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: is_gupta@yahoo.com 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].
Mathematical Journal of Interdisciplinary Sciences Vol. 3, No. 1, September 2014 pp. 23–35
Gupta, IS
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 fibrereinforced 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])
σij = λe kk δij + 2µt e ij + α(a k a m e km δij + e kk a i a j ) +2(µL − µt )(a i a k e k j + a ja k e ki ) + βa k a m e km a i a j ,
(2.1)
where σij are components of stress; eij are components of strain; λ,μT are elastic constants; α, β, (μL-μT) are reinforcement parameters and a = (a1 , a 2 , a 3 ), a12 + a22 + a32 = 1 . We choose the fibre-direction as a = (1, 0, 0) . The strain components can be expressed in terms of the displacements ui as ∂u ∂u j . e ij = i + ∂x j ∂x i
Equation (2.1) then yields
σ11 = a11
∂u ∂u1 ∂u + a12 2 + a12 3 , ∂x1 ∂x 2 ∂x 3
σ22 = a12
∂u ∂u1 ∂u + a 22 2 + a 23 3 , ∂x 3 ∂x1 ∂x 2
σ33 = a12
∂u ∂u1 ∂u + a 23 2 + a 22 3 , ∂x1 ∂x 2 ∂x 3
∂u ∂u σ21 = σ12 = µL 1 + 2 , ∂x 2 ∂x1
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(2.2)
∂u ∂u σ13 = σ31 = µL 1 + 3 , ∂x 3 ∂x1
∂u ∂u σ23 = σ23 = µt 2 + 3 , ∂x 3 ∂x 2
(2.3)
where A11= λ + 2α+4μL–2μT+β, A12=λ+α, A22 = λ + 2µT , A23 = λ.
(2.4)
The equations of motion without body forces are ∂σij ∂2 u = ρ 2 i (ij=1,2,3), (2.5) ∂ xj ∂t where ρ is the density of the elastic medium. Using equation (2.3) in equation (2.5) 2 2 2 2 2 2 a ∂ u1 + ( a + µ ) ∂ u 2 + (a + µ ) ∂ u3 + µ ∂ u1 + µ ∂ u1 = ρ ∂ u1 , 11 L L L L 12 12 2 2 2 2
∂x1
a 22 a 22
∂ 2 u2 ∂x 22 ∂ 2 u3 ∂x 32
∂x1∂x 2
+ ( a12 + µL ) + ( a 23 + µt )
∂ 2 u1 ∂x1∂x 2 ∂ 2 u2 ∂x 2 ∂x 3
∂x1∂x 3
+ (a 23 + µt ) + (a12 + µL )
∂ 2 u3 ∂x1∂x 2 ∂ 2 u1 ∂x1∂x 3
∂x 2
+ µL + µL
∂ 2 u2 ∂x12 ∂ 2 u3 ∂x12
∂t
∂x 3
+ µL + µt
∂ 2 u2 ∂x 32 ∂ 2 u3 ∂x 22
=ρ =ρ
∂ 2 u2 ∂t 2 ∂ 2 u3 ∂t 2
, ,
(2.6)
∂ For plane strain deformation in the x1x2-plane,put = 0, u3 = 0 in first two ∂ x 3 equations of (2.6) we get;
a11
∂2 u ∂2v ∂2 u ∂2 u ρ , = B B + + 2 1 ∂x∂y ∂t 2 ∂y 2 ∂x 2
(2.7)
∂2 u ∂2v ∂2v ∂2v + B1 2 = ρ 2 . a 22 2 + B2 ∂x∂y ∂y ∂t ∂x The third equation of (2.6) is identically satisfied and here we have used the notation x1 = x,x 2 = y,u1 = u,u 2 = v,B1 = µL , B2 = a12 + µL . 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 < ∞;0 ≤ y < ∞ . For Rayleigh waves of circular frequency ω,wave number k and phase velocity CR
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Note on Surface Wave in FibreReinforced Medium
Gupta, IS
propagating in the x-direction through the fibre-reinforced anisotropic halfspace. We may assume the solution of equation (2.7) as u=ue−kqy e ik ( Cr t -x) ,
(3.1)
u=Ve−kqy e ik ( Cr t -x) , 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
(ρC
2 r
− a11 + B1q 2 ) u + iqB2 V = 0,
iqB2 u + (ρC2r − B1 + a 22 q 2 ) V = 0.
(3.2)
The values of q may be obtained from ρC2r -a11 +B1q 2 iqB2
iqB2 =0, ρC -B1 +a 22 q 2 2 r
which on simplification, we get
−{( A22 + B1 )ρC R2 − A11 A22 − B12 + B22 } ± − {( A22 + B1 )ρC R2 − A11 A22 − B12 + B22 } − 4 B1 A22 (ρC R2 − A11 )(ρC R2 − B1 ) 2
q12 , q 22
2 B1 A22
(3.3)
u = (u11e−q1 ky + u12 e−q2 ky )e−ik ( Cr t−x ) , v = (V11e
−q1 ky
+ V12 e
−q 2 ky
.
Therefore the solution (3.1) can be written as
½
)e
− ik ( Cr t −x )
(3.4)
,
( u11 , u12 ) and (V11 , V12 ) are not independent but are connected by equation
(3.2) for q = q1 and q 2 . Taking second member of equaton (3.2), we get where
u11 u = m1 , 12 = m 2 , V11 V11 m1 = iM1 , m 2 = iM 2 M1 =
ρC2r − B1 + a 22 q12 , q1B2
and
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(3.5)
(3.6) (3.7)
M2 =
ρC2r − B1 + a 22 q 22 , q 2 B2
(3.8)
where q1 and q2 are real and positive quantities defined in equation (3.3). Therefore the dispalcements in equation (3.4) can be written as
( v = (V e
)
u = m1V11e−q1 ky + m 2 V12 e−q2 ky e 11
−q1 ky
)
+ V12 e−q2 ky e
ik(Cr t −x )
ik(Cr t −x )
,
(3.9)
.
4. Boundary Conditions The displacements in equation (3.9) must satisfy the boundary conditions,
σ21 = σ22 = 0,at y = 0,
(4.1)
where σ21 and σ22 are defined in equation (2.3). For plane deformation in the ∂ x1x2-plane, = 0, u3 = 0 . Equation (2.3) then yields ∂x 3 ∂u ∂v σ21 = µL + , ∂y ∂x (4.2) ∂u ∂v σ22 = a12 , + a 22 ∂x ∂y where we have used the notation x1 = x, x2 = y, u1= u and u2 = v. From equations (3.9) and (4.2), we get
(m1q1 + i) V11 + (m 2 q 2 + i) V12 = 0, (a12 m1i + a 22 q1 ) V11 + (m 2 a12 i + a 2 q 2 ) V12 = 0.
(4.3)
Eliminating V11 and V12 from equation (4.3) ,we get
m1q1 + i a12 m1i + a 22 q1
m2q2 + i = 0. m 2 a12 i + a 2 q 2
(4.4)
On simplification , equation (4.4) becomes
(m1q1 + i)(m 2 a12 i + a 22 q 2 ) − (m 2 q 2 + i)(a12 m1i + a 22 q1 ) = 0. (4.5)
This is the velocity equation for Rayleigh-waves in a fibre-reinforced elastic medium.
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Note on Surface Wave in FibreReinforced Medium
Gupta, IS
5. Particular Case If we put α = β = 0 and µL = µt = µ , the elastic cofficients become
A11 = λ + 2µ, A12 = λ, B1 = µ,
A22 = λ + 2µ, A23 = λ, B2 = λ + µ. From equations (3.7) and (3.8) we get 1 M1 = q1 , M 2 = , then q2 C 2 2 C 2 2 1 − R , q = 1 − R , q = 1 C 2 2 C 2 S p where
C p2 =
λ + 2µ µ and CS2 = . ρ ρ
(5.1)
(5.2)
(5.3)
λ, μ are Lame’s constants. Using equations (5.1)-(5.3), the velocity equation (4.5) reduces to 2 2 C 2 C 2 2 C 2 R R (5.4) 2 − 2 = 4 1 − 2 1 − R2 . C p CS CS 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 x2 = 0 and x2 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=u2 = 0, u3 = w,x1 = x,x2 = y,x3 = z and = 0 , as ∂xs
µ´L
2 ´ 2 ´ ∂ 2 w´ ´ ∂ w ´ ∂ w = + µ ρ , T ∂t 2 ∂y 2 ∂x 2
(6.1)
where ρ’ is density, w’ is displacement along z-axis and (µ´L , µ´t ) are reinforced anisotropic elastic parameters for the layer.The equation of motion for the half space is ∂2 w ∂2 w ∂2 w = µL 2 + µT ρ , (6.2) ∂t 2 ∂y 2 ∂x
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Note on Surface Wave in FibreReinforced Medium
where µL , µt and ρ 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 x w´= W´e iω t − − p´ y, (7.1) CL 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 ω2 ´ (7.2) µL − µT´ p´ 2 = ρ´ω 2 . CL2
From equation(7.2), p’ can be written as p´= ±ip1´ where
´ 1
p =
ω CL
ρ´C L2 − µ´L ρ´C L2 − µ´L ω = K ,K = . ´ ´ CL µT µT
(7.3)
Therefore equation (7.1) can be written as
w ( x, y, t ) = W e ´
´ 1
x iωt − −ip1´ y CL
´ 2
+W e
x iωt − +ip1´ y CL
.
(7.4)
,
(7.5)
Similarly the solution of equation (6.2) for half-space is
w ( x, y, t ) = W1e
x iωt − −ip1 y CL
+ W2 e
x iωt − +ip1 y CL
where
ρC 2 − µL . p12 = K 2 L µT
(7.6)
For the existence of Love type surface waves it is necessary that in equation (7.5), w (x,y,t ) → 0 as y → ∞ .Therefore p1 hould be purely imaginary.
µ − ρC2L Let p1 = ip 2 = i K L . µt
(7.7)
Equation (7.5) then shows that
W1 = 0.
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(7.8)
Gupta, IS
8. Boundary Conditions The following boundary conditions must be satisfied ´ I. σ23 = 0 at y = − h,
II. w´ = w3 at y = 0,
(8.1)
´ 23
III. σ = σ23 at y = 0. σ23 is defined inequation (2.3) and can be written as (putting, ∂ u 2 = u1 = 0, u3 = w, x1 = x,x 2 = y,x 3 = z and =0) ∂x 3 ∂w σ23 = µT ; ∂y ∂w´ Similarly σ ´23 = µ´T (8.2) . ∂y From equations (7.4) – (8.2), we get ´
´
W1´ eip1H − W2´ e−ip1H = 0,
W1´ + W2´ − W2 = 0,
(8.3)
µT´ p1´ (W1´ − W2´ ) + µT p1W2 = 0. Eliminating W1´ , W2´ ,and W2 from equation (8.3). On simplification, we obtain µ p i tan( p1´ H ) = T´ ´1 . (8.4) p1µT Substituting for p1’ and p1 from equations (7.4) and (7.8) in equation (8.4), we get 1/ 2 ´ 2 (µ − ρC 2 ) ½ µ ρ CL − µ´L L = L KH . T´ . tan (8.5) ´ 2 ´ ´ µT ρ CL − µL µT ½ µT´ This is the frequency equation for Love waves in an anisotropic fibre-reinforced elastic medium. 9 . Particular Case If we put μL = μT = μ, in equation (8.5), we get C 2 1 − L ½ 1/ 2 2 2 CL µ CS KH tan ’ − 1 , = ⋅ 2 CS ρ CL ’2 ½ CS 30
(9.1)
µ´ µ and Cs2 = , ´ ρ ρ which is the classical frequency equation of Love waves in a homogeneous elastic layer over a isotropic half space. where Cs´ 2 =
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 = 5.65×10 9 Nm−2 ,
µL = 5.65×109 Nm−2 ,
µT = 2.46 ×109 Nm−2 ,
α = −1.28 ×10 9 Nm−2 ,
β = 220.90 ×10 9 Nm−2
ρ = 7800 kg m−3 .
We obtain the numerical values of constants from equation (2.4) using above values as A11 = 241.71×10 9 Nm−2 ,
A12 = 4.387 ×10 9 Nm−2 ,
A22 = 10.57 ×10 9 Nm−2 ,
A23 = 5.65×10 9 Nm−2 ,
B1 = 5.66 ×10 9 Nm−2 ,
B2 = 10.03×10 9 Nm−2 .
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 2
ρC 2 B A ρC 2 ρC 2 A A A A22 R − 1 R + 2 12 − 12 − 11 − B1 R − 12 B1 B1 A22 A22 B1 B1 B1 A22 (10.1) 2 ρC R2 A12 ρC R2 B2 A11 + − − 1 = 0. − × B A B B B 1
22
1
1
1
Using these values in equation (10.1), we obtain the value of wave velocity as ρC R2 (dimensionless) = 42. 38574 . µL
ρC R2 = 6.51043 µL
C R2 = 307.56832 ×10 5 m / s,
C R = 5.545884 km / s,
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Note on Surface Wave in FibreReinforced Medium
Gupta, IS
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).
ρC 2
Figure 1: Variation of dimensionless quantity R with fibre reinforced µL parameter β.
2 Figure 1 Shows increase in β shows increase in ρC R others parameter remains µL 2 constant. There is linear relationship between dimensionless quantity ρC R µL and fibre reinforced parameter β in m–2. 2 Figure 2 shows the variation of dimensionless quantity ρC R with elastic µL parameter µL in Nm–2 exponentially . Equation (8.5) shows that CL is dependent on the particular value of k and µ µ’ not a fixed constant. If k is small CL2 → L and while if k is large CL2 → L’ . ρ ρ
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Note on Surface Wave in FibreReinforced Medium
2 Figure 2: Variation of dimensionless quantity ρC R with elastic parameter µL.
µL
It is clear for the existence of Love waves p1 should be imaginary and p1’ is real satisfied if (µL’ / ρ ’) < CL2 < (µL / ρ) depends upon the reinforced parameters µL and µL’ .The upper limit of CL for the existence of Love waves in different elastic media is given as. Fibre reinforced medium µL / ρ = 0.851 km / s Aluminum Steel
µL / ρ = 4.5513 km / s
µ / ρ = 3.145km / s
Gold(24 c) Copper(cast)
µ / ρ = 4.6382 km / s µ / ρ = 2.947 km / s
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Gupta, IS
Copper (ore) Silicon
µ / ρ = 4.238 km / s
µ / ρ = 5.649 km / 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 µL.
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
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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 fibrereinforced 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
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Note on Surface Wave in FibreReinforced Medium