Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 8s (2024) 

 

483 
https://internationalpubls.com 

Rayleigh Waves Propagation in a Micropolar Viscothermoelastic Half-

Space with Impedance Boundary Conditions 

 

Mamatha Kumari1, E. Rama2, Rajneesh Kumar3, P Vikas Singh4 

1Assistant Professor, St. Martin’s Engineering College, Hyderabad. kumari4bu@gmail.com 

2Associate Professor, Osmania University, Hyderabad. ramamathsou@gmail.com 

3Department of Mathematics, Kurukshetra University, Kurukshetra 136119, India 

rajneesh_kuk@rediffmail.com 

4Department of Mathematics, Osmania University, pulsinghvikas@gmail.com 

 

Article History: 

Received: 10-05-2024 

Revised: 28-06-2024 

Accepted: 12-07-2024 

Abstract 

This paper deals with the propagation of Rayleigh waves in a micropolar viscothermoelastic 

half space with impedance boundary conditions. The boundary of the half space is 

thermally insulated/isothermal and it is assumed that normal traction, shear traction and 

shear couple traction at the surface, varies linearly with normal, tangential components of 

displacement and microrotation respectively. The secular equation for Rayleigh wave with 

impedance boundary conditions is obtained and this equation is in agreement with the 

classical secular equation for elastic solid with traction free boundary conditions when 

micropolar, thermal and impedance parameters are removed. The non-dimensional speed 

of Rayleigh wave is computed as a function of impedance parameters and presented 

graphically for a particular micropolar thermoelastic material. 

Keywords: Micropolar thermoelasticity, Rayleigh waves, Impedance boundary conditions, 

Secular equation. 

 

1.Introduction 

Eringen’s [1]micropolar theory of elasticity is now well known due to its possible utility in examining 

the deformation properties of materials such as cellular solids, polymers, composite fibrous, granular 

material, masonry, bones and many more with microstructures. This theory takes into account the 

intrinsic rotation along with linear displacement in the materials possess microstructure and the motion 

is governed by six degrees of freedom, three of microrotation and three of classical translation. The 

classical theory of elasticity, which ignores the microrotation degrees of freedoms, can explain the 

behaviour of common solid materials like coal, concrete etc. This theory is inadequate to explain the 

behaviour of materials with inner microstructure such as polycrystalline and materials with fibrous or 

coarse grain. Therefore, micropolar theory was developed to explain the microscopic motion and long-

range interactions in solids.   

As mechanical and thermal fields are associated in almost all practical engineering problems where, 

the application of mechanical forces can change the temperature of the system.  Keeping this important 

interaction in view, Nowacki[2] and Eringen[3]extended the micropolar theory by including thermal 

effects and presented linear theory of micropolar thermoelasticity. Tauchert, Claus Jr and Ariman[4] 

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Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 8s (2024) 

 

484 
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developed the linear theory of micropolar thermoelasticity and formulate the constitutive equations. 

Various problems on micropolar thermoelasticity have been investigated extensively by researcher due 

its applications in various fields like earthquake, nuclear reactors, aeronautics, astronautics and modern 

sensor devices. 

The generalized theory of thermoelasticity is a modified version of classical uncoupled and coupled 

theory of thermoelasticity and has been developed in order to remove the paradox of impossible 

phenomena of infinite velocity of thermal signals in the classical coupled theory of thermoelasticity. 

Lord and Shulman [5] and Green and Lindsay[6]includes the concept of thermal relaxation time and 

eliminate this paradox of infinite velocity of thermal signals. Based upon generalized theory of 

thermoelasticity given by Green and Lindsay, Boschi and Ieşan [7] proposed a generalized theory of 

linear micropolar thermoelasticity that admits the possibility of second sound effect. 

Ciarletta[8]established the finite speed of thermal waves by using theory of micropolar 

thermoelasticity without energy dissipation. Based upon Lord and Shulman theory[5]Sherief, Hamza 

and EI-Sayed [9],derived the generalized equation for the linear theory of micropolar thermoelasticity. 

