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Rayleigh wave propagation in transversely isotropic magnetothermoelastic medium with threephaselag heat transfer and diffusion
International Journal of Mechanical and Materials Engineering volume 14, Article number: 12 (2019)
Abstract
The present research deals with the propagation of Rayleigh wave in transversely isotropic magnetothermoelastic homogeneous medium in the presence of mass diffusion and threephaselag heat transfer. The wave characteristics such as phase velocity, attenuation coefficients, specific loss, and penetration depths are computed numerically and depicted graphically. The normal stress, tangential stress components, temperature change, and mass concentration are computed and drawn graphically. The effects of threephaselag heat transfer, GN typeIII, and LS theory of heat transfer are depicted on the various quantities. Some particular cases are also deduced from the present investigation.
Introduction
There are two types of surface waves namely Rayleigh wave and Love wave. These waves have primary importance in earthquake engineering. Rayleigh (1885) first investigated the waves that exist near the surface of a homogeneous elastic halfspace and named it as Rayleigh waves. Rayleigh wave exists in a homogeneous, elastic halfspace whereas Love wave requires a surficial layer of lowers wave velocity than the underlying halfspace. The propagation of waves in thermoelastic materials has numerous applications in various fields of science and technology, earthquake engineering, seismology, nuclear reactors, aerospace, submarine structures, and in the nondestructive evaluation in material process control and fabrication.
Green and Naghdi (1992, 1993) dealt with the linear and the nonlinear theories of thermoelastic body with and without energy dissipation. Three new thermoelastic theories were proposed by them, based on entropy equality. Their theories are known as thermoelasticity theory of type I, the thermoelasticity theory of type II (i.e., thermoelasticity without energy dissipation), and the thermoelasticity theory of type III (i.e., thermoelasticity with energy dissipation). On linearization, type I becomes the classical heat equation whereas on linearization typeII as well as typeIII, theories give a finite speed of thermal wave propagation.
The effects of heat conduction upon the propagation of Rayleigh surface waves in a semiinfinite elastic solid is studied for transversely isotropic thermoelastic (TIT) materials by Sharma, Pal, and Chand (2005) and Sharma and Singh (1985). Marin (1997) had proved the Cesaro means of the kinetic and strain energies of dipolar bodies with finite energy. Ting (2004) explored a surface wave propagation in an anisotropic rotating medium. Othman and Song (2006, 2008) presented different hypotheses about magnetothermoelastic waves in a homogeneous and isotropic medium. Kumar and Kansal (2008a) investigated the effect of rotation on the characteristics of Rayleigh wave propagation in a homogeneous, isotropic, thermoelastic diffusive halfspace in the context of different