note on surface wave in fibre-reinforced medium

Mathematical Journal of Interdisciplinary Sciences

Vol. 3, No. 1, September 2014

pp. 23–35

DOI: 10.15415/mjis.2014.31003

Note on Surface Wave in Fibre-Reinforced Medium

INDer SINgh gupta

Department of Mathematics, J.V.M.g.r.r (p.g) College, Charkhi Dadri-127306, India

email: [email protected]

received: October 14, 2013| revised: March 26, 2014| accepted: april 16, 2014

published online: September 20, 2014 the author(s) 2014. this article is published with open access at www.chitkara.edu.in/publications

Abstract: the wave velocity equations are derived for the particular cases of surface waves –rayleigh, and Love types using direct method along with numerical results. results are compared with results of classical theory when reinforced parameters tends to zero.

Keywords: Decoupling, fibre-reinforced media, transversely isotropic media, surface waves, half space.

1. Introduction

after the pioneer work of rayleigh [9], many investigators have studied the problem extensively under different conditions. they have contributed in a wide range towards its application in various fields e.g. Seismology, geophysics, acoustics, telecommunications and environmental sciences etc. a good amount of literature is to be found in the standard books of Ben-menahem and Singh [3], Bullen [4], ewing et al. [6] and Love [7].

In most of the previous investigations the effect of reinforcement has been neglected. this concept was introduced by Belfield et al. [2]. the characteristic property of a reinforced composite is that its components act together as a single anisotropic unit as long as they remain in the elastic conditions. the reinforcement introduces anisotropy in the medium which becomes transversely isotropic. For wave propagation in an isotropic homogeneous medium; the introduction of displacement potentials leads to the decoupling of p, SV & Sh motions. the decoupling cannot be achieved for wave propagation in transversely isotropic media (see, e.g., rahman & ahmad [8]). Stresses produced in a fibre-reinforced half-space due to moving load was discussed by Chattopadhyay and Venkateswarlu [5].

gupta, IS

24

For studying the propagation of surface waves in fiber -reinforced anisotropic elastic media, Sengupta & Nath [10] used the method of potential to decouple the p & SV motions which is not justified. therefore Singh [11] pointed out that the results of Sengupta and Nath (2001) are in error regarding rayleigh and Stoneley waves. Same mistake was done by abd-alla et al. [1] to investigate the surface waves in fibre-reniforced anisotropic elastic medium under gravity field.

In this paper, the author studies the propagation of surface waves in fibre-reinforced anisotropic elastic solid media for rayleigh waves and Love waves using direct method . Numerical results for rayleigh and Love surface waves are derived for specific materials and graphs are also drawn to show the effect of reinforced parameters on the speed of surface waves.

2. Basic Equations

the constitutive equations for a fibre-reinforced linearly elastic anisotropic medium with respect to the reinforcement direction

a are (Belfield et al. [2])

σ λ δ µ α δ

µ µij kk ij t ij k m km ij kk i j

L t i k k j

e e a a e e a a

a a e

= + + +

+ − +

2

2

( )

( )( aa a e a a e a aj k ki k m km i j) ,+β (2.1)

where σij are components of stress; e

ij are components of strain; λ,μ

t

are elastic constants; α, β, (μL-μ

t) are reinforcement parameters and

a a a a= + + =( , , ),1 2 3 12

22

32 1a a a . We choose the fibre-direction as

a= ( , , )1 0 0 .

the strain components can be expressed in terms of the displacements ui as

eu

x

u

xiji

j

j

i

=∂∂+∂

∂

. (2.2)

equation (2.1) then yields

σ

σ

11 111

112

2

212

3

3

22 121

122

2

2

=∂∂+

∂∂+

∂∂

=∂∂+

∂∂+

au

xa

u

xa

u

x

au

xa

u

x

,

aau

x

au

xa

u

xa

u

x

uL

233

3

33 121

123

2

222

3

3

21 121

∂∂

=∂∂+

∂∂+

∂∂

= =∂

,

,σ

σ σ µ∂∂+∂∂

x

u

x2

2

1

,

Note on Surface Wave in Fibre-

reinforced Medium

25

σ σ µ

σ σ µ

13 311

3

3

1

23 232

3

3

= =∂∂+∂∂

= =∂∂+∂∂

L

t

u

x

u

x

u

x

u

x

,

22

,

(2.3)

where a11

= λ + 2α+4μL–2μ

t+β, a

12=λ+α,

A AT22 232= + =λ µ λ, . (2.4)

the equations of motion without body forces are

∂

∂=∂∂( )

σρij

xj

iu

tij=1,2,3

2

2, (2.5)

where ρ is the density of the elastic medium. using equation (2.3) in equation (2.5)

au

xa

u

x xa

u

x x

u

xL L L11

2

1

1

2 12

2

2

1 2

12

2

3

1 3

2

1∂

∂+ +

∂

∂ ∂+ +

∂

∂ ∂+

∂

∂( )( )µ µ µ

22

2

2

1

3

2

2

1

2

22

2

2

2

2 12

2

1

1 2

23

+∂

∂=∂

∂

∂

∂+ +

∂

∂ ∂+

µ ρ

µ

L

L

u

x

u

t

au

xa

u

x xa

,

( ) ++∂

∂ ∂+

∂

∂+

∂

∂=∂

∂

∂

∂

( )µ µ µ ρt L L

u

x x

u

x

u

x

u

t

au

x

2

3

1 2

2

2

1

2

2

2

3

2

2

2

2

22

2

3

3

,

22 23

2

2

2 3

12

2

1

1 3

2

3

1

2

2

3+ +∂

∂ ∂+ +

∂

∂ ∂+

∂

∂+

∂( )( )a

u

x xa

u

x x

u

x

ut L L tµ µ µ µ

∂∂=∂

∂x

u

t2

2

2

3

2ρ ,

(2.6)

For plane strain deformation in the x1x

2-plane,put

∂∂= =

xu

330 0, in first two

equations of (2.6) we get;

a

u

xB

v

x yB

u

y

u

t

av

yB

u

x yB

11

2

2 2

2

1

2

2

2

2

22

2

2 2

2

∂∂+

∂∂ ∂

+∂∂=∂∂

∂∂+

∂∂ ∂

+

ρ ,

11

2

2

2

2

∂∂=∂∂

v

x

v

tρ .

(2.7)

the third equation of (2.6) is identically satisfied and here we have used the notation x x,x y,u u,u v,B B aL L1 2 1 2 1 2 12= = = = = = +µ µ, .

3. Propagation of Rayleigh Waves

We consider a fibre-reniforced elastic half-space with free surface as x-z plane y-axis pointing into the half-space such that -−∞< <∞ ≤ <∞x z y, ;0 . For rayleigh waves of circular frequency ω,wave number k and phase velocity C

r

gupta, IS

26

propagating in the x-direction through the fibre-reinforced anisotropic half-space. We may assume the solution of equation (2.7) as

u=ue e

u=Ve e

kqy ik C t x)

kqy ik C t x)

r

r

−

−

( -

( -

,

, (3.1)

where u and V are the amplitude factors, i= −1 and q is assumed to be real and positive.putting the values of the displacements in equation (2.7), we get

ρ

ρ

C a B q u iqB V

iqB u C B a q V

r

r

211 1

22

22

1 222

0

0

− +( ) + =

+ − +( ) =,

. (3.2)

the values of q may be obtained from

ρ

ρC -a +B q iqB

iqB C -B +a q=0,r

211 1

22

2 r2

1 222

which on simplification, we get

− + − − + ±

− + −

( ){ }

( )

A B C A A B B

A B C A A

R

R

22 1

2

11 22 1

2

2

2

1

2

2

222 1

2

11 22

ρ

ρq q,

−− + − − −{ } ( )( )

B B B A C A C B

B A

R R1

2

2

2 2

1 22

2

11

2

1

1 22

4

2

ρ ρ½

.

(3.3)therefore the solution (3.1) can be written as

u (u e u e e

v (V e V e

q ky q ky ik C t x

q ky q ky

r= +

= +

− − − −

− −

11 12

11 12

1 2

1 2

) ,

)

( )

ee ik C t xr− −( ) , (3.4)

u u11 12,( ) and V V11 12,( ) are not independent but are connected by equation (3.2) for q q= 1 and q2 . taking second member of equaton (3.2), we get

u

Vm

u

Vm11

111

12

112= =, , (3.5)

where m iM m iM1 1 2 2= =, (3.6)

MC B a q

q Br

1

21 22 1

2

1 2

=− +ρ

, (3.7)

and


reinforced Medium

27

MC B a q

q Br

2

21 22 2

2

2 2

=− +ρ

, (3.8)

where q1 and q

2 are real and positive quantities defined in equation (3.3).

therefore the dispalcements in equation (3.4) can be written as

u m V e m V e e

v V e V e

q ky q ky ik C t x

q ky q k

r= +( )= +

− − −( )

− −

1 11 2 12

11 12

1 2

1 2

,

yy ik C t xe r( ) −( ). (3.9)

4. Boundary Conditions

the displacements in equation (3.9) must satisfy the boundary conditions,

σ σ21 22 0= = =,at y 0, (4.1)

where σ21

and σ22

are defined in equation (2.3). For plane deformation in the

x1x

2-plane, ∂

∂= =

xu

330 0, . equation (2.3) then yields

σ µ

σ

21

22 12 22

=∂∂+∂∂

=∂∂+

∂∂

L

u

y

v

x

au

xa

v

y

,

,

(4.2)

where we have used the notation x1 = x, x

2 = y, u

1= u and u

2 = v.

From equations (3.9) and (4.2), we get

m q i V m q i V

a m i a q V m a i a q V

1 1 11 2 2 12

12 1 22 1 11 2 12 2 2

0+( ) + +( ) =

+( ) + +( ),

112 0= . (4.3)

eliminating V11

and V12

from equation (4.3) ,we get

m q i m q i

a m i a q m a i a q1 1 2 2

12 1 22 1 2 12 2 2

0+ ++ +

= . (4.4)

On simplification , equation (4.4) becomes

m q i m a i a q m q i a m i a q1 1 2 12 22 2 2 2 12 1 22 1 0+( ) +( )− +( ) +( )

= . (4.5)

this is the velocity equation for rayleigh-waves in a fibre-reinforced elastic medium.

gupta, IS

28

5. Particular Case

If we put α β= = 0 and µ µ µL t= = , the elastic cofficients become

A A B

A A B11 12 1

22 23 2

2

2

= + = =

= + = = +

λ µ λ µ

λ µ λ λ µ

, , ,

, , . (5.1)

From equations (3.7) and (3.8) we get

M q Mq1 1 2

2

1= =, , then

qC

Cq

C

CR

S

R

p12

2

2 22

2

21 1= −

= −

, , (5.2)

where

C Cp S2 22=+

=λ µρ

µρ

and . (5.3)

λ, μ are Lame’s constants. using equations (5.1)-(5.3), the velocity equation (4.5) reduces to

C

C

C

C

C

CR

S

R

p

R

S

2

2

2 2

2

22

22 4 1 1−

= −

−

2

. (5.4)

which is the rayleigh wave velocity equation in isotropic media.

6. Propagation of Love Waves

Consider an isotropic layer of fibre-reinforced elastic medium of thickness h over a homogeneous anisotropic fibre-reinforced elastic half space. the surface of contact in plane x

2 = 0 and x

2 axis is directed vertically downwards. the wave

is assumed to propagate along the x1 – direction. an antiplane strain equation

of motion for Love waves, in the upper layer is obtained from third equation of

(2.6) by putting u1=u

2 = 0, u

3 = w,x

1 = x,x

2 = y,x

3 = z and

∂∂=

xs

0 , as

µ µ ρL T

w

x

w

y

w

t´

´´

´´

´∂∂+

∂∂=∂∂

2

2

2

2

2

2, (6.1)

where ρ’ is density, w’ is displacement along z-axis and µ µL t´ ´,( ) are reinforced

anisotropic elastic parameters for the layer.the equation of motion for the half space is

µ µ ρL T

w

x

w

y

w

t

∂∂+

∂∂=∂∂

2

2

2

2

2

2, (6.2)


reinforced Medium

29

where µ µ ρL t and, are the corresponding quantities in the half space.

7. Solution of the Equation of Motion

We assume a solution ofequation (6.1) for plane harmonic waves propagating in the upper layer along the x –direction in the form

w W ex

Cp y,i

L

´ ´ ´= −

−ω t (7.1)

where W’ is the amplitude factor, ω and CL are circular frequency and phase

velocity respectively in the layer.Substituting equation (7.1) in equation (6.1) we get

ωµ µ ρ ω

2

22 2

Cp

LL T´ ´ ´ ´− = . (7.2)

From equation(7.2), p’ can be written as p ip´ ´=± 1 where

p K1

2 2´

´

´

´

´

´ ´=

−=

−=

ω ρ µµ

ρ µµ

ωC

C CK

CL

L L

T

L L

T L

, . (7.3)

therefore equation (7.1) can be written as

w x y t W e W ei t

x

Cip y i t

x

CL L´ ´ ´´

, ,( )= +−

− −

1 2

1ω ω

+ip y1

´

. (7.4)

Similarly the solution of equation (6.2) for half-space is

w x y t W e W ei t

x

Cip y i t

x

CL L, ,( )= +−

− −

1 2

1ω ω +ip y1

, (7.5)

where

p KCL L

T12 2

2

=−

ρ µµ

. (7.6)

For the existence of Love type surface waves it is necessary that in equation (7.5), w x,y,t asy( )→ →∞0 .therefore p

1 hould be purely imaginary.

Let p ip i KCL L

t1 2

2

= =−

µ ρµ

. (7.7)

equation (7.5) then shows that

W1 0= . (7.8)

gupta, IS

30

8. Boundary Conditions

the following boundary conditions must be satisfied

I. at y h,

II. w w at y

III. at y

σ

σ σ

23

3

23 23

0

0

0

´

´

´

= =−

= =

= =

,

.

(8.1)

σ23 is defined inequation (2.3) and can be written as (putting,

u u u w x x,x y,x z2 1 3 1 2 30= = = = = =, , and ∂∂=

x3

0 )

σ µ23 =∂∂T

w

y;

Similarlyσ µ23

´ ´´

=∂∂T

w

y. (8.2)

From equations (7.4) – (8.2), we get

W e W e

W W W

p W W p W

ip H ip H

T T

1 2

1 2 2

1 1 2 1 2

1 1 0

0

´ ´

´ ´

´ ´ ´ ´

´ ´

− =

+ − =

−( )+

− ,

,

µ µ == 0.

(8.3)

eliminating W W1 2´ ´, ,and W2 from equation (8.3). On simplification, we obtain

i p Hp

pT

T

tan( ) .11

1

´´ ´

=µµ

(8.4)

Substituting for p1’ and p

1 from equations (7.4) and (7.8) in equation (8.4), we

get

tan/

ρ µµ

µ ρ

ρ

´ ´

´ ´

CKH

C

CL L

T

L L

L

2 1 2 2

2

−

=

−( )½−−

µµ

µµ

L

T

T

T´

´

´

½

. . (8.5)

this is the frequency equation for Love waves in an anisotropic fibre-reinforced elastic medium.

9 . Particular Case

If we put μL = μ

T = μ, in equation (8.5), we get

tan’

/C

CKH

C

CL

S

L

S2 1 2

2

2

1

1

−

= ⋅

−

µρ

½

½C

CL

S

2

2’

, (9.1)


reinforced Medium

31

where C Cs s´

´

´2 2= =µρ

µρ

and ,

which is the classical frequency equation of Love waves in a homogeneous elastic layer over a isotropic half space.

10. Numerical Results and Discussion

to study the effect of reinforcement on the velocity of rayleigh waves we use the following numerical values for the physical constant (Chattopadhyay, [5])

λ µ

µ α

= × = ×

= × =− ×

− −

−

5 65 10 5 65 10

2 46 10 1 28 10

9 2 9 2

9 2 9

. , . ,

. , .

Nm Nm

Nm

L

T NNm

Nm kg m

−

− −= × =

2

9 2 3220 90 10 7800

,

. .β ρ

We obtain the numerical values of constants from equation (2.4) using above values as

A Nm A Nm

A Nm A

119 2

129 2

229 2

23

241 71 10 4 387 10

10 57 10

= × = ×

= ×

− −

−

. , . ,

. , == ×

= × = ×

−

− −

5 65 10

5 66 10 10 03 10

9 2

19 2

29 2

. ,

. , . .

Nm

B Nm B Nm

Making use of equations (3.3) and (3.5)-(3.8) in equation (4.5), the velocity equation for rayleigh waves can be modified as

AC

B

C

B

B A

B A

A

AR R

22

2

1

2

1

2 12

1 22

12

2

1ρ ρ−

+ −

22

11

1

2

1

2

1

12

22

2

1

−

− −

×

A

BB

C

B

A

A

C

B

R

R

ρ

ρ−−

+ − −

=

A

A

C

B

B

B

A

BR12

22

2

1

2

1

11

1

2

1 0ρ

.

(10.1)

using these values in equation (10.1), we obtain the value of wave velocity as ρµCR

L

2

(dimensionless) = 42. 38574 .

ρµCR

L

2

6 51043= .

C m sR2 5307 56832 10= ×. / ,

C km sR = 5 545884. / ,

gupta, IS

32

the values calculated by Sengupta and Nath [10] for velocity of rayleigh waves are not coincide of our results however same values of physical quantities are used for numerical calculations. It is clear the above numerical results of rayleigh waves velocity in a fibre reinforced elastic medium is considerably higher than the rayleigh waves velocity in isotropic medium. In this reference terrestrial rayleigh waves is about 3 km /s(Love [7], p.160).

Figure 1: Variation of dimensionless quantity

ρµCR

L

2

with fibre reinforced parameter β.

Figure 1 Shows increase in β shows increase in ρµCR

L

2

others parameter remains

constant. there is linear relationship between dimensionless quantity ρµCR

L

2

and fibre reinforced parameter β in m–2.

Figure 2 shows the variation of dimensionless quantity ρµCR

L

2 with elastic

parameter μL in Nm–2 exponentially .

equation (8.5) shows that CL is dependent on the particular value of k and

not a fixed constant. If k is small CLL2 →µρ

and while if k is large CLL2 →µρ

’

’.


reinforced Medium

33

It is clear for the existence of Love waves p1 should be imaginary and p1

’ is real satisfied if µ ρ µ ρL L LC’ ’/ /( )< <( )2 depends upon the reinforced parameters µ µL Land ’ .the upper limit of C

L for the existence of Love waves in different

elastic media is given as.Fibre reinforced medium µ ρL km s/ . /= 0 851

aluminum µ ρL km s/ . /= 4 5513

Steel µ ρ/ . /= 3 145km s

gold(24 c) µ ρ/ . /= 4 6382km s

Copper(cast) µ ρ/ . /= 2 947km s

Figure 2: Variation of dimensionless quantity ρµCR

L

2

with elastic parameter μL.

gupta, IS

34

Copper (ore) µ ρ/ . /= 4 238km s

Silicon µ ρ/ . /= 5 649km s

It is clear from the numerical values of upper limit of CL for the existence

of propagation of Love waves in the fibre- reinforced medium decreases in compression with the upper limits in the other elastic medium.

Figure 3 shows that upper limit of CL in km/s for the existence of Love

waves in the medium increases as μL increases, density remains constant .

Figure 3: Variation of upper limit of CL with elastic constant μ

L.

11. Conclusion

We conclude that the rayleigh waves velocity in a fibre-reinforced elastic medium is higher than the rayleigh wave velocity in isotropic media. the upper limit for the existence of Love waves in fibre-reinforced medium is less than other elastic medium .hence the speed of rayleigh waves and Love waves are affected by reinforced parameters. In the absence of reinforced parameters the velocity equation of each surface waves coincide with the classical results for isotropic elastic medium.

12. Acknowledgment

the author wants to express his gratitude to the reviewer for his valuable comments and suggestions for improving this paper. the author also shows sense of gratitude for mentor prof. Sarva Jit Singh, emeritus Scientist for their


reinforced Medium

35

abiding inspiration and guidance . the author also thanks the ugC, New Delhi for providing financial support through minor research project.

References

[1] abd-alla,a.M., Nofal1, t.a., abo-Dahab,S.M., and al-Mullise,a,(2013),Surface waves propagation in fibre-reinforced anisotropic elastic media subjected to gravity field, International Journal of Physical Sciences, 8(14): 574-584.

[2] Belfield, a J., rogers t.g. and Spencer, a.J.M., (1983), Stress in elastic plates reinforced by fibres lying in concentric circles. J. Mech. Phys. Solids, 31:25-54. http://dx.doi.org/10.1016/0022-5096(83)90018-2

[3] Ben-Menahem, a and Singh,S.J., (1981), Seismic waves and sources, Springer verlag, New York.

[4] Bullen, K.e., (1965), an introduction to the theory of seismology, Cambridge university press,London,85-89.

[5] Chattopadhyay. a andVenkateshwarlu, r.L.K,(1998), Stress produced in a fibre-reinforced half –space due to a moving load, Bull. Cal. Math. Soc., 90: 337-342.

[6] ewing, W.M., Jardetzky,W.S and press, F,(1957), elastic waves in layered media, Mcgrow hill., London, 348-350.

[7] Love, a.e. h.,(1911), Some problems of geodynamics, Cambridge university press, Cambridge.

[8] rahman, a and ahmad, F,(1998),representation of the displacement in terms of scalar function for use in transversely isotropic materials, Acoust. Soc. Am., 104: 3675-3676. http://dx.doi.org/10.1121/1.423974

[9] rayleigh, L.,(1885),On wave propagation along the plane surface of an elastic solid, Proc. London, Math. Soc., 17:4-11.

http://dx.doi.org/10.1112/plms/s1-17.1.4

[10] Sengupta,p.r and Nath, S, (2001) Surface waves in fibre-reinforced anisotropic elastic media, Sadhana 26: 363-370. http://dx.doi.org/10.1007/BF02703405

[11] Singh, S.J,(2002),Comments on “Surface waves in fibre-reinforced anisotropic elastic media” by Sengupta and Nath [Sadhana 26: 363–370 (2001)], Sadhana, 27(3): 405-407. http://dx.doi.org/10.1007/BF02703661

[12] Stoneley,r.(1924), the elastic waves at the surface of separation of two solids, Proc. R. Soc., London, A106, 416-420. http://dx.doi.org/10.1098/rspa.1924.0079

note on surface wave in fibre-reinforced medium

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