A comprehensive study is available on the phenomenon of wave propagation in micropolar-

generalized thermoelastic solid because of their practical applicability in the various fields of science 

and technology such as, seismology, acoustics, aerospace and submarine structures. Surface waves 

due to their destructive nature during earthquake are of particular importance in the study of 

seismology. Lord Rayleigh[10]was the first to study the wave propagating along the isotropic elastic 

solid and such waves after his name are known as Rayleigh waves. Several researchers have explored 

the concept of Rayleigh waves in different type of elastic materials. For example, Lockett 

[11]discussed the effects of thermal properties of an isotropic thermoelastic material on velocity of 

Rayleigh waves. Kumar and Singh[12] discussed about the existence Rayleigh wave in micropolar 

generalized thermoelastic half space with stretch. Rao and Reddy[13]studied the Rayleigh type wave 

propagation in a micropolar cylindrical surface. Kumar, Kaur and Rajvanshi[14]investigated the 

propagation of Lamb waves in micropolar-generalized thermoelastic solid with two temperatures 

bordered with layer of inviscid liquid. Kumar and Partap[15] studied propagation of Rayleigh Lamb 

waves in a micropolar elastic cylindrical plate. Sharma and Khator [22,24] examined some problems 

of power generation due to renewable sources. M.Marin [23,25,] explored some problems in bioheat 

thermoelastic, Cosserat thermoelastic media and non-local thermoelastic materials. Kaushal et al [26] 

investigated boundary value problem in frequency domain by considering modified Green- Lindsay 

thermoelastic medium. Kumar and Devi [27] analysed interaction due to hall current and rotation in 

modified couple stress elastic half-space subjected to ramp-type loading.  

The boundary conditions in almost all the problems related to Rayleigh waves are considered as a 

traction free surface that is stresses vanishes on the surface. The possibilities of other type of boundary 

conditions are rarely consider in seismology or geophysics but there are other fields of physics like 

electromagnetism and acoustics, where it is common to use impedance boundary conditions. The 

impedance boundary conditions prescribed on the boundary is the linear combination of the unknown 

function and their derivatives. Tiersten [16]encountered these types of boundary conditions while 

studying the wave propagation in an isotropic elastic solid coated with thin film of different material. 

Malischewsky [17]modified the Tiersten’s conditions in terms of stresses and displacement and 

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Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 8s (2024) 

 

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obtained the secular equation for Rayleigh waves. Godoy, Duran and Nedelec [18]proved the existence 

of surface waves in an elastic half space with impedance boundary conditions and derived the secular 

equation with these conditions. Vinh and Hue[19] used impedance boundary conditions to investigate 

Rayleigh waves in an orthotropic and monoclinic half space. Recently Singh[20] studied about the 

Rayleigh wave in a thermoelastic solid half space subjected to impedance boundary conditions. 

Sharma and Khator [22,24] examined some problems of power generation due to renewable sources. 

M.Marin [23,25,27] explored some problems in bioheat thermoelastic, Cosserat thermoelastic media 

and non-local thermoelastic materials. Kaushal et al [31] investigated boundary value problem in 

frequency domain by considering modified Green- Lindsay thermoelastic medium. Kumar and Devi 

[26] analyzed interaction due to hall current and rotation in modified couple stress elastic half-space 

subjected to ramp-type loading. 

Rayleigh waves are extremely useful for material characterization and to remove defects in the objects, 

as these are very sensitive to surface defects. Very few papers on Rayleigh waves with impedance 

boundary conditions are available but this concept has not been used in micropolar thermoelastic 

material. In this paper the propagation Rayleigh waves in a micropolar thermoelastic half space with 

impedance boundary condition has been investigated. Secular equation for thermally insulated and 

isothermal surface is obtained and this equation coincides with the secular equation of Rayleigh waves 

in thermoelastic solid when the micropolar effect is removed. On removing the micropolar effects, 

impedance parameters and thermal effects this equation reduces to famous secular equation of 

Rayleigh wave in isotropic elastic solid with traction free boundary conditions. Effect of micropolarity 

present in the medium on the phase velocity is highlighted through comparative study with respect 

impedance parameter. 

2. Basic equations 

Following Eringen[1], the governing equations  for homogeneous, isotropic micropolar  

viscothermoelastic solid in absence of body forces and body couples  are 

(𝜇∗ + 𝐾∗)∇2�⃗� + (𝜆∗ + 𝜇∗)∇(∇. �⃗� ) + 𝑘∗(∇ × 𝜙)⃗⃗ ⃗⃗ − 𝜈∇T = ρ(
𝜕2�⃗� 

𝜕𝑡2
) 

(1)  

(α∗ + β∗ + γ∗) ∇(∇. �⃗� ) − γ∗∇ × (∇ × �⃗� ) + 𝑘∗(∇ × �⃗� ) − 2𝑘∗�⃗� =  ρj (
𝜕2�⃗� 

𝜕𝑡2
) 

(2) 

where�⃗�  is the displacement vector, 𝜌 is the density of the material, j is the microinertia,�⃗�  is the 

microrotation vector, 𝜆 , 𝜇 , 𝑘 , 𝛼 , 𝛽 , 𝛾, 𝜆∗, 𝜇∗ , 𝑘∗ , α∗ , β∗ , γ∗are material constants and 

                         𝜆∗ = 𝜆 + 𝜆𝜈
𝜕

𝜕𝑡
,    𝜇∗ = 𝜇 + 𝜇𝜈

𝜕

𝜕𝑡
,  

                          𝑘∗ = 𝜅 + 𝜅𝜈
𝜕

𝜕𝑡
, α∗ = α + α𝜈

𝜕

𝜕𝑡
,  

                         β∗ = β + β𝜈
𝜕

𝜕𝑡
, γ∗ = γ + γ𝜈

𝜕

𝜕𝑡
 

 

 

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Communications on Applied Nonlinear Analysis 

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The constitutive relations are given by 

𝜎𝑖𝑗 = 𝜆
∗𝑢𝑟,𝑟𝛿𝑖𝑗 + 𝜇

∗(𝑢𝑖,𝑗 + 𝑢𝑗,𝑖 ) + 𝑘
∗(𝑢𝑗,𝑖 − 𝜖𝑖𝑗𝑟𝜙𝑟) − 𝜈𝑇𝛿𝑖𝑗 (3) 

𝑚𝑖𝑗 = α
∗𝜙𝑟,𝑟𝛿𝑖𝑗 + β

∗𝜙𝑖,𝑗 + γ
∗𝜙𝑗,𝑖 (4) 

where (𝑖, 𝑗, 𝑟 = 1,2,3), 𝜎𝑖𝑗 is the stress tensor,𝑚𝑖𝑗 is the couple stress tensor and 𝛿𝑖𝑗 is the kronecker 

delta.  

Following Lord and Shulman (1967), the heat conduction equation is 

𝐾∗∇2𝑇 = 𝜌𝐶∗ (
𝜕

𝜕𝑡
+ 𝜏0

𝜕2

𝜕𝑡2
)𝑇 + 𝜈𝑇0 (

𝜕

𝜕𝑡
+ 𝜏0

𝜕2

𝜕𝑡2
) ∇. �⃗�  

(5) 

 

where 𝐾∗ is the coefficient of thermal conductivity, 𝜈 = (3𝜆 + 2 𝜇 + 𝐾)𝛼𝑡 , 𝐶
∗  is the specific heat at 

constant strain, 𝛼𝑡 is the coefficient of thermal linear expansion, T is the change in temperature of the 

medium at any time, 𝑇0 is the reference temperature of the body and𝜏0is the thermal relaxation time 

3. Formulation of the problem 

We consider a homogeneous and isotropic micropolar viscothermoelastic half space at uniform 

temperature 𝑇0 in the undeformed state. Origin is placed at the plane surface and 𝑦-axis pointing 

vertically downward into the half space. The direction of propagation of the waves is considered along 

𝑥-axis so that all particles vibrating on a line parallel to z-axis are equally displaced. Therefore, all the 

field quantities will be independent of z-coordinates. For the two-dimensional problem, we assume the 

components of the displacement �⃗�  and microrotation vector �⃗�   of the form   

�⃗� = (𝑢 , 𝑣 , 0)  ,  �⃗� = (0 , 0 , 𝜙) (6) 

Using (6), equation (1) and (2) can be written as  

(𝜆∗ + 2𝜇∗ + 𝑘∗)
𝜕2𝑢

𝜕𝑥2
+ (𝜇∗ + 𝑘∗)

𝜕2𝑢

𝜕𝑦2
+ (𝜆∗ + 𝜇∗)

𝜕2𝑣

𝜕𝑥𝜕𝑦
+ 𝑘∗

𝜕𝜙

𝜕𝑦
− 𝜈

𝜕𝑇

𝜕𝑥
= 𝜌

𝜕2𝑢

𝜕𝑡2
 

 

(7) 

(𝜆∗ + 2𝜇∗ + 𝑘∗)
𝜕2𝑣

𝜕𝑦2
+ (𝜇∗ + 𝑘∗)

𝜕2𝑣

𝜕𝑥2
+ (𝜆∗ + 𝜇∗)

𝜕2𝑢

𝜕𝑥𝜕𝑦
− 𝑘∗

𝜕𝜙

𝜕𝑥
− 𝜈

𝜕𝑇

𝜕𝑦
= 𝜌

𝜕2𝑣

𝜕𝑡2
 

 

(8) 

𝛾∗ (
𝜕2𝜙

𝜕𝑥2
+ 
𝜕2𝜙

𝜕𝑦2
) + 𝑘∗ (

𝜕𝑣

𝜕𝑥
−
𝜕𝑢

𝜕𝑦
) − 2𝑘∗𝜙 = 𝜌𝑗

𝜕2𝜙

𝜕𝑡2
 

(9) 

Using Helmholtz’s representation, the displacement components 𝑢  and 𝑣can be written in terms of 

potential functions as  

𝑢 =
𝜕𝜙1

𝜕𝑥
+
𝜕𝜓1

𝜕𝑦
 ,  𝑣 =

𝜕𝜙1

𝜕𝑦
−
𝜕𝜓1

𝜕𝑥
 (10) 

Substituting (10) in equations (5) and (7)-(9), we obtained  

(𝜆∗ + 2𝜇∗ + 𝑘∗)∇2𝜙1 − 𝜈𝑇 = 𝜌
𝜕2𝜙1
𝜕𝑡2

 
(11) 



Communications on Applied Nonlinear Analysis 

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(𝜇∗ + 𝑘∗)∇2𝜓1 + 𝑘
∗𝜙 = 𝜌

𝜕2𝜓1
𝜕𝑡2

 
(12) 

𝛾∗∇2𝜙 − 2𝑘∗𝜙 − 𝑘∗∇2𝜓1 = 𝜌𝑗
𝜕2𝜙

𝜕𝑡2
 

(13) 

𝐾∗∇2𝑇 = (
𝜕

𝜕𝑡
+ 𝜏0

𝜕2

𝜕𝑡2
) (𝜌𝐶∗𝑇 + 𝜈𝑇0∇

2𝜙1) 
(14) 

4. Solution of the Problem 

The surface wave solutions of the equations (11)-(14) may be consider as  

{𝜙1, 𝜓1 , 𝑇, 𝜙} = {𝜙1̅̅̅̅ (𝑦), 𝜓1̅̅̅̅ (𝑦) , �̅�(𝑦), �̅�(𝑦)}𝑒
𝑖𝐾(𝑥−𝑐𝑡) (15) 

where𝑐 is the phase velocity, 𝐾  is the wave number, 𝜔 = 𝐾𝑐 is the circular frequency. It is assumed 

that the Rayleigh surface waves possibly damped in time, propagating along 𝑥-axis with wave speed 

𝑅𝑒(𝑐) = 𝑉 > 0and 𝐼𝑚(𝑐) ≤ 0. 

Using (15) in the equations (11) – (14), we have  

[𝐷4 − 𝐴𝐷2 + 𝐵](𝜙1̅̅̅̅ (𝑦), �̅�(𝑦)) =0 (16) 

[𝐷4 − 𝐴′𝐷2 + 𝐵′](𝜓1̅̅̅̅ (𝑦), �̅�(𝑦)) =0 (17) 

Here 

𝐷 =
𝑑

𝑑𝑦
  , 𝐴 = 𝐾2 [2 −

𝑐2(1+𝐴2+
𝐴1

𝑐1
2)

𝐴1
] , 𝐵 = 𝐾4 [

𝐴1−𝑐
2(1+𝐴2+

𝐴1

𝑐1
2)+

𝑐4

𝑐1
2

𝐴1
] 

𝐴′ = 𝐾2 (1 −
𝑐2

𝑐2
2) + 𝐾

2 −
𝐾2𝑐2𝜌𝑗

𝛾
+
2𝑘∗

𝛾
−

𝑘∗2

𝛾(𝜇 + 𝐾)
 

𝐵′ = 𝐾2 (𝐾2 −
𝐾2𝑐2𝜌𝑗

𝛾
+
2𝑘∗

𝛾
)(1 −

𝑐2

𝑐2
2) −

𝐾2𝑘∗2

𝛾(𝜇 + 𝐾)
 

𝑐1
2 =

𝜆∗+2𝜇∗+𝑘∗

𝜌
  ,  𝑐2

2 =
𝜇∗+𝑘∗

𝜌
 , 𝜏∗ = 𝜏0 +

𝑖

𝜔
 ,  𝐴1 =

𝐾∗

𝜌𝐶∗𝜏∗
 ,  𝐴2 =

𝜈2𝑇0

𝜌2𝑐1
2𝐶∗

 

(𝜆∗, 𝜇∗ , α∗, β∗, γ∗, 𝜅∗) = (𝜆, 𝜇 , α, β, γ, 𝜅) ∗ (1 − 𝜄𝑄𝑖)     (𝑖 = 1 − 6) 

𝑄1 = 𝜔(
𝜆𝜈
𝜆
) , 𝑄2 = 𝜔 (

𝜇𝜈
𝜇
) , 

𝑄3 = 𝜔 (
𝛼𝜈

𝛼
),   𝑄4 = 𝜔 (

𝛽𝜈

𝛽
), 

𝑄5 = 𝜔 (
𝛾𝜈

𝛾
) , 𝑄6 = 𝜔 (

𝜅𝜈

𝜅
), 

 

 

 

(18) 

Using the radiation conditions 𝜙1̅̅̅̅ (𝑦), 𝜓1̅̅̅̅ (𝑦) , �̅�(𝑦), �̅�(𝑦) → 0 as 𝑦 → ∞ on the general solutions of 

the equations (16) and (17) and using (15), we obtained  



Communications on Applied Nonlinear Analysis 

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𝜙1 = (𝐵1𝑒
−𝐾𝑏1𝑦 + 𝐵2𝑒

−𝐾𝑏2𝑦)𝑒𝑖𝐾(𝑥−𝑐𝑡) (19) 

𝜓1 = (𝐵3𝑒
−𝐾𝑏3𝑦 + 𝐵4𝑒

−𝐾𝑏4𝑦)𝑒𝑖𝐾(𝑥−𝑐𝑡) (20) 

𝑇 = (𝑟1𝐵1𝑒
−𝐾𝑏1𝑦 + 𝑟2𝐵2𝑒

−𝐾𝑏2𝑦)𝑒𝑖𝐾(𝑥−𝑐𝑡) (21) 

𝜙 = (𝑟3𝐵3𝑒
−𝐾𝑏3𝑦 + 𝑟4𝐵4𝑒

−𝐾𝑏4𝑦)𝑒𝑖𝐾(𝑥−𝑐𝑡) (22) 

where 

𝑏1
2 + 𝑏2

2 =
𝐴

𝐾2
,  𝑏1

2𝑏2
2 =

𝐵

𝐾4
 , 𝑏3

2 + 𝑏4
2 =

𝐴′

𝐾2
,  𝑏3

2𝑏4
2 =

𝐵′

𝐾4
  

(23) 

{
 
 

 
 𝑟𝑖 = 𝐾2 [

(𝑏𝑖
2 − 1)(𝜆∗ + 2𝜇∗ + 𝑘∗) + 𝜌𝑐2

𝜈
]  ,   (𝑖 = 1,2)

𝑟𝑗 =
𝐾2(𝜇∗ + 𝑘∗)

𝑘∗
[1 −

𝑐2

𝑐2
2 − 𝑏𝑗

2] ,                    (𝑗 = 3,4) 

 

 

 

(24) 

and𝐵1 , 𝐵2 , 𝐵3 and 𝐵4 are arbitrary constants. 

5. Boundary conditions and secular equation 

The general form of impedance boundary conditions in two dimensions in terms of displacements and 

stresses given by Malischewsky (1988) can be written as 𝜎𝑖2 + 𝜖𝑖𝑢𝑖 = 0 , for 𝑦 = 0,where 𝜖𝑖are the 

impedance parameters and have the dimensions of stress/length. For elastic half space, Godoy, Duran 

and Nedelec (2012), expressed 𝜖𝑖as𝜖𝑖 = 𝜔𝑍𝑖. Here 𝑍𝑖 are impedance real valued parameters, has 

dimensions of stress/velocity and  𝜔 = 𝑘𝑐is the circular frequency. Here the impedance boundary 

conditions at the surface 𝑦 = 0of a micropolar thermoelastic solid are consider as 𝜎2𝑖 + 𝜔𝑍𝑖𝑢𝑖 =

0 ,which can be written as  

𝜎21 + 𝜔𝑍1𝑢 = 0 ,   𝜎22 + 𝜔𝑍2𝑣 = 0  ,  𝑚23 + 𝜔𝑍3𝜙 = 0 ,  
𝜕𝑇

𝜕𝑦
+ ℎ𝑇 = 0 (25) 

 

where ℎ → 0 corresponds to thermally insulated surface and ℎ → ∞ corresponds to isothermal surface.   

Imposing boundary conditions (25) on the surface 𝑦 = 0, we get a system of four homogeneous 

equations. For a non-trivial solution, the determinant of the coefficients 𝐵1 , 𝐵2 , 𝐵3 and 𝐵4 must 

vanishes which yields the following secular equation for the velocity of propagation of the Rayleigh 

waves 

𝑚1[𝑇1(𝑙2𝑛4 − 𝑛2𝑙4) − 𝑇2(𝑙1𝑛4 − 𝑛1𝑙4)]

= 𝑚2[𝑇1(𝑙2𝑛3 − 𝑛2𝑙3) − 𝑇2(𝑙1𝑛3 − 𝑛1𝑙3)] 

(26) 

where 

𝑙𝑖 = 𝑉1𝑍1
∗ − 𝑏𝑖 − (1 +

𝑘∗

𝜇
) 𝑏𝑖 ,          (𝑖 = 1,2) 

𝑙𝑗 = 𝑉1𝑍1
∗𝑏𝑗 − 1 − (1 +

𝑘∗

𝜇∗
) (1 −

𝑐2

𝑐2
2)  ,    (𝑗 = 3,4) 



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𝑛𝑖 = 2 +
𝑘∗

𝜇∗
− 𝑉1

2 − 𝑉1𝑍2
∗𝑏𝑖  ,                     (𝑖 = 1,2) 

𝑛𝑗 = (2 +
𝑘∗

𝜇∗
) 𝑏𝑗 − 𝑉1𝑍2

∗   ,                      (𝑗 = 3,4) 

𝑚1 = (𝜇
∗𝑉1𝑍3

∗ − 𝛾∗𝑏3) (1 −
𝑐2

𝑐2
2 − 𝑏3

2) , 𝑚2 = (𝜇𝑉1𝑍3
∗ − 𝛾∗𝑏4) (1 −

𝑐2

𝑐2
2 − 𝑏4

2) 

𝑉1 = √
𝜌𝑐2

𝜇
  , 𝑍𝑖

∗ =
𝑍𝑖

√𝜌𝜇 
  ,(𝑖 = 1,2,3) 

For thermally insulated surface  

𝑇𝑖 = 𝑏𝑖 [(2 +
𝜆∗+𝑘∗

𝜇∗
) (𝑏𝑖

2 − 1) + 𝑉1
2]  , (𝑖 = 1,2) 

For isothermal surface  

𝑇𝑖 = [(2 +
𝜆∗+𝑘∗

𝜇∗
) (𝑏𝑖

2 − 1) + 𝑉1
2]  , (𝑖 = 1,2) 

6. Particular cases  

1) In the absence of micropolar and viscous effects, the equation (26) reduces to the secular equation 

for the phase velocity of Rayleigh waves in a thermoelastic half space with impedance boundary 

conditions. Neglecting micropolar constants (𝐾 = 𝑗 = 0) in the condition (18), we obtained   

𝐴′ = 𝑘2 (1 −
𝑐2

𝑐2
2) + 𝑘

2 ,  𝐵′ = 𝑘4(1 −
𝑐2

𝑐2
2) 

Using equation (23), we get 

𝑏3
2 = 1 −

𝑐2

𝑐2
2,𝑏4

2 = 1 , 𝑚1 = 0 and 𝑚2 be a non-zero value   

Consequently, the secular equation (26) reduces to  

𝑙3(𝑛1𝑇2 − 𝑛2𝑇1) − 𝑛3(𝑙1𝑇2 − 𝑙2𝑇1) = 0 (27) 

  

The equation (27) coincides with the secular equation, obtained by author Singh[20] for Rayleigh 

waves in thermoelastic solid half space with impedance boundary conditions. 

2) Further equation (27) reduces to secular equation for Rayleigh wave velocity with traction free 

boundary conditions when 𝑍𝑖
∗ = 0  , (𝑖 = 1,2,3) 

3) If we neglect the impedance parameter, micropolarity, viscosity and thermal effects from the model 

i.e. 𝑘∗ = 𝑗 = 𝑍1
∗ = 𝑍2

∗ = 𝑍3
∗ = 𝜈 = 0 ,the equation (26) reduces to  

(2 −
𝑐2

𝑐2
2)

2

= 4√1 −
𝑐2

𝑐1
2
√1 −

𝑐2

𝑐2
2 

(28) 

where 𝑐1
2 =

𝜆∗+2𝜇∗

𝜌
  ,  𝑐2

2 =
𝜇∗

𝜌
 

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Equation (28) is the well-known dispersion equation for the phase velocity of Rayleigh waves in 

classical elastic half space 

 

7. Numerical results and discussions 

To illustrate the theoretical results numerical computations have been carried out and non- 

dimensional Rayleigh wave speed has been calculated in a micropolar thermoelastic solid. The 

aluminium epoxy composite is taken as a micropolar thermoelastic solid and following 

Gauthier[21],the values of relevant physical constants of this material are  

𝜌 = 2.19 × 103𝑘𝑔/𝑚3 ,  𝜆 = 7.59 × 1010𝑁/𝑚2 ,  𝜇 = 1.89 × 1010𝑁/𝑚2 , 

𝐾 = 0.0149 × 1010𝑁/𝑚2 

𝛼 = 0.01 × 106𝑁, 𝛽 = 0.015 × 106𝑁, 𝛾 = 0.268 × 106𝑁 , 𝑗 = 0.196 × 104𝑚2 , 𝐾∗ =

0.492 × 102𝑊/𝑚 𝐾 , 𝐶∗ = 1.89 × 1010𝐽/𝑘𝑔. 𝐾. , 𝜏0 = 0.5 × 10−10𝑠, 𝑇0 = 298𝐾, 𝛼𝑡 =

2.36 × 10−6𝐾−1 

secular equation (26) using the functional iteration method, assuming that c is a complex constant 

parameter with Re(c)=V ≥0. The graphic representations of the effects of viscosity, impedance 

parameters, and the Rayleigh wave speed's dependence on wave number are illustrated in Fig. 1 

through Fig. 6.Figs. 1–3 illustrate the impact of viscosity on the non-dimensional Rayleigh wave 

speed 𝑉1 in relation to impedance parameters Z. Figures 4 through 6 illustrate the variations in wave 

speed 𝑉1with regard to the impedance parameter in a micropolar thermoelastic half space under 

thermally insulated and isothermal boundary conditions.  

 

Fig. 1.  Viscosity effects w.r.t.  Impedance parameter 𝑍2 on non-dimensional speed 𝑉1of Rayleigh 

wave. 

The effect of viscosity parameter in the non-dimensional speed 𝑉1 is significant pertinent and is 

noticed evidently from the figs (1)- (3) 

In fig (1) the non-dimensional wave speed 𝑉1 has been depicted against the non-dimensional 

impedance parameter  𝑍1  at constant frequency 𝜔 = 10 𝑟𝑎𝑑/𝑠  and retaining the boundary free of 

normal and tangential couple traction 𝑍2 = 0,  𝑍3 = 0. It is evident that due to viscosity 𝑉1 contains 

higher magnitude. 

Impedance parameter Z
1

N
on

-d
im

en
si

on
al

 w
av

e 
sp

ee
d

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Micropolar thermoelastic 

Micropolar viscothermoelastic 

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Fig. 2.  Viscosity effects w.r.t. impedance 

parameter Z2  on non-dimensional speed 𝑉1 of 

Rayleigh wave 

 

Fig. 3. Viscosity effects w.r.t.  Impedance 

parameter Z3  on non-dimensional speed 

𝑉1Rayleigh wave 

Fig (2) and Fig (3) reveals the variations of wave speed 𝑉1 with respect to non-dimensional 

impedance parameters 𝑍2𝑎𝑛𝑑 𝑍3 respectively keeping the other two impedance parameters fixed at 

zero value. From both the figs it is evidently visible that viscosity decreases and increases the wave 

speed respectively. 

 

Fig4.Variation of non-dimensional wave 

speed 𝑉1w.r.t. impedance parameter 𝑍1 in a 

micropolar viscothermally insulated and 

isothermal half space. 

 

Fig.5. Variation of non-dimensional wave 

speed  𝑉1w.r.t. impedance parameter  𝑍2 in a 

micropolar thermally insulated and isothermal 

half space 

Fig 4. depicts the non-dimensional wave speed of Rayleigh wave as a function of impedance parameter 

𝑍1 when the boundary is free from normal and couple traction (𝑍2 = 0, 𝑍3 = 0). Comparison of 𝑉1 has 

been determined when the solid half space is due to thermally insulated and Isothermal boundary 

restrictions. It is evident that wave speed 𝑉1 contracted in case of isothermal surface(curve-2) as 

contrast to thermally insulated surface(curve-1) for some value of impedance parameter 𝑍1 where 

(𝑍2 = 𝑍3 = 0,𝜔 = 10 𝑟𝑎𝑑/sec ). It is noticed from the graph that wave speed contracted gradually 

with the increase in impedance parameters  𝑍1 for the region 0≤ 𝑍1 ≤ 1 for both the conditions. The 

similar stencil of variant of wave speed is noticed with respect to impedance parameters 𝑍2 𝑎𝑛𝑑𝑍3 as 

Impedance parameter Z
2

N
on

-d
im

en
si

on
al

 s
pe

ed

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1.75

1.8

1.85

1.9

1.95

Micropolar thermoelastic

Micropolar viscothermoelastic 

Impedance parameter Z
3

N
o
n
-d

im
e
n
s
io

n
a
l 
s
p
e
e
d

 

 

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
1.9

1.905

1.91

1.915

1.92

1.925

1.93

1.935

1.94

1.945

1.95

Micropolar thermoelastic 

Micropolar viscothermoelastic

Impedance parameter Z
1

N
on

-d
im

en
si

on
al

 s
pe

ed

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Isothermal

Insulated

Impedance paramater Z
2

N
o
n
 d

im
e
n
s
io

n
a
l 
s
p
e
e
d

 

 

0 0.5 1 1.5
1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

Insulated 

Isothermal 



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revealed in Fig5 and Fig6.The wave speed is more in case of thermally insulated state as contrast to 

isothermal condition.  

 

Fig.6 Variation of non-dimensional wave speed 𝑉1w.r.t. impedance parameter 𝑍3 in a micropolar 

thermally insulated and isothermal half space 

7. Conclusion  

In this investigation, the Rayleigh waves in a micropolar viscothermoelastic half space with impedance 

boundary conditions for a thermally insulated and isothermal surface are examined. The secular 

equation for Rayleigh waves that satisfies impedance boundary conditions is obtained in the explicit 

form. The secular equation is consistent with the secular equation of Rayleigh waves for thermoelastic 

half space with impedance boundary conditions when the micropolar parameters are eliminated. 

Moreover, the classical equation for an elastic solid, as determined by Lord Rayleigh [10], is obtained 

by eliminating impedance and viscothermal effects from this equation. The numerical analysis yields 

the following conclusion: 

 • Rayleigh waves are present in a micropolar viscothermoelastic material with impedance boundary 

conditions.  

• Rayleigh waves non-dimensional speed is affected by the viscosity of micropolar thermoelastic solids 

in relation to all impedance parameters, with both increases and decreases occurring.  

The degree of dispersion of the non-dimensional Rayleigh wave speed is contingent upon the wave 

number and the impedance parameter range. 

• The non-dimensional wave speed increases in the presence of a thermally insulated boundary when 

compared to isothermal boundary conditions when calculated as a function of impedance parameters.  

Rayleigh waves are particularly significant in the field of earthquake engineering due to their 

destructive character during earthquakes, and the investigation of waves in micropolar 

viscothermoelastic material is quite significant. Waves in certain rock behave like micropolar 

viscothermoelastic solids. As a consequence, the results of the present study, despite being a theoretical 

modal, are of paramount importance to researchers in the fields of geophysics, composite materials, 

geological materials, and seismology.  

References 

[1] A.C. Eringen // Journal of Mathematics and Mechanics 15 (1966) 909. 

[2] A.C. Eringen, Foundations of micropolar thermoelasticity, International Centre for Mechanical Science, Udline 

Course and Lectures 23 (Springer-Verlag, Berlin, 1970). 

Impedance parameter Z
3

N
on

-d
im

en
si

on
al

 s
pe

ed
 

 

 

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
1.9

1.905

1.91

1.915

1.92

1.925

1.93

1.935

1.94

1.945

1.95

Insulated

Isothermal 



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