Published: 23 December 2023

# Plane waves in an isotropic thermoelastic diffusive material using strain gradient theory

Baljeet Singh1
Himanshu Singla2
1Department of Mathematics, Post Graduate Government College, Sector-11, Chandigarh, 160011, India
2Department of Mathematics, Panjab University, Chandigarh, 160014, India
Corresponding Author:
Baljeet Singh
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#### Abstract

The governing equations of motion for an isotropic strain-gradient thermoelastic material with diffusion are formulated in context of Lord and Shulman generalization of thermoelasticity and are further specialised for a two dimensional plane. Plane harmonic solution of the governing equations in two-dimension suggests the existence of five plane waves which include four coupled longitudinal waves and a shear vertical wave. A numerical example is considered to illustrate graphically the effect of frequency, measure constant of diffusion, measure constant of thermo-diffusion, thermal relaxation time, diffusive relaxation time and the coefficients of hyperstress tensor on the phase speed and attenuation coefficients of the plane waves.

## 1. Introduction

Biot [1] gave the theory of coupled thermoelasticty, in which equation of motion and heat conduction equation are coupled. This theory eliminates the first shortcoming of uncoupled theory that elastic changes have no effect on the temperature. To overcome the second shortcoming of uncoupled theory, the non-classical theories of Lord and Shulman [2], Green and Lindsay [3], Green and Naghdi [4] and Tzou [5] came in to existence which are also referred as generalized thermoelastic theories in the literature. Ezzat and his coworkers [6-9] deveoped various new thermoelastic theories by considering various factors including the fractional order, memory-dependent, variable thermal conductivity and phase lag.

A spontaneous movement of particles, from high region concentration to low region concentration is called as diffusion. The coupling of temperature, mass diffusion and strain parameters in an elastic solid is called as thermodiffusion. The diffusion process has many application in fields of oil extractions, integrated circuit fabrication, geophysics and semiconductor industry. The process of thermodiffusion is applied in exploration of the field associated with the advent of semiconductor devices and the advancement of microelectronics. Nowacki [10-12] considered the coupling of elastic, thermal and diffusion parameters and developed a coupled theory of thermodiffusion. Further, Sherief et al. [13] derived the governing equations for a theory of generalized thermodiffusion in elastic solids. Aoudai [14] considered the uniqueness and reciprocity theorem in elastic media with generalized thermoelastic diffusion. Ezzat and Fayik [15] applied fractional calculus to derive an another theory of thermodiffusion in elastic solids. Karamany and Ezzat [16] introduced a theory of generalized thermodiffusion which was based on the memory-dependent derivatives. Recently, Li et al. [17] and Abouelregal [18] studied the effects of thermodiffusion in various anisotropic devices.

The gradient theory of elasticity is considered to be adequate for investigating important problems related to size effects, chiral materials and nanotechnology. The origin of the strain gradient theories of nonsimple elastic solids goes back to works of Toupin [19] and Mindlin and Eshel [20]. Aifantis [21] formulated gradient elasticity theories for infinitesimal deformations. Recently, Iesan and Quintanilla [22] developed the strain gradient theory of thermoviscoelasticity. There are various applications of strain gradient theories in literature including Ahmadi and Firoozbakhsh [23], Aifantis ([24], [25]), Forest et al. [26], Forest and Aifantis [27], Iesan and Quintanilla ([28] and [29]). Aouadi et al. [30] developed a consistent nonlinear theory of strain gradient theory for thermoelastic diffusion materials and derived the basic equation of nonlinear strain gradient theory of thermoelasticity with mass diffusion effect.

Wave propagation in thermoelastic materials is a topic of interest for many researchers since last many decades. Chadwick and Sneddon [31] studied the plane wave in a thermoelastic solid. Puri [32] studied the plane waves in generalized thermoelastic solid in context of Lord Shulman theory. Agarwal [33] studied the plane waves in generalized thermoelastic solid in context of both Green Lindsay and Lord-Shulman theories. Sharma et al. [34] studied the plane harmonic waves in orthorhombic heat conducting materials. Verma and Hasebe [35] studied the wave propagation in plates of general anisotropic media. Singh ([36], [37]) explored the plane wave characteristics in a generalized thermoelastic medium with diffusion in context of Lord-Shulman, Green-Lindsay and Green-Nagdhi theories. Various other research works on wave propagation in thermoelastic diffusive materials are also available in literature, which include Othman et al. [38], Kumar and Gupta [39], Deswal et al. [40].

