set of equations reduces to

An Einstein summation convention for repeated indices upon k is applied.

Eliminating the constantsM1,M2andM3we get the wave velocity equation for Rayleigh waves in the rotating thermoelastic fibre-reinforced viscoelastic media of order n asunder,

The Eq. (5.1) is the magnitude of the frequency equationfor Rayleigh wave for the medium M1. For k = 0, that is, our results are similar to Abd-Alla et. al. [18]. For a non-rotating media, we have to put?=0, then for k = 0 our results are same as that of[9]. If one ignores the fiber-reinforced parameters, then the results are same as Rayleigh[5].

Figure 3: Variation of the magnitude of the frequency and attenuation coefficient for Rayleigh waves with respect to ?with variation of c,ω,k,d

6 Numerical results and discussion

The following values of elastic constants are considered Chattopadhyay et al. (2002) and Singh (2006), for mediumsMandM1respectively.

Taking into consideration Green-Lindsay theory, the numerical technique outlined above was used to obtain secular equation, surface wavevelocity and attenuation coefficients under the effects of rotation in two models. For the sake ofbrevity, some computational results are being presented here.The variations are shown in Figs. 1-3 respectively.

7 Conclusions

The analysis of graphs permits us some concluding remarks.

1.The surface waves in a homogeneous, anisotropic, fibre-reinforced viscoelastic solid media under the rotation and higher orderkof nth order including time rate of strain are investigated.

2.Love waves do not depend on temperature; these are only affected by viscosity,rotation, frequency, higher orderkof net order, including time rate of strain, phase velocity and thicknessof the medium. In the absence of all fields, the dispersion equation is incomplete agreement with the corresponding classical result.

3.Rayleigh waves in a homogeneous, general thermo viscoelastic solid medium of higher order, including time rate of change of strain we find that the wave velocity equation,proves that there is a dispersion ofwaves due to the presence of rotation, temperature,frequency, phase velocity and viscosity. The results are incomplete agreement with the corresponding classical results in the absence of all fields.

4.The wave velocity equation of Stoneley waves is very similar to the corresponding problem in the classical theoryof elasticity. The dispersion of waves is due to the presence of rotation, phase velocity, frequency, temperatureand viscosity of the solid.Also, wave velocity equation of this generalized type of surface waves is incompleteagreement with the corresponding classical result in the absence of all fields.5.The results presented in this paper will be very helpful for researchers in geophysics,designers of new materials and the study of the phenomenon of rotation is also used to improve the conditions of oil extractions.

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1Math. Dept., Faculty of Science, Taif University 888, Saudi Arabia.

2Math. Dept., Faculty of Science, Sohag University, Egypt.

3Department of Mathematics, COMSATS, Institute of Information, Park Road, Chakshahzad, Islamabad,Pakistan.

4Math. Dept., Faculty of Science, SVU, Qena 83523, Egypt.

*Corresponding author: A.M.Abd-Alla, E-mail:mohmrr@yahoo.com