PROPAGATION OF STONELEY WAVES AT THE BOUNDARY

SURFACE OF THERMOELASTIC DIFFUSION SOLID AND

MICROSTRETCH THERMOELASTIC DIFFUSION SOLID Rajneesh Kumar

Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India

e-mail: rajneesh_kuk@rediffmail.com

Abstract. This present investigation is focus on the propagation of Stoneley waves at the interface of thermoelastic diffusion solid half space and microstretch thermoelastic diffusion solid half space. The secular equation of Stoneley waves is derived in the compact form by using the appropriate boundary conditions. Phase velocity and attenuation coefficients are obtained numerically. Also the components of normal displacement, normal stress and temperature distribution are obtained. The effect of diffusion and relaxation times on the phase velocity, attenuation coefficient, normal displacement, stress components and temperature distribution are depicted graphically for a particular model. Some particular cases of interest are also deduced. Keywords: microstretch; thermoelastic diffusion; Stoneley waves, phase velocity; attenuation coefficient.

1. IntroductionThe exact nature of the layers beneath the earth’s surface is not known. One has, therefore to consider various appropriate models for the purpose of theoretical investigations. These problems not only provide better information about the internal composition of the earth but also helpful in exploration of valuable materials beneath the earth surface.

Eringen [1, 2] developed the theory of micromorphic bodies by considering a material point as endowed with three deformable directions. Subsequently, he developed the theory of microstretch elastic solid [3], which is a generalization of micropolar elasticity [4]. The material points in microstretch elastic body can stretch and contract independently of the translational and rotational processes. The difference between these solids and micropolar elastic solids stems from the presence of scalar microstretch and a vector first moment. These solids can undergo intrinsic volume change independent of the macro volume change and is accompanied by a non-deviatoric stress moment vector.

Eringen [5] also developed the theory of thermo microstretch elastic solids. The microstretch continuum is a model for Bravias lattice with a basis on the atomic level and a two-phase dipolar solid with a core on the macroscopic level. For example, composite materials reinforced with chopped elastic fibres, porous media whose pores are filled with gas or inviscid liquid, asphalt or other elastic inclusions and ‘solid-liquid’ crystals, etc., should be characterizable by microstretch solids. A comprehensive review on the micropolar continuum theory has been given in his book by Eringen[6]. Rayleigh [7] discussed the surface wave propagation along the free boundary of an elastic half-space, non-attenuated in their direction of propagation and damped normal to the boundary.

Materials Physics and Mechanics 35 (2018) 87-100 Received: September 26, 2015

http://dx.doi.org/10.18720/MPM.3512018_12 © 2018, Peter the Great St. Petersburg Polytechnic University © 2018, Institute of Problems of Mechanical Engineering RAS

Stoneley [8] studied the propagation of waves along the plane interface between two distinct elastic solid half spaces in perfect contact and derived the dispersion equation. Tajuddin [9] investigated the existence of Stoneley waves at an interface between two micropolar elastic half spaces. Iesan and Pompei [10] discussed the equilibrium theory of microstretch elastic solids. Tomar and Singh [11] discussed the propagation of Stoneley waves at an interface between two microstretch elastic half-spaces. Markov [12] discussed the propagation of Stoneley elastic wave at the boundary of two fluid-saturated porous media and determined the velocity and attenuation of the Stoneley surface waves. Ahmed and Abo-Dahab [13] studied the propagation of Rayleigh and Stoneley waves in a thermoelastic orthotropic granular half-space supporting a different layer under the influence of initial stress and gravity field.

Diffusion can be defined as the random walk to accumulate the particles from region of high concentration to that of low concentration. At the present time, there is a great deal of interest in the study of this phenomenon due to its application in geophysics and electronic industry. Study of phenomenon of diffusion is utilized to enhance the conditions of oil extraction (searching ways of more efficiently recovering oil from its deposits).

Nowacki [14-17] in a series of papers presented the theory of thermoelastic diffusion by using coupled thermoelastic model. Uniqueness and reciprocity theorems for the equations of generalized thermoelastic diffusion problem, in isotropic media, was proved by Sherief et al. [18] on the basis of the variational principle equations, under restrictive assumptions on the elastic coefficients. Kumar and Kansal [19] derived the basic equation of anisotropic thermoelastic diffusion based upon Green-Lindsay model and discussed the Lamb waves. Kumar and Chawla [20] discussed the wave propagation at the imperfect boundary between transversely isotropic themoelastic diffusive half space and an isotropic elastic layer. Kumar and Kansal [21] construct the fundamental solution of system of differential equations in the theory of thermomicrostretch elastic diffusive solids in case of steady oscillations in terms of elementary functions. Sharma [22, 23] discussed the plane harmonic generalized thermoelastic diffusive waves and elasto-thermodiffusive surface waves in heat-conducting solids. Recently Kumar et al [24] studied the reflection and transmission of plane waves at the interface between a microstretch thermoelastic diffusion solid half-space and elastic solid half space.

Keeping in view of these applications, dispersion equation for Stoneley waves at the interface of thermoelastic diffusion solid half space and microstretch thermoelastic diffusion solid half space is derived. Numerical computations are performed for a particular model to study the variations of phase velocity and attenuation coefficient with respect to wave number. Also normal stress, normal displacement and temperature distribution are depicted graphically.

2. Basic EquationsFollowing Eringen [6], Sherief et al. [18] and Kumar & Kansal [19], the equations of motion and the constitutive relations in a homogeneous isotropic microstretch thermoelastic diffusion solid in the absence of body forces, body couples, stretch force, and heat sources are given by: ( ) ( ) ( ) *2 OK u K u Kλ µ µ ϕ λ ϕ+ + ∇ ∇⋅ − + ∇×∇× + ∇× + ∇ −

2

11 1 2 21 1 uT C

t t tβ τ β τ ρ∂ ∂ ∂ − + ∇ − + ∇ = ∂ ∂ ∂

, (1)

( ) ( ) ( )2

2. 2 ,K u K jtϕα β γ ϕ γ ϕ ϕ ρ ∂

+ + ∇ ∇ − ∇× ∇× + ∇× − =∂

(2)

( ) ( )2 *

2 * 1 * 00 1 1 2 1 0 2.

2jT T C C u

tρ ϕα ϕ ν τ ν τ λϕ λ ∂

∇ + + + + − − ∇ =∂

, (3)

( )* 2 * *1 0 0 1 0 0 0 0 11 . 1 1 ,K T T u T C T aT C C

t t tβ ετ ν ετ ϕ ρ τ γ∂ ∂ ∂ ∇ = + ∇ + + + + + + ∂ ∂ ∂

(4)

88 Rajneesh Kumar

( ) ( ) ( )2 2 * 2 0 2 12 2 1( . ) 0,D u D Da T T C C Db C Cβ ν ϕ τ ετ τ∇ ∇ + ∇ + ∇ + + + − ∇ + =

(5) and constitutive relations are:

( ) ( ) * 1, , , , 1 1 2(1 ) (1 ) ,ij r r ij i j j i j i ijr r o ij ij ijt u u u K u T C

t tλ δ µ ε ϕ λ δ ϕ β τ δ β τ δ∂ ∂

= + + + − + − + − +∂ ∂

(6) *

, , , 0 , ,ij r r ij i j j i mji mm bαϕ δ βϕ γϕ ε ϕ= + + + (7) * *

0 , 0 , ,i i ijm j mbλ α ϕ ε ϕ= + (8) where λ ,µ , 1, , , , , , , ,o o oK bα β γ λ λ α are material constants, ρ is the mass density,

( )1 2 3, ,u u u u= is the displacement vector,

( )1 2 3, ,ϕ ϕ ϕ ϕ= is the microrotation vector,

*ϕ is the scalar microstretch function, T and 0T are the small temperature increment and the reference temperature of the body chosen

such that � 𝑇𝑇𝑇𝑇0� ≪ 1,

C is the concentration of the diffusion material in the elastic body, *K is the coefficient of the thermal conductivity, *C is the specific heat at constant strain,

D is the thermoelastic diffusion constant, a and b are, respectively, coefficients describing the measure of thermodiffusion and of mass diffusion effects,

( )1 13 2 ,tKβ λ µ α= + + ( )2 13 2 ,cKβ λ µ α= + + ( )1 23 2 ,tKν λ µ α= + + ( )2 23 2 cKν λ µ α= + +

1 2,t tα α are coefficients of linear thermal expansion,

1 2,c cα α are the coefficients of linear diffusion expansion, j is the microintertia, oj is the microinertia of the microelements,

ijt and ijm are components of stress and couple stress tensors respectively, *iλ is the microstress tensor,

( ), ,12ij i j j ie u u = +

are components of infinitesimal strain,

kke is the dilatation,

ijδ is the Kronecker delta, 0 1,τ τ are diffusion relaxation times with 1 0 0τ τ≥ ≥ ,

and 0 1,τ τ are thermal relaxation times with 1 0 0τ τ≥ ≥ . Here 0 1

0 1 1 0τ τ τ τ γ= = = = = for Coupled Thermoelasitc (CT) model, 11 0,τ τ= =

1 01,ε γ τ= = for Lord-Shulman (L-S) model and 0,ε = 01γ τ= , where 0 0τ > for Green-

Lindsay (G-L) model. In the above equations, a comma followed by a suffix denotes spatial derivative and a

superposed dot denotes the derivative with respect to time respectively. Thermoelastic with diffusion. Following Kumar and Kansal [19], the basic equations

for a homogeneous isotropic thermoelastic diffusion solid in the absence of body forces, heat sources and mass diffusion sources are given by:

Propagation of Stoneley waves at the boundary surface of thermoelastic diffusion solid... 89

( ) ( ) ( )2

11 1 2 22 . 1 1 ,uu u T C

t t tλ µ µ β τ β τ ρ∂ ∂ ∂ + ∇ ∇ − ∇×∇× − + ∇ − + ∇ = ∂ ∂ ∂

(9)

( )* 2 *1 0 0 0 0 11 . 1 ,K T T u C T aT C C

t tβ ε τ ρ τ γ∂ ∂ ∇ = + ∇ + + + + ∂ ∂

(10)

( ) ( ) ( )2 2 0 2 12 1( . ) 0,D u Da T T C C Db C Cβ τ ε τ τ∇ ∇ + ∇ + + + − ∇ + =

(11)

and constitutive relations are:

( ) 1, , , 1 1 2(1 ) (1 ) ,ij r r ij i j j i ij ijt u u u T C

t tλ δ µ β τ δ β τ δ∂ ∂

= + + − + − +∂ ∂

(12)

where symbols have their usual meaning as defined above for microstretch thermoelastic diffusion solid.

3. Formulation of the ProblemWe consider a homogeneous isotropic generalized thermoelastic diffusion half-space 1Moverlying a homogeneous isotropic microstretch generalized thermoelastic diffusion half-space

2M connecting at the interface 3 0x = . The origin of the coordinate system 1 2 3( , , )x x x is taken at any point on the plane horizontal surface 3 0.=x We choose the 1x -axis in the direction of wave propagation in such a way that all the particles on a line parallel to the

2x -axis are equally displaced. Therefore, all field quantities are independent of the

2x -coordinate. Medium 2M occupies the region 3 0−∞ < ≤x and the region 30 ≤ < ∞x is occupied by the half-space (medium 1M ). The plane 3 0x = represents the interface between two media 1M and 2M .

We define all the quantities with attached bar for medium 1M and without bar for medium 2.M

For the two dimensional problem, we take: For medium 2M :

*1 3 1 3 2 1 3 1 3 1 3( , , ) ( ,0, ), (0, ,0), ( , , ), ( , , ), ( , , ),u x x t u u x x t x x t C x x tϕ ϕ ϕ= = Τ

(13) and, for medium 1M :

1 3 1 3 1 3 1 3( , , ) ( ,0, ), ( , , ), ( , , ),u x x t u u x x t C x x t= Τ

(14)

Fig. 1. Geometry of the problem.

90 Rajneesh Kumar

We define the following dimensionless quantities:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

'

'

* 2 2*' ' ' ' ' * * ' '1 1 11 3 1 3 1 3 1 3 2 2

1 1 1 1 1* **

' * ' ' ' ' ' *22

1 1 1 1 1

1, , , , , , , , , ,

1, , , , , , , , ,

ij ij ij ijo o o o

iij ij i

o o o

c c cx x x x u u u u t t t tc T T T T

m m T T T T C C C C t tc T c T T c

ρ ω ρ ρω ϕ ϕ ϕ ϕβ β β β

λ ωω βλ ωβ β ρ

= = = = =

= = = = =

( ) ( ) ( ) ( )*

' 0' ' 1' * 0 1 ' ' 11 1 1 3 1 3

1

, , , , , , , , , ,o oo

cu u u uT

ρ ωτ τ τ τ ω τ τ τ τβ

= = (15)

where * 2

* 1* ,C c

Kρω = 2

12 Kc λ µρ

+ += , *ω is the characteristic frequency of the medium.

We introduce the potential functions , ,φ ψ φ and ψ through the relations: For medium M1:

1 31 3 3 1

, ,u ux x x xφ ψ φ ψ∂ ∂ ∂ ∂

= − = +∂ ∂ ∂ ∂

(16)

and, for medium M2:

1 31 3 3 1

,u ux x x xφ ψ φ ψ∂ ∂ ∂ ∂

= − = +∂ ∂ ∂ ∂

. (17)

Substituting the values of 1u and 3u from (16) in the equations (9)-(11) and with the aid of (14) and (15), we obtain:

2 1 13 ,t cT a Cφ τ τ φ∇ − − = (18)

( )2 21 ,δ ψ ψ− ∇ = (19) 2 0 2 0 0

11 13 ,e t cT a T a Cτ φ τ τ∇ = ∇ + + (20) 4 1 2 1 2 0

14 15 16 0t c fa a T a C Cφ τ τ τ∇ + ∇ − ∇ + = (21) Substituting the values of 1u and 3u from (17) in the equations (1)-(5) and with the aid of

(13) and (15), we obtain: 2 * 1 1

2 3 ,t ca T a Cφ ϕ τ τ φ∇ + − − = (22)

( )2 21 21 ,aδ ψ ϕ ψ− ∇ + = (23)

( )2 24 6 2 5 2 ,a a aϕ ψ ϕ∇ − − ∇ = (24)

( )2 2 * 2 1 1 *1 7 8 9 10 ,t ca a a T a Cδ ϕ φ τ τ ϕ∇ − − ∇ + + = (25)

( )2 0 2 * 0 011 12 13 ,e t cT a a T a Cτ φ ϕ τ τ∇ = ∇ + + + (26)

4 2 * 1 2 1 2 014 21 15 16 0t c fa a a T a C Cφ ϕ τ τ τ∇ + ∇ + ∇ − ∇ + = , (27)

where

( ) ( ) ( )2

211 2 0 3 4 5 62 2 *2 *2 2

1 1 0 1 1

1 1 2, , , , , , , , , ,c K Ka a K a a a ac T j c c

ρ γ λ µλ δρ β ρ ω ω ρ

+= = = =

( )2 4

0 1 1 0 1 111 12 13 * *

2

1, , , ,T T c aa a aK

β β ν ρω ρ ρ β

=

, ( )

2*1 2

14 15 16 2 21 1 1

, , , , ,aDa a a bc c

β βωρ β

=

( )2 4 2 *

2 20 01 1 1 2 1 2 2 27 8 9 10 1 2 21*2 2 4

0 1 1 2 0 1 0 1

22, , , , , , , , ,c c c Da a a a c aj T c j c

λ αλ ν ν ρ ν β ωδω ρ ρ β β β ρ ρ

= = = =

,

Propagation of Stoneley waves at the boundary surface of thermoelastic diffusion solid... 91

1 1 1 0 0 01 01 , 1 , 1 , 1 ,t c f tt t t t

τ τ τ τ τ ετ τ τ∂ ∂ ∂ ∂= + = + = + = +

∂ ∂ ∂ ∂2 2

0 0 2310 1 2 2

1 3 1 3

1 , 1 , ,e cuue

t t x x x xτ ετ τ γ ∂∂∂ ∂ ∂ ∂

= + = + = + ∇ = +∂ ∂ ∂ ∂ ∂ ∂

.

4. Solution of the ProblemWe assume the solutions of the form:

{ }( ) { }( ) ( )1* *2 1 3 2 3, , , , , , , , , , , , , , , , , , , , i x ctT C T C x x t T C T C x e ξφ ψ φ ψ φ ϕ φ ψ φ ψ φ ϕ −=

, (28)

where ξ is the wave number, cω ξ= is the angular frequency, and c is phase velocity of the wave. Using (28) in equations (18), (20) and (21) and satisfying the radiation condition

, , 0φ →T C as 3 →−∞x , we obtain the values of , ,φ T C for medium 1M ,

{ } { } ( )3 13

1 21

, , 1, , ,pm x i x ctp p p

pT C A n n e e ξφ − −

=

= ∑ (29)

where , ( 1, 2,3)pA p = are arbitrary constants, the coupling constants 1 2,p pn n given in appendix A.

Similarly, we write the appropriate values of *, , ,T Cφ ϕ for 2 3( 0)M x < satisfying the radiation conditions as:

{ } { } ( )3 14

*1 2 3

1, , , 1, , , pm x i x ct

p p p pp

T C A n n n e e ξφ ϕ −

=

= ∑ (30)

where 2 ( 1, 2,3)pm p = are the roots of the equation: 6 * 4 * 2 *

1 1 1 0D A D B D C+ + + = , (31) and 2 ( 1, 2,3, 4)pm p = are the roots of the equation:

8 * 6 * 4 * 2 *1 1 1 1 0D A D B D C D D+ + + + = , (32)

where 3 ,D d dx= the coefficients * * * * * *1 1 1 1 1 1, , , , ,A B C A B C and *

1D are given in appendix B. The appropriate values of ψ for medium 1 3( 0)M x < and 2,ψ ϕ for 2 3( 0)M x > satisfying

the appropriate radiation conditions are: ( )14 3

4i x ctm xA e e ξψ −−= , (33)

{ } { } ( )3 16

2 45

, 1, pm x i x ctp p

pA n e e ξψ ϕ −

=

= ∑ , (34)

where 2 2 2 24 (1 / (1 ))m cξ δ= − − , (35)

and 2 ( 5,6)pm p = are the roots of the equation 4 * 2 *

2 2 0D A D B+ + = , (36) where the coefficients *

2A and *2B are given in appendix B.

The roots of equation (31) in the descending order corresponds to the velocities of propagation of three possible waves, namely longitudinal wave (P), thermal wave (T) and mass diffusion wave (MD), respectively. Also, the roots of equation (32) in the descending order corresponds to the velocities of propagation of four possible waves, namely longitudinal displacement wave (LD), thermal wave (T), mass diffusion wave (MD), and longitudinal microstretch wave (LM), respectively. Similarly, two roots of the equation (36) in the

92 Rajneesh Kumar

descending order corresponds to the velocities of propagation of two coupled transverse displacement and transverse microrotational waves (CD I, CD II), respectively.

(i) In the absence of diffusion effect, equation (32) leads to sixth order differential equation:

6 * 4 * 2 *3 3 3 0, D A D B D C+ + + = (37)

where the coefficients *3A , *

3B and *3C are given in appendix B.

The roots of the equation (37) 2 ( 1, 2,3)pm p = correspond to the LD, T and LM waves, respectively.

Clearly, on neglecting the diffusion effect, the wave corresponding to this parameter namely mass diffusion wave (MD) become deceased.

Therefore, it is observed from the equation (32) and (37), that there exist a new type of wave namely MD wave.

(ii) On neglecting the diffusion, micropolarity and microstretch effects, equation (32) and (36) simultaneously leads to the forth and second order differential equations as

4 * 2 *4 4 0,D A D B+ + = (38)

and ( )2 2

26 / 1 0,D b δ+ − = (39)

where the coefficients *4A and *

4B are given in appendix B. The roots of the equation (38) correspond to the Longitudinal wave (P-wave), and

T waves, and (39) relate to the SV-wave, respectively. Therefore, it is again observed that there exist new type of wave in (32) namely Longitudinal microstretch wave (LM) and transverse microrotational waves (CD II) in (36) which become decoupled in this case. Substituting the values of , ,φ ψ φ and ψ from equations (29), (33) and (30), (34) in equations (16) and (17), we obtained displacement components for medium 1M

( ) ( )

( ) ( )

11 3 2 3 3 3 4 3

11 3 2 3 3 3 4 3

1 1 2 3 4 4

3 1 1 2 2 3 3 4

,

,

ξ

ξ

ξ

ξ

−− − − −

−− − − −

= + + + = − + + +

i x ctm x m x m x m x

i x ctm x m x m x m x

u i A e A e A e m A e e

u m A e m A e m A e i A e e(40)

for medium 2M

( ) ( ) ( )

( ) ( ) ( )

11 3 2 3 3 3 4 3 5 3 6 3

11 3 2 3 3 3 4 3 5 3 6 3

1 1 2 3 4 5 5 6 6

3 1 1 2 2 3 3 4 4 5 6

,

.

i x ctm x m x m x m x m x m x

i x ctm x m x m x m x m x m x

u i A e A e A e A e m A e m A e e

u m A e m A e m A e m A e i A e A e e

ξ

ξ

ξ

ξ

−

−

= + + + − + = + + + + +

(41)

5. Boundary ConditionsThe appropriate boundary conditions are the continuity of stress components, displacement components, temperature change, mass concentration, normal heat flux vector, normal mass diffusion flux vector and vanishing of tangential couple stress component, microstretch component, microrotation component. Mathematically, these can be written (at the surface

3 0x = ) as: (i) 33 33 ,t t= (42) (ii) 31 31 ,t t= (43) (iii) 32 0,m = (44) (iv) *

3 0,λ = (45) (v) 3 3 ,u u= (46)

Propagation of Stoneley waves at the boundary surface of thermoelastic diffusion solid... 93

(vi) 1 1 ,u u= (47) (vii) T T= , (48) (viii) C C= , (49)

(ix) * *

3 3,T TK Kx x

∂ ∂= ∂ ∂ (50)

(x) * *

3 3

C CD Dx x∂ ∂=∂ ∂ . (51)

6. Derivations of the secular equationsMaking use of equations (29),(30),(33),(34),(40) and (41) in the equations (42)-(51) with the aid of (6)-(8),(12) and (15), we obtain a system of eight simultaneous linear equations:

4 6

, 41 1

0,+= =

+ =∑ ∑qp p q p pp p

k A k A for ( 1, 2,....,10),q = (52)

where the values of , , (1, 2,3,.......,10)ijk for i j = are:

( ) ( )( )( ) ( )

( )

2 2 11 2 1 2 1 3

11

2 2 14 1 2 1, 4 2, 4 1 3, 4

4 1

1 1 , ( 1, 2,3)

1 , ( 4),

1 1 , ( 5,6,7,8)

1 , ( 9,10)

p p p p

pp

p p p p

p

m b a n n i c n i c for p

i m b for pk

m b a n n i c n i c for p

i m b for p

ξ ξ τ ξ τ

ξ

ξ ξ τ ξ τ

ξ− − − −

−

− + + − + − = − ==

− + + − + − =

− =

( )( )( )

( )

2 3

2 22 3

2

4 2 3

2 22 4 3 1 4, 4

, ( 1, 2,3)

, ( 4),

, ( 5,6,7,8)

, ( 9,10)

ξ

ξ

ξ

ξ−

− −

− + = − + ==

+ =− + + =

p

pp

p

p p

i m b b for p

b m b for pk

i m b b for p

b m b a n for p

35 1, 4

4 4, 6 4

0, ( 1, 2,3)0, ( 4)

,, ( 5,6,7,8)

, ( 9,10)

pp

p p

for pfor p

ki b n for pb n m for pξ −

− −

= == − = − =

46 4 1, 4

5 4, 4

0, ( 1, 2,3)0, ( 4)

,, ( 5,6,7,8)

, ( 9,10)

pp p

p

for pfor p

kb m n for p

i b n for pξ− −

−

= == = − =

54

0, ( 1, 2,3)0, ( 4)

,, ( 5,6,7,8), ( 9,10)

pp

for pfor p

km for p

i for pξ−

= == = =

6

4

, ( 1, 2,3), ( 4)

,, ( 5,6,7,8)

, ( 9,10)

pp

p

i for pm for p

ki for pm for p

ξ

ξ

−

= == − = =

2

72, 4

, ( 1, 2,3)0, ( 4)

,, ( 5,6,7,8)

0, ( 9,10)

p

pp

n for pfor p

kn for p

for p−

= == − = =

3

83, 4

, ( 1, 2,3)0, ( 4)

,, ( 5,6,7,8)

0, ( 9,10)

p

pp

n for pfor p

kn for p

for p−

= == − = =

*2

9 *4 2, 4

, ( 1, 2,3)0, ( 4)

,, ( 5,6,7,8)

0, ( 9,10)

p p

pp p

K m n for pfor p

kK m n for p

for p− −

= ==

= =

94 Rajneesh Kumar

*3

10 *4 3, 4

, ( 1, 2,3)0, ( 4)

,, ( 5,6,7,8)

0, ( 9,10)

p p

pp p

D m n for pfor p

kD m n for p

for p− −

= ==

= =

2 220 0

1 2 3 4 5 62 2 2 2 2 21 1 1 1 1 1

, , , , , .ω ω αλ µ µ ω γρ ρ ρ ρ ρ ρ

+= = = = = =

bKb b b b b bc c c c c c

The system of equations (52) has a non-trivial solution if the determinant of amplitudes,p pA A vanishes which leads to the secular equation:

10 100 , (1,2,3,.......,10)ijk for i j

×= = (53)

Equation (53) is the dispersion equation for the propapgation of Stoneley waves at the interface between thermoelastic diffusion and microstretch thermoelastic diffusion solid half spaces. This equation has complete information about the phase velocity, wave number, and attenuation coefficient of the surface waves propagating in such a medium.

7. Particular Cases(i) In the absence of diffusion effect, the dispersion equation for the propagation of Stoneley waves at an interface between thermoelastic and microstretch thermoelastic solid half spaces is obtained as:

8 80 , (1, 2,3,.......,8)ijk for i j

×= = (54)

with the values of ijk as:

( )( )

( )

2 21 2 1

11 2 2

3 1 2 1, 3 2, 3 1

3 1

, ( 1, 2)

1 , ( 3),

1 , ( 4,5,6)1 , ( 7,8)

p p

pp

p p p

p

m b a n for p

i m b for pk

m b a n n i c for pi m b for p

ξ

ξ

ξ ξ τξ

− − −

−

− + =

− == − + + − =

− =

( )( )( )

( )

2 3

2 22 3

2

3 2 3

2 22 3 3 1 4, 3

, ( 1, 2)

, ( 3),

, ( 4,5,6)

, ( 7,8)

p

pp

p

p p

i m b b for p

b m b for pk

i m b b for p

b m b a n for p

ξ

ξ

ξ

ξ−

− −

− + = − + ==

+ =− + + =

33

0, ( 1, 2)0, ( 3)

,, ( 4,5,6), ( 7,8)

pp

for pfor p

km for p

i for pξ−

= == = =

4

3

, ( 1, 2), ( 3)

,, ( 4,5,6)

, ( 7,8)

pp

p

i for pm for p

ki for p

m for p

ξ

ξ

−

= == − = =

55 1, 3

4 4, 3 3

0, ( 1, 2)0, ( 3)

,, ( 4,5,6)

, ( 7,8)

pp

p p

for pfor p

ki b n for pb n m for pξ −

− −

= == − =− =

Propagation of Stoneley waves at the boundary surface of thermoelastic diffusion solid... 95

66 5 1, 3

5 4, 3

0, ( 1, 2)0, ( 3)

,, ( 4,5,6), ( 7,8)

pp p

p

for pfor p

kb m n for p

i b n for pξ− −

−

= == = − =

7

4, 3

0, ( 1, 2)0, ( 3)

,0, ( 4,5,6)

, ( 7,8)

p

p

for pfor p

kfor p

n for p−

= == =− =

8 *3 2, 3

0, ( 1, 2)0, ( 3)

., ( 4,5,6)

0, ( 7,8)

pp p

for pfor p

kK m n for p

for p− −

= == = =

(ii) In the absence of thermal and diffusion effects, the dispersion equation (53) reduced to the propagation of Stoneley waves at an interface of elastic/microstretch elastic solid half spaces.

(iii) Take 0 0,τ > 0ε = and 01γ τ= in equation (53), yield the expression of secular

equation for the propagation of Stoneley waves at an interface between thermoelastic diffusion and microstretch thermoelastic diffusion solid half spaces with two relaxation times.

(iv) Using 11 1 00,τ τ γ τ= = = and 1ε = in equations (53), gives the corresponding results

for the propagation of Stoneley waves at an interface between thermoelastic diffusion and microstretch thermoelastic diffusion solid half spaces with with one relaxation time.

(v) On taking 0 10 1 1 0τ τ τ τ γ= = = = = in equations (53), provide the corresponding

expression of secular equation for the propagation of Stoneley waves at an interface between thermoelastic diffusion and microstretch thermoelastic diffusion solid half spaces with Coupled Thermoelastic (CT) theory.

8. Numerical results and discussionThe analysis is conducted for a magnesium crystal-like material. Following Eringen [25], the values of micropolar parameters for medium M1 are given by 10 -29.4 10 Nmλ = × ,

10 -24.0 10 Nmµ = × , 3 31.74 10 Kgmρ −= × . Thermal and diffusion parameters are given by * 3 1 11.04 10C JKg K− −= × ,

* 6 1 1 1K 1.7 10 Jm s K− − −= × , 5 -1t1 2.33 10 Kα −= × , 5 -1

t2 2.48 10 Kα −= × , 30 0.298 10 KT = × ,

1 0.01τ = , 0 0.02τ = , 4 3 -1c1 2.65 10 m Kgα −= × , 4 3 -1

c2 2.83 10 m Kg ,α −= × 4 2 2 -12.9 10 m s Ka −= × , 5 -1 5 232 10 Kg m sb −= × , 1 0.04τ = , * 1.5η = , 0 0.03τ = , 8 30.85 10D Kgm s− −= × .

For medium M2 micropolar parameters are given by 10 -20.759 10 Nmλ = × , 10 -20.189 10 Nmµ = × , 10 21.49 10 ,K Nm−= × 3 32.190 10 Kgmρ −= × , 19 20.196 10j m−= × , 90.268 10 Nγ −= × .

Thermal and diffusion parameters are given by * 3 1 11.18 10C JKg K− −= × , * 6 1 1 1K 1.5 10 Jm s K− − −= × , 5 -1

t1 2.22 10 Kα −= × , 5 -1t2 2.38 10 Kα −= × , 3

0 0.198 10 KT = × ,

1 0.009τ = , 0 0.01τ = , 4 3 -1c1 2.34 10 m Kgα −= × , 4 3 -1

c2 2.61 10 m Kgα −= × , 4 2 2 -12.32 10 m s Ka −= × , 5 -1 5 230.61 10 Kg m sb −= × , 1 0.03τ = , * 1.48η = , 0 0.02τ = ,

8 30.63 10D Kgm s− −= × and the microstretch parameters are taken as 19 20.165 10oj m−= × , 90.61 10o Nα −= × , 90.25 10ob N−= × , 10 20.37 10o Nmλ −= × , 10 2

1 0.37 10 Nmλ −= × . MATLAB software 7.04 has been used for numerical computation of the resulting

quantities.

96 Rajneesh Kumar

Fig. 2. Variation of phase velocity with wavenumber.

Fig. 3. Variation of attenuation with wavenumber.

Fig. 4. Variation of normal displacement u3 with wavenumber.

Fig. 5. Variation of normal stress t33 with wavenumber.

Figure 2 shows that phase velocity initially decreases sharply, attains minima and then shows a stationary behavior. Also the values of phase velocity decreases under the effect of diffusion.

Figure 3 shows that in absence of diffusion effect, attenuation coefficient increases smoothly with increase in wave number for LS and GL theories. While under the effect of diffusion, minimum variation is observed in the magnitude values of attenuation coefficient, which appears to be stationary.

Figure 4 shows that initially small variation is observed in the magnitude values of normal displacement u3 with wave number. However, amplitude of variation increases with increase in wave number. Maximum value is observed under the effect of diffusion.

Figure 5 depicts that normal stress t33 fluctuates with wave number. This fluctuation increases with increase in the wave number. Maximum value is observed in absence of the diffusion effect. Initially under the effect of diffusion the values of displacement starts with an initial decrease.

It is clear from Fig. 6 that magnitude values of temperature for LS and GL theories under the effect of diffusion initially decreases and then shows a stationary behavior. But if there is no diffusion, a sharp increase and smooth decrease in values is observed till it becomes stationary.

0 5 10 15 20-1.0x10-1

0.0

1.0x10-1

2.0x10-1

3.0x10-1

4.0x10-1

5.0x10-1

6.0x10-1

7.0x10-1

8.0x10-1

PHAS

E VE

LOCI

TY

WAVE NUMBER

LSWD LSWTD GLWD GLWTD

0 5 10 15 20

0.0

2.0x101

4.0x101

6.0x101

8.0x101

1.0x102

1.2x102

ATTE

NUAT

ION

WAVE NUMBER

LSWD LSWTD GLWD GLWTD

0 5 10 15 20-3x105

-2x105

-1x105

0

1x105

2x105

3x105

u 3

WAVE NUMBER

LSWD LSWTD GLWD GLWTD

0 5 10 15 20-3x106

-2x106

-1x106

0

1x106

2x106

3x106

t 33

WAVE NUMBER

LSWD LSWTD GLWD GLWTD

Propagation of Stoneley waves at the boundary surface of thermoelastic diffusion solid... 97

Fig. 6. Variation of temperature distribution with wavenumber.

9. ConclusionsIn the present article, frequency equation for the Stoneley waves at bounded interface is derived in the compact form by using appropriate boundary conditions. It is found that Stoneley waves in the considered model are dispersive. Numerical computations are performed for a particular model to study the variation of phase velocity and attenuation coefficients with respect to wavenumbers. It is seen that for small values of non-dimensional wavenumber, the effect of different relaxation times have a significant effect on dispersion curve and negligible effect is observed for higher value. Also normal displacement, normal stress and temperature distribution are obtained and depicted graphically to show the effect of diffusion and thermal relaxation times for a particular model. Some particular cases of interest are also deduced.

References [1] A.C. Eringen, Mechanics of micromorphic materials, In: Proceedings of the II International

Congress of Applied Mechanics, ed. by H. Gortler (Springer, Berlin, 1966), p. 131. [2] A.C. Eringen, Mechanics of micromorphic continua, In: Mechanics of Generalized

Continua, ed. by E. Kroner (IUTAM Symposium, Freudenstadt-Stuttgart, Springer, Berlin, 1968), p. 18.

[3] A.C. Eringen, Micropolar elastic solids with stretch, ed. by M.I. Anisina (Ari Kitabevi Matbassi, Istanbul, Turkey, 1971), p. 18.

[4] A.C. Eringen // Journal of Mathematics and Mechanics 15(6) (1966) 909. [5] A.C. Eringen // International Journal of Engineering Science 28(12) (1990) 1291. [6] A.C. Eringen, Microcontinuum Field Theories I: Foundations and Solids (Springer-Verlag,

New York, 1999). [7] L. Rayleigh // Proceedings of the London Mathematical Society 17(1) (1885) 4. [8] R. Stoneley // Proceedings of Royal Society of London 106(738) (1924) 416. [9] M. Tajuddin // Journal of Applied Mechanics 62(1) (1995) 255. [10] D. Iesan, A. Pompei // International Journal of Engineering Science 33(3) (1995) 399. [11] S.K. Tomar, Dilbag Singh // Journal of Vibration and Control 12(9) (2006) 995. [12] M.G. Markov // Geophysical Journal International 177(2) (2009) 603. [13] S.M. Ahmed, S.M. Abo-Dahab // Mathematical Problems in Engineering 2012 (2012)

Article ID 245965. [14] W. Nowacki // Bulletin of Polish Academy of Sciences Series, Science and Technology 22

(1974) 55.

0 5 10 15 20

0

1x104

2x104

3x104

4x104

5x104

TEMP

ARAT

URE

WAVE NUMBER

LSWD LSWTD GLWD GLWTD

98 Rajneesh Kumar

[15] W. Nowacki // Bulletin of Polish Academy of Sciences Series, Science and Technology 22 (1974) 129.

[16] W. Nowacki // Bulletin of Polish Academy of Sciences Series, Science and Technology 22 (1974) 275.

[17] W. Nowacki // Engineering Fracture Mechanics 8(1) (1976) 261. [18] H.H. Sherief, F.A. Hamza, H.A. Saleh // International Journal of Engineering Science

42(5-6) (2004) 591. [19] R. Kumar, T. Kansal // International Journal of Solids and Structures 45(22) (2008) 5890. [20] R. Kumar, V. Chawla // Journal of Technical Physics 50(2) (2009) 121. [21] R. Kumar, T. Kansal // International Scholarly Research Network (2011), Article ID

764632. [22] J.N. Sharma // Journal of Sound and Vibration 301 (2007) 979. [23] J.N. Sharma, Y.D. Sharma, P.K. Sharma // Journal of Sound and Vibration 315 (2008)

927. [24] R. Kumar, S.K. Garg, S. Ahuja // Latin Journal of Solid and Structure 10(1) (2013) 1081. [25] A.C. Eringen // International Journal of Engineering Science 22(8-10) (1984) 1113.

Appendix A ( ) ( )

( ) ( )( ) ( )

6 2 4 2 2 2 6 4 21 22 22 23 23 24 24 11 12 13 14

6 2 4 2 2 2 6 4 22 15 15 16 16 17 17 11 12 13 14

8 2 6 2 4 218 18 19 19 20 20 2

3

,

,

p p p p p p p

p p p p p p p

p p pp

n l m l l m l l m l l m l m l m l

n l m l l m l l m l l m l m l m l

l m l l m l l m l ln

ξ ξ ξ

ξ ξ ξ

ξ ξ ξ

= − − + − + − + + + + = − + − + − + + + +

− + − + − + −= −

( )

( ) ( )

2 21 21

6 4 211 12 13 14

6 2 4 2 2 6 4 21 22 22 23 23 24 11 12 13

,

,

p

p p p

p p p p p p p

m l

l m l m l m l

n l m l l m l l m l m l m l m

ξ

ξ ξ

+ + + +

= − − + − + − + +

( ) ( )6 2 4 2 2 2 6 4 22 15 15 16 16 17 17 11 12 13 ,p p p p p p pn l m l l m l l m l l m l m l mξ ξ ξ = − + − + − + + + for ( )1,2,3,4p= .

Appendix B *1 12 11 ,A d d= *

1 13 11 ,B d d= *1 14 11 ,C d d= *

3 32 31 ,A d d= *3 33 31B d d= , *

3 34 31C d d= , *4 42 41 ,A d d= *

4 43 41 ,B d d= *1 12 11 ,A d d= *

1 13 11 ,B d d= *1 14 11 ,C d d= *

1 15 11 ,D d d=

( )* 2 2

2 4 26 27 1 5 4(1 ) (1 ) ,A a b b a a aδ δ= + − + − ( )* 2 22 26 27 1 5 4 (1 ) ,B b b a a aξ δ= − −

( )11 11 13 18 ,d l b l= − + ( )212 12 11 11 12 15 13 18 19 2 22 ,d l b l b l b l l a lξ= − + − + − −

( ) ( )2 2 213 13 11 12 12 15 16 13 19 20 2 22 2 23,d l b l b l l b l l a l a lξ ξ ξ= − + + − + − − −

( ) ( )2 214 14 11 13 12 16 17 13 20 21 2 23

2 2 215 11 14 12 17 13 21 2 24 2 24

,

,

d l b l b l l b l l a l

d b l b l b l a l a l

ξ ξ

ξ ξ ξ

= − + + − + − +

= + + + −

( ) ( )( ) ( )( ) [ ]

2 231 1 32 2 8 16 1 18 11 12 17

2 2 233 1 11 18 12 17 16 18 11 12 17 20 8 12 14 2 14 17 8 18 8

234 11 14 20 16 18 2 8 18 14 17 12 8 20 16 17

41 42 18 11 12

, ,

( ) ( ),

( ) ( ) ,

1,

d d a a b b b b b

d b b b b b b b b b b a b b a b b a b a

d b b b b b a a b b b b a b b b

d d b b b b

δ δ

δ ξ ξ

ξ

= − = − + + −

= − + + + − + − + − −

= − + − − −

= = − − +( ) ( )217 43 11 18 12 17, ,d b b b b ξ= −

Propagation of Stoneley waves at the boundary surface of thermoelastic diffusion solid... 99

( ) ( )( ) ( )( ) ( )

211 11 13 18 12 12 11 11 12 15 13 18 19 2 22

2 2 213 13 11 12 12 15 16 13 19 20 2 22 2 23

2 214 14 11 13 12 16 17 13 20 21 2 23

, ,

,

,

= − + = − + − + − −

= − + + − + − − −

= − + + − + − +

d l b l d l b l b l b l l a l

d l b l b l l b l l a l a l

d l b l b l l b l l a l

ξ

ξ ξ ξ

ξ ξ

( )( ) ( )

2 2 2 211 23 1 12 1 19 22 12 23 14 13 22 19 1 11 12 14 23 13 21 15 15

2 2 214 22 11 2 15 12 13 21 15 15 15 1 14 19 17 23

2 2 216 1 12 17 14 19 14 16 18 23 21 19 17

, ( ) , ( ) ,

, ( ),

( ) ,

= = − + = − + − + + −

= − − − + = +

= − − + + =

l b l b b g b g l b b g g g b g a g b

l b g a b g g a g b l a b b b

l g b a b a g g b a g l

δ δ ξ δ

ξ ξ δ

δ ξ ξ ( )14 16 21 19 12 18

2 2 2 2 218 14 1 19 1 14 17 22 14 14 20 17 22 1 14 14 14 13 22 18 21 20

2 221 14 13 22 18 21 20 22 8 22 14 15 23 14 15 21 22 22 8 23 22 24

224

,

, ( ) , ( ) ,( ), , ( ) ,(

− −

= − = − − = + − + +

= − − = − = − + − −

=

a g a g g g

l a l a b b a g l b b a g a g b g a gl a g b g a g l a b a b l a b g b g a g b gl

δ δ ξ ξ δ

ξ ξ

ξ 14 21 22 22 23 24) ,− +a g b g g g

( )( ) ( )

2 2 2 211 23 1 12 1 19 22 12 23 14 13 22 19 1 11 12 14 23 13 21 15 15

2 2 214 22 11 21 15 12 13 2 15 15 15 1 14 19 17 23

2 2 216 1 12 17 14 19 14 16 18 23 21 19 17

, ( ) , ( ) ,

, ( ),

( ) ,

l b l b b g b g l b b g g g b g a g b

l b g a b g g a g b l a b b b

l g b a b a g g b a g l

δ δ ξ δ

ξ ξ δ

δ ξ ξ

= = − + = − + − + + −

= − − − + = +

= − − + + = ( )14 16 21 19 12 18

2 2 2 2 218 14 1 19 1 14 17 22 14 14 20 17 22 1 14 14 14 13 22 18 21 20

2 221 14 13 22 18 21 20 22 8 22 14 15 23 14 15 21 22 22 8 23 22 24

,

, ( ) , ( ) ,

( ), , ( ) ,

a g a g g g

l a l a b b a g l b b a g a g b g a g

l a g b g a g l a b a b l a b g b g a g b g

δ δ ξ ξ δ

ξ ξ

− −

= − = − − = + − + +

= − − = − = − + − −2 2

11 15 20 16 19 12 23 24 13 14 20 16 18 14 16 18 1 15 14 19 15 18

16 15 20 16 19 18 8 20 16 17 19 8 19 15 17 20 8 18 14 17 21 14 19 15 182

22 8 19 15 17 23 22 2

, , , , ,, , , , ,, (

g b b b b g b b g b b b b g b b g b b b bg b b b b g a b b b g a b b b g a b b b g b b b bg a b b b g b b

ξ δ

ξ

= − = − = − = − = −= − = − = − = − = −

= − = − 4 24 8 18 14 17), ,g a b b b= −2 2

11 15 20 16 19 12 23 24 13 14 20 16 18 14 16 18 1 15 14 19 15 18

16 15 20 16 19 18 8 20 16 17 19 8 19 15 17 20 8 18 14 17 21 14 19 15 182

22 8 19 15 17 23 22 2

, , , , ,

, , , , ,

, (

g b b b b g b b g b b b b g b b g b b b b

g b b b b g a b b b g a b b b g a b b b g b b b b

g a b b b g b b

ξ δ

ξ

= − = − = − = − = −

= − = − = − = − = −

= − = − 4 24 8 18 14 17), ,g a b b b= −

( ) ( ) ( ) ( )( ) ( )

( )

2 2 1 111 12 1 13 3 14 9 1 15 10

2 2 2 2 2 2 2 2 216 1 7 17 11 0 18 0 19 13 1

2 2 220 12 0 21 14 22 15 1 23 16

1 , 1 , 1 , 1 , 1 ,

, ( ), 1 , ( ),

( ), 2 , 1 , 1

b c b i c b a i c b a i c b a i c

b c a b a i c c b c i c b a i c c

b a i c c b a b a i c b a i

ξ ξ τ ξ τ ξ τ ξ τ

ξ δ ξ ετ ξ ξ τ ξ ξ γ ξ

ξ ετ ξ ξ ξ τ

= − = − = − = − = −

= − − = + = − − = − +

= − + = = − − = −( )1 024, (1 ),c b i c i cξ τ ξ ετ ξ= −

( ) ( ) ( ) ( )( ) ( )

( )

2 2 1 111 12 1 13 3 14 9 1 15 10

2 2 2 2 2 2 2 2 216 1 7 17 11 0 18 0 19 13 1

2 2 220 12 0 21 14 22 15 1 23 16

1 , 1 , 1 , 1 , 1 ,

, ( ), 1 , ( ),

( ), 2 , 1 , 1

b c b i c b a i c b a i c b a i c

b c a b a i c c b c i c b a i c c

b a i c c b a b a i c b a i

ξ ξ τ ξ τ ξ τ ξ τ

ξ δ ξ ετ ξ ξ τ ξ ξ γ ξ

ξ ετ ξ ξ ξ τ

= − = − = − = − = −

= − − = + = − − =− +

= − + = =− − = −( )1 024, (1 ).c b i c i cξ τ ξ ετ ξ= −

100 Rajneesh Kumar