No theoretical/numerical study is traced in literature yet which investigated the characteristics of plane waves in a strain-gradient generalized thermoelastic medium with the effects of diffusion/mass transfer. Motivated by the strain gradient theory of thermoelastic diffusive material given by Aouadi et al. [30], the plane wave characteristics are investigated in an isotropic strain-gradient thermoelastic diffusive material. In Section 2, the governing equations of an isotropic model are derived with the help of Lord and Shulman [2] and Aouadi et al. [30]. In Section 3, the governing equations are specialized for a plane. These specialized equations are solved in Section 4 to explore the possibility of plane waves in the model. Two velocity equations for plane waves are derived which suggest the propagation of four coupled longitudinal waves and a shear vertical wave. Some special cases of these velocity equations are also derived in absence of diffusion parameters or hyperstress coefficients or both parameters. A numerical example of Magnesium is considered in Section 5 to illustrate graphically the phase speeds and attenuation coefficients of plane waves. The theoretical and numerical findings of the present work are presented in the last section.

## 2. Governing equations of an isotropic model

In absence of body forces and external heat sources, the governing equations of an isotropic strain-gradient thermoelastic diffusive material are formulated in context of Aouadi et al. [30] and Lord and Shulman [2] theories as.

(a) Equations of motion:

1
$\rho \frac{{\partial }^{2}{u}_{i}}{\partial {t}^{2}}=\mu {\nabla }^{2}{u}_{i}-{\nu }_{1}{\nabla }^{4}{u}_{i}+\left(\mu +{\lambda }_{0}\right)\frac{\partial {u}_{j}}{\partial {x}_{j}\partial {x}_{i}}-{\nu }_{2}{\nabla }^{2}\frac{\partial {u}_{j}}{\partial {x}_{j}\partial {x}_{i}}-{\gamma }_{1}\frac{\partial \mathrm{\Theta }}{\partial {x}_{i}}-{\gamma }_{2}\frac{\partial P}{\partial {x}_{i}}.$

(b) Heat conduction equation:

2
$\left(1+{\tau }_{0}\frac{\partial }{\partial t}\right)\left({\gamma }_{1}\frac{\partial {e}_{kk}}{\partial t}+{C}_{e}\frac{\partial \mathrm{\Theta }}{\partial t}+d\frac{\partial P}{\partial t}\right)=\kappa {\nabla }^{2}\theta .$

(c) Mass diffusion equation:

3
$\left(1+{\tau }_{1}\frac{\partial }{\partial t}\right)\left({\gamma }_{2}\frac{\partial {e}_{kk}}{\partial t}+d\frac{\partial \mathrm{\Theta }}{\partial t}+r\frac{\partial P}{\partial t}\right)=h{\nabla }^{2}P,$

where:

${\gamma }_{1}={\beta }_{1}+\frac{a{\beta }_{2}}{b},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\gamma }_{2}=\frac{{\beta }_{2}}{b},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\lambda }_{0}=\lambda -\frac{{\beta }_{2}^{2}}{b},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{C}_{e}=\frac{\rho {c}_{E}}{{T}_{0}}+\frac{{a}^{2}}{b},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}d=\frac{a}{b},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}r=\frac{1}{b},$
${\beta }_{1}=\left(3\lambda +2\mu \right){\alpha }_{t},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\beta }_{2}=\left(3\lambda +2\mu \right){\alpha }_{c},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\nabla }^{2}=\frac{{\partial }^{2}}{\partial {x}_{1}^{2}}+\frac{{\partial }^{2}}{\partial {x}_{2}^{2}}+\frac{{\partial }^{2}}{\partial {x}_{3}^{2}},$

in which $\rho$ is the density of material, $\lambda$, $\mu$ are the Lame’s constants, $\kappa$ is coefficient of thermal conductivity, $h$ is the coefficient of mass diffusion conductivity, ${\nu }_{1}$, ${\nu }_{2}$ are constitutive coefficients of hyperstress tensor, $a$ is the measure constant of thermo-diffusive effect, $b$ is the measure constant of diffusive effect, ${\tau }_{0}$ is the thermal relaxation time, ${\tau }_{1}$ is the diffusion relaxation time, ${c}_{E}$ is the specific heat at constant strain, ${\alpha }_{t}$ is coefficient of linear thermal expansion, ${\alpha }_{c}$ is the coefficient of linear diffusion expansion, $P$ is the chemical potential per unit mass of diffusive material, $\mathrm{\Theta }$ is increment in temperature, ${T}_{0}$ is the temperature of the body in natural state, ${u}_{i}$ is the displacement vector, ${e}_{ij}$ are the components of strain tensor, ${e}_{kk}={u}_{k,k}$, ${x}_{i}$ and ${x}_{j}$ are dependent variables and ${\nabla }^{2}$ is the Laplace operator.

