Int. J. Pure Appl. Sci. Technol., 14(2) (2013), pp. 39-49
International Journal of Pure and Applied Sciences and Technology
ISSN 2229 - 6107
Available online at www.ijopaasat.in
Research Paper
Torsional Vibrations in a Non-Homogeneous Medium
over a Viscoelastic Dissipative Medium
Rajneesh Kakar*1, Kanwaljeet Kaur2 and Kishan Chand Gupta2
1
Principal, DIPS Polytechnic College, Hoshiarpur-146001, India
2
Faculty of Applied Sciences, BMSCE, Muktsar-152026, India
* Corresponding author, e-mail: ([email protected])
(Received: 2-12-12; Accepted: 11-1-13)
Abstract: The aim of this paper is to study the propagation of torsional surface wave in a
non-homogeneous isotropic medium lying over a dissipative viscoelastic half space. The
inhomogeneity taken for the upper layer is hyperbolic. Whittaker’s function is used to
obtain dispersion equation for the upper free layer. For lower viscoelastic nonhomogeneous layer, the analytical solutions for the dispersion relation is obtained by
using the method of separation of variables. The obtained dispersion equations are in
agreement with the classical result. The numerical calculations have been presented
graphically by using MATLAB. This study is helpful to collect the data for the seismic
waves produced by superficial explosions.
Keywords: Torsional Wave, Hyperbolic Inhomogeneity, Viscoelasticity, Voigt
Equation.
1. Introduction:
To understand the mechanical shocks and vibrations in the earth, the wave propagation in an elastic
media plays an important role. The propagation of torsional waves in an inhomogeneous layer is of
considerable importance in earth quakes engineering and seismology on account of occurrence of in
homogeneities in the crust of the earth. Elastic Waves in layered media has been studied by Ewing
et.al. [1] Biot [3-4] studied the theory of deformation of a porous viscoelastic anisotropic Solid.
Sharma [5-6] investigated propagation of Love waves in initially-stressed medium and surface wave
propagation in a transversely isotropic elastic layer. Dey and Gupta [8] solved the problem of
propagation of torsional surface waves in a homogeneous substratum over a heterogeneous half-space.
By the virtue of Biot’s theory, many researchers studied surface waves of types Love and Rayleigh, in
an inhomogeneous media. Chattopadhyay [9] studied reflection of elastic waves under initial stress at
a free surface. Also Chadwick [10], Zhang [11] also studied the propagation of Love Waves through
non-homogeneous media. Recently, Kakar et al. [12] discussed the effects of various inhomogeneities
on torsional surface waves propagating in a viscoelastic medium.
Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
40
In this paper, the possibility of existence of torsional surface wave in a non-homogeneous isotropic
medium lying over a dissipative viscoelastic half space is studied. We have assumed the hyperbolic
non-homogeneity for the upper layer. The effect of inhomogeneity present in the medium on the
velocity of torsional wave is studied mathematically and numerically.
2. Research Methodology:
For upper space, the hyperbolic inhomogeneity is assumed i.e.
I.
II.
z
z
µ = µ 0 cosh 2   , ρ = ρ 0 cosh 2  
ζ 
ζ 
Where, ζ > 0 is constant having dimension equal to length, ρ density and µ rigidity vary with
space variable z, which is orthogonal to the x-axis i.e. direction of wave.
For lower half space, the medium is homogeneous µ = µ0 , ρ = ρ 0 , F = F0 , body free, isotropic
and viscoelastic of Voigt type. Where ρ=density, µ= rigidity and F= friction of the medium.
Fig. 1: Geometry of the problem
3. Basic Equations:
The dynamical equations of motion are (Biot [7])
∂srr 1 ∂srθ ∂srz 1
∂ 2u
+
+
+ ( srr − sθθ ) + TR = ρ 2
∂r r ∂θ
∂z r
∂t
∂srθ 1 ∂sθθ ∂sθ z 2 srθ
∂ 2v
+
+
+
+ Tθ = ρ 2 ,
∂r r ∂θ
∂z
r
∂t
∂srz 1 ∂sθ z ∂szz srz
∂2w
+
+
+
+ TZ = ρ 2 ,
∂r r ∂θ
∂z
r
∂t
(1)
Where, srr , srθ , srz , srr .sθθ , sθ z , szz are the respective stress components, TR , Tθ , TZ are the respective
body forces and u , v, w are the respective displacement components.
Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
41
Now, the stress-strain relations are given by
srr = δ110 err + δ120 eθθ + δ130 ezz ,
sθθ = δ 210 err + δ 220 eθθ + δ 230 ezz ,
szz = δ 310 err + δ 320 eθθ + δ 330 ezz ,
srz = δ 440 erz ,
(2)
sθ z = δ 550 eθ z ,
srθ = δ 660 erθ .
Where, δ ij = elastic constants ( ij = 1,2……6).
The strain components are given by
err =
1 ∂u
1  1 ∂v u 
1 ∂w
, eθθ = 
+  , ezz =
,
2 ∂r
2  r ∂θ r 
2 ∂z
1  1 ∂w ∂v 
1  ∂w ∂u 
eθ z = 
+  , erz = 
+ ,
2  r ∂r ∂z 
2  ∂r ∂z 
1  1 ∂u ∂v v 
erθ = 
+ − .
2  r ∂θ ∂r r 
(3)
The rotational components are given by
Ωr =
1  1 ∂w ∂v 
− ,

2  r ∂θ ∂z 
1  1 ∂u ∂w 
Ωθ = 
−
,
2  r ∂z ∂r 
1  ∂ (rv) ∂u 
Ωz = 
−

∂θ 
r  ∂r
(4)
The stress-strain relations for general isotropic, viscoelastic medium according to Voigt are
srr = λ∆δ ij + 2µ eij
(5)
Where, λ and µ are elastic constants and
∆=
∂ u ∂ν ∂ w
+
+
∂x ∂y ∂z
(6)
4. Formulation of the Problem:
Let M be a non-homogeneous elastic layer of finite thickness d and M ' is homogeneous viscoelastic
dissipative mantle of earth. The cylindrical co-ordinate system ( r , θ , z ) is located at the interface
separating the layer from the half-space and the z-axis is directed downwards (as shown in Fig.1)
The displacements for torsional wave is given by u = 0, w = 0, v = v (r, z, t)
From Eq. (1), Eq. (5) and Eq. (7), we get
(7)
Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
42
∂srθ ∂sθ z 2 srθ
∂ 2v
+
+
=ρ 2
∂r
∂z
r
∂t
(8)
Stress and displacement components are related as
 ∂v 
 ∂v v 
eθ z = µ   , erθ = µ  − 
 ∂z 
 ∂r r 
(9)
From Eq. (8) and Eq. (9), we get
 ∂2 1 ∂ 1 
∂  ∂v 
∂ 2v
+
−
v
+
µ
=
ρ



2
r ∂r r 2 
∂z  ∂z 
∂t 2
 ∂r
µ
(10)
Let the solution of Eq. (10) is
v = ψ ( z )Y1 (Gr )eiΩt
(11)
ψ ( z ) is the solution of the following equation

ψ // ( z ) + ς 2ψ / ( z ) − G 2  1 −

β2 
ψ ( z ) = 0
β /2 
µ/
Ω
µ
Where, ς =
, β = and β / 2 =
and
µ
k
ρ
Y1 (Gr ) is Bessel function of first kind.
5. Boundary Conditions:
1. At the free surface z = −d, stress = 0
 ∂v 
=0
 
 ∂z  z =− d
2. At the interface z = 0,
 ∂v 
1 ∂v
 =µ
∂z
 ∂z  z =0
µ0 
1
3. At the interface z = 0, continuity of the displacement requires
( v ) z =0 = v1
4. For z → ∞
lim v1 ( z ) = 0
z →∞
Where, v and v1 are the displacements in the layer and the half space respectively.
6. Solution for the Upper Layer:
We assumed the inhomogeneities for the upper layer are
(12)
Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
z
µ = µ 0 cosh 2 
ζ

2 z 
 , ρ = ρ 0 cosh  ζ 

 
ξ
Put v =
43
(13)
Θ
From Eq. (10), Eq. (12) and Eq. (13), we get
1
 1 //
Θ ( z) −
Θ/
4Θ 2
 2Θ
( )
ξ // ( z ) − 
2

