Skip to main content

Rayleigh wave propagation in transversely isotropic magneto-thermoelastic medium with three-phase-lag heat transfer and diffusion

Abstract

The present research deals with the propagation of Rayleigh wave in transversely isotropic magneto-thermoelastic homogeneous medium in the presence of mass diffusion and three-phase-lag 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 three-phase-lag heat transfer, GN type-III, 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 half-space and named it as Rayleigh waves. Rayleigh wave exists in a homogeneous, elastic half-space whereas Love wave requires a surficial layer of lowers wave velocity than the underlying half-space. 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 non-destructive 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 type-II as well as type-III, theories give a finite speed of thermal wave propagation.

The effects of heat conduction upon the propagation of Rayleigh surface waves in a semi-infinite 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 magneto-thermoelastic 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 half-space 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 half-space in the context of dual-phase-lag mode. Abd-Alla, Abo-Dahab, and Hammad (2011); Abd-Alla, Abo-Dahab, Hammad, and Mahmoud (2011); and Abd-Alla 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 micro-temperatures 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 non-homogeneous orthotropic medium. Singh, Kumari, and Singh (2014) solved the basic equations for the Rayleigh wave on the surface of TIT dual-phase-lag material under magnetic field. Kumar and Kansal (2013) investigated the propagation of Rayleigh waves in a TIT diffusive solid half-space. 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 half-space in the context of three-phase-lag models of thermoelasticity. Kumar, Sharma, Lata, and Abo-Dahab (2017) and Lata, Kumar, and Sharma (2016) investigated the Rayleigh waves in a homogeneous transversely isotropic magneto-thermoelastic (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 AI-Bary (2016, 2017); Ezzat, El-Karamany, and El-Bary (2017); Ezzat, El-Karamany, 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 magneto-thermoelastic medium with fractional order three-phase-lag heat transfer. In this paper, we have attempted to study the Rayleigh wave propagation with fractional order three-phase-lag heat transfer in a transversely isotropic magneto-thermoelastic 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

$$ {t}_{ij}={c}_{ij kl}{e}_{kl}+{a}_{ij}T+{b}_{ij}C, $$
(1)
$$ \left(1+{\tau}_q\frac{\partial }{\partial t}+{\tau}_q^2\frac{\partial^2}{{\partial t}^2}\right){\dot{q}}_i=-\left[{K}_{ij}\left(1+{\tau}_T\frac{\partial }{\partial t}\right){\dot{T}}_{,j}+{K}_{ij}^{\ast}\left(1+{\tau}_v\frac{\partial }{\partial t}\right){T}_{,j}\right], $$
(2)
$$ \rho {ST}_0=\rho {C}_ET+{aT}_0C-{a}_{ij}{e}_{ij}{T}_0, $$
(3)
$$ \mathrm{P}={b}_{kl}{e}_{kl}+ bC- aT $$
(4)
$$ {\eta}_i=-{\alpha}_{ij}^{\ast }{P}_{,j} $$
(5)

Here, Cijkl are elastic parameters and having symmetry (Cijkl = Cklij = Cjikl = Cijlk). The basis of these symmetries of Cijkl is due to

  1. 1.

    The stress tensor is symmetric, which is only possible if (Cijkl = Cjikl)

  2. 2.

    If a strain energy density exists for the material, the elastic stiffness tensor must satisfy Cijkl = Cklij

  3. 3.

    From stress tensor and elastic stiffness, tensor symmetries infer (Cijkl = Cijlk) and Cijkl = Cklij = Cjikl = Cijlk

The simplified Maxwell’s linear equation (Rafiq et al. 2019) of electrodynamics for a slowly moving and perfectly conducting elastic solid are

$$ \operatorname{curl}\overrightarrow{\ h}=\overrightarrow{j}+{\varepsilon}_0\frac{\partial \overrightarrow{E}}{\partial t};\operatorname{curl}\overrightarrow{E}=-{\mu}_0\frac{\partial \overrightarrow{h}}{\partial t};\overrightarrow{\ E}=-{\mu}_0\left(\frac{\partial \overrightarrow{u}}{\partial t}\times \overrightarrow{H}\right);\operatorname{div}\overrightarrow{h}=0. $$
(6)

From Eq. (6), we obtain

$$ \overrightarrow{E}={\mu}_0{H}_0\left(\dot{w},0,-\dot{u}\right) $$
(7)
$$ \overrightarrow{h}=\left(0,-{H}_0e,0\right), $$
(8)
$$ \overrightarrow{j}=\left(-{h}_{,z}-{\varepsilon}_0{\mu}_0{H}_0\ddot{w},0,-{h}_{,x}-{\varepsilon}_0{\mu}_0{H}_0\ddot{u}\right) $$
(9)

The equation of motion, entropy equation, and mass conservation equation following Kumar and Kansal (2009) are

$$ {t}_{ij,j}+{F}_i=\rho {\ddot{u}}_i, $$
(10)
$$ {q}_{i,i}+\rho {T}_0\dot{S}-\rho M+P{\eta}_{i,i}=0, $$
(11)
$$ {\eta}_{i,i}=\dot{C+\rho N} $$
(12)

where

$$ {\displaystyle \begin{array}{c}{F}_i={\mu}_0{\left(\overrightarrow{j}\times \overrightarrow{H}\right)}_i\\ {}\overrightarrow{F}=\left({F}_x,{F}_y,{F}_z\right)=\left({\mu}_0{H}_0^2{e}_{,x}-{\varepsilon}_0{\mu}_0^2{H}_0^2\ddot{u},0,{\mu}_0{H}_0^2{e}_{,z}-{\varepsilon}_0{\mu}_0^2{H}_0^2\ddot{w}\right)\end{array}} $$

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 half-space (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

$$ {K}_{ij}\left(1+{\tau}_t\frac{\partial }{\partial t}\right){\dot{T}}_{, ji}+{K}_{ij}^{\ast}\left(1+{\tau}_v\frac{\partial }{\partial t}\right){T}_{, ji}=\left(1+{\tau}_q\frac{\partial }{\partial t}+{\left({\tau}_q\right)}^2\frac{\partial^2}{\partial {t}^2}\right)\left[{\rho C}_E\ddot{T}+{a}_{ij}{T}_0{\ddot{\mathrm{e}}}_{ij}+a{T}_0\ddot{C}\right], $$
(13)

where

$$ {a}_{ij}=-{a}_i{\delta}_{ij},\kern0.5em {b}_{ij}=-{b}_i{\delta}_{ij},\kern0.75em {\alpha}_{ij}^{\ast }={\alpha}_i^{\ast }{\delta}_{ij},\kern0.5em {K}_{ij}^{\ast }={K}_i^{\ast }{\delta}_{ij},\kern0.5em {K}_{ij}={K}_i{\delta}_{ij} $$