## 3. Specialization in two-dimension

A half-space of an isotropic thermoelastic material with diffusion in strain gradient theory is considered in rectangular Cartesian coordinate system (${x}_{1},{x}_{2},{x}_{3}$) having the surface of the half-space as the plane ${x}_{3}=0.$ Therefore, the present analysis is restricted to ${x}_{1}-{x}_{3}$ plane. The following components of displacement vector are taken as Eq. (4):

4
${u}_{1}={u}_{1}\left({x}_{1},{x}_{3},t\right),\mathrm{}\mathrm{}{u}_{3}={u}_{3}\left({x}_{1},{x}_{3},t\right),\mathrm{}\mathrm{}{u}_{2}=0.$

Using the Helmholtz decomposition theorem on vectors, the components of displacement vectors are written in terms of scalar potentials functions ${\varphi }_{1}$, ${\varphi }_{3}$ as Eq. (5):

5
${u}_{1}=\frac{\partial {\varphi }_{1}}{\partial {x}_{1}}-\frac{\partial {\varphi }_{3}}{\partial {x}_{3}},{u}_{3}=\frac{\partial {\varphi }_{1}}{\partial {x}_{3}}+\frac{\partial {\varphi }_{3}}{\partial {x}_{1}}.$

Using Eqs. (4) and (5), the Eqs. (1) to (3) are specialized in ${x}_{1}-{x}_{3}$ plane as Eqs. (6-9):

6
$\left(\left({\lambda }_{0}+2\mu \right){\nabla }^{2}-\rho \frac{{\partial }^{2}}{\partial {t}^{2}}\right){\varphi }_{1}-{\nu }_{1}{\nabla }^{4}{\varphi }_{1}-{\nu }_{2}{\nabla }^{4}{\varphi }_{1}-{\gamma }_{1}\mathrm{\Theta }-{\gamma }_{2}P=0,$
7
$\mu {\nabla }^{2}{\varphi }_{3}-{\nu }_{1}{\nabla }^{4}{\varphi }_{3}=\rho \frac{{\partial }^{2}{\varphi }_{3}}{\partial {t}^{2}},$
8
$\left(1+{\tau }_{0}\frac{\partial }{\partial t}\right)\left({\gamma }_{1}{T}_{0}{\nabla }^{2}\frac{\partial {\varphi }_{1}}{\partial t}+{C}_{e}{T}_{0}\frac{\partial \mathrm{\Theta }}{\partial t}+d{T}_{0}\frac{\partial P}{\partial t}\right)=\kappa {\nabla }^{2}\theta ,$
9
$\left(1+{\tau }_{1}\frac{\partial }{\partial t}\right)\left({\gamma }_{2}{\nabla }^{2}\frac{\partial {\varphi }_{1}}{\partial t}+d\frac{\partial \mathrm{\Theta }}{\partial t}+r\frac{\partial P}{\partial t}\right)=h{\nabla }^{2}P.$

Here, the Eqs. (6), (8) and (9) are coupled in ${\varphi }_{1}$, $\mathrm{\Theta }$ and $P$ and Eq. (7) is uncoupled in ${\varphi }_{3}$.

## 4. Plane wave solution

To solve the Eqs. (6) to (9), the plane harmonic waves propagating with wave normal lying in the ${x}_{1}-{x}_{3}$ plane and making an angle $\theta$ with the ${x}_{3}$-axis is considered. Then, the following form of plane harmonic solutions are considered as Eq. (10):

10
$\left\{{\varphi }_{1},{\varphi }_{3},\mathrm{\Theta },P\right\}=\left\{{\stackrel{-}{\varphi }}_{1},{\stackrel{-}{\varphi }}_{3},\stackrel{-}{\mathrm{\Theta }},\stackrel{-}{P},\right\}\mathrm{exp}\left(ik\left({x}_{1}\mathrm{s}\mathrm{i}\mathrm{n}\theta +{x}_{3}\mathrm{c}\mathrm{o}\mathrm{s}\theta \right)-i\omega t\right),$

where, $i=\sqrt{-1}$, $k$ is wave number, $\omega$ is angular frequency, ${\stackrel{-}{\varphi }}_{1}$, $\stackrel{-}{\mathrm{\Theta }}$, $\stackrel{-}{P}$ and ${\stackrel{-}{\varphi }}_{3}$ are arbitrary constants and the pair ($\mathrm{s}\mathrm{i}\mathrm{n}\theta$, $\mathrm{c}\mathrm{o}\mathrm{s}\theta$) denote the projection of the wave normal on to ${x}_{1}-{x}_{3}$ plane. The homogeneous system of Eqs. (11-14) is obtained by inserting Eq. (10) into Eqs. (6) to (9):

