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Surface Wave Propagation in a Generalized Thermoelastic

Material with Voids

Baljeet Singh1, Raj Pal2

1Department of Mathematics, Post Graduate Government College, Chandigarh, India 2Department of Mathematics, Government Post Graduate College, Hisar, India

E-mail: bsinghgc11@gmail.com

Received February 21,2011; revised March 6, 2011; accepted March 10, 2011

Abstract

In the present paper, the propagation of surface wave in a generalized thermoelastic solid with voids is con-sidered. The governing equations are solved to obtain the general solution in x-z plane. The appropriate boundary conditions at an interface between two dissimilar half-spaces are satisfied by appropriate particular solutions to obtain the frequency equation of the surface wave in the medium. Some special cases are also discussed.

Keywords:Thermoelasticity, Surface Waves, Boundary Conditions, Voids

1. Introduction

Theory of linear elastic materials with voids is an impor-tant generalization of the classical theory of elasticity. The theory is used for investigating various types of geo- logical and biological materials for which classical theo- ry of elasticity is not adequate. The theory of linear elas-tic materials with voids deals the materials with a distri-bution of small pores or voids, where the volume of void is included among the kinematics variables. The theory reduces to the classical theory in the limiting case of vo-lume of void tending to zero. Non-linear theory of elastic materials with voids was developed by Nunziato and Cowin [1]. Cowin and Nunziato [2] developed a theory of linear elastic materials with voids to study mathe-matically the mechanical behavior of porous solids. Puri and Cowin [3] studied the behavior of plane waves in a linear elastic material with voids. Iesan [4] developed the linear theory of thermoelastic materials with voids.

Dhaliwal and Wang [5] formulated the heat-flux de-pendent thermoelasticity theory for an elastic material with voids. This theory includes the heat-flux among the constitutive variables and assumes an evolution equation for the heat-flux. Ciarletta and Scalia [6] developed a nonlinear theory of non-simple thermoelastic materials with voids. Ciarletta and Scarpetta [7] studied some re-sults on thermoelasticity for dielectric materials with vo- ids. Marin [8-9] studied uniqueness and domain of influ- ence results in thermoelastic bodies with voids. Chirita

and Scalia [10] studied the spatial and temporal behavior in linear thermoelasticity of materials with voids. A theo- ry of thermoelastic materials with voids and without en-ergy dissipation is developed by Cicco and Diaco [11]. Ciarletta et al. [12] presented a model for acoustic wave propagation in a porous material which also allows for propagation of a thermal displacement wave. Singh [13] studied the wave propagation in a homogeneous, iso-tropic generalized thermoelastic half space with voids in context of Lord and Shulman theory. Ciarletta et al. [14] studied the linear theory of micropolar thermoelasticity for materials with voids. Recently, Aoudai [15] derived the equations of the linear theory of thermoelastic diffu-sion in porous media based on the concept of volume fraction.

Lord Rayleigh [16] investigated the surface wave along the plane surface of an elastic solid. Chandrasekharaiah [17] discussed the effect of voids on Rayleigh waves in an elastic solid with voids and on Rayleigh-lamb waves in homogeneous elastic plate with voids. Many research-ers have studied the surface waves in various theories of thermoelasticity. For example, Chadwick and Windle [18], Agarwal [19], Sharma and Singh [20], Mayer [21], Semerak [22], Chandrasekharaiah [23], Sharma et al. [24], Sharma and Kaur [25] and many others.

(2)

these equations are solved for general solutions. In Sec-tions 4 and 5, the particular soluSec-tions are obtained and applied at required boundary conditions to obtain the fre- quency equation of surface waves in thermoelastic mate-rial with voids. In Section 6, some limiting cases of the problem are discussed. In last section, some concluding remarks are given.

2. Governing Equations

Following, Iesan [4] and Lord and Shulman [26], the constitutive equations and field equations in terms of the displacement, volume fraction and temperature, for ho-mogeneous isotropic generalized thermo-elastic material with voids in the absence of the body forces, heat sources and extrinsic equilibrated body forces are

2 δ

ij eij ij ekk b

         (1)

0

i i

q  q  K ,i (2)

,

i

h   i (3) kk

e a m

      (4)

kk

g  be     m (5)

0 i i

T q,

  (6)

, , , ,

i jj j ij i i i

u u b u

          (7)

0 0 , 0 ,

0 0 ,

E k k k k

ii

c T u

mT K

   

    

     

  

 

u

(8)

,ii buk k, m

        (9)

where  , are Lame’s constants. 0 0 is the temperature of the medium in its natural state as-sumed to be such that

,

T T T

  

0 1, T is the absolute

temperatu ij

T

 

re, are the components of the stress tensor,

, ,

1

, 2

ij i j j i i

euu u are the components of the displa- cement vector, η is the entropy per unit mass, Κ is the coefficient of thermal conductivity, 0 is the thermal relaxation time.  , ,b are void material parameters, m is thermo-void coefficient,  

32

t, t

δ

  is the coefficient of linear thermal expansion, ij is Kroneck-er delta qi are the components of heat flux vector, hi are the components of equilibrated stress tensor,

,

 is change in volume fraction field, g is the intrinsic equili-brated body force and a is thermal constant.

The Equations (7) to (9) are written in x-z plane as

2 1

2 3 2

2 2

1 2

2

, u

u u

x z

2 3

2 1 2 3

2 2

2 3 2

2 u u u

z x

z x

u b

z z t

    

 

  

   

 

 

 

  

  

(11)

2 2

3 1

0 0

2 2

2 2 ,

E

u u

c T mT

t x t z t

K

x z

      

t



  

   

   

 

 

(12)

2 2

3 1

2 2

2

2 ,

u u b

x z

x z

m

t

 

      

 

 

      

(13)

where 1 0

t   

 .

Now the displacement components 1 and 3 are

written in terms of potential function

u u

 and  as

1 , 3

u u

x z z x

   

   

   

    (14)

Using Equation (14) into Equations (10) to (13), we have

2 2 2

2 2 2 ,

x z t

 

   

  

 

 (15)

2

x22 2z2 b 2t2

  

         

  

  , (16)

2 2

* 2

1

2 2

t t x z t

 

        

    

   

 , (17)

2 2 2 2 2

2 2 2 2 2 ,

            

 

x zbx zm

 

 

(18) t 

where,

* 0 0

1

, ,

E E E

T m

K

c c

   T

c

  

   .

Here Equation (15) is uncoupled, whereas Equations (16), (17) and (18) are coupled in ,  and .

3

1 2

x z

u b

x x t

    

 

 

   

  

 

  

  

. Solutions

o solve the Equations (16) to (18), we consider

(10)

T

(3)

 

 

 

2 2 2

1 1

1 0,

D k    z  z  b z (2

  0)

 

 

 

2 2 * 2 2

1 0,

i k D z k D i

i z

   

 

 

     

  

(2 z

1)

 

 

 

2 2

2 2 2 0

b k D z m z

k D z

  

    

 

      (22)

where,

2 2

2 2

1 2

1

d

, ,

2 2

d , 2

c D

z b

b kc

   

 

   

 

 

and

The non-trivial solutions of Equations (20) to (22) ex-is

(23) where

t if

6 4 2

0 1 2 3 0

L DL DL DL

 

*

* 2 2 2 * 2

1 1 1

* *

2 2 * 2 * *

2

* 4 * 2 * 2 2 2

3 2

1 3

1 1 1 1

2 2 * 2

1 1 1

3

1 2

1 2

2

L k i bb k

i

L k k i

k b k i k i

i im k

i i i b ib m

bb k bb k ib b

L k

 

     

    

      

0

L

        

    

        

   

     

  

      

    

  

   

  



2 2 * 4 * 2 * 2 2

2 3

1

2 2 2

1 1 1

2 2 4 * 2

1 1 1

1 k b b

i k i i im i k

i k i k i bk

ib m k b b k ibb k

     

4

       

       

  

   

    

  

  

Let be the roots of the auxiliary Equation

(2 1 2 3

, ,

m m m hen the g

3), t eneral solutions ,  and  are written as



 

3

1 2

3 2

1 2 3 4

5 6

e e e e

e e e

m z

m z m z m z

ik x ct m z m z

A A A A

A A

 

  

 

1

(24)

1

e

(25)

1

e

(26)

where

 

3

1 2

3 2

1 1 2 2 3 3 1 4

2 5 3 6

e e e

e e e

m z

m z m z m z

ik x ct m z m z

A A A A

A A

   

 

 

    

 

 

3

1 2

3 2

1 1 2 2 3 3 1 4

2 5 3 6

e e e

e e e

m z

m z m z m z

ik x ct m z m z

A A A A

A A

   

 

 

    

 

2 2 2 2 2

, 1, 2,3

i i i

i

i

m k b G G

i G mb

  

 

 

 

2 2

2

1

2 ,

i b k mi b i i

1, 2,3

   

 

 

2 2 2

i i

G   km 

The general solution  of Equation (15) is written as

* 4 * 4

 

0e 1e e

ik x ct m z m z

B B

, (27)

where, 2 2 2

4 1 .

m k c

 

 

4. Formulation of the Problem

[image:3.595.48.544.48.739.2] [image:3.595.304.537.341.710.2]

Let us consider two semi-infinite half-spaces of ther-ic solid with voids welded contact as shown in

igure 1. The particular solutions in half-spaces M and

moelast in

F

M are as follows: For medium M,

1 2 3

 

1e 2e 3e e

ik x ct m z m z m z

A A A

  (28)

1 2 3

 

1 1e 2 2e 3 3e e

ik x ct m z

m z m z

A A A

      

    (29)

2 3

 

2 2e 3 3e e

ik x ct m z m z

A A

    

1

1 1e

m z A

 

   (30)

where,

  4

1e

m z ik x ct B

   (31)

2 2 2 1

1 2 3

0

2 2 2 2 2 2 2 2 2 3

1 2 3

L m m   2

1 2 2 3 3 1

0 0

,

,

L

m m m

L

L

m m m m m m m

L L

   

  

 

: , , , , , M       a b m Thermo-elastic solid half-space with voids

 

: , , , , , M    a b m Thermo-elastic solid half-space with voids

(4)

Similarly, for medium M

(32)

(33)

(34)

(35) Here, the symbols with primes in the following sec-tions correspond to medium M.

5. Boundary Conditions

The boundary conditions at the interface z = 0 are e

continuity placement

co perature and volume

frac-tional field, i.e.

1 2 3

 

1e 2e 3e e

ik x ct m z m z m z

A A A

 

1 2 3

 

1 1e 2 2e 3 3e e

ik x ct m z m z m z

A A A

      

      

   

1 2 3

 

1 1e 2 2e 3 3e e

ik x ct m z m z m z

A A A

                    4 1e

m z ik x ct B

  

th of force stress components, dis

mponents, heat flux, tem

1 2

3 3 1 1

, ,

, ,

, , ,

zz zz zx zx

z z z z

u u u u

                               .    where, (36) 3 3

1 , 1 ,

z x z x

u u

x z x z

                        0 1 2 0 2 , , 1 , 1 zx b e u u ikc K K ikc                                

The particular solutions (29) to (35) satisf boundary conditions if

2 ,

2

zz zz kk

zz zz kk

e e b

e e                     2 ,

zx ezx zx

   

 

 

 

 

y the above

11 21

a

12 13 14 15 16 17 18

22 23 24 25 26 27 28

31 32 33 34 35 36 37 38

41 42 43 44 45 46 47 48

51 52 53 54 55 56 57 58

61 62 63 64 65 66 67 68

71 72 73 74 75 76 77 78

81 82 83 84 85 86 87 88

0

a a a a a a a a

a a a a a a a

a a a a a a a a

a a a a a a a a

a a a a a a a a

a a a a a a a a

a a a a a a a a

a a a a a a a a

 (37)

where

2 2

11 2 1 1 1

a k m b

2 2

12 2 2 2 2

a k m b

                     

2

3 3 3

14 4

2 2

15 1 1 1

2 2

16 2 2 2

2 2

17 3 3 3

18 4 21 1

22 2 23 3

2 2

24 4 25

, 2 ,

2 ,

2 ,

2 ,

2 , 2 ,

2 , 2 ,

,

m b

a i km

a k m b

a k m b

a k m b

a i km a i km

a i km a i km

a k m a

                                                                                  

1 26 2

2 2

3 28 4

31 1 1 32 2 2 33 3 3 34

35 1 1 1 36 2 1 2 37 3 1 3

38

41 1 1 42 2 2 43 3 3 44

45 1 1 2 46 2 2 2 47 3 3

2 , 2

2 , .

, , , 0,

, , ,

0.

, , , 0,

, ,

i km a i km

i km a k m

a m a m a m a

a m a m a m

a

a m a m a m a

a m a m a m

                                                          

   2

48

51 1 52 2 53 3 54

55 1 56 2 57 3 58

61 62 63 64 4

65 66 67 68 4

71 1 72 2 73 3 74

75 1 76 2 77 3 78

81 1 82

, 0. , , , , , , , . , , , , , , , .

, , , 0,

, , , 0.

,

a

a m a m a m a ik

a m a m a m a ik

a ik a ik a ik a m

a ik a ik a ik a m

a a a a

a a a a

a a                                               

  2 83 3 84

85 1 86 2 87 3 88

, , 0,

, , , 0.

a a

a a a a

                2 13 2

a  k   

27

a  

,

The Equation (37) gives the frequency equation for surface wave in a generalized thermo-elastic medium with voids.

6. Limiting Cases

1) If we neglect the void parameters, then the Equation (37) reduces to

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

a a a a a a

a a a a a a

a a a a a a

a a a a a a

a a a a a a

a a a a a a

                                    (3 ) w 8 here,

2 2 11

a   k   m

1 1

2 2

2 2 13 4

2

14 1 1

2 ,

2 , 2

,

m a i km

m                          

12 2 2 a k

a k  

(5)

,

,

The Equation (38) gives the frequency equation for surface waves in a generalized thermo-elastic medium.

2) If we neglect thermal parameters, then the Equation (37) reduces to

2 2

22 2 23 4

2 2

24 1 25 2 26 4

31 1 1 32 2 2 33

34 1 1 1 35 1 2 2 36

41 1 4

2 , 2

2 , 2 ,

, , 0,

, , 0,

,

a k m a i km

a im k a im k a k m

a m a m a

a m a m a

a m a

     

  

 

   

              

             

     

         

  2 2 43

44 1 45 2 46

51 52 53 4,

54 55 56 4,

61 1 62 2 63

64 1 65 2 66

, ,

, , .

, ,

, ,

, , 0,

, , 0.

m a ik

a m a m a ik

a ik a ik a m ik a ik a m

a a a

a a a

 

 

  

       

     

  

   

     

         

15 2 2 16 4

2 2

21 2 1 , 2 , ,

a   im k a    im k a    km

a 

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

0

b b b b b b

b b b b b b

b b b b b b

b b b b b b

b b b b b b

b b b b b b

 (39)

where

,

,

.

The Equation (39) gives the frequency equation for surface wave in an elastic medium with voids.

3) If we neglect void and thermal parameters, then the Equation (37) reduces to

2 2

11 1 1

2 2

12 3 3 13 4

2 2

22 3 23 4

2 2

24 1 25 3 26 4

2 ,

2 , 2

, 2 ,

2 , 2 ,

b k m b

b k m b b i km

k b im k b k m

b im k b im k b k m

b

   

    

  

    

      

     

     

31 1 1 32 3 3 33

34 2 1 1 35 2 3 3 36

41 1 42 3 43

44 1 45 3 46

52 53 4,

54 55 56 4

61 1 62 3 63

64 1 65 3 66

, , 0,

, , 0.

, , ,

, , .

, ,

, ,

, , 0,

, , 0.

m b m b

b m b m b

b m b m b ik

b m b m b ik

b ik b ik b m b ik b ik b m

b b b

b b b

 

   

 

 

  

   

  

   

 

  

  

    

  

 

    

14 1 1

2 2

2 ,

2 , 2 ,

b k m b

b k m b b i km

   

    

     

   

       

     

15 3 3 16 4

2 2

21 2 1

b   im      

51

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

0

c c c c

c c c c

c c c c

c c c c

 (40)

where

2 2

11 1 12 4

2 2

13 1 14 4

2 2

21 1 22 4

2 2

23 1 24 4

31 1 32

33 1 34

41 42 4

43 44 4

2 , 2

2 , 2

2 , ,

2 , .

, ,

,

, ,

, .

c k m c i km

c k m c i km

c im k c k m

c im k c k m

c m c ik

c m c ik c ik c m

c ik c m

   

   

 

 

      ,

,

     

   

    

   

   

  

 

 

  

The Equation (40) gives the frequency equation for surface wave in an elastic medium

4) If we remove the upper half-space, then the Equa-tion (37) reduces to

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

0

d d d d

d d d d

d d d d

d d d d

 (41)

where

2 2

11 1 1 1

2 2

12 2 2 2

2 2

13 3 3 3

14 4 21 1 22

2 2

23 3 24 4

31 1 1 32 2 2 33 3 3 34

41 1 1 42 2 2 43 3 3 44

2 ,

2 ,

2 ,

2 2 , 2

2 , .

, , ,

, , ,

d k m b

d k m b

d k m b

d i km d i km d i km

d i km d k m

d m d m d m d

d m d m d m d

    

    

    

  

 

  

  

     

     

     

     

    

   

    .

2,

0. 0

The Equation (41) gives the frequency equation of a Ra

The frequency equation of surface waves in generalized thermoelastic material with voids is ob

quency equation of Rayleigh surface wav

limiting case. The theoretical results indicate that the speed of surface wave depends on various material

pa-yleigh surface wave in a half-space of a generalized thermo-elastic material with void.

7. Conclusions

(6)

s. Present analytical solutions can be used to numerically the speed of surface wave for a particular material modeled as thermoelastic material with voids.

8. References

[1] J. W. Nunziato and S. C. Cowin, “A Nonlinear Theor Elastic Materials with Voids,” Archive for Rational chanics and Analysis, Vol. 72, No. 2, 1979, pp. 175-201. [2] S. C. Cowin and J. W. Nunziato, “Linear Elastic

Materi-als with Voids,” Journal of Elasticity, Vol. 13, No. 2 1983, 125-147. doi:10.1007/BF00041230

[3]

oelastic Materials

echanica, Vol. 60, No. 1-2, 1986, pp.

67- r-nal of Applied Mathematics and Mechanics, Vol. 75, No.

7-714.

] M. Marin, “A Uniqueness Result for Body with Voids in

and Temporal

in Elastic Materials

ation in a Generalized

Thermoe-s with VoidThermoe-s,” European

Zeitschrift für Angewandte Mathe-

gs of the London

ol. 25, No.

aries,”

ty of America, Vol. 280,

Surface

tion of Surface

of Thermal Relaxation on

h, “Thermoelastic Rayleigh Waves

d

9

99-309.

rameter find [14] M. Ciarletta, M. Svanadze and L. Buonanno, “Plane

Waves and Vibrations in the Theory of Micropolar Thermoelasticity for Material

y

M of e- matik und Physik, Vol. 61, No. 2, 2010, pp. 357-379.

, Mat

P. Puri and S. C. Cowin, “Plane Waves in Linear Elastic Materials with Voids,” Journal of Elasticity, Vol. 15, No. 2, 1985, pp. 167-183. doi:10.1007/BF00041991

] D. Iesan, “A Theory of Therm

[4 with 2, 1987, pp. 205-211.

[18] P. Chadwick and D. W. Windle, “Propagation of Ray-leigh Waves along Isothermal and Insulated Bound

Proceedings of the Royal Socie

Voids,” Acta M

89. doi:10.1007/BF01302942

[5] R. S. Dhaliwal and J. Wang, “A Heat-Flux Dependent Theory of Thermoelasticity with Voids,” Acta Mechanica, Vol. 110, No. 1-4, 1993, pp. 33-39.

[6] M. Ciarletta and A. Scalia, “On the Nonlinear Theory of Nonsimple Thermoelastic Materials with Voids,” Journal of Applied Mathematics and Mechanics, Vol. 73, No. 2, 1993, pp. 67-75.

[7] M. Ciarletta and E. Scarpetta, “Some Results on Ther-Moelasticity for Dielectric Materials with Voids,” Jou

Wave

9, 1995, pp. 70 [8

Linear Thermoelasticity,” Rendiconti di Matematica, Vol. 17, No. 1, 1997, pp. 103-113.

[9] M. Marin, “On the Domain of Influence in Thermoelas-ticity of Bodies with Voids,” Archiv der Mathematik, Vol. 33, No. 4, 1997, pp. 301-308.

[10] S. Chirita and A. Scalia, “On the Spatial

Rayl

Behavior in Linear Thermoelasticity of Materials with Voids,” Journal of Thermal Stresses, Vol. 24, No. 5, 2001, pp. 433-455.

[11] S. D. Cicco and M. Diaco, “A Theory of Thermoelastic Materials with Voids without Energy Dissipation,” Jour-nal of Thermal Stresses, Vol. 25, No. 5, 2002, pp. 493- 503. doi:10.1080/01495730252890203

[12] M. Ciarletta, B. Straughan and V. Zampoli, “Thermo- poroacoustic Acceleration Waves

with Voids without Energy Dissipation,” International Journal of Engineering Science, Vol. 45, No. 9, 2007, pp. 736-743. doi:10.1016/j.ijengsci.2007.05.001

[13] B. Singh, “Wave Propag

lastic Material with Voids,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 698-709. doi:10.1016/j.amc.2006.11.123

Journal of Mechanics-A-Solids, Vol. 28, No. 4, 2009, pp. 897-903. doi:10.1016/j.euromechsol.2009.03.008 [15] M. Aoudai, “A Theory of Thermoelastic Diffusion Ma-

terial with Voids,”

doi:10.1007/s00033-009-0016-0

[16] L. Rayleigh, “On Waves Propagating along the Plane Surface of an Elastic Solid,” Proceedin

hematical Society, Vol. 17, No. 1, 1885, pp. 4-11. doi:10.1112/plms/s1-17.1.4

[17] D. S. Chandrasekharaiah, “Effects of Surface Stresses and Voids on Rayleigh Waves in an Elastic Solid,” In-ternational Journal of Engineering Science, V

No. 1380, 1964, pp. 47-71. doi:10.1098/rspa.1964.0130 [19] V. K. Agarwal, “On Surface Waves in Generalized

Thermoelasticity,” Journal of Elasticity, Vol. 8, No. 2, 1978, pp. 171-177. doi:10.1007/BF00052480

[20] J. N. Sharma and H. Singh, “Thermoelastic s in a Transversely Isotropic Half-Space with Thermal Relaxation,” Indian Journal of Pure and Applied Mathematics, Vol. 16, No. 10, 1985, pp. 1202-1219. [21] A. P. Mayer, “Thermoelastic Attenua

Acoustic Waves,” International Journal of Engineering Science, Vol. 28, No. 10, 1990, pp. 1073-1082.

doi:10.1016/0020-7225(90)90135-6 [22] F. V. Semerak, “The Effect

eigh Surface Waves in a Thermoelastic Medium,”

Journal of Mathematical Sciences, Vol. 88, No. 3, 1997, pp. 396-399.

[23] D. S. Chandrasekharaia

without Energy Dissipation,” Mechanics Research Com-munication, Vol. 24, No. 1, 1997, pp. 93-102.

doi:10.1016/S0093-6413(96)00083-3

[24] J. N. Sharma, D. Singh and R. Kumar, “Generalize Thermoelastic Waves in Homogeneous Isotropic Plates,”

Journal of the Acoustical Society of America, Vol. 108, No. 2, 2000, pp. 848-851. doi:10.1121/1.42961

[25] J. N. Sharma and D. Kaur, “Rayleigh Waves in Rotating Thermoelastic Solids with Voids” International Journal of Applied Mathematics and Mechanics, Vol. 6, No. 3, 2010, pp. 43-61.

[26] H. W. Lord and Y. Shulman, “A Generalized Dynamical Theory of Thermoelasticity,” Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, 1967, pp. 2

Figure

Figure 1. Geometry of the problem.

References

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