β2
+ G 2 1 − / 2
 β
 
 ξ ( Z ) = 0
 
ξ // ( z ) − Λ 2ξ ( Z ) = 0

(15)
2
2
Where, Λ = G 1 +

(14)
1
β2 
−

ζ 2G 2 β / 2 
Solution of Eq. (15) with boundary condition no. 4 becomes
ξ ( Z ) = Εe −Λz
(16)
Therefore, the displacement for the torsional wave in the inhomogeneous half-space using boundary
condition no. 4 becomes
v =
1
Εe −Λz
z
µ0 cosh  
ζ 
Y1 (Gr )eiΩt
(17)
Now using the boundary conditions no. 3 and no.4, we get
−3
−1


 2χG   ε   
  ε   2 /  2χG   ε   
 1    ε   2  ε 
1−
Gd 
Ζ1  −  1−   Gd   Wη ,0 
1−   Gd   + 2χ 1−   Gd  Wη ,0 
ε   G   
 G 
2 
 2    G    G  2  ε   G   


−3
−1


 2χG   ε   
  ε   2 /  2χG   ε   
 1    ε   2  ε 
+Ζ2  −  1−   Gd   W−η ,0  −
1
−
Gd
−
2
1
−
Gd
W
−
1
−
Gd
χ
   
  G   −η ,0  ε   G   
   
    
2

 2    G    G  2  ε   G   


1
 µ 
1
β 2 2
−Ε   1 +
− /2  = 0
 µ0   ζ G β 
(18)
Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
44
 1  ε 
 2χ G  
−
W
−
η
,0



−
 2 G
ε 
 1  ε 

χ
χ
2
G
2
G








2
/ 
Ζ1  −  Wη ,0 
+ Ζ 

 + 2χWη ,0 
ε  2 
2 
/  2χ G 
 2  G  2  ε 

−2χW−η ,0  −



 ε 
2
1
2
 µ 
1 β 
−Ε  1+
− /2  = 0
 µ0   ζ G β 
2
(19)
and
 2χG 
 2χ G 
Ζ1Wη ,0 
 + Ζ2W−η ,0  −
−Ε = 0
ε
ε




2
2
(20)
Put Whittaker’s function up to linear term in Eqs. (18), (19), and (20) so as to remove
Ζ1 , Ζ 2 , and Ε , we have
1−
Sre
Ure
β2 ε
Gd
/2 ε
β
β2 ε
1− / 2 Gd
β ε
− Sr e
− 1−
1
β2 ε
Gd
/2 ε
β
β2 ε
− 1− / 2 Gd
β ε
1
+ Ur e
=
µ
µ0
1+
1
β2
− /2
2
ζ G β
2
(21)
β2 ε
1− /2
β ε
Eq. (21) show is the dispersion equation of torsional wave in an inhomogeneous upper layer.
Special case:
For homogeneous medium, i.e. ε → 0, ε → 0 and ζ → ∞ , the Eq. (21) reduces to

tan 


 β2
  µ 
β2
 /2 − 1Gd  =  1 − /2

 

β
β
  µ0 


−
1
/2

β

β2
(22)
When the non-homogeneity is neglected, the frequency equation is in well agreement with the
corresponding classical result. Eq. (22) is well known classical result given by Love.
7. Solution for Viscoelastic Layer:
Equation of motion for viscoelastic voigt type can be written by using eqs. (1), (5), (7) may be written
as (Biot [7])
∂   ∂2 1 ∂ 1 
∂ 2v

µ
+
F
+
−
v
=
ρ



∂t   ∂r 2 r ∂r r 2 
∂t 2

(23)
Where, µ= rigidity and F= friction of the medium.
Let the solution of eq. (23) be
v = Φ ( z )ℵ1 ( Lr )eiΩt
(24)
Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
45
Put Eq. (24) in Eq. (23)


β12
Φ ( z ) − L 1 − 2
 Φ( z) = 0
 β 2 (1 + iC ) 
//
2
Where, β1 =
ΩF
Ω
,C=
and β1 =
k
µ
(25)
µ
ρ
1/ 2
  k 2η 1/ 2  θ  
  k 2η 1/ 2  θ   − β 2 (1k +ηC2 ) 


Φ = L cos   2
sin   z  − M sin   2
sin   z  e  2
2 
2 
  β2 (1+ C ) 
  β2 (1+ C ) 
 2  
 2  




2
θ 
cos  z
 2
(26)
1/ 2
 k 2η

θ 
θ 
Let α =  2
, β = sin   , γ = cos   , η =  β 22 (1 + C 2 ) − β12  ,
2 
2
2
 β 2 (1 + C ) 