11
$\left[{k}^{4}\mathrm{}\nu +\left({k}^{2}{c}_{1}^{2}-{\omega }^{2}\right)\right]\mathrm{}{\stackrel{-}{\varphi }}_{1}+{\stackrel{-}{\gamma }}_{1}\mathrm{}{\stackrel{-}{\theta }}_{1}+{\stackrel{-}{\gamma }}_{2}\mathrm{}\stackrel{-}{P}=0,$
12
${K}_{1}\mathrm{}{k}^{2}\mathrm{}{\omega }^{2}\mathrm{}{\stackrel{-}{\varphi }}_{1}+\left({K}_{2}\mathrm{}{k}^{2}-{\omega }^{2}\right)\mathrm{}\stackrel{-}{\theta }-{K}_{3}\mathrm{}{\omega }^{2}\mathrm{}\stackrel{-}{P}=0,$
13
${d}_{1}\mathrm{}{k}^{2}\mathrm{}{\omega }^{2}\mathrm{}{\stackrel{-}{\varphi }}_{1}-{d}_{2}\mathrm{}{\omega }^{2}\mathrm{}\stackrel{-}{\theta }+\left({d}_{3}\mathrm{}{k}^{2}-{\omega }^{2}\right)\mathrm{}\stackrel{-}{P}=0,$
14
$\left({b}_{1}{k}^{4}+{c}_{2}^{2}{k}^{2}-{\omega }^{2}\right)\mathrm{}{\stackrel{-}{\varphi }}_{3}=0,$

where:

${c}_{1}^{2}=\frac{\lambda +2\mu }{\rho },\mathrm{}\mathrm{}{c}_{2}^{2}=\frac{\mu }{\rho },\mathrm{}\mathrm{}{\stackrel{-}{\gamma }}_{1}=\frac{\gamma }{\rho },\mathrm{}\mathrm{}{\stackrel{-}{\gamma }}_{2}=\frac{{\gamma }_{2}}{\rho },\mathrm{}\mathrm{}{K}_{1}=\frac{{\gamma }_{1}}{{C}_{e}},\mathrm{}\mathrm{}{K}_{2}=\frac{\kappa }{{C}_{e}\mathrm{}{T}_{0}\mathrm{}{\tau }_{0}^{\mathrm{*}}},\mathrm{}$
$\nu ={\nu }_{1}+{\nu }_{2},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{K}_{3}=\frac{d}{{C}_{e}},\mathrm{}\mathrm{}{d}_{1}=\frac{{\gamma }_{2}}{r},\mathrm{}\mathrm{}{d}_{2}=\frac{d}{r},\mathrm{}\mathrm{}{d}_{3}=\frac{h}{r\mathrm{}{\tau }_{1}^{\mathrm{*}}},\mathrm{}\mathrm{}{\tau }_{0}^{\mathrm{*}}={\tau }_{0}+\frac{\iota }{\omega },\mathrm{}\mathrm{}$
${\tau }_{1}^{\mathrm{*}}={\tau }_{1}+\frac{\iota }{\omega },\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{b}_{1}=\frac{{\nu }_{1}}{\rho }.$

The system of Eqs. (11) to (13) admits non-trivial solution if and only if determinant of coefficient matrix vanishes, i.e.:

15
$\left|\begin{array}{ccc}{k}^{4}\nu +\left({k}^{2}{c}_{1}^{2}-{\omega }^{2}\right)& {\stackrel{-}{\gamma }}_{1}& {\stackrel{-}{\gamma }}_{2}\\ {K}_{1}{k}^{2}{\omega }^{2}& \left({K}_{2}{k}^{2}-{\omega }^{2}\right)& -{K}_{3}{\omega }^{2}\\ {d}_{1}{k}^{2}{\omega }^{2}& -{d}_{2}{\omega }^{2}& \left({d}_{3}{k}^{2}-{\omega }^{2}\right)\end{array}\right|=0,$

which is expanded as following bi-quadratic equation in ${k}^{2}$:

16
${S}_{0}\left({k}^{2}{\right)}^{4}+{S}_{1}\left({k}^{2}{\right)}^{3}+{S}_{2}\left({k}^{2}{\right)}^{2}+{S}_{3}\left({k}^{2}\right)+{S}_{4}=0,$

where:

${S}_{0}={d}_{3}\mathrm{}{K}_{2}\mathrm{}\nu ,{S}_{1}={c}_{1}^{2}\mathrm{}{d}_{3}\mathrm{}{K}_{2}-\left({d}_{3}+{K}_{2}\right)\mathrm{}{\omega }^{2}\mathrm{}\nu -{K}_{2}\mathrm{}{\omega }^{2}\mathrm{}\nu ,$
${S}_{2}=\left({\omega }^{2}{\right)}^{2}\mathrm{}\nu -{c}_{1}^{2}\mathrm{}{d}_{3}\mathrm{}{\omega }^{2}-{c}_{1}^{2}\mathrm{}{K}_{2}\mathrm{}{\omega }^{2}-{d}_{3}\mathrm{}{K}_{2}\mathrm{}{\omega }^{2}-{d}_{1}\mathrm{}{\stackrel{-}{\gamma }}_{2}\mathrm{}{K}_{2}\mathrm{}{\omega }^{2}-{d}_{3}\mathrm{}{\stackrel{-}{\gamma }}_{1}\mathrm{}{K}_{1}\mathrm{}{\omega }^{2}-{d}_{2}\mathrm{}{K}_{3}\mathrm{}\left({\omega }^{2}{\right)}^{2}\nu ,$
${S}_{3}={c}_{1}^{2}\mathrm{}\left({\omega }^{2}{\right)}^{2}+{d}_{3}\mathrm{}\left({\omega }^{2}{\right)}^{2}+{K}_{2}\mathrm{}\left({\omega }^{2}{\right)}^{2}+{d}_{1}\mathrm{}{\stackrel{-}{\gamma }}_{2}\mathrm{}\left({\omega }^{2}{\right)}^{2}+{\stackrel{-}{\gamma }}_{1}\mathrm{}{K}_{1}\mathrm{}\left({\omega }^{2}{\right)}^{2}-{c}_{1}^{2}\mathrm{}{d}_{2}\mathrm{}{K}_{3}\mathrm{}\left({\omega }^{2}{\right)}^{2}-{d}_{1}\mathrm{}{\stackrel{-}{\gamma }}_{1}\mathrm{}{K}_{3}\mathrm{}\left({\omega }^{2}{\right)}^{2}-{d}_{2}\mathrm{}{\stackrel{-}{\gamma }}_{2}\mathrm{}{K}_{1}\mathrm{}\left({\omega }^{2}{\right)}^{2},$
${S}_{4}={\omega }^{6}\mathrm{}\left({d}_{2}\mathrm{}{K}_{3}-1\right).$

The four roots of velocity Eq. (16) correspond to four coupled longitudinal plane waves. The phase speed ${V}_{i}$ ($i=$ 1, 2,..., 4) and the attenuation coefficients ${Q}_{i}$ ($i=$ 1, 2,..., 4) of the plane waves ${P}_{i}$ ($i=$1, 2,…, 4) may be obtained by using following formulae:

17
${V}_{i}=\frac{\omega }{Re\left({k}_{i}\right)},{Q}_{i}=|Imag\left({k}_{i}\right)|\mathrm{},$

where $Re\left({k}_{i}\right)$ and $Imag\left({k}_{i}\right)$ are the real and imaginary part of ${k}_{i}.$ The solution of Eq. (14) correspond to the SV wave with speed:

18
${V}_{5}=\sqrt{{c}_{2}^{2}+{b}_{1}{k}^{2}}.$

### 4.1. In absence of hyperstress coefficients

In absence of hyperstress coefficients, the velocity Eq. (15) reduces to the Eq. (19):

19
$\left|\begin{array}{ccc}\left({k}^{2}{c}_{1}^{2}-{\omega }^{2}\right)& {\stackrel{-}{\gamma }}_{1}& {\stackrel{-}{\gamma }}_{2}\\ {K}_{1}{k}^{2}{\omega }^{2}& \left({K}_{2}{k}^{2}-{\omega }^{2}\right)& -{K}_{3}{\omega }^{2}\\ {d}_{1}{k}^{2}{\omega }^{2}& -{d}_{2}{\omega }^{2}& \left({d}_{3}{k}^{2}-{\omega }^{2}\right)\end{array}\right|=0.$

The cubic Eq. (19) indicates the propagation of three coupled longitudinal waves. In absence of hyperstress coefficients, the speed of SV wave given in Eq. (18) will reduce to the speed of classical SV wave as given by Eq. (20):

20
${V}_{5}=\sqrt{\frac{\mu }{\rho }}.$

The reduced velocity Eqs. (19) and (20) are in agreement with Singh [36].

### 4.2. In absence of diffusion parameters

In absence of diffusion parameters, the velocity Eq. (15) reduces to Eq. (21):

21
$\left|\begin{array}{cc}{k}^{4}\nu +\left({k}^{2}{c}_{1}^{2}-{\omega }^{2}\right)& {\stackrel{-}{\gamma }}_{1}\\ {K}_{1}{k}^{2}{\omega }^{2}& \left({K}_{2}{k}^{2}-{\omega }^{2}\right)=0\end{array}\right|=0.$

The cubic Eq. (21) indicates the propagation of three coupled longitudinal waves. The speed of SV wave given by equation (18) is independent of diffusion parameters.

### 4.3. In absence of both diffusion and hyperstress coefficients

In absence of diffusion and hyperstress coefficients, the Eq. (15) reduces to Eq. (22):

22
$\left|\begin{array}{cc}\left({k}^{2}{c}_{1}^{2}-{\omega }^{2}\right)& {\stackrel{-}{\gamma }}_{1}\\ {K}_{1}{k}^{2}{\omega }^{2}& \left({K}_{2}{k}^{2}-{\omega }^{2}\right)\end{array}\right|=0.$

The quadratic Eq. (22) suggests the propagation of two coupled plane waves. In absence of diffusion and hyperstress coefficients, the speed of SV wave given in Eq. (18) will reduce to the speed of classical SV wave as given by Eq. (20).

## 5. Numerical results and discussion

The objective of present numerical simulations is to check the variations of phase speeds and attenuation coefficients for a specific material. The following material constants of Magnesium at ${T}_{0}=$ 300 K from Singh [41] are considered to compute the phase speeds and attenuation coefficients as in Table 1.

Using above above physical constants and using Eq. (17), the velocity Eq. (16) is solved to obtain the speeds and attenuations coefficients of four coupled longitudinal waves. The velocity Eq. (18) is also solved to obtain the speed of the $SV$ wave. The wave speeds and attenuation coefficients of four coupled longitudinal waves (${P}_{1}$, ${P}_{2}$, ${P}_{3}$ and ${P}_{4}$) are plotted in Fig. 1 against the circular frequency $\omega$ for three different values of measure constant of thermo-diffusive effect $a$ For all values of $a$, the illustrations in figure 1 shows that the speeds and attenuation coefficients of all coupled longitudinal increase nonlinearly as the circular frequency $\omega$ increases. For increasing value of frequency, the rate of increase in speeds of ${P}_{1}$ and ${P}_{2}$ waves become fast and the rate of increase in speeds of ${P}_{3}$ and ${P}_{4}$ waves becomes slow. The effect of measure constant of thermodiffusion $a$ is observed on ${P}_{1}$ and ${P}_{3}$ waves as these waves become faster as value of $a$ is increased at a given frequency $\omega$. This effect of thermodiffusion on ${P}_{1}$ and ${P}_{3}$ waves becomes more considerable as the frequency $\omega$ increases. For the selected range of frequency, the speeds of ${P}_{2}$ and ${P}_{4}$ waves are not influenced considerably by changing the values of constant $a$. The variations of the attenuation coefficients of the coupled longitudinal waves except ${P}_{1}$ wave are also affected due to change in values of constant $a$ at a given frequency.

Fig. 1The wave speeds and attenuation coefficients of the P1, P2, P3 and P4 waves against frequency ω, when measure constant of thermo-diffusive effect a = 0.005 (solid line), a = 0.05 (dotted line) and a = 0.5 (dashed line)

Table 1Material constants of magnesium

 Symbol Value Unit Symbol Value unit $\lambda$ 3.17$×1{0}^{10}$ Nm–2 $\mu$ 1.639×1010 Nm–2 $\rho$ 1740 Kgm–3 ${C}_{E}$ 2361 JKg–1deg–1 $\kappa$ 49.2 Wm–1deg–1 $h$ 0.85 Kgsm–3 ${\alpha }_{t}$ 0.005 deg–1 ${\alpha }_{c}$ 0.005 m3Kg–1 $a$ 0.005 m–2s–2deg–1 $B$ 0.005 m5kg–1s–2 ${\tau }_{0}$ 0.005 s ${\tau }_{1}$ 0.005 s ${\nu }_{1}$ 0.002 Kgm–1s–2 ${\nu }_{2}$ 0.002 Kgm–1s–2

Fig. 2The wave speeds and attenuation coefficients of the P1, P2, P3 and P4 waves against frequency ω, when measure constant of diffusive effect b = 0.005 (solid line), b = 0.05 (dotted line) and b = 0.5 (dashed line)

The wave speeds and attenuation coefficients of four coupled longitudinal waves (${P}_{1}$, ${P}_{2}$, ${P}_{3}$ and ${P}_{4}$) are plotted in Fig. 2 against the circular frequency $\omega$ for three different values of measure constant of diffusive effect $b$. The effect of measure constant of diffusion $b$ is observed on ${P}_{1},{P}_{2}$ and ${P}_{3}$ waves as these waves become faster as value of $b$ is increased at a given frequency $\omega$. The diffusion effect on ${P}_{1}$, ${P}_{2}$ and ${P}_{3}$ waves becomes more considerable as the frequency $\omega$ increases. For the selected range of frequency, the speed ${P}_{4}$ wave is not influenced considerably by changing the values of constant $b$. The attenuation coefficients of the coupled longitudinal waves except ${P}_{1}$ and ${P}_{4}$ waves are also affected due to change in diffusion constant $a$ at a given frequency.

Fig. 3The wave speeds and attenuation coefficients of the P1, P2, P3 and P4 waves against frequency ω, when thermal relaxation time τ0 = 0.005 (solid line), τ0 = 0.05 (dotted line) and τ0 = 0.5 (dashed line)

The wave speeds and attenuation coefficients of four coupled longitudinal waves (${P}_{1}$, ${P}_{2}$, ${P}_{3}$ and ${P}_{4}$) are plotted in Fig. 3 against the circular frequency $\omega$ for three different values of thermal relaxation time ${\tau }_{0}$. The effect of thermal relaxation is observed on the speed and attenuation coefficient of ${P}_{4}$ wave only. The speed of ${P}_{4}$ wave becomes slow as the value of ${\tau }_{0}$ increases at a given frequency. The effect of thermal relaxation time on speed and attenuation of ${P}_{4}$ wave becomes more considerable as the frequency increases.

Fig. 4The wave speeds and attenuation coefficients of the P1, P2, P3 and P4 waves against frequency ω, when diffusion relaxation time τ1 = 0.005 (solid line), τ1 = 0.05 (dotted line) and τ1 = 0.5 (dashed line)

The wave speeds and attenuation coefficients of four coupled longitudinal waves (${P}_{1}$, ${P}_{2}$, ${P}_{3}$ and ${P}_{4}$) are plotted in Fig. 4 against the circular frequency $\omega$ for three different values of diffusion relaxation time ${\tau }_{1}$. The effect of diffusion relaxation time is observed on the speeds of ${P}_{1}$ and ${P}_{3}$ waves and on attenuation coefficient of ${P}_{2}$ and ${P}_{3}$ waves. For a given frequency, the speed of ${P}_{1}$ becomes faster as the value of ${\tau }_{1}$ increases, whereas the speed of ${P}_{3}$ waves becomes slower. This effect of diffusion relaxation time on these coupled waves becomes more considerable as the value of circular frequency increases.

Fig. 5The wave speeds and attenuation coefficients of the P1, P2, P3 and P4 waves against frequency ω, when the coefficient of hyperstress tensor ν = 0.005 (solid line), ν = 0.05 (dotted line) and ν = 0.5 (dashed line)

The wave speeds and attenuation coefficients of four coupled longitudinal waves (${P}_{1}$, ${P}_{2}$, ${P}_{3}$ and ${P}_{4}$) are plotted in Fig. 5 against the circular frequency $\omega$ for three different values of hyperstress coefficient $\nu$. The effect of hyperstress is observed on the speeds and attenuations of ${P}_{1}$ and ${P}_{2}$ waves. For a given frequency, the speeds of ${P}_{1}$ and ${P}_{2}$ becomes faster as the value of $\nu$ increases. This effect of hyperstress on these coupled waves becomes more considerable as the value of circular frequency increases.

The wave speeds and attenuation coefficients of coupled longitudinal waves are also plotted in Fig. 6 (in absence of hyperstress) where ${P}_{2}$ wave will not propagate in absence of hyperstress. The wave speeds and attenuation coefficients of coupled longitudinal waves are plotted in Fig. 7 (in absence of diffusion) where ${P}_{3}$ wave will not propagate in absence of diffusion. In Fig. 8, the wave speeds and attenuation coefficients of coupled longitudinal waves are shown graphically in absence of both diffusion and hyperstress, where the ${P}_{1}$ and ${P}_{4}$ waves will propagate.

Fig. 6The wave speeds and attenuation coefficients of the P1, P3 and P4 waves against frequency ω (in absence of hyperstress)

The wave speed of $SV$ wave is plotted in Fig. 9 against the circular frequency $\omega$ for three different values of hyperstress coefficient $\nu$. In presence of hyperstress, the speed of $SV$ wave increases nonlinearly as the frequency increases. For a given frequency, the speed of $SV$ becomes faster as the value of $\nu$ increases. This effect of hyperstress on the $SV$ wave is observed more considerable as the value of circular frequency increases.

Fig. 7The wave speeds and attenuation coefficients of the P1, P2 and P4 waves against frequency ω (in absence of diffusion)

Fig. 8The wave speeds and attenuation coefficients of the P1 and P4 waves against frequency ω (in absence of hyperstress and diffusion)

Fig. 9The wave speed of the SV wave against frequency ω, when coefficient of hyperstress tensor ν1 = 0(solid line), ν = 0.005 (dotted line) and ν = 0.05 (dashed line)

## 6. Conclusions

A problem on propagation of plane waves is considered in an isotropic, linear and homogeneous strain-gradient thermoelastic diffusive medium. The governing equations are formulated in context of Aoudai et al. [30] and Lord and Shulman [2] theories and are specialized in two-dimension. Two velocity equations for plane waves are derived, which suggest the propagation of four coupled longitudinal waves and a shear vertical wave. The numerical computations in MATLAB for a specific material (Magnesium) ensure the existence of these waves. The graphical illustrations show that the wave speeds and attenuation coefficients of these waves are affected by the circular frequency, measure constants of diffusion and thermo-diffusion, thermal and diffusion relaxation times and the coefficient of hyperstress tensor. Following some important observations are made from the present numerical results:

1) The speeds and attenuation coefficients of coupled longitudinal waves increase nonlinearly as the frequency increases.

2) For a given value of frequency, the ${P}_{1}$ and ${P}_{3}$ waves travel faster as the measure constant $a$ increases and the ${P}_{1}$, ${P}_{2}$ and ${P}_{3}$ waves travel faster as the measure constant $b$ increases.

3) The effect of thermal relaxation is observed on the speed and attenuation coefficient of ${P}_{4}$ wave only. The speed of ${P}_{4}$ wave becomes slow as the value of ${\tau }_{0}$ increases at a given frequency. The effect of diffusion relaxation time is observed on the speeds of ${P}_{1}$ and ${P}_{3}$ waves and on attenuation coefficient of ${P}_{2}$ and ${P}_{3}$ waves. For a given frequency, the speed of ${P}_{1}$ becomes faster as the value of ${\tau }_{1}$ increases, whereas the speed of ${P}_{3}$ waves becomes slower.

4) The effect of hyperstress is observed on the speeds and attenuations of ${P}_{1}$ and ${P}_{2}$ waves. For a given frequency, the speeds of ${P}_{1}$ and ${P}_{2}$ becomes faster as the value of $\nu$ increases.

5) The effects of measure constants of diffusion and thermo-diffusion, thermal and diffusion relaxation times and the coefficients of hyperstress tensor becomes more considerable as the frequency increases.

6) In absence of hyperstress or diffusion parameters, the number of coupled longitudinal waves will reduce to three. In absence of hyperstress coefficients, the ${P}_{2}$ wave will not propagate and the ${P}_{3}$ wave will not propagate in the absence of diffusion.

7) In presence of hyperstress, the speed of $SV$ wave increases nonlinearly against the frequency. For a given frequency, the the $SV$ wave travels faster as the value of $\nu$ increases. This effect of hyperstress on the $SV$ wave is observed more considerable as the frequency increases.

#### References

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10 March 2023
Accepted
18 September 2023
Published
23 December 2023
Keywords
plane wave
phase velocity
attenuation coefficient
thermal and diffusion relaxation times
hyperstress coefficients
Acknowledgements

One of the author Himanshu Singla is thankful for the financial assistance provided by the Council of Scientific and Industrial Research, New Delhi, India, in the form of Senior Research Fellowship through Grant No. 09/135(0792)/2017-EMR-1.

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Author Contributions

Baljeet Singh has formulated the governing equations for the material under study. Himanshu Singla solved these equations for plane waves. Himanshu Singla also illustrated the numerical results as per numerical plan suggested by Baljeet Singh. Himanshu Singla wrote the whole manuscript and author Baljeet Singh finalized the manuscript after giving inputs in each section of the manuscript.

Conflict of interest

The authors declare that they have no potential conflict of interest.