Method and solution of the problem

We consider a perfectly conducting homogeneous transversely isotropic magneto-thermoelastic medium in the context of the three-phase-lag model of thermoelasticity initially at a uniform temperature T0, an initial magnetic field \( {\overrightarrow{H}}_0=\left(0,{H}_0,0\right) \) towards y-axis. Moreover, we considered x, y, z taking origin on the surface (z = 0) along the z-axis directing vertically downwards inside the medium. For the 2D problem in the xz-plane, we take

$$ \boldsymbol{u}=\left(u,0,w\right) $$

Now using the transformation on Eqs. (79) following Slaughter (2002) is as under:

$$ {C}_{11}\frac{\partial^2u}{\partial {x}^2}+{C}_{13}\frac{\partial^2w}{\partial x\partial z}+{C}_{44}\left(\frac{\partial^2u}{\partial {z}^2}+\frac{\partial^2w}{\partial x\partial z}\right)-{a}_1\frac{\partial T}{\partial x}-{b}_1\frac{\partial C}{\partial x}+\left({\mu}_0{H}_0^2\frac{\partial e}{\partial x}-{\epsilon}_0{\mu}_0^2{H}_0^2\frac{\partial^2u}{\partial {t}^2}\right)=\rho \left(\frac{\partial^2u}{\partial {t}^2}\right), $$
(14)
$$ \left({C}_{13}+{C}_{44}\right)\frac{\partial^2u}{\partial x\partial z}+{C}_{44}\frac{\partial^2w}{\partial {x}^2}+{C}_{33}\frac{\partial^2w}{\partial {z}^2}-{a}_3\frac{\partial T}{\partial z}-{b}_3\frac{\partial C}{\partial z}+\left({\mu}_0{H}_0^2\frac{\partial e}{\partial z}-{\epsilon}_0{\mu}_0^2{H}_0^2\frac{\partial^2w}{\partial {t}^2}\right)=\rho \left(\frac{\partial^2w}{\partial {t}^2}\right), $$
(15)
$$ {K}_1\left(1+{\tau}_t\frac{\partial }{\partial t}\right)\frac{\partial^2\dot{T}}{\partial {x}^2}+{K}_3\left(1+{\tau}_t\frac{\partial }{\partial t}\right)\frac{\partial^2\dot{T}}{\partial {z}^2}+{K}_1^{\ast}\left(1+{\tau}_v\frac{\partial }{\partial t}\right)\frac{\partial^2T}{\partial {x}^2}+{K}_3^{\ast}\left(1+{\tau}_v\frac{\partial }{\partial t}\right)\frac{\partial^2T}{\partial {z}^2}=\left(1+{\tau}_q\frac{\partial }{\partial t}+{\left({\tau}_q\right)}^2\frac{\partial^2}{\partial {t}^2}\right)\left[{\rho C}_E\frac{\partial^2T}{\partial {t}^2}+{T}_0\left\{{a}_1\frac{\partial \ddot{u}}{\partial x}+{a}_1\frac{\partial \ddot{w}}{\partial z}\right\}+a{T}_0\ddot{C}\right], $$
(16)
$$ {\alpha}_1^{\ast}\left[{b}_1\frac{\partial^3u}{\partial {x}^3}+{b}_3\frac{\partial^3w}{\partial {x}^2\partial z}\right]+{\alpha}_3^{\ast}\left[{b}_1\frac{\partial^3u}{\partial x\partial {z}^2}+{b}_3\frac{\partial^3w}{\partial {z}^3}\right]-{\alpha}_1^{\ast }b\frac{\partial^2C}{\partial {x}^2}-{\alpha}_3^{\ast }b\frac{\partial^2C}{\partial {z}^2}+{\alpha}_1^{\ast }a\frac{\partial^2T}{\partial {x}^2}+{\alpha}_3^{\ast }a\frac{\partial^2T}{\partial {z}^2}=-\left(\dot{C}\right). $$
(17)

and

$$ {t}_{xx}={C}_{11}{e}_{xx}+{C}_{13}{e}_{xz}-{a}_1T, $$
(18)
$$ {t}_{zz}={C}_{13}{e}_{xx}+{C}_{33}{e}_{zz}-{a}_3T, $$
(19)
$$ {t}_{xz}=2{C}_{44}{e}_{xz}, $$
(20)

where

$$ {\displaystyle \begin{array}{c}{a}_1=\left({C}_{11}+{C}_{12}\right){\alpha}_1+{C}_{13}{\alpha}_{3,}\\ {}{a}_3=2{C}_{13}{\alpha}_1+{C}_{33}{\alpha}_3,\\ {}{b}_1=\left({C}_{11}+{C}_{12}\right){\alpha}_{1c}+{C}_{13}{\alpha}_{3c,}\end{array}}. $$

Using dimensionless quantities,

$$ {\displaystyle \begin{array}{c}\left({x}^{\prime },{z}^{\prime },{u}^{\prime },{w}^{\prime}\right)=\frac{\omega_1^{\ast }}{C_1}\left(x,z,u,w\right),\rho {C}_1^2={C}_{11},{\omega}_1^{\ast }=\frac{\rho {C}_1^2{C}_E}{K_1}\\ {}{T}^{\prime }=\frac{a_1T}{\rho {C}_1^2},{C}^{\prime }=\frac{b_1C}{\rho {C}_1^2},\left(\ {t}^{\prime },{\tau}_0^{\prime },{\tau}^{0\hbox{'}},{\tau}_T^{\prime },{\tau}_v^{\prime },{\tau}_q^{\prime}\right)={\omega}_1^{\ast}\left(t,{\tau}_0,{\tau}^0,{\tau}_T,{\tau}_v,{\tau}_q\right).\end{array}} $$
(21)

Making use of (21) in Eqs. (1417), after suppressing the primes, yield

$$ \left(1+{\delta}_4\right)\frac{\partial^2u}{\partial {x}^2}+\left({\delta}_1+{\delta}_4\right)\frac{\partial^2w}{\partial x\partial z}+{\delta}_2\frac{\partial^2u}{\partial {z}^2}-\frac{\partial T}{\partial x}-\frac{\partial C}{\partial x}=\left(1+{\delta}_5\right)\frac{\partial^2u}{\partial {t}^2}, $$
(22)
$$ \left({\delta}_1+{\delta}_4\right)\frac{\partial^2u}{\partial x\partial z}+{\delta}_2\frac{\partial^2w}{\partial {x}^2}+\left({\delta}_3+{\delta}_4\right)\frac{\partial^2w}{\partial {z}^2}-{\delta}_7\frac{\partial T}{\partial z}-{\delta}_8\frac{\partial C}{\partial z}=\left(1+{\delta}_5\right)\frac{\partial^2w}{\partial {t}^2} $$
(23)
$$ \left(1+{\tau}_T\frac{\partial }{\partial t}\right)\left({\delta}_9\frac{\partial^2\dot{T}}{\partial {x}^2}+{\delta}_{12}\frac{\partial^2\dot{T}}{\partial {z}^2}\right)+\left(1+{\tau}_v\frac{\partial }{\partial t}\right)\left({\delta}_{10}\frac{\partial^2T}{\partial {x}^2}+{\delta}_{11}\frac{\partial^2T}{\partial {z}^2}\right)=\left(1+{\tau}_q\frac{\partial^{\alpha }}{\partial {t}^{\alpha }}+{\tau_q}^2\frac{\partial^2}{\partial {t}^2}\right)\left[{\delta}_9\ddot{T}+{\delta}_{13}\frac{\partial \ddot{u}}{\partial x}+{\delta}_{14}\frac{\partial \ddot{w}}{\partial z}+{\delta}_{15}\ddot{C}\right]. $$
(24)
$$ {q}_1\frac{\partial^3u}{\partial {x}^3}+{q}_2\frac{\partial^3w}{\partial {x}^2\partial z}+{q}_3\frac{\partial^3u}{\partial x\partial {z}^2}+{q}_4{\frac{\partial^3w}{\partial {z}^3}}_3^{\ast }+{q}_5\frac{\partial^2C}{\partial {x}^2}+{q}_6\frac{\partial^2C}{\partial {z}^2}+{q}_7\frac{\partial^2T}{\partial {x}^2}+{q}_8\frac{\partial^2T}{\partial {z}^2}+{q}_9\frac{\partial C}{\partial t}=0 $$
(25)

where

$$ {\delta}_1=\frac{c_{13}+{c}_{44}}{c_{11}},{\delta}_2=\frac{c_{44}}{c_{11}},{\delta}_3=\frac{c_{33}}{c_{11}},{\delta}_4=\frac{\mu_0{H}_0^2}{\rho {C}_1^2},{\delta}_5=\frac{\varepsilon_0{\mu}_0^2{H}_0^2}{\rho },{\delta}_7=\frac{a_3}{a_1},{\delta}_8=\frac{b_3}{b_1},{\delta}_9=\frac{\rho {\omega}_1^{\ast 3}}{a_1},{\delta}_{10}=\frac{\rho {\omega}_1^{\ast 2}{K}_1^{\ast }}{a_1{K}_1}, $$
$$ {\delta}_{11}=\frac{\rho {\omega}_1^{\ast 2}{K}_3^{\ast }}{a_1{K}_1},{\delta}_{12}=\frac{\rho {\omega}_1^{\ast 3}{K}_3}{a_1{K}_1},{\delta}_{13}=\frac{T_0{\omega}_1^{\ast 2}{a}_1}{K_1},{\delta}_{14}=\frac{T_0{\omega}_1^{\ast 2}{a}_3}{K_1},{\delta}_{15}=\frac{a\rho {C}_1^2{T}_0{\omega}_1^{\ast 2}}{K_1{b}_1}. $$

Rayleigh wave propagation

We pursue Rayleigh wave solution of the equations of the form

$$ \left(\begin{array}{c}u\\ {}w\\ {}T\\ {}C\end{array}\right)=\left(\begin{array}{c}1\\ {}W\\ {}S\\ {}R\end{array}\right)U{e}^{i\xi \left(x+ mz- ct\right)} $$
(26)

where \( c=\frac{\upomega}{\upxi} \) is the non-dimensional 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. (2225), we get

$$ U\left[{l}_1+{l}_6+{l}_2{m}^2\right]+W\left[{l}_3m\right]+S\left[{l}_5\right]+R\left[{l}_5\right]=0, $$
(27)
$$ U\left[{l}_3m\right]+W\left[{l}_2+{l}_6+{l}_7{m}^2\right]+S\left[{l}_8m\right]+R\left[{l}_9m\right]=0, $$
(28)
$$ U\left[{l}_{12}\right]+W\left[{l}_{13}m\right]+S\left[{l}_{10}+{l}_{11}{m}^2\right]+R\left[{l}_{14}\right]=0, $$
(29)
$$ U\left[{l}_{15}+{l}_{16}{m}^2\right]+W\left[{l}_{17}m+{l}_{18}{m}^3\right]+S\left[{l}_{21}+{l}_{22}{m}^2\right]+R\left[{l}_{19}+{l}_{20}{m}^2\right]=0, $$
(30)

where

$$ {\displaystyle \begin{array}{c}{l}_1=-{\xi}^2\left(1+{\delta}_4\right),{l}_2=-{\delta}_2{\xi}^2,{l}_3=-{\xi}^2\left({\delta}_1+{\delta}_4\right),{l}_5=- i\xi, {l}_6=\left(1+{\delta}_5\right){\xi}^2{c}^2,{l}_7=-{\xi}^2\left({\delta}_3+{\delta}_4\right),\\ {}{l}_8=-i{\xi \delta}_{7,}{l}_9=-i{\xi \delta}_{8,}{l}_{10}=-{\delta}_{10}\left(1- i\xi c{\tau}_v\right){\xi}^2+{\delta}_9\left(1- i\xi c{\tau}_T\right)i{\xi}^3c-{\delta}_9{\xi}^2{c}^2\left(1- i\xi c{\tau}_q-\frac{{\tau_q}^2{\xi}^2{c}^2}{2}\right),\\ {}{l}_{11}=-{\delta}_{11}\left(1- i\xi c{\tau}_v\right){\xi}^2+{\delta}_{12}\left(1- i\xi c{\tau}_T\right)i{\xi}^3c,{l}_{12}=-{\delta}_{13}i{\xi}^3{c}^2\left(1- i\xi c{\tau}_q-\frac{{\tau_q}^2{\xi}^2{c}^2}{2}\right),\\ {}{l}_{13}=-{\delta}_{14}i{\xi}^3{c}^2\left(1- i\xi c{\tau}_q-\frac{{\tau_q}^2{\xi}^2{c}^2}{2}\right),{l}_{14}=-{\delta}_{15}{\xi}^2{c}^2\left(1- i\xi c{\tau}_q-\frac{{\tau_q}^2{\xi}^2{c}^2}{2}\right),{l}_{15}=-{q}_1i{\xi}^3,\\ {}{l}_{16}=-{q}_3i{\xi}^3,{l}_{17}=-{q}_2i{\xi}^3,{l}_{18}=-{q}_4i{\xi}^3,{l}_{19}=-{q}_5{\xi}^2-{q}_9 i\xi c,{l}_{20}=-{q}_6{\xi}^2,{l}_{21}=-{q}_7{\xi}^2,\\ {}{l}_{22}=-{q}_8{\xi}^2,{q}_1=\frac{\alpha_1^{\ast }{b}_1{\omega}_1^{\ast 2}}{c_1^2},{q}_2=\frac{\alpha_1^{\ast }{b}_3{\omega}_1^{\ast 2}}{c_1^2},{q}_3=\frac{\alpha_3^{\ast }{b}_1{\omega}_1^{\ast 2}}{c_1^2},{q}_4=\frac{\alpha_3^{\ast }{b}_3{\omega}_1^{\ast 2}}{c_1^2},{q}_5=-\frac{\alpha_1^{\ast }b{\omega}_1^{\ast 2}\rho }{b_1}\\ {}{q}_6=-\frac{\alpha_3^{\ast }b{\omega}_1^{\ast 2}\rho }{b_1},{q}_7=-\frac{\alpha_1^{\ast }a{\omega}_1^{\ast 2}\rho }{a_1},{q}_8=\frac{\alpha_3^{\ast }a{\omega}_1^{\ast 2}\rho }{a_1},{q}_9=-\frac{\omega_1^{\ast }{c}_1^2\rho }{b_1}.\end{array}} $$

and from (2730), the characteristic equation is a biquadratic equation in m2 given by

$$ {m}^8+\frac{B}{A}{m}^6+\frac{C}{A}{m}^4+\frac{D}{A}{m}^2+\frac{E}{A}=0, $$
(31)

where

$$ A={l}_2{l}_7{l}_{11}{l}_{20}-{l}_2{l}_9{l}_{18}{l}_{11}, $$
$$ B={l}_1{l}_7{l}_{11}{l}_{20}-{l}_1{l}_9{l}_{18}{l}_{11}+{l}_2{l}_6{l}_{11}{l}_{20}+{l}_2{l}_7{l}_{10}{l}_{20}-{l}_{14}{l}_2{l}_{22}{l}_7+{l}_{14}{l}_2{l}_8{l}_{18}+{l}_2{l}_9{l}_{13}{l}_{22}-{l}_2{l}_9{l}_{13}{l}_{22}-{l}_{10}{l}_9{l}_2{l}_{18}+{l}_2{l}_{11}{l}_{17}{l}_9-{l}_3{l}_3{l}_{11}{l}_{20}+{l}_3{l}_9{l}_{16}{l}_{11}+{l}_5{l}_3{l}_{11}{l}_{18}+{l}_5{l}_7{l}_{11}{l}_{15}+{l}_2{l}_8{l}_{13}{l}_{20}, $$
$$ C={l}_1{l}_6{l}_{11}{l}_{20}+{l}_1{l}_7{l}_{19}{l}_{11}+{l}_1{l}_7{l}_{10}{l}_{20}-{l}_1{l}_7{l}_{14}{l}_{22}-{l}_{14}{l}_1{l}_8{l}_{18}-{l}_{10}{l}_1{l}_9{l}_{18}+{l}_1{l}_9{l}_{13}{l}_{22}-{l}_1{l}_9{l}_{11}{l}_{17}+{l}_2{l}_6{l}_{11}{l}_{19}+{l}_2{l}_6{l}_{10}{l}_{20}-{l}_2{l}_6{l}_{14}{l}_{22}+{l}_1{l}_7{l}_{10}{l}_{19}-{l}_2{l}_7{l}_{14}{l}_{21}-{l}_2{l}_8{l}_{13}{l}_{19}+{l}_1{l}_8{l}_{13}{l}_{20}-{l}_2{l}_8{l}_{14}{l}_{17}-{l}_2{l}_9{l}_{13}{l}_{21}-{l}_2{l}_9{l}_{10}{l}_{17}-{l}_3^2{l}_{11}{l}_{19}-{l}_3^2{l}_{10}{l}_{20}+{l}_3^2{l}_{14}{l}_{22}+{l}_3{l}_8{l}_{12}{l}_{20}-{l}_3{l}_8{l}_{14}{l}_{16}-{l}_3{l}_9{l}_{12}{l}_{22}-{l}_5{l}_3{l}_{13}{l}_{22}+{l}_5{l}_3{l}_{10}{l}_{18}+{l}_5{l}_3{l}_{11}{l}_{17}-{l}_5{l}_6{l}_{16}{l}_{11}-{l}_5{l}_7{l}_{12}{l}_{22}+{l}_5{l}_7{l}_{10}{l}_{16}+{l}_5{l}_7{l}_{11}{l}_{16}-{l}_5{l}_8{l}_{12}{l}_{18}+{l}_5{l}_8{l}_{16}{l}_{13}+{l}_5{l}_3{l}_{13}{l}_{20}-{l}_5{l}_3{l}_{14}{l}_{18}-{l}_5{l}_7{l}_{12}{l}_{20}+{l}_5{l}_7{l}_{14}{l}_{16}+{l}_5{l}_9{l}_{12}{l}_{18}-{l}_5{l}_9{l}_{13}{l}_{16}, $$
$$ D={l}_1{l}_6{l}_{11}{l}_{19}+{l}_1{l}_6{l}_{10}{l}_{20}-{l}_1{l}_6{l}_{14}{l}_{22}+{l}_1{l}_7{l}_{10}{l}_{19}+{l}_5{l}_7{l}_{10}{l}_{15}-{l}_1{l}_7{l}_{14}{l}_{21}-{l}_1{l}_8{l}_{13}{l}_{19}-{l}_1{l}_8{l}_{14}{l}_{17}+{l}_1{l}_9{l}_{13}{l}_{21}-{l}_5{l}_8{l}_{12}{l}_{17}-{l}_1{l}_9{l}_{10}{l}_{17}+{l}_2{l}_6{l}_{10}{l}_{19}-{l}_2{l}_6{l}_{14}{l}_{21}-{l}_3^2{l}_{10}{l}_{19}+{l}_5{l}_8{l}_{13}{l}_{15}+{l}_3^2{l}_{14}{l}_{21}+{l}_3{l}_8{l}_{12}{l}_{19}-{l}_3{l}_8{l}_{14}{l}_{15}-{l}_3{l}_9{l}_{12}{l}_{21}+{l}_5{l}_3{l}_{13}{l}_{19}+{l}_3{l}_9{l}_{10}{l}_{15}-{l}_3{l}_5{l}_{13}{l}_{21}+{l}_3{l}_5{l}_{10}{l}_{17}+{l}_5{l}_6{l}_{12}{l}_{22}-{l}_5{l}_6{l}_{12}{l}_{20}-{l}_{10}{l}_5{l}_6{l}_{16}-{l}_5{l}_3{l}_{14}{l}_{17}-{l}_5{l}_6{l}_{10}{l}_{16}-{l}_5{l}_6{l}_{15}{l}_{11}-{l}_5{l}_7{l}_{12}{l}_{21}+{l}_5{l}_6{l}_{14}{l}_{16}-{l}_5{l}_7{l}_{12}{l}_{19}+{l}_5{l}_7{l}_{14}{l}_{15}+{l}_5{l}_9{l}_{12}{l}_{17}-{\mathrm{l}}_5{l}_9{l}_{13}{l}_{15}+{l}_5{l}_6{l}_{14}{l}_{15}, $$
$$ E={l}_1{l}_6{l}_{10}{l}_{19}-{l}_1{l}_6{l}_{14}{l}_{21}-{l}_5{l}_6{l}_{12}{l}_{21}-{l}_5{l}_6{l}_{10}{l}_{15}-{l}_5{l}_6{l}_{12}{l}_{19}. $$

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 half-space, hence, satisfy the radiation conditions Re(mp) ≥ 0.

The displacements, temperature change, and concentration can be written as

$$ \left(\begin{array}{c}u\\ {}w\\ {}T\\ {}C\end{array}\right)=\sum \limits_{p=1}^4\left(\begin{array}{c}1\\ {}{n}_{1p}\\ {}{n}_{2p}\\ {}{n}_{3p}\end{array}\right){A}_p{e}^{i\xi \left(x+{i m}_pz- ct\right)} $$
(32)

where Ap (p = 1, 2, 3, 4) are arbitrary constants and coupling constants are

Boundary conditions

The boundary conditions at z = 0 are given by

$$ {t}_{zz}=0,{t}_{zx}=0,\frac{\partial T}{\partial z}+ hT=0,P=0. $$
(33)

After applying dimensionless quantities from Eq. (21), the above boundary conditions reduces to

$$ {\displaystyle \begin{array}{c}\left({\delta}_1-{\delta}_2\right)\frac{\partial u}{\partial x}+{\delta}_3\frac{\partial w}{\partial z}-{\delta}_7T-{\delta}_8C=0,\\ {}{\delta}_2\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)=0,\\ {}\frac{\partial T}{\partial z}+ hT=0,\\ {}\frac{\partial u}{\partial x}+{\epsilon}_2\frac{\partial w}{\partial z}-{\eta}_2C+{\eta}_1T=0,\end{array}} $$

where

$$ {\eta}_1=\frac{a{C}_{11}}{a_1{b}_1},{\eta}_2=\frac{b{C}_{11}}{b_1^2}, $$

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

$$ \sum \limits_{p=1}^4{Q}_{jp}{A}_p=0,j=1,2,3,4. $$
(34)

where

$$ {\displaystyle \begin{array}{c}{Q}_{1p}=\left({\delta}_1-{\delta}_2\right)+{\delta}_3{im}_p{n}_{1p}+\frac{i{\delta}_7{n}_{2p}}{\xi }+\frac{i{\delta}_8{n}_{3p}}{\xi },\\ {}{Q}_{2p}={im}_p+{n}_{1p},\\ {}{Q}_{3p}=\left(-\xi {m}_p+h\right){n}_{2p},\\ {}{Q}_{4p}=1+i{\epsilon}_2{m}_p{n}_{1p}-\frac{i{\eta}_1{n}_{2p}}{\xi }+\frac{i{\eta}_2{n}_{3p}}{\xi }.\end{array}} $$

Secular equations are

$$ {\displaystyle \begin{array}{c}\left[\begin{array}{c}{Q}_{11}\\ {}{Q}_{21}\\ {}{Q}_{31}\\ {}{Q}_{41}\end{array}\begin{array}{c}{Q}_{12}\\ {}{Q}_{22}\\ {}{Q}_{32}\\ {}{Q}_{42}\end{array}\begin{array}{c}{Q}_{13}\\ {}{Q}_{23}\\ {}{Q}_{33}\\ {}{Q}_{43}\end{array}\begin{array}{c}{Q}_{14}\\ {}{Q}_{24}\\ {}{Q}_{34}\\ {}{Q}_{44}\end{array}\right]=0,\kern0.5em \mathrm{or}\\ {}-{Q}_{31}{D}_1+{Q}_{32}{D}_2-{Q}_{33}{D}_3+{Q}_{34}{D}_4=0,\end{array}} $$
(35)

where

$$ {\displaystyle \begin{array}{c}{D}_1=\left[\begin{array}{ccc}{Q}_{12}& {Q}_{13}& {Q}_{14}\\ {}{Q}_{22}& {Q}_{23}& {Q}_{24}\\ {}{Q}_{42}& {Q}_{43}& {Q}_{44}\end{array}\right],\\ {}{D}_1={Q}_{12}\left({Q}_{23}{Q}_{44}-{Q}_{24}{Q}_{43}\right)-{Q}_{13}\left({Q}_{22}{Q}_{44}-{Q}_{24}{Q}_{42}\right)+{Q}_{14}\left({Q}_{22}{Q}_{43}-{Q}_{23}{Q}_{42}\right),\\ {}{D}_2=\left[\begin{array}{ccc}{Q}_{11}& {Q}_{13}& {Q}_{14}\\ {}{Q}_{21}& {Q}_{23}& {Q}_{24}\\ {}{Q}_{41}& {Q}_{43}& {Q}_{44}\end{array}\right],\\ {}{D}_2={Q}_{11}\left({Q}_{23}{Q}_{44}-{Q}_{24}{Q}_{43}\right)-{Q}_{13}\left({Q}_{21}{Q}_{44}-{Q}_{24}{Q}_{42}\right)+{Q}_{14}\left({Q}_{21}{Q}_{43}-{Q}_{23}{Q}_{41}\right),\\ {}{D}_3=\left[\begin{array}{ccc}{Q}_{11}& {Q}_{12}& {Q}_{14}\\ {}{Q}_{21}& {Q}_{22}& {Q}_{24}\\ {}{Q}_{41}& {Q}_{42}& {Q}_{44}\end{array}\right],\\ {}{D}_3={Q}_{11}\left({Q}_{22}{Q}_{44}-{Q}_{24}{Q}_{42}\right)-{Q}_{12}\left({Q}_{21}{Q}_{44}-{Q}_{24}{Q}_{41}\right)+{Q}_{14}\left({Q}_{21}{Q}_{42}-{Q}_{22}{Q}_{41}\right),\\ {}{D}_4=\left[\begin{array}{ccc}{Q}_{11}& {Q}_{12}& {Q}_{13}\\ {}{Q}_{21}& {Q}_{22}& {Q}_{23}\\ {}{Q}_{31}& {Q}_{32}& {Q}_{33}\end{array}\right],\\ {}{D}_4={Q}_{11}\left({Q}_{22}{Q}_{33}-{Q}_{23}{Q}_{32}\right)-{Q}_{12}\left({Q}_{21}{Q}_{33}-{Q}_2{Q}_{31}\right)+{Q}_{13}\left({Q}_{21}{Q}_{32}-{Q}_{22}{Q}_{31}\right).\end{array}} $$

These secular equations have entire information regarding the wavenumber, phase velocity, and attenuation coefficient of Rayleigh waves in the transversely isotropic magneto-thermoelastic medium. Moreover, If we write

$$ {c}^{-1}={v}^{-1}+F{i\omega}^{-1}, $$
(36)

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 mp = Qp + ipq, so that the exponent in Rayleigh wave solutions (28) becomes

$$ iE\left(x-i{m}_p^iz- vt\right)-E\left(\frac{F}{E}x+{m}_p^rz\right), $$
(37)

where

$$ {m}_p^r={Q}_p-{p}_q\frac{F}{E},{m}_p^i={p}_q+{Q}_p\frac{F}{E}. $$

Equation (28) can be written as

$$ \left(\begin{array}{c}u\\ {}w\\ {}T\\ {}C\end{array}\right)=\sum \limits_{p=1}^4\left(\begin{array}{c}1\\ {}{n}_{1p}\\ {}{n}_{2p}\\ {}{n}_{3p}\end{array}\right){A}_p{e}^{\left(- Fx-{\chi}_p^r\right)}\times {e}^{i\left[E\left(x- vt\right)-{\chi}_p^i\right]}, $$
(38)

where

$$ {\displaystyle \begin{array}{c}{\left|{\chi}_p^r\right|}^2-{\left|{\chi}_p^i\right|}^2={E}^2\left\{{\left({m}_p^r\right)}^2-{\left({m}_p^i\right)}^2\right\},\\ {}\left|{\chi}_p^r\right|\left|{\chi}_p^i\right| cos\theta =\frac{1}{2}{E}^2{m}_p^r{m}_p^i,\end{array}} $$

θ is the angle between the real and imaginary part of the vector χp.

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

$$ V=\frac{\omega }{\mathit{\operatorname{Re}}\left(\xi \right)} $$
(39)

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

$$ Q= Img\left(\xi \right), $$
(40)

Specific loss

The specific loss is the most direct way of defining internal resistance for a material. The specific loss W is given by

$$ W=\left(\frac{\Delta W}{W}\right)=4\uppi \left|\frac{Img\left(\xi \right)}{\mathit{\operatorname{Re}}\left(\xi \right)}\right|, $$
(41)

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

$$ S=\frac{1}{Img\left(\xi \right)} $$
(42)

Particular cases

  1. 1.

    If τT ≠ 0, τv ≠ 0, τq ≠ 0, we obtain results for Rayleigh wave propagation in transversely isotropic magneto-thermoelastic solid with diffusion and with and without energy dissipation and TPL (three-phase-lag) effects.

  2. 2.

    If τT = 0, τv = 0, τq = 0, and K ≠ 0, we obtain results for Rayleigh wave propagation in magneto-thermoelastic transversely isotropic solid with diffusion and GN-III theory (thermoelasticity with energy dissipation).

  3. 3.

    If τT = 0, τv = 0, τq = 0, and K = 0, we obtain results for Rayleigh wave propagation in magneto-thermoelastic transversely isotropic solid with diffusion and GN-II theory (generalized thermoelasticity without energy dissipation).

  4. 4.

    If τT ≠ 0, τv ≠ 0, τq ≠ 0 , and K = 0, we obtain results for Rayleigh wave propagation in magneto-thermoelastic transversely isotropic solid with diffusion and GN-II theory with TPL effect

  5. 5.

    If τT = 0, τv = 0, τq = τ0 > 0, and K = 0, and ignoring \( {\tau}_q^2 \), we obtain results for Rayleigh wave propagation in magneto-thermoelastic transversely isotropic solid with diffusion and Lord-Shulman (L-S) model.

  6. 6.

    If τT = 0, τv = 0, and τq = 0 and if the medium is not permitted with the magnetic field, i.e., μ0 = H0 = 0, then we obtain results for Rayleigh wave propagation in transversely isotropic thermoelastic solid with diffusion and without TPL effect

  7. 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 magneto-thermoelastic 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

$$ {c}_{11}=3.07\times {10}^{11}N{m}^{-2},{c}_{33}=3.581\times {10}^{11}N{m}^{-2},{c}_{13}=1.027\times {10}^{10}N{m}^{-2},{c}_{44}=1.510\times {10}^{11}N{m}^{-2},{\beta}_1=7.04\times {10}^6N{m}^{-2}{\mathit{\deg}}^{-1},{\beta}_3=6.90\times {10}^6N{m}^{-2}{\mathit{\deg}}^{-1},\rho =8.836\times {10}^3K{gm}^{-3},{C}_E=4.27\times {10}^2 jK{g}^{-1}{\mathit{\deg}}^{-1},{K}_1=0.690\times {10}^2W{m}^{-1}{Kdeg}^{-1},{K}_3=0.690\times {10}^2W{m}^{-1}{K}^{-1},{T}_0=298\ \mathrm{K},{H}_0=1\mathrm{J}{\mathrm{m}}^{-1}\mathrm{n}{\mathrm{b}}^{-1},{\upvarepsilon}_0=8.838\times {10}^{-12}\mathrm{F}{\mathrm{m}}^{-1},L=1. $$

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 three-phase-lag, GN-III, and LS theory of thermoelasticity and demonstrated graphically as

  1. 1.

    The solid line relates to the three-phase lag theory τT ≠ 0, τv ≠ 0, τq ≠ 0,

  2. 2.

    The dashed line relates to GN-III theory τT = 0, τv = 0, τq = 0, and K ≠ 0,

  3. 3.

    The dotted line relates to LS theory τT = 0, τv = 0, τq = τ0 > 0, and K = 0.

Figure 1 illustrates the deviations of tangential stress tzx with wave number. From the graph, we observe that tangential stress tzx decreases with wave number in all the three theories with a little difference in magnitude. Figure 2 shows the deviations of normal stress tzz with wavenumber. Here, we observe that the normal stress tzz 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 GN-III theory, the attenuation coefficient increases in the form of a curve with an increase in wavenumber, while for L-S 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 GN-II theories, while for L-S 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 GN-III theory, specific loss increases with increase in wavenumber, while for L-S 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 GN-III theory case and no change for TPL case, while for L-S 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 three-phase-lag GN-III 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 magneto-thermoelastic medium.

Fig. 1
figure 1

Variations of tangential stress tzx with wavenumber

Fig. 2
figure 2

Variations of normal stress tzz with wavenumber

Fig. 3
figure 3

Variations of attenuation coefficient with wavenumber

Fig. 4
figure 4

Variations of penetration depth with wavenumber

Fig. 5
figure 5

Variations of specific loss with wavenumber

Fig. 6
figure 6

Variations of concentration C with wavenumber

Fig. 7
figure 7

Variations of Rayleigh wave velocity with wave number

Fig. 8
figure 8

Variations of temperature T with wave number

Conclusion

From the above study, we conclude the following:

  • A mathematical model to study the Rayleigh wave propagation in the homogeneous transversely isotropic magneto-thermoelastic medium in the presence of mass diffusion and the three-phase-lag 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 magneto-thermoelastic medium has been derived. The comparison of different theories of thermoelasticity, i.e., TPL, GN-III, and L-S theories are carried out.

  • From the graphs, we observe a significant influence of three-phase-lag, GN-III 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 magneto-thermoelastic 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 magneto-thermoelastic medium carries information about transversely isotropic magneto-thermoelastic medium properties and is important for the design of geophysics and seismic investigations.

  • Significant resemblance and non-resemblance among the results under TPL, GN-III, and L-S 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

δij Kronecker delta

Cijkl Elastic parameters

βij Thermal elastic coupling tensor

T Absolute temperature

T0 Reference temperature

φ Conductive temperature

tij Stress tensors

eij Strain tensors

ui Components of displacement

ρ Medium density

CE Specific heat

aij Tensor of thermal moduli

αij Linear thermal expansion coefficient

Kij Materialistic constant

\( {K}_{ij}^{\ast } \) Thermal conductivity

ω Angular frequency

μ0 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

Fi Components of the Lorentz force

τ0 Relaxation time

ε0 Electric permeability

δ(t) Dirac’s delta function

τt Phase lag of heat flux

τv Phase lag of temperature gradient

τq Phase lag of thermal displacement

α Fractional-order derivative

ξ Wavenumber

bij Tensor of diffusion moduli

C The concentration of the diffusion material

\( {\alpha}_{ij}^{\ast } \) Diffusion parameters

ηi The flow of diffusion mass vector

qi 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

C1 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).

References

  • Abd-Alla, A. M., Abo-Dahab, S. M., & Hammad, H. A. (2011). Propagation of Rayleigh waves in generalized magnetothermoelastic orthotropic material under initial stress and gravity field. Applied Mathematical Modelling, 35, 2981–3000.

    Article  MathSciNet  MATH  Google Scholar 

  • Abd-Alla, A. M., Abo-Dahab, S. M., Hammad, H. A., & Mahmoud, a. S. (2011). On generalized magneto-thermoelastic Rayleigh waves in a granular medium under the influence of a gravity field and initial stress. Journal of Vibration and Control, 17(1), 115–128.

    Article  MathSciNet  MATH  Google Scholar 

  • Abd-Alla, A. M., & Ahmed, S. M. (1996). Rayleigh waves in an orthotropic thermoelastic medium under gravity and initial stress. Earth, Moon, and Planets, 75, 185–197.

    Article  MATH  Google Scholar 

  • Abouelregal, A. E. (2011). Rayleigh waves in a thermoelastic solid half space using dual-phase-lag model. International Journal of Engineering Science, 49, 781–791.

    Article  MathSciNet  MATH  Google Scholar 

  • Biswas, S., Mukhopadhyay, B., & Shaw, S. (2017). Rayleigh surface wave propagation in orthotropic thermoelastic solids under three-phase-lag model. Journal of Thermal Stresses, 40(4), 403–419.

    Article  Google Scholar 

  • Chauthale, S., & Khobragade, N. W. (2017). Thermoelastic response of a thick circular plate due to heat generation and its thermal stresses. Global Journal of Pure and Applied Mathematics, 13, 7505–7527.

  • Dhaliwal, R. S., & Sherief, H. H. (1980). Generalized thermoelasticity for anisotropic media. Quarterly of Applied Mathematics, XXXVII(1), 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  • Ezzat, M., & AI-Bary, A. (2016). Magneto-thermoelectric viscoelastic materials with memory dependent derivatives involving two temperature. International Journal of Applied Electromagnetics and Mechanics, 50(4), 549–567.

    Article  Google Scholar 

  • Ezzat, M., & AI-Bary, A. (2017). Fractional magneto-thermoelastic materials with phase lag Green-Naghdi theories. Steel and Composite Structures, 24(3), 297–307.

    Google Scholar 

  • Ezzat, M. A., El-Karamany, A. S., & El-Bary, A. A. (2017). Two-temperature theory in Green–Naghdi thermoelasticity with fractional phase-lag heat transfer. Microsystem Technologies- Springer Nature, 24(2), 951–961.

    Article  Google Scholar 

  • Ezzat, M. A., El-Karamany, A. S., & Ezzat, S. M. (2012). Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer. Nuclear Engineering and Design (Elsevier), 252, 267–277.

    Article  Google Scholar 

  • Green, A., & Naghdi, a. P. (1992). On undamped heat waves in an elastic solid. Journal of Thermal Stresses, 15(2), 253–264.

    Article  MathSciNet  Google Scholar 

  • Green, A., & Naghdi, P. (1993). Thermoelasticity without energy dissipation. Journal of Elasticity, 31(3), 189–208.

    Article  MathSciNet  MATH  Google Scholar 

  • Hassan, M., Marin, M., Ellahi, R., & Alamri, S. (2018). Exploration of convective heat transfer and flow characteristics synthesis by Cu–Ag/water hybrid-nanofluids. Heat Transfer Research, 49(18), 1837–1848. https://doi.org/10.1615/HeatTransRes.2018025569.

    Article  Google Scholar 

  • Kumar, R., & Gupta, V. (2015). Effect of phase-lags on Rayleigh wave propagation in thermoelastic medium with mass diffusion. Multidiscipline Modeling in Materials and Structures, 11, 474–493.

    Article  Google Scholar 

  • Kumar, R., & Kansal, T. (2008a). Effect of rotation on Rayleigh waves in an isotropic generalized thermoelastic diffusive half-space. Archives of Mechanics, 65(5), 421–443.

    MathSciNet  MATH  Google Scholar 

  • Kumar, R., & Kansal, T. (2008b). Rayleigh waves in transversely isotropicthermoelastic diffusive half-space. Canadian Journal of Physics, 86, 133–1143. https://doi.org/10.1139/P08-055.

    Article  Google Scholar 

  • Kumar, R., & Kansal, T. (2009). Propagation of Rayleigh waves in transversely isotropic generalized thermoelastic diffusion. Journal of Engineering Physics and Thermophysics, Springer, 82(6), 1199–1210.

    Article  Google Scholar 

  • Kumar, R., & Kansal, T. (2013). Propagation of cylindrical Rayleigh waves in a transversely isotropic thermoelastic diffusive solid half-space. Journal of Theoretical and Applied Mechanics, 43(3), 3–20.

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar, R., Kaushal, P., & Sharma, R. (2018). Transversely isotropic magneto-visco thermoelastic medium with vacuum and without energy dissipation. Journal of Solid Mechanics, 10(2), 416–434.

    Google Scholar 

  • Kumar, R., Sharma, N., & Lata, a. P. (2016a). Effects of Hall current in a transversely isotropic magnetothermoelastic with and without energy dissipation due to normal force. Structural Engineering and Mechanics, 57(1), 91–103.

    Article  Google Scholar 

  • Kumar, R., Sharma, N., & Lata, P. (2016b). Effects of thermal and diffusion phase-lags in a plate with axisymmetric heat supply. Multidiscipline Modeling in Materials and Structures(Emerald), 12(2), 275–290.

    Article  MATH  Google Scholar 

  • Kumar, R., Sharma, N., & Lata, P. (2016c). Thermomechanical interactions due to hall current in transversely isotropic thermoelastic with and without energy dissipation with two temperatures and rotation. Journal of Solid Mechanics, 8(4), 840–858.

    Google Scholar 

  • Kumar, R., Sharma, N., Lata, P., & Abo-Dahab, S. (2017). Rayleigh waves in anisotropic magnetothermoelastic medium. Coupled Systems Mechanics, 6(3), 317–333.

    Google Scholar 

  • Lata, P., & Kaur, I. (2019a). Transversely isotropic thick plate with two temperature and GN type-III in frequency domain. Coupled Systems Mechanics-Techno Press, 8(1), 55–70.

    Google Scholar 

  • Lata, P., & Kaur, I. (2019b). Study of transversely isotropic thick circular plate due to ring load with two temperature & Green Nagdhi theory of type-I, II and III. In International conference on sustainable computing in science, Technology & Management (SUSCOM-2019), − Elsevier SSRN (pp. 1753–1767). Jaipur: Amity University Rajasthan.

    Google Scholar 

  • Lata, P., & Kaur, I. (2019c). Thermomechanical interactions in transversely isotropic thick circular plate with axisymmetric heat supply. Structural Engineering and Mechanics, 69(6), 607–614.

    Google Scholar 

  • Lata, P., & Kaur, I. (2019d). Transversely isotropic magneto thermoelastic solid with two temperature and without energy dissipation in generalized thermoelasticity due to inclined load. SN Applied Sciences, 1, 426. https://doi.org/10.1007/s42452-019-0438-z.

    Article  Google Scholar 

  • Lata, P., & Kaur, I. (2019e). Effect of rotation and inclined load on transversely isotropic magneto thermoelastic solid. Structural Engineering and Mechanics, 70(2), 245–255.

    Google Scholar 

  • Lata, P., Kumar, R., & Sharma, N. (2016). Plane waves in an anisotropic thermoelastic. Steel and Composite Structures, 22(3), 567–587.

    Article  Google Scholar 

  • Mahmoud, S. R. (2011). Effect of rotation, gravity field and initial stress on generalized magneto-thermoelastic Rayleigh waves in a granular medium. Applied Mathematical Sciences, 41(5), 2013–2032.

    MathSciNet  MATH  Google Scholar 

  • Mahmoud, S. R. (2014). Effect of non-homogenity, magnetic field and gravity field on Rayleigh waves in an initially stressed elastic half-space of orthotropic material subject to rotation. Journal of Computational and Theoretical Nanoscience, 11(7), 1627–1634.

    Article  Google Scholar 

  • Marin, M. (1997). Cesaro means in thermoelasticity of dipolar bodies. Acta Mechanica, 122(1–4), 155–168.

    Article  MathSciNet  MATH  Google Scholar 

  • Marin, M. (2009). On the minimum principle for dipolar materials with stretch. Nonlinear Analysis Real World Applications, 10(3), 1572–1578.

    Article  MathSciNet  MATH  Google Scholar 

  • Marin, M. (2010). A partition of energy in thermoelasticity of microstretch bodies. Nonlinear Analysis: Real World Applications, 11(4), 2436–2447.

    Article  MathSciNet  MATH  Google Scholar 

  • Marin, M., Baleanu, D., & Vlase, S. (2017). Effect of microtemperatures for micropolar thermoelastic bodies. Structural Engineering and Mechanics, 61(3), 381–387.

    Article  Google Scholar 

  • Marin, M., & Craciun, E. (2017). Uniqueness results for a boundary value problem in dipolar thermoelasticity to model composite materials. Composites Part B: Engineering, 126, 27–37.

    Article  Google Scholar 

  • Marin, M., Ellahi, R., & Chirilă, A. (2017). On solutions of Saint-Venant’s problem for elastic dipolar bodies with voids. Carpathian Journal of Mathematics, 33(2), 219–232.

    MathSciNet  MATH  Google Scholar 

  • Marin, M., & Nicaise, S. (2016). Existence and stability results for thermoelastic dipolar bodies with double porosity. Continuum Mechanics and Thermodynamics, 28(6), 1645–1657.

    Article  MathSciNet  MATH  Google Scholar 

  • Othman, M. I. A., & Marin, M. (2017). Effect of thermal loading due to laser pulse on thermoelastic porous medium under G-N theory. Results in Physics, 7, 3863–3872.

    Article  Google Scholar 

  • Othman, M. I., & Said, S. M. (2018). Effect of diffusion and internal heat source on a two-temperature thermoelastic medium with three-phase-lag model. Archives of Thermodynamics, 39(2), 15–39.

  • Othman, M. I., & Song, Y. Q. (2006). The effect of rotation on the reflection of magneto-thermoelastic waves under thermoelasticity without energy dissipation. Acta Mechanica, 184, 89–204.

    Article  MATH  Google Scholar 

  • Othman, M. I., & Song, Y. Q. (2008). Reflection of magneto-thermoelastic waves from a rotating elastic half-space. International Journal of Engineering Science, 46, 459–474.

    Article  MathSciNet  MATH  Google Scholar 

  • Rafiq, M., Singh, B., Arifa, S., Nazeer, M., Usman, M., Arif, S., et al. (2019). Harmonic waves solution in dual-phase-lagmagneto-thermoelasticity. Open Physics, 17, 8–15. https://doi.org/10.1515/phys-2019-0002.

    Article  Google Scholar 

  • Rayleigh, L. (1885). On waves propagated along the plane surface of an elastic solid. Proceedings of the London Mathematical Society, s1-17(1), 4–11.

    Article  MathSciNet  MATH  Google Scholar 

  • Sharma, J. N., & Kaur, D. (2010). Rayleigh waves in rotating thermoelastic solids with voids. International Journal of Applied Mathematics and Mechanics, 6(3), 43–61.

    Google Scholar 

  • Sharma, J. N., Pal, M., & Chand, D. (2005). Propagation characteristics of Rayleigh waves in transversely isotropic piezothermoelastic materials. Journal of Sound and Vibration, 284, 227–248.

    Article  Google Scholar 

  • Sharma, J. N., & Singh, H. (1985). Thermoelastic surface waves in a transversely isotropic half space with thermal relaxations. Indian Journal of Pure and Applied Mathematics, 16, 1202–1212.

    MathSciNet  MATH  Google Scholar 

  • Singh, B., Kumari, S., & Singh, J. (2014). Propagation of the Rayleigh wave in an initially stressed transversely isotropic dual-phase-lag magnetothermoelastic half-space. Journal of Engineering Physics and Thermophysics, 87(6), 1539–1547.

    Article  Google Scholar 

  • Slaughter, W. S. (2002). The linearised theory of elasticity. Boston: Birkhausar.

  • Ting, T. C. (2004). Surface waves in a rotating anisotropic elastic half-space. Wave Motion, 40, 329–346.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Not applicable.

Funding

No fund/grant/scholarship has been taken for the research work.

Author information

Affiliations

Authors

Contributions

The work is carried by the corresponding author under the guidance and supervision of PL. Both authors read and approved the final manuscript.

Corresponding author

Correspondence to Iqbal Kaur.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kaur, I., Lata, P. Rayleigh wave propagation in transversely isotropic magneto-thermoelastic medium with three-phase-lag heat transfer and diffusion. Int J Mech Mater Eng 14, 12 (2019). https://doi.org/10.1186/s40712-019-0108-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s40712-019-0108-3

Keywords

  • Transversely isotropic, magneto-thermoelastic, three-phase-lag heat transfer
  • Wave propagation