theories of thermoelastic diffusion, including the Coriolis and Centrifugal forces. Sharma and Kaur (2010) considered Rayleigh waves in rotating thermoelastic solids with the void. Mahmoud (2011) investigated the Rayleigh wave velocity under the effect of rotation, initial stress, magnetic field, and gravity field in a granular medium. Abouelregal (2011) studied Rayleigh wave propagation in thermoelastic halfspace in the context of dualphaselag mode. AbdAlla, AboDahab, and Hammad (2011); AbdAlla, AboDahab, Hammad, and Mahmoud (2011); and AbdAlla and Ahmed (1996) studied Rayleigh waves in an orthotropic thermoelastic medium under the influence of gravity, magnetic field, and initial stress.
Marin, Baleanu, and Vlase (2017) have discussed the effect of microtemperatures for micropolar thermoelastic bodies. Mahmoud (2014) studied the effect of the magnetic field, gravity field, and rotation on the propagation of Rayleigh waves in an initially stressed nonhomogeneous orthotropic medium. Singh, Kumari, and Singh (2014) solved the basic equations for the Rayleigh wave on the surface of TIT dualphaselag material under magnetic field. Kumar and Kansal (2013) investigated the propagation of Rayleigh waves in a TIT diffusive solid halfspace. Kumar and Gupta (2015) investigated the effect of phase lags on Rayleigh wave propagation in the thermoelastic medium. Biswas, Mukhopadhyay, and Shaw (2017) dealt with the propagation of Rayleigh surface waves in a homogeneous, orthotropic thermoelastic halfspace in the context of threephaselag models of thermoelasticity. Kumar, Sharma, Lata, and AboDahab (2017) and Lata, Kumar, and Sharma (2016) investigated the Rayleigh waves in a homogeneous transversely isotropic magnetothermoelastic (TIM) medium with two temperatures, Hall current, and rotation. Despite this, several researchers worked on a different theory of thermoelasticity as Chauthale and Khobragade (2017); Ezzat and AIBary (2016, 2017); Ezzat, ElKaramany, and ElBary (2017); Ezzat, ElKaramany, and Ezzat (2012); Hassan, Marin, Ellahi, and Alamri (2018); Kumar, Kaushal, and Sharma (2018); Kumar, Sharma, and Lata (2016a, 2016b, 2016c); Lata and Kaur (2019a, 2019b, 2019c, 2019d, 2019e); Lata et al. (2016); Marin (2009, 2010); Marin and Craciun (2017); Marin, Ellahi, and Chirilă (2017); Marin and Nicaise (2016); and Othman and Marin (2017).
Inspite of these, not much work has been carried out in the study of the Rayleigh wave propagation in a transversely isotropic magnetothermoelastic medium with fractional order threephaselag heat transfer. In this paper, we have attempted to study the Rayleigh wave propagation with fractional order threephaselag heat transfer in a transversely isotropic magnetothermoelastic medium.
Basic equations
The basic governing equations for homogeneous, anisotropic, generalized thermodiffusive elastic solids in the absence of body forces, heat and mass diffusion sources following Kumar and Kansal (2008b) are
Here, C_{ijkl} are elastic parameters and having symmetry (C_{ijkl} = C_{klij} = C_{jikl} = C_{ijlk}). The basis of these symmetries of C_{ijkl} is due to

1.
The stress tensor is symmetric, which is only possible if (C_{ijkl} = C_{jikl})

2.
If a strain energy density exists for the material, the elastic stiffness tensor must satisfy C_{ijkl} = C_{klij}

3.
From stress tensor and elastic stiffness, tensor symmetries infer (C_{ijkl} = C_{ijlk}) and C_{ijkl} = C_{klij} = C_{jikl} = C_{ijlk}
The simplified Maxwell’s linear equation (Rafiq et al. 2019) of electrodynamics for a slowly moving and perfectly conducting elastic solid are
From Eq. (6), we obtain
The equation of motion, entropy equation, and mass conservation equation following Kumar and Kansal (2009) are
where
are the components of the Lorentz force that appeared due to initially applied a magnetic field, the total magnetic field is given by \( \overrightarrow{H}={\overrightarrow{H}}_0+\overrightarrow{h} \), \( {\overrightarrow{H}}_0 \) is the external applied magnetic field intensity vector, and M and N are the strengths of the heat source and mass diffusion source per unit mass.
The medium is supposed to be perfectly electrically conducting and is halfspace (x, 0, z) such that all the variables are independent of the dimension y. Let \( {\overrightarrow{H}}_0=\left(0,{H}_0,0\right). \)
The heat conduction equation following Othman and Said (2018), we have
where
Method and solution of the problem
We consider a perfectly conducting homogeneous transversely isotropic magnetothermoelastic medium in the context of the threephaselag model of thermoelasticity initially at a uniform temperature T_{0}, an initial magnetic field \( {\overrightarrow{H}}_0=\left(0,{H}_0,0\right) \) towards yaxis. Moreover, we considered x, y, z taking origin on the surface (z = 0) along the zaxis directing vertically downwards inside the medium. For the 2D problem in the xzplane, we take
Now using the transformation on Eqs. (7–9) following Slaughter (2002) is as under:
and
where
Using dimensionless quantities,
Making use of (21) in Eqs. (14–17), after suppressing the primes, yield
where
Rayleigh wave propagation
We pursue Rayleigh wave solution of the equations of the form
where \( c=\frac{\upomega}{\upxi} \) is the nondimensional phase velocity and m is an unknown parameter. 1, W, S, and R are the amplitude ratios of displacements u, w, temperature change T, and concentration C, respectively.
Upon using Eq. (26) in Eqs. (22–25), we get
where
and from (27–30), the characteristic equation is a biquadratic equation in m^{2} given by
where
The characteristic in Eq. (27) gives four roots \( {m}_p^2\mathrm{where} \) p = 1, 2, 3, 4. Since we consider only the surface waves, therefore, motion is restricted to the free surface z = 0 of the halfspace, hence, satisfy the radiation conditions Re(m_{p}) ≥ 0.
The displacements, temperature change, and concentration can be written as
where A_{p} (p = 1, 2, 3, 4) are arbitrary constants and coupling constants are
Boundary conditions
The boundary conditions at z = 0 are given by
After applying dimensionless quantities from Eq. (21), the above boundary conditions reduces to
where
Derivations of the secular equations
By using the values of u, w, T, and C from (28) in (29), we get four linear equations as
where
Secular equations are
where
These secular equations have entire information regarding the wavenumber, phase velocity, and attenuation coefficient of Rayleigh waves in the transversely isotropic magnetothermoelastic medium. Moreover, If we write
then ξ = E + iF, where \( E=\frac{\omega }{v},v \) (velocity), and F (attenuation coefficient) are real.
The roots of the characteristic in Eq. (27) are complex and therefore, we assume that m_{p} = Q_{p} + ip_{q}, so that the exponent in Rayleigh wave solutions (28) becomes
where
Equation (28) can be written as
where
θ is the angle between the real and imaginary part of the vector χ_{p}.
Phase velocity
Phase velocity defines the speed at which waves oscillating at a particular frequency propagate and it depends on the real component of the wave number. The phase velocities are given by
Attenuation coefficient
The attenuation coefficient is the gradual loss of flux intensity through a medium, and it depends on the imaginary component of the wavenumber. The attenuation coefficient is defined as
Specific loss
The specific loss is the most direct way of defining internal resistance for a material. The specific loss W is given by
Penetration depth
Penetration depth describes how deep a wave can penetrate into a material and describes the decay of waves inside of a material. The penetration depth S is defined by
Particular cases

1.
If τ_{T} ≠ 0, τ_{v} ≠ 0, τ_{q} ≠ 0, we obtain results for Rayleigh wave propagation in transversely isotropic magnetothermoelastic solid with diffusion and with and without energy dissipation and TPL (threephaselag) effects.

2.
If τ_{T} = 0, τ_{v} = 0, τ_{q} = 0, and K^{∗} ≠ 0, we obtain results for Rayleigh wave propagation in magnetothermoelastic transversely isotropic solid with diffusion and GNIII theory (thermoelasticity with energy dissipation).

3.
If τ_{T} = 0, τ_{v} = 0, τ_{q} = 0, and K^{∗} = 0, we obtain results for Rayleigh wave propagation in magnetothermoelastic transversely isotropic solid with diffusion and GNII theory (generalized thermoelasticity without energy dissipation).

4.
If τ_{T} ≠ 0, τ_{v} ≠ 0, τ_{q} ≠ 0 , and K^{∗} = 0, we obtain results for Rayleigh wave propagation in magnetothermoelastic transversely isotropic solid with diffusion and GNII theory with TPL effect

5.
If τ_{T} = 0, τ_{v} = 0, τ_{q} = τ_{0} > 0, and K^{∗} = 0, and ignoring \( {\tau}_q^2 \), we obtain results for Rayleigh wave propagation in magnetothermoelastic transversely isotropic solid with diffusion and LordShulman (LS) model.

6.
If τ_{T} = 0, τ_{v} = 0, and τ_{q} = 0 and if the medium is not permitted with the magnetic field, i.e., μ_{0} = H_{0} = 0, then we obtain results for Rayleigh wave propagation in transversely isotropic thermoelastic solid with diffusion and without TPL effect

7.
If \( \kern0.50em {C}_{11}={C}_{33}=\lambda +2\mu, {C}_{12}={C}_{13}=\lambda, {C}_{44}=\mu, {\alpha}_1={\alpha}_3={\alpha}^{\prime },{a}_1={a}_3=a,{b}_1={b}_3=b,{K}_1={K}_3=K,{K}_1^{\ast }={K}_3^{\ast }={K}^{\ast } \), we obtain expressions for Rayleigh wave propagation in magnetothermoelastic isotropic materials with diffusion and with and without energy dissipation with TPL effect.
Numerical results and discussion
In order to illustrate our theoretical results in the proceeding section and to show the effect of Hall current and fractional order parameter, we now present some numerical results. Following Dhaliwal and Sherief (1980), cobalt material has been taken for thermoelastic material as
Using the above values, the graphical representations of stress components, temperature change, and concentration, Rayleigh wave velocity, attenuation coefficient, specific loss, and penetration depth of Raleigh wave in the transversely isotropic thermoelastic medium have been investigated with threephaselag, GNIII, and LS theory of thermoelasticity and demonstrated graphically as

1.
The solid line relates to the threephase lag theory τ_{T} ≠ 0, τ_{v} ≠ 0, τ_{q} ≠ 0,

2.
The dashed line relates to GNIII theory τ_{T} = 0, τ_{v} = 0, τ_{q} = 0, and K^{∗} ≠ 0,

3.
The dotted line relates to LS theory τ_{T} = 0, τ_{v} = 0, τ_{q} = τ_{0} > 0, and K^{∗} = 0.
Figure 1 illustrates the deviations of tangential stress t_{zx} with wave number. From the graph, we observe that tangential stress t_{zx} decreases with wave number in all the three theories with a little difference in magnitude. Figure 2 shows the deviations of normal stress t_{zz} with wavenumber. Here, we observe that the normal stress t_{zz} increases with increase in wavenumber with a small magnitude difference in all the three theories. Figure 3 illustrates the deviations of the attenuation coefficient with wavenumber. For the TPL theory, we observe that increase in attenuation coefficient is a gradually increasing which shows that for TPL theory attenuation coefficient is directly proportional to wavenumber. For GNIII theory, the attenuation coefficient increases in the form of a curve with an increase in wavenumber, while for LS theory, the value of the attenuation coefficient decreases with increase in wavenumber. Figure 4 shows the deviations of penetration depth with wavenumber. From the graphs, we observe that the penetration depth decreases for TPL and GNII theories, while for LS theory, it first increases and then starts decreasing with increase in wavenumber and hence shows the influence of three different theories on penetration depth. Figure 5 illustrates the variations of specific loss with wavenumber. From the graphs, we observe that the value of specific loss first decreases and then becomes stationary with an increase in wavenumber for TPL theory. In GNIII theory, specific loss increases with increase in wavenumber, while for LS theory, the value of specific loss first increases and then starts decreasing after attaining a maximum value at wavenumber = 2.5. Figure 6 shows variations of concentration C with wavenumber. From the graph, we observe that the concentration C increases with increase in wavenumber for all the three theories with a little magnitude difference. Figure 7 shows variations of Rayleigh wave velocity with wavenumber. The Rayleigh wave velocity increases for the GNIII theory case and no change for TPL case, while for LS theory, it first decreases and then remains the same with an increase in wavenumber. Figure 8 shows variations of temperature T with wavenumber. From the graph, we observe that the temperature T increases with increase in wavenumber for all the three theories with a little magnitude difference. Thus, we conclude that there is a significant influence of threephaselag GNIII and LS on the deformation wave parameter attenuation coefficients, specific loss, wave velocity, penetration depth, temperature, concentration, tangential stress, normal stress components, and of the transversely isotropic magnetothermoelastic medium.
Conclusion
From the above study, we conclude the following:

A mathematical model to study the Rayleigh wave propagation in the homogeneous transversely isotropic magnetothermoelastic medium in the presence of mass diffusion and the threephaselag heat transfer has been developed, and various wave characteristics, i.e., attenuation coefficients, specific loss, wave velocity, penetration depth, temperature, concentration, tangential stress, and normal stress components have been derived and represented graphically. The secular equation of Rayleigh waves in the presence of the effect of diffusion in a transversely isotropic magnetothermoelastic medium has been derived. The comparison of different theories of thermoelasticity, i.e., TPL, GNIII, and LS theories are carried out.

From the graphs, we observe a significant influence of threephaselag, GNIII and LS theories on the various wave characteristics, i.e., attenuation coefficients, specific loss, wave velocity, penetration depth, temperature, concentration, tangential stress, and normal stress components in transversely isotropic magnetothermoelastic medium. Attenuation of waves increases, whereas the penetration depth decreases with the increase in wavenumber.

The study of elastic wave attenuation particularly in transversely isotropic magnetothermoelastic medium carries information about transversely isotropic magnetothermoelastic medium properties and is important for the design of geophysics and seismic investigations.

Significant resemblance and nonresemblance among the results under TPL, GNIII, and LS theory of thermoelasticity have been identified.

However, the problem is theoretical, but it can deliver useful information for experimental researchers working in the field of geophysics and earthquake engineering and seismologist working in the field of mining tremors and drilling into the Earth crust.
Nomenclature
δ_{ij} Kronecker delta
C_{ijkl} Elastic parameters
β_{ij} Thermal elastic coupling tensor
T Absolute temperature
T_{0} Reference temperature
φ Conductive temperature
t_{ij} Stress tensors
e_{ij} Strain tensors
u_{i} Components of displacement
ρ Medium density
C_{E} Specific heat
a_{ij} Tensor of thermal moduli
α_{ij} Linear thermal expansion coefficient
K_{ij} Materialistic constant
\( {K}_{ij}^{\ast } \) Thermal conductivity
ω Angular frequency
μ_{0} Magnetic permeability
Ω Angular velocity of the solid and equal to Ωn, where n is a unit vector
\( \overrightarrow{u} \) Displacement vector
\( {\overrightarrow{H}}_0 \) Magnetic field intensity vector
\( \overrightarrow{j} \) Current density vector
F_{i} Components of the Lorentz force
τ_{0} Relaxation time
ε_{0} Electric permeability
δ(t) Dirac’s delta function
τ_{t} Phase lag of heat flux
τ_{v} Phase lag of temperature gradient
τ_{q} Phase lag of thermal displacement
α Fractionalorder derivative
ξ Wavenumber
b_{ij} Tensor of diffusion moduli
C The concentration of the diffusion material
\( {\alpha}_{ij}^{\ast } \) Diffusion parameters
η_{i} The flow of diffusion mass vector
q_{i} Components of heat flux vector
P Chemical potential per unit mass
S Entropy per unit mass
k Material constant
\( {\omega}_1^{\ast } \) Characteristics frequency of the medium
C_{1} Longitudinal wave velocity
Availability of data and materials
For the numerical results, cobalt material has been taken for thermoelastic material from Dhaliwal and Sherief (1980).
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Kaur, I., Lata, P. Rayleigh wave propagation in transversely isotropic magnetothermoelastic medium with threephaselag heat transfer and diffusion. Int J Mech Mater Eng 14, 12 (2019). https://doi.org/10.1186/s4071201901083
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DOI: https://doi.org/10.1186/s4071201901083
Keywords
 Transversely isotropic, magnetothermoelastic, threephaselag heat transfer
 Wave propagation