1. 15SHODH, SAMIKSHA AUR MULYANKAN
International Indexed & Refereed Research Journal, ISSN 0974-2832, (Print), E- ISSN- 2320-5474, Aug-Oct, 2013 ( Combind ) VOL –V * ISSUE – 55-57
Introduction
In this paper the problem of shear waves in
linearnon-homogeneousviscoelasticinfiniteandfinite
plates have been discussed. Laplace transfor and the
WKBJ-method are applied to solve the problem. The
cases of the non-homogeneity have been considered
and their effects on the wave speed and propagation of
the wave front are analysed. The numerical results hae
been obtained for the instaneous shearing traction,
applied uniformely at the boundary of the infinite
plate and the outer boundary of the finite plate. An
approximatesolutionoftheviscoelasticwaveproblem
has been obtained by the application of the Rayleigh-
Ritz and Laplace transform. A quantitative of the
approximate solutions for the displacements and
stresses between the non-homogeneous viscoelastic
plates at various times have even made.
2. Formulation of The Problemn and Basic
Equations.
We use cylindrical polar coordinates (r, θ, z)
and taketheorigin atthecentreoftheinfiniteplate.Let
the plate under consideration occupy th eregion r0
< r
< ∞, where r0
is the radius of the circular opening in
the plate, and let t be the time. For radially are σrθ
(r,t)
≡ σ(r,t),erθ
(r,t) ≡ e(r,t)anduθ(r,t) ≡ u(r,t)respectively..
Sincethebehaviouroftheviscoelasticmaterial
conforms to the simple Biot's model with a continuous
spectrum of relaxation times,the constitutiveequation
is:
(2.1)
whereµistheusualLeme'scoefficient,takenasafunction
oftheradialdistancerfromthecentreofthecylindrical
opening and the time t.
The equation of motion and the strain -
displacement relation for this case are
∂σ
∂
σ
ρ
∂
∂r r
u
t
+ =
2 2
2
, (2.2)
e
u
r
u
r
= −






1
2
∂
∂ , (2.3)
whereρistheinitialdensitytakenasconstant.
Multiplyingbothsidesof(2.1),(2.2)and(2.3)byexp(-
pt) and integrating from θ to ