REFLECTION AND TRANSMISSION OF PLANE …In the present paper, the reflection and refraction phenomenon of plane waves at a loosely boundary interface between an elastic solid half-space

$Page 1: REFLECTION AND TRANSMISSION OF PLANE …In the present paper, the reflection and refraction phenomenon of plane waves at a loosely boundary interface between an elastic solid half-space$
REFLECTION AND TRANSMISSION OF PLANE WAVES

AT THE LOOSELY BONDED INTERFACE OF AN ELASTIC SOLID

HALF-SPACE AND A MICROSTRETCH THERMOELASTIC

DIFFUSION SOLID HALF-SPACE

Rajneesh Kumar1*

, Sanjeev Ahuja2**

, S.K. Garg3***

1Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India

2University Institute of Engineering & Technology, Kurukshetra University, Kurukshetra, Haryana, India

3Department of Mathematics, Deen Bandhu Chotu Ram Uni. of Science & Technology., Sonipat, Haryana, India

*e-mail: [email protected],

**e-mail: [email protected],

***e-mail: [email protected]

Abstract. The problem of reflection and refraction phenomenon due to plane waves incident

obliquely at the loosely bonded interface between an elastic solid half-space and the

microstretch thermoelastic diffusion solid half-space is discussed. It is assumed that the

interface behaves like a dislocation, which preserves the continuity of traction while allowing

a finite amount of slip. The amplitude ratios of various reflected and transmitted waves to that

of incident wave are obtained. These amplitude ratios are further used to find the expressions

of energy ratios of various reflected and refracted waves to that of incident wave. Numerical

results have been obtained for a particular model to study the variation of energy ratios with

respect to angle of incidence. The effect of relaxation times and loosely bonding parameters

are shown with energy ratios for a specific model. The law of conservation of energy at the

loosely boundary interface is verified. Some special cases of interest have been deduced from

the present investigation.

1. Introduction

Due to the unidentified nature of the layers beneath the earth’s surface and for the purpose of

theoretical investigations, various appropriate mathematical models are to be considered. In

the problem of reflection and refraction of plane waves at the boundaries between two half-

spaces, it is generally considered that the half-spaces are in welded contact at the interface.

Therefore, it is reasonable approximation to consider the presence of a thin layer of viscous

liquid at the interface and two half-spaces are caused to be loosely bounded. Murty [1, 2]

studied reflection and refraction of elastic waves through a loosely bonded interface, i.e., an

interface at which the two media are not perfectly bonded, by assuming the continuity of the

traction and that a finite amount of slip can take place at the interface.

Eringen [3] developed the theory of a micropolar elastic solid with stretch in which he

has taken the effect of axial stretch during the rotation of molecules. Eringen [4, 5] also

developed the theory of a thermo-microstretch elastic solid and fluids, in which he included

microstructural expansions and contractions. Microstretch continuum is a model for Bravais

lattice with a basis on the atomic level, and a two-phase dipolar solid with a core on the

Materials Physics and Mechanics 21 (2014) 147-167 Received: July 2, 2014

© 2014, Institute of Problems of Mechanical Engineering

mailto:[email protected]

macroscopic level. For example, composite materials reinforced with chopped elastic fibres,

porous media whose pores 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].

The linear theory of micropolar thermoelasticity was developed by extending the theory

of micropolar continua to include the thermal effect by Eringen [7] and Nowacki [8]. Now it

is known as the micropolar coupled thermoelasticity. Dost and Tabarrok [9] have presented

the generalized micropolar thermal elasticity by using Green-Lindsay theory.

Chanderashekhariah [10] developed a heat-flux dependent generalized theory of micropolar

thermoelasticity.

The thermodiffusion in elastic solids is due to coupling of fields of temperature, mass

diffusion and that of strain in addition to heat and mass exchange with the environment.

Nowacki [11-14] developed the theory of thermoelastic diffusion by using coupled

thermoelastic model. Dudziak and Kowalski [15] and Olesiak and Pyryev [16], respectively,

discussed the theory of thermodiffusion and coupled quasi-stationary problems of thermal

diffusion for an elastic layer. Gawinecki and Szymaniec [17] proved a theorem about global

existence of the solution for a nonlinear parabolic thermoelastic diffusion problem.

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. Recently, Kumar

and Kansal [19] developed the basic equation of anisotropic thermoelastic diffusion based

upon Green-Lindsay model.

Many researches have investigated the problems of reflection and transmission of plane

waves at the interface of two dissimilar media. Notable among them are Borejko et al. [20],

Wu and Lundberg [21], Sinha and Elsibai [22], Singh [23], Kumar and Singh [24] discussed a

problem on reflection and transmission of elastic waves at the loosely bonded interface

between an elastic solid half-space and a micropolar elastic solid half-space, Kumar and

Panchal [25] studied the problem of reflection and transmission of plane waves incident on

the interface between the loosely bonded elastic solid and micropolar cubic crystal half-

spaces. In recent times, Sharma and Bhargava [26] investigated the propagation of

thermoelastic plane waves at an imperfect boundary of thermal conducting viscous

liquid/generalized thermoelastic solid.

In the present paper, the reflection and refraction phenomenon of plane waves at a

loosely boundary interface between an elastic solid half-space and a microstretch

thermoelastic diffusion solid half-space has been analyzed. In microstretch thermoelastic

diffusion solid medium, potential functions are introduced to represent four coupled

longitudinal waves and two coupled transverse waves. The amplitude ratios of various

reflected and refracted waves to that of incident wave are derived. The expressions for the

energy ratios of reflected and transmitted waves are obtained. The graphical representation is

given for these energy ratios for different values of the bonding parameter.

2. Basic equations

Following Sherief et al. [18], Eringen [6], Kumar and 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 K

21

1 1 2 21 1 ,

uT C

t t t

(2.1)

148 Rajneesh Kumar, Sanjeev Ahuja, S.K. Garg

2

2. 2 ,K u K j

t

(2.2)

2 *

2 * 1 * 00 1 1 2 1 0 2

.2

jT T C C u

t

(2.3)

* 2 * *

1 0 0 1 0 0 0 0 11 . 1 1 ,K T T u T C T aT C Ct t t

(2.4)

* 0 1

2 , 2 , 1 , ,0,kk ii ii ii ii

D D Da T T C C Db C C (2.5)

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 Ct t

(2.6)

*

, , , 0 , ,ij r r ij i j j i mji mm b (2.7)

* *

0 , 0 , ,i i ijm j mb (2.8)

where , , 1, , , , , , ,o o oK b are material constants; is the mass density;

1 2 3, ,u u u u is the displacement vector and 1 2 3, , is the microrotation vector; *

is the scalar microstretch function; T and 0T are the small temperature increment and the

reference temperature of the body chosen such that 0 1T T ; 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, b are,

respectively, coefficients describing the measure of thermodiffusion and of mass diffusion

effects; 1 13 2 tK ; 2 13 2 cK ; 1 23 2 tK ;

2 23 2 cK ; 1 2,t t are coefficients of linear thermal expansion; and 1 2,c c 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;

*

i is the microstress tensor; 1, ,2ij i j j ie u u are components of infinitesimal strain; kke is

the dilatation; ij is the Kronecker delta; 0 1, are diffusion relaxation times with 1 0 0

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

0 1 1 0 for

Coupled Thermoelasitc (CT) model, 1

1 0, 1, 1 0 for Lord-Shulman (L-S)

model and 0, 0

1 , where 0 0 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.

The basic equation of motion and stress-strain relation in a homogeneous isotropic

elastic solid are written as

2

2

2.

ee e e e e e u

u ut

, (2.9)

,2= ij

e

kk

ee

ij

ee

ij eet (2.10)

149Reflection and transmission of plane waves at the loosely bonded interface...

where e , e are Lame’s constants, eu is the displacement vector, e is density, e

kke is the

dilatation, e

ijt and 1, ,2

e e e

ij i j j ie u u are components of stress and strain tensors respectively.

3. Formulation of the problem

We consider an isotropic elastic solid half-space (M1) lying over a homogeneous isotropic,

microstretch generalized thermoelastic diffusion solid half-space (M2). The origin of the

cartesian coordinate system 1 2 3( , , )x x x is taken at any point on the plane surface (interface)

and 3x -axis point vertically downwards into the microstretch thermoelastic diffusion solid

half-space. The elastic solid half-space (M1) occupies the region 3 0x and the region 3 0x

is occupied by the microstretch themoelastic diffusion solid half-space (M2) as shown in

Fig. 1. We consider plane waves in the 1 3x x -plane with wave front parallel to the 2x -axis. For

two- dimensional problem, we take

1 3( ,0, )u u u , 2(0, ,0) , 1 3( ,0, )e e eu u u . (3.1)

Fig. 1. Geometry of the problem.

We define the following dimensionless quantities

*

' '

1 3 1 3

1

, , ,x x x xc

*' ' 11 3 1 3

1

, , ,o

cu u u u

T

'

1

,ij

ij

o

tt

T

'

1

,

e

ije

ij

o

tt

T ' ,

o

TT

T ' * ,t t

' * ,o o 0' * 0 , ' '

*

11 3 1 3

1

, , ,e e e e

o

cu u u u

T

' *

1 1 , 1' * 1, '

2* *1

1

,o

c

T

'* *

*

1 1

,ii

oc T

2' 12 2

1

,o

c

T

' 2

2

1

CC

c

,

*'

1 1

ij ij

o

m mc T

, (3.2)

where * 2

* 1

*,

C c

K

2

1

2 Kc

, * is the characteristic frequency of the medium,


Making use of (3.1) in equations (2.1)-(2.5), with the aid of dimensionless quantities defined

by (3.2) and after suppressing the primes, we obtain

2*

2 2 2 * * 1 * 12 11 1 3 2 2

1 3 1 1 1

1 ,t c

ue T Cu

x x x x x t

(3.3)

2*

2 2 2 * * 1 * 1 323 1 3 2 2

3 1 3 3 3

1 ,t c

ue T Cu

x x x x x t

(3.4)

22 31 2

1 2 2 3 2 2

3 1

,uu

x x t

(3.5)

2 *

2 2 * * * * 1 * 1

1 1 2 3 4 2,t ce T C

t

(3.6)

*2 * 0 * 0 0 * 0

1 2 3 ,e e t c

e T CT

t t t t

(3.7)

* 2 * 2 * * 1 2 0 * 1 2

1 4 2 3 0,t f c

Cq e q q T q C

t

(3.8)

where

1 2

1

,j c

2 *2,

K

j

3 *2

2,

K

j

*

1 2

1

,K

c

2* 12

1 0

,c

T

* 0

3 2

1

,c

2

2

1

,c

2* 0 11 * *

,T

K

* 1 0 1

2 * *,

T

K

4* 13 * *

2

c a

K

,

* 2* 11 4

1

,D

qc

** 22 2

1 1

,D a

qc

**

3 2

1

,Db

qc

** 2 24 4

1

,D

qc

*

1 *2

0

2,

j

* 0

2 *2

0

2,

j

2* 1 13 *2

0 1

2,

c

j

4* 2 14 *2

0 1 2 0

2,

c

j T

22 2

1 2

1

c

c ,

2 02

0

2,c

j

1

11 ,tt

1 11 ,ct

0 01 ,ft

0

01 ,tt

0

01 ,et

0

11 ,ct

1 3

1 3

,u u

ex x

2 2

2

2 2

1 3x x

.

We introduce the potential functions and through the relations

1 3

1 3 3 1

, ,u ux x x x

(3.9)

in the equations (3.3)-(3.8), we obtain

2 * * 1 * 1

3 2 ,t cT C (3.10)

2 2 *

1 21 , (3.11)

2 2

1 3 2 2 2 , (3.12)

2 2 * * * 2 * 1 * 1 *

1 1 2 3 4 ,t cT C (3.13)


2 0 * 2 * * 0 * 0

1 2 3 ,e t cT T C (3.14)

* 4 * 2 * * 1 2 * 1 2 0

1 4 2 3 0,t c fq q q T q C C (3.15)

For the propagation of plane waves in 1 3x x plane, we assume

* *

2 1 3 2, , , , , ( , , ) , , , , , i tT C x x t T C e , (3.16)

where is the angular frequency.

Substituting the values of *

2, , , , ,T C from equation (3.16) in the equations (3.10)-

(3.15), we obtain

2 2 * * '' * ''

3 2 0,t cT C (3.17)

2 2 2 *

1 21 0, (3.18)

2 2 2

2 1 3 2 0, (3.19)

* 2 * '' *

2 3 1 2 0,tT r r C (3.20)

* 10 2 10 2 * 10 * * 10

1 2 3 0,e t e cT C (3.21)

* 4 * 2 * * '' 2 10 * '' 2

1 4 2 3 0t f cq q q T q C , (3.22)

where 2 2 * 2

1 1 1 ,r *

2 4(1 ),r i 11

1(1 ),t i 11 1(1 ),c i

10

0(1 ),e i i 10

0(1 ),t i i 10 (1 ),c i i 10 0(1 )f i i .

The system of equations (3.17), (3.20)-(3.22) has a non-trivial solution if the determinant of

the coefficients *, , ,T

T C vanishes, which yields to the following polynomial

characteristic equation

8 6 4 2

1 2 3 4 0B B B B , (3.23)

where i iB A A for ( 1,2,3,4),i * *

1 14 14,A g a g * * 2 * * *

1 2 1 12 6 13 9 14 12,A g g a g a g a g * * 2 * * *

2 3 2 12 7 13 10 14 13,A g g a g a g a g

* * 2 * *

3 4 3 12 8 13 11,A g g a g a g * 2

4 4 ,A g

and * 2 * 2

1 1 46 2 23 46 24 43 1 32 46 45 34 42, ( ),g a g a a a a a a a a a *

4 45 22 33 23 32( ),g a a a a a * 2

3 33 22 46 24 42 23 32 46 45 34 42 43 24 32 22 34 1 32 45( ) ( ) ( ) ,g a a a a a a a a a a a a a a a a a a

* 2

6 1 31 46 41 34( ),g a a a a * 2

7 33 21 46 24 41 23 31 46 34 41 24 31 21 34 43 1 31 45( ) ( ) ,g a a a a a a a a a a a a a a a a a

*

8 45 23 31 21 33( ),g a a a a a *

9 24 41 21 46,g a a a a

*

10 21 32 46 45 34 42 22 31 46 34 41 24 31 42 32 41( ) ( ) ( )g a a a a a a a a a a a a a a a a ,

*

11 45 21 32 22 31( ),g a a a a a * 2

12 23 41 21 43 1 31 42 41 32( ),g a a a a a a a a 10

45 ,fa * 11

46 3 ca q ,

*

13 33 22 41 21 42 23 32 41 31 42 43 21 32 22 31( ) ( ) ( ),g a a a a a a a a a a a a a a a * 2

14 1 41,g a 2 2

11 ,a


*

21 2 ,a * 10

31 1 ,ea *

41 1 ,a q 11

12 ,ta * 11

22 3 ,ta 10

32 ,ta * 11

42 2 ,ta q *

13 3 ,a

* 2

23 1 ,a * 10

33 2 ,ea * 11

14 2 ,ca 24 2,a r * 10

34 3 ,ca * 2

43 4 ,a q 2

44 45 46( )a a a .

The general solution of equation (3.23) can be written as

4

1

i

i

, (3.24)

where the potentials , 1,2,3,4i i are solutions of wave equations, given by

22

20, 1,2,3,4i

i

iV

(3.25)

Here 2 , 1,2,3,4iV i are the velocities of four longitudinal waves in the descending order

and derived from the roots of the biquadratic equation in 2V , named as, longitudinal

displacement wave (LD), thermal wave (T), mass diffusion wave (MD) and longitudinal

microstretch wave (LM), given by

8 2 6 4 4 6 2 8

4 3 2 1 0B V B V B V B V . (3.26)

Using the equation (3.24) in the equations (3.17), (3.20)-(3.22), with the aid of equations

(3.16) and (3.25), we obtain the general solutions for , ,T C , and * as

4*

1 2 3

1

( , , , ) (1, , , )i i i iT C k k k , (3.27)

where * 6 * 4 2 * 2 4

1 6 7 8( ) ,d

i i ik g g V g V k * 8 * 6 2 * 4 4 2

2 14 12 13( ) ( ),d

i i i ik g g V g V V k * 6 * 4 2 * 2 4

3 9 10 11( ) ,d

i i ik g g V g V k * 6 * 4 2 * 2 4 * 6

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

i i ik g g V g V g V i .

The system of equations (3.18)-(3.19) has a non-trivial solution if the determinant of the

coefficients 2,T

vanishes, which yields the following polynomial characteristic equation

4 * 2 * 0A B , (3.28)

where

* 2 * 2 2 2

1 1 2 3 11 1 ,A * 2 2 2

3 11B .

The general solution of equation (3.28) can be written as

6

5

i

i

, (3.29)

where the potentials , 1,2i i are solutions of wave equations, given by

22

20, 5,6i

i

iV

. (3.30)

Here 2 , 5,6iV i are the velocities of two coupled transverse waves in the descending order

and derived from the root of quadratic equation in 2V , named as, transverse displacement

wave (CD I) and transverse microrotational wave (CD II), given by


* 4 * 2 2 4 0B V A V . (3.31)

Making use of equation (3.29) in the equations (3.18)-(3.19) with the aid of equations (3.16)

and (3.30), the general solutions for and 2 are obtained as

6

2 1

5

, 1, i i

i

n

, (3.32)

where

2

21 2 2 2

3 1

i

i

nV

for 5,6i .

Making the use of (3.1) in the equation (2.9), with the aid of dimensionless quantities (3.2)

and after suppressing the primes, yield

2 2 2

2

1 12 2

1 1 1

,e e e e

e eeu u

c x c

(3.33)

2 2 2

2

3 32 2

1 3 1

,e e e e

e eeu u

c x c

(3.34)

where 31

1 3

,ee

e uue

x x

and 2 ,e e e e

e e e are velocities of

longitudinal wave (P-wave) and transverse wave (SV-wave) corresponding to M1,

respectively.

The components of 1

eu and 3

eu are related by the potential functions as:

1

eu =1

e

x

-

3

e

x

, 3

eu =3

e

x

+

1

e

x

, (3.35)

where e and e satisfy the wave equations as

e2 =2

e, e2 =

2

e, (3.36)

and 1

e c , 1

e c .

4. Reflection and refraction We consider a plane harmonic wave (P or SV) propagating through the isotropic elastic solid

half-space and is incident at the interface 3 0x as shown in Fig. 1. Corresponding to each

incident wave, two homogeneous waves (P and SV) are reflected in an isotropic elastic solid

and six inhomogeneous waves (LD, T, MD, LM, CD I and CD II) are transmitted in isotropic

microstretch thermoelastic diffusion solid half-space.

In elastic solid half-space, the potential functions satisfying equation (3.36) can be

written as

1 0 3 0 1 1 3 1sin cos / sin cos /

0 1

i x x t i x x te e eA e A e

, (4.1)


1 0 3 0 1 2 3 2sin cos / sin cos /

0 1

i x x t i x x te e eB e B e

. (4.2)

The coefficients 0

eA ( 0

eB ), 1

eA and 1

eB are amplitudes of the incident P (or SV), reflected P and

reflected SV waves respectively.

Following Borcherdt [27], in a homogeneous isotropic microstretch thermoelastic

diffusion half-space, potential functions satisfying equations (3.25) and (3.30) can be written

as

4

( . )( . )*

1 2 3

1

, , , 1, , , iii P r tA r

i i i i

i

T C k k k B e e

, (4.3)

6

( . )( . )

2

5

, {1, } iii P r tA r

ip i

i

n B e e

, (4.4)

The coefficients iB , 1,2,3,4,5,6i are the amplitudes of refracted waves. The propagation

vector iP

, 1,2,3,4,5,6i and attenuation iA

factor ( 1,2,3,4,5,6i ) are given by

iP

= 1 3ˆ ˆ

R i Rx dV x , iA

= 1 3ˆ ˆ

I i Ix dV x , 1,2,3,4,5,6i (4.5)

where

IiRii dVidVdV = p.v.

122

2

2

iV

, 1,2,3,4,5,6i (4.6)

and IR i is the complex wave number. The subscripts R and I denote the real and

imaginary parts of the corresponding complex number and p.v. stands for the principal value

of the complex quantity derived from square root. ξR ≥ 0 ensures propagation in positive 1x -

direction. The complex wave number ξ in the microstretch thermoelastic diffusion medium is

given by

)sin(sin iiiii AiP

, 1,2,3,4,5,6i (4.7)

where i , 1,2,3,4,5,6i is the angle between the propagation and attenuation vector and

i , 1,2,3,4,5,6i is the angle of refraction in medium II.

5. Boundary conditions

Murti [2] proposed boundary conditions for a loosely bonded interface with the assumption

that there is a thin interface layer of viscous liquid between two half-spaces.

Following Murty [2] and Kumar et al. [25], the appropriate boundary conditions at the

loosely bonded interface 3 0x are defined by

Continuity of the normal stress component

33 33=et t . (5.1)1

Continuity of the tangential stress component

31 31=et t . (5.1)2

Continuity of the normal displacement component


3 3

eu u . (5.1)3

Vanishing of the tangential couple stress component

32 0.m (5.1)4

Vanishing of the microstress component *

3 0 . (5.1)5

Vanishing the temperature gradient

3

0T

x

. (5.1)6

Vanishing the mass concentration

3

0C

x

. (5.1)7

Proportionality between the shear stress and the slip

131

3

ut

x

. (5.1)8

Following Kumar et al. [25], boundary condition (5.1)8 may be rewritten as

*

031 1 1*

0sin 1

eVt ik u u

, (5.1)9

where be the coefficient of viscosity, 1 00 sin

kV

is the angle of the incidence, and 0V

is the velocity of the incident wave. Here, 0 0,V is replaced with ,j jV , where j takes

values from 1 to 2 depending upon the type of the incident wave. It is convenient to introduce

a variable * *0 1 such that, * 0 corresponds to a smooth, surface * 1

corresponds to a welded interface, and any intermediate value corresponds to a loosely

bonded interface between two half-spaces. This * may be considered as a bonding constant.

Making use of potentials given by equations (4.1)-(4.4), we find that the boundary conditions

are satisfied if and only if

R =

0

0sin

V

=

1sin=

2sin (5.2)

and

I = 0. (5.3)

where

0V =, for incident wave

, for incident wave

P

SV

, (5.4)


It means that waves are attenuating only in 3x -direction. From equation (4.7), it implies that if

0iA , then '

i i , 1,2,3,4,5,6i , that is, attenuated vectors for the six refracted waves

are directed along the 3x -axis.

Using equations (4.1)-(4.4) in the boundary conditions (5.1) and with the aid of

equations (3.9), (3.35), (5.2)-(5.4), we get a system of eight non-homogeneous equations

which can be written as

8

1

ij j i

j

d Z g

, (5.5)

where *

,ji

j jZ Z e

,jZ *,j 1,2,3,4,5,6,7,8j represent amplitude ratios and phase shift

of reflected P-, reflected SV-, refracted LD-, refracted T-, refracted MD-, refracted LM-,

refracted CD I -, refracted CD II - waves to that of amplitude of incident wave, respectively. 2

2

11 12 e eRd c

, 12 2 e R

dVd

, 5

17 (2 ) RdV

d K

,

22 '' ''

1 3 0 22 2

1 1 1 2 2

j j t j c jRj

dV k k kd c c

for ( 3,4,5,6)j

618 (2 ) R

dVd K

,

21 2 e RdV

d

,

2 2

22

e RdV

d

,

2 (2 )jR

j

dVd K

for ( 3,4,5,6)j ,

22

5527 2

( )pR

KndVd K

,

22

6628 2

( )pR

KndVd K

, 31

dVd

, 32

Rd

, 2

3

j

j

dVd

for

( 3,4,5,6)j , 37

Rd

,

38Rd

, 4 0jd for ( 1,2),j

4 2( 2)R

j o jd b k

for

( 3,4,5,6)j , 2

4 ( 2)

j

j j p

dVd n

for ( 7,8)j , 5 0jd for ( 1,2)j , 5 jd

2

2( 2)

j

O j

dVk

for ( 3,4,5,6)j , 6 0jd for ( 1,2,7,8)j , 5 ( 2)R

j o j pd b n

for ( 7,8)j ,

2

6 1( 2)

j

j j

dVd k

for ( 3,4,5,6)j , 7 0jd for ( 1,2,7,8)j , 2

7 3( 2)

j

j j

dVd k

for ( 3,4,5,6)j ,

81Rd

, 82

dVd

, *

8 1 *

11 2

jR Rj

dVd c K

for

( 3,4,5,6)j , 2 2*

5 587 1 *

1 R RdV dV

d c K

,

2 2*

6 688 1 *

1,R R

dV dVd c K


1/21/2 22

0

2 2

0

sin1 1RdV

V

,

1/21/2 22

0

2 2

0

sin1 1RdV

V

,

1/22

0

2

0

sin1. .

j

j

dVp v

V V

for

( 1,2,3,4,5,6)j .

Here p.v. is evaluated with restriction dVjI ≥ 0 to satisfy decay condition in the microstretch

thermoelastic diffusion medium. The coefficients , ( 1,2,3,4,5,6,7,8)ig for i on the right

side of the equation (5.5) are given by

(i) For incident P-wave

1( 1)i

i ig d for ( 1,2) ,i 1

1( 1)i

i ig d for ( 3,8),i 0ig for ( 4,5,6,7),i (5.6)

(ii) For incident SV-wave

1

2( 1)i

i ig d for ( 1,2) ,i 2i ig d for ( 3,8),i 0ig for ( 4,5,6,7).i (5.7)

Now we consider a surface element of unit area at the interface between two media. The

reason for this consideration is to calculate the partition of energy of the incident wave

among the reflected and refracted waves on the both sides of surface. Following Achenbach

[28], the energy flux across the surface element, that is, rate at which the energy is

communicated per unit area of the surface is represented as

P = lmlm ult , (5.8)

where lmt is the stress tensor, ml are the direction cosines of the unit normal l̂ outward

to the surface element and lu are the components of the particle velocity. The time average

of P over a period, denoted by < P >, represents the average energy transmission per unit

surface area per unit time. Thus, on the surface with normal along 3x -direction, the average

energy intensities of the waves in the elastic solid are given by

<e

P > = Re 13

et .Re ( 1

eu ) + Re 33

et .Re ( 3

eu ). (5.9)

Following Achenbach [28], for any two complex functions f and g, we have

< Re (f).Re (g) > = ½ Re (f g ). (5.10)

The expressions for energy ratios Ei, i = 1, 2 for the reflected P and reflected SV are given by

0

( 1,2)e

ii e

PE for i

P

, (5.11)

where 4 2

211 1 1Re(cos )

2

ee c

P Z

, 4 2

212 2 2Re(cos )

2

ee c

P Z

and

(i) For incident P- wave

4 2

1 00

cos

2

ee c

P

, (5.12)


(ii) For incident SV- wave

4 2

1 00

cos

2

ee c

P

, (5.13)

are the average energy intensities of the reflected P-, reflected SV-, incident P- and incident

SV-waves respectively. In equation (5.11) negative sign is taken because the direction of

reflected waves is opposite to that of incident wave.

For microstretch thermoelastic diffusion medium, the average energy intensities of the waves

on the surface with normal along 3x -direction, are given by

< *

ijP >=Re ( )

13

it Re( ( )

1

ju )+Re ( )

33

it Re( ( )

3

ju )+Re ( )

32

im Re( ( )

2

j )+Re * ( )

3

i Re(( )* j

).

(5.14)

The expressions for the energy ratios ( , 1,2,3,4,5,6)ijE for i j for the refracted waves are

given by

0

ij

ij e

PE

P

for ( , 1,2,3,4,5,6)i j , (5.15)

where

22

2

142

2 2 *

2 24 2'' ''1

2 1 3 0 21 2 2

*Re 2

2

R

j i jR Ri j

i t i c i

dVic

dVdV k k dVoi iP K Z Zij c

k k kc

for ( , 1, 2,3, 4)i j ,

22

24

2 22

*

4 21

*Re

2

(2 )

jipR

i j

p jpR R i

dVdV KniK

P Z ZijdV n n dVii

Kc

for ( , 5, 6)i j ,

2 2

221

* 255

74 2'' ''1

2 1 3 0 21 2 2

4

Re5 22

2

R

ipR i R R

i t i c i

dVic

dV dVP Z Zi i

k k kc

b n koK

c

for ( 1, 2, 3, 4)i ,

22

524

5 2 72

5

4 21

*Re

2*

(2 )

ipR

i i

ip pi R R i

dV Kn dViK

P Z Z

n ndV dVK

c

for ( 5, 6)i ,


2 2

2

1 24 * 2666 2 84 2'' ''

12 1 3 0 21 2 2

Re

2

*2

R i

o ipR i R Ri i

i t i c i

dVc

dV dVP Z Z

k k kc

b n kK

c

for ( 1, 2, 3, 4)i ,

2 2

624

6 2 82

6

4 21

*Re

2 *(2 )

ipR i

i i

ip pi R R i

KndV dVK

P Z Z

n ndV dVK

c

for ( 5, 6)i ,

2 2

552

4

5 7 22

255

4 21

*Re

2 *

2

pR R

j j

jj pR R

dV

P Z Z

dVdV

K nK

b n koK

c

for ( 1, 2, 3, 4)j ,

22

5

24

5 7 22

55 54 21

*Re

2 *

(2 )

j jpR

j j

jppR R

dV dVKnK

P Z Z

n ndV dVK

c

for ( 5, 6)j ,

2 2

6624

6 8 22

2664 21

*Re

2 *2

pR R

j j

jj pR R

dV

P Z Z

dVdV

KnK

b n koK

c

for ( 1, 2, 3, 4)j ,

22

4

6 8 22

66 64 21

*Re

2 *(2 )

j jRjp

j j

p jpR R

dV dVK Kn

P Z Z

n ndV dVK

c

for ( 5, 6)j . (5.16)

The diagonal entries of energy matrix Eij in equation (5.15) represent the energy ratios of the

waves, whereas sum of the non-diagonal entries of Eij give the share of interaction energy

among all the refracted waves in the medium and is given by

ERR =6 6

1 1

( )ij ii

i j

E E

. (5.17)

The energy ratios Ei, i = 1, 2, diagonal entries and sum of non-diagonal entries of energy

matrix Eij, that is, 11 22 33 44 55 66, , , , ,E E E E E E , and ERR yield the conservation of incident

energy across the interface, through the relation

1 2 11 22 33 44 55 66 1RRE E E E E E E E E . (5.18)


6. Particular cases

(i) Take 0 0, 0 and 0

1 in equation (5.17), along with (5.11) and (5.15)

yield the expressions of energy ratios for elastic/ microstretch thermoelastic

diffusion solid half-spaces with two relaxation times.

(ii) Using 1

1 0, 1 0 and 1 in equations (5.17), together with (5.11) and

(5.15) gives the corresponding results for elastic/ microstretch thermoelastic

diffusion solid half-spaces with one relaxation time.

(iii) On taking 0 1

0 1 1 0 in equations (5.17), (5.11) and (5.15) provide

the corresponding expression at elastic/ microstretch thermoelastic diffusion solid

half-spaces with Coupled Thermoelastic (CT) theory.

(iv) If we neglect the stretch effect, the boundary conditions will be reduced to the

(5.1)1-(5.1)4, (5.1)6-(5.1)8 and we obtain the energy ratios for elastic/ micropolar

thermoelastic diffusion solid half-spaces and these results will be identical, if we

directly solve the problem independently.

(v) If mass concentration effect is neglected, the boundary conditions in present

problem will be reduced to (5.1)1-(5.1)6, (5.1)8 and we obtain the energy ratios for

elastic/microstretch generalized thermoelastic solid half-spaces. The resulting

expressions obtained due to these changes will be similar, if we investigate the

problem directly.

7. Numerical results and discussion

The analysis is conducted for a magnesium crystal-like material. Following Eringen [29], the

values of micropolar parameters are 10 -29.4 10 Nm , 10 -24.0 10 Nm , 10 21.0 10 Nm ,K 3 31.74 10 Kgm , 19 20.2 10 m ,j 90.779 10 N . Thermal

and diffusion parameters are given by * 3 -1 11.04 10 JKg K ,C * 6 1 1 1K 1.7 10 Jm s K , 5 -1

t1 2.33 10 K , 5 -1

t2 2.48 10 K , 3

0 .298 10 K,T 1 0.01, 0 0.02,

4 3 -1

c1 2.65 10 m Kg , 4 3 -1

c2 2.83 10 m Kg , 4 2 2 -12.9 10 m s K ,a 5 -1 5 232 10 Kg m s ,b

1 0.04, 0 0.03, 8 30.85 10 Kgm sD and, the microstretch parameters are taken as 19 20.19 10 m ,oj 90.779 10 N,o

90.5 10 N,ob 10 20.5 10 Nm ,o

10 2

1 0.5 10 Nm . Following Bullen [30], the numerical data of granite for elastic medium

is given by 3 32.65 10 Kgm ,e 3 15.27 10 ms ,e 3 13.17 10 mse .

The MATLAB software 7.04 has been used to determine the values of energy ratios

, 1,2iE i and energy matrix , , 1,2,3,4,5,6ijE i j for some particular values of bonding

parameter * 0,0.75 &1 . The energy ratios of reflected and transmitted waves are

calculated in the previous section for different values of incident angle ( o ) ranging from 00

to 090 for fixed frequency 2 100 Hz . Corresponding to incident P wave, the

variation of energy ratios with respect to angle of incident have been plotted in Figures

(2)‐(10). Similarly, corresponding to SV waves, the variation of energy ratios with respect to

angle of incident have been plotted in Figures (11)‐(19). In all figures, the words LS and GL

symbolize the graphs of L-S and G-L theories in microstretch thermoelastic diffusion medium

for the smooth surface * 0 , loosely bonded interface * 0.75 and welded interface

* 1 and are represented by , , and , ,

respectively.


Incident P-wave.

Figures 2-10 depict the variation of energy ratios with the angle of incidence )( 0 for P

waves.

Figure 2 exhibits the variation of energy ratio 1E with the angle of incidence ( o ). For

each bonding parameter, the values of 1E for both cases LS and GL decrease monotonically

for the range 0 0

00 40 and consistently for 0 0

040 80 and then increase sharply at

080o , with difference in their magnitude values. Figure 3 depicts the variation of energy

ratio 2E with o and it shows nearly opposite behavior to that of 1E , the values of 2E increase

with the increase in o from 00 to 050 and then decrease monotonically for the range

0 0

080 90 for all the cases. Figure 4 depicts the variation of energy ratio 11E with o and

it shows that the behavior and trend of variation are similar for LS and GL cases, but the

values 11E for the GL case are higher than LS case for smooth surface, loosely bonded

interface and welded interface. Figure 5 exhibits the variation of energy ratio 22E with o and

it shows that the values of 22E are higher corresponding to * 0 for both LS and GL cases

which is also true for * 0.75 &1 , but, trends of the graph in case of loosely bonded and

welded interface are contrary to that of smooth surface.

Figure 6 depicts the variation of energy ratio 33E with o and it indicates the values of

33E in case of LS are very large as compared to the GL for the bonding parameter * 0.75 ,

while, the trend of the graphs for all other cases are similar which increase consistently within

the range 0 0

00 80 and decrease sharply near the end values of o considered. Figure 7

depicts the variation of energy ratio 44E with o and it shows that the values of 44E oscillates

for the whole range of o for all particular values of the bonding parameter and a significant

difference in the magnitude values for both cases LS and GL can be perceived. Figure 8

exhibits the variation of energy ratio 55E with o and it indicates the behavior of the graphs in

case of loose bonding is closely same to that of * 1 and nearly opposites to that of * 0

for both LS and GL cases. In case of smooth surface the value of 55E first decrease suddenly

within the range 0 0

00 15 , shows a consistent variation in the values and then decrease

further for 0 0

080 90 . Figure 9 shows the variation of 66E with o and it indicates that

the behavior of the curves for value of 66E is nearly same to that of the 33E as shown in

Fig. 6, however, the corresponding values are different in magnitude. For smooth surface, the

values of 66E in case of GL are higher as compared to LS case, but the trend of variation is

opposite from loose bonded and welded interface. Figure 10 shows the variation of interaction

energy ratio RRE with o and it indicates the values of RRE for all the cases show a similar

trend for different values of bonding parameter which decrease for 0 0

00 80 and suddenly

shows a sharp increase as o increases further. It is further noticed that the interaction energy

in both LS and GL cases attain the stationary values near the end of range for * 0.75 &1 .

Incident SV-wave

Figures 11-19 depict the variation of energy ratios with the angle of incidence )( 0 for

SV waves.


Fig. 2. Variation of energy ratio E1 w.r.t.

angle of incidence P-wave.











Figure 11 represents the variation of energy ratio 1E with 0 and it indicates that the

behavior of the curves for the values of 1E for both cases LS and GL are similar, however, the

corresponding values are different in magnitude.






Fig. 10. Variation of energy ratio ERR w.r.t.


The values of energy ratios increase monotonically for smaller values of 0 , whereas for

higher values of ,0 the values of 1E decrease and finally become dispersionless.

Fig. 11. Variation of energy ratio E1 w.r.t. angle

of incidence SV-wave.

Figure 12 shows the variation of energy ratio 2E with .0 and it indicates that the

behavior of the curves is opposite to that of the Fig. 11. In this figure the values for both cases


LS and GL decrease when 00 20 which sharply increase for

020 50 and for higher

values of 0 the values 2E become dispersionless. Figure 13 shows that for the whole range

of 0 the values of 11E for different cases LS and GL show an oscillatory behavior with

respect to different value of * . For the loosely bonded interface the values of 11E in case of

LS are higher as compared to GL and the same trend is noticed for smooth and welded

interface. Figure 14 exhibits the variation of energy ratio 22E with 0 and it indicates that the

values of 22E oscillates for smaller values of 0 although for higher values of 0 , the values

of 22E become constant. For higher values of 0 , it is noticed that the values of 22E in case of

LS remain more for smooth interface in comparison with other cases. Figure 15 depicts that

the values of energy ratio 33E oscillates for both LS and GL cases for all values of 0 and it is

noticed that, for higher values of 0 , the values of 33E in case of GL remain more for welded

interface in comparison with other cases. Figure 16 shows the variation of energy ratio 44E

with 0 and it is evident that the behavior and variation of 44E is similar as 33E whereas

magnitude values of 44E are different.


angle of incidence SV-wave.







Figures 17-18 show that the values of 55E and 66E depict an oscillatory behavior for all

values of 0 . A significant effect of loosely bonded interface can be seen on energy

ratio 66E for higher values of 0 . Figure 19 represents the variation of RRE with 0 and it

shows that the values of RRE increases for smaller values of 0 , whereas for higher values of


0 , the values of RRE slightly decreases. For smooth interface, it is noticed in GL case the

values of RRE remain more for higher values of 0 in comparison with other cases.







Fig. 19. Variation of energy ratio ERR w.r.t.


Conclusion

The reflection and transmission of elastic waves at the loosely bonded interface between an

elastic solid half-space and a microstretch thermoelastic diffusion solid half-space has been

investigated. The six waves in microstretch thermoelastic diffusion medium are identified and

explained through different wave equations in terms of displacement potentials. The energy

ratios of different reflected and transmitted waves to that of incident wave are computed

numerically and presented graphically with respect to the angle of incidence. An appreciable

effect of looseness of the boundary is observed on energy ratios of various reflected and

transmitted waves.

Also, from the numerical results we conclude that, for incident P-wave, the energy

ratios 11E , 22E , and 44E are more influenced due to the effect of loosely boundary and in case

of incident SV-wave, a significant effect of loosely boundary is noticed on the energy ratios

11E , 33E and 66E . Moreover, in majority of cases, the magnitude of energy ratios for G-L

theory is more as compared to L-S theory. The sum of all energy ratios of the reflected waves,

transmitted waves and interference between transmitted waves is verified to be always unity

which ensures the law of conservation of incident energy at the loosely bonded interface.


References

[1] G.S. Murty // Physics of the Earth and Planetary Interiors 11 (1975) 65.

[2] G.S. Murty // Geophysical Journal International 44(2) (1976) 389.

[3] A.C. Eringen, In: Micropolar Elastic Solids with Stretch, ed. by Prof. Dr. Mustafa Inan

Anisma, (Ari Kitapevi Matbaasi, Istanbul, Turkey, 1971), p. 1.

[4] A.C. Eringen // International Journal of Engineering Science 28 (1990) 1291.


[6] A.C. Eringen, Microcontinuum Field Theories I: Foundations and Solids (Springer-

Verlag, New York, 1999).

[7] A.C. Eringen, Foundations of Micropolar Thermoelasticity. Course of Lectures No. 23-

CISM Udine (Springer, 1970).

[8] W. Nowacki // Archives of Mechanics (Archiwum Mechaniki Stosowanej) 22 (1971) 3.

[9] S. Dost, B. Tabarrok // International Journal of Engineering Science 16 (1978) 173.

[10] D.S. Chandrasekharaiah // International Journal of Engineering Science 24 (1986) 1389.

[11] W. Nowacki // Academie Polonaise des Sciences, Bulletin, Serie des Sciences Techniques

22 (1974) 55.


22 (1974) 129.


22 (1974) 275.

[14] W. Nowacki // Engineering Fracture Mechanics 8 (1976) 261.

[15] W. Dudziak, S.J. Kowalski // International Journal of Heat and Mass Transfer 32 (1898)

2005.

[16] Z.S. Olesiak, Y.A. Pyryev // International Journal of Engineering Science 33 (1995)

773.

[17] J.A. Gawinecki, A. Szymaniec // Proceedings in Applied Mathematics and Mechanics 1

(2002) 446.

[18] H.H. Sherief, F.A. Hamza, H.A. Saleh // International Journal of Engineering Science 42

(2004) 591.

[19] R. Kumar, T. Kansal // International Journal of Solids and Structures 45 (2008) 5890.

[20] P. Borejko // Wave Motion 24 (1996) 371.

[21] C.M. Wu, B. Lundberg // Journal of Sound and Vibration 190 (1996) 645.

[22] S.B. Sinha, K.A. Elsibai // Journal of Thermal Stresses 20 (1997) 129.

[23] B. Singh // International Journal of Engineering Science 39 (2001) 583.

[24] R. Kumar, B. Singh // Indian Journal Pure and Applied Mathematics 28 (1997) 1133.

[25] R. Kumar, M. Panchal // Applied Mathematics and Mechanics 31 (2010) 605.

[26] K. Sharma, R. Bhargava // Afrika Matematika 25 (2014) 81.

[27] R.D. Borcherdt // Geophysical Journal of the Royal Astronomical Society 70 (1982) 621.

[28] J.D. Achenbach, Wave propagation in elastic solids (North-Holland Pub. Co.,

Amsterdam, 1973).


[30] K.E. Bullen, An introduction to the theory of seismology (Cambridge University Press,

England, 1963).


Documents

REFLECTION AND TRANSMISSION OF PLANE …In the present paper, the reflection and refraction phenomenon of plane waves at a loosely boundary interface between an elastic solid half-space