C
 Therefore, the final solution of Eq. (23) is
θ = tan −1  2
 β2
β2 
 β 2 + C β −1 
1
 1

Φ = [ L cos(αγ z ) − M sin(αγ z )] e−αβ z
(27)
8. Boundary Conditions:
1. At the free surface z = −d, stress = 0
 ∂v 
=0

 ∂z  z =− d
µ0 
∂   ∂v1 
 ∂v 

=
µ
+
F




∂t   ∂z  z =0
 ∂z  z =0 
1
3. ( v ) z =0 = v
µ0 
2.
∂   ∂v1 v1 

−  =0
4.  µ + F  
∂t   ∂r r  z =0

Using Eq. (27) and boundary condition no. 1, we have
− 1−
Ne
β2
β
/2
Ld
− Oe
1−
β2
β
/2
Ld
=0
(28)
Using Eq. (17), Eq. (27) and boundary condition no. 2, we have

β2
β 2 
 N µ 1 − / 2 L − O µ 1 − / 2 L  = {[ L ( − µ 0 Γ1Γ 2 − iΩΓ1Γ 2 F ) − M ( − µ 0 Γ1Γ 3 − iΩΓ1Γ 3 F ) ]}
β
β


(29)
From boundary condition no. 3 and 4, we have
N +O = L
(30)
θ 
cot   L + M = 0
2
(31)
To eliminate N, O, L and M, we get |aij|= 0 where i, j = 1, 2, 3 ... 4, 4 × 4 determinantal equation
T11
T12
T13
T14
T21
T31
T22
T32
T23
T33
T24
=0
T34
T41
T42
T43
T44
Where,
− 1−
T11 = e
β2
β
/2
1−
Ld
T21 = µ0 1 −
,T12 = −e
β2
β
/2
β2
β
/2
Ld
,T13 = 0,T14 = 0
L, T22 = µ0 1 −
β2
β
/2
L, T23 = ( − µ0Γ1Γ 2 − iΩΓ1Γ 2 F ) ,
T24 = ( − µ0Γ1Γ3 − iΩΓ1Γ3 F ) , T31 = 1, T32 = 1, T33 = 1, T34 = 0, T41 = 0, T42 = 0,
θ
T 4 3 = co t 
 2

 , T44 = 1

Expanding the determinant we get,



θ 
θ 
 + Γ 2  + iF Γ 1Ω  Γ 3 cot   + Γ 2 
2
2




 β

tan  /2 − 1  Gd =
β



2
µα  Γ 3 cot 

µ0k
β2
β
/2
(32)
−1
Equating the real part of Eq. (32) we get the dispersion equation as
2
2

β2   β2  
2
 1 + C − 2  +  C 2  
β1   β1  
 β2


µ 

tan  /2 − 1 Gd =
2
β

µ0
β


1+ C2
−1
(
1/ 4
)
 θ 
θ 
 θ 
sin  2  cot  2  + cos  2  
 
 
  
β12
(33)
Special case:
For homogeneous elastic medium, F → 0, then Eq. (33) reduces to
Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
 β2

µ
β2
tan  /2 − 1Gd =  1 − /2
β

β
µ0 


47


−
1
/2

β

β2
(34)
When the dissipation in the medium is neglected, the frequency equation is in well agreement with the
corresponding classical result.
1
0.9
P H A SE V E LO C IT Y
0.8
CURVE 3
(ANGLE=180`)
0.7
0.6
CURVE 2
(ANGLE=120`)
0.5
0.4
0.3
CURVE 1
(ANGLE=90`)
0.2
0.1
0
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
DEPTH (KH)
Fig. 2: Variation of phase velocity with depth at various angle of projection
9. Numerical Analysis:
The phase velocity is calculated numerically for torsional surface waves and effect of dissipation is
studied from Eq. (33). The curves are obtained for Eq. (33). The curves in fig.2 are plotted between
β
β
/
v/s KH i.e. between phase velocity and depth at various values angles, the phase velocity of
torsional wave first increases then becomes constant at various values of angle of projection.
10. Overall Conclusions:
I.
II.
III.
IV.
In case of 1st and 2nd layer when inhomogeneity factor and dissipation is removed, the
dispersion equation reduces to the classical dispersion equation given by Love.
The phase velocity increases with the decrease in dissipation in 2nd viscoelastic layer
The phase velocity increases with the decrease in wave number in 2nd viscoelastic layer.
In the 1st layer, the phase velocity directly depends on the inhomogeneity factor.
Acknowledgements
The authors are thankful to the referees for their valuable comments.
Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
48
Appendix:
Sr = ( A1 B1 + A2 B3 − A3 B5 − A4 B7 ) , Sr1 = ( A1 B2 − A2 B4 + A3 B6 − A4 B8 ) ,
Ur1 = ( − A5 B10 + A6 B12 ) , A 1 = 1  1   ε   1 − ε G d 
4  χ  G  
G

2
−1
−1
−3
2
Ur = ( A5 B9 + A6 B11 ) ,
, A 2 =  ε   1 − ε G d 
G
G

−3

2
2
 ε  ε
2
ε
1
ε
ε
A3 =  1 − Gd  , A4 = 4 χ  1 − Gd  , A5 =  −   1 − Gd  
G
G
 G  G



 2 
 G
ε


A6 = 2 χ  1 −
Gd 
G


B
1
 τ
= 
 τ



0
η
2
−1
2
ε


, A6 = 2 χ  1 −
Gd 
G



1 
 η
−



2
2 

 1 −
τ




 η − 1


 τ 
2
1 −  2
B2 = 

τ
τ





2

η − 1  
η


 
2 2   1
 τ  2
B3 =  0   1 − 
 −
τ
 2
τ  




η
0
 τ 
0 
τ 
2
τ 

τ 


1 +



η
2
B4 = 
τ 

 τ 
0
η
2
B6 = 
 τ 
0 
τ 
η
2
B7 = 
τ 

 τ 
0
B8 = 







 − η −


2
1 + 
τ



2

−η − 1 



η 
2 2
1+ 
0 
0
2τ  
τ





2
η
2
 η 1
− − 
 2 2
τ
2
,
,


,

η
1 

−

 −

2
2 

 1 +
τ









2


−η − 1  



  η
1

 1 +  2 2    
−  1 +
τ

   2τ 2  






η
B5 = 
0
2
2
−1
2

−3
2
1 

2 


−









,

 ,




2
−η − 1  

 
 2 2  ,
2
(τ 0 ) 

2
η − 1  

 
 2 2 −
τ




  η 1 2
 η
  − 
1 
2 2
  
−  1+ 
0
τ0
2 
   2τ




2
2
2
η − 1  

 
 2 2  ,
2

(τ )


  η 1 2
  − 
 −  2 22 

(τ 0 )




,



2
2
2


1  
1    η
1  
η
 η
−
−
−
−




 −   1




η 
2  
2    2
2  
 2
1 −  2
−
1
+
−
,


2

τ0
τ
2τ  

 2

(τ )










2


−η − 1  





  η − 1   1 +  2 2   +
  2τ 2  
τ







η 1 
 − 
2 2
2
(τ 0 )
2
2
2


−η − 1   −η − 1  








   η + 1   1 +  2 2   −  2 2   ,
2
   2τ 0 2  
τ0

(τ 0 ) 







2
2


1  
1  
 η
η
−
−
−








2  
2
2   η
  1 − η   1 +  2
− 

2
 2
τ
2τ  

(τ 0 )    2 τ







0
2
2

1  
1  
η
η
−
−







1 
2  
2
2  
 2
−
+ 
,
 1 −
2
0

τ
2 

τ 0)
(






Int. J. Pure Appl. Sci. Technol., 14(2) (2013), 39-49
B
B
9
1 0
 τ
 τ
= 
 τ
= 
 τ
 τ 
0 
τ 
τ0 

 τ 
0
η
B11 = 
B12 = 



0
2
η
2

 η − 1 



2 
 1 −  2
τ



η
2



η


1 +



2


 1 −



 − η

2

−
τ
0
2






1 

2 
49

 − η − 1 



2
2 
 1 + 
0
τ



2








 1 +



 η

 2
−
τ


 ,



2
1 

2 
2






,
2
2
2

−η − 1  
η − 1   η − 1 


 

  

 2 2     η − 1  1 −  2 2   +  2 2 


2
τ0
τ
   2τ 2  

(τ )






2

η 1 
−



1 −  2 02 
τ







   − 1 − η   1 +
   2 2τ  




 η 1
− − 
 2 2
τ
2


−





,



2
 η 1 
−
−

 
 2 2  .
2

(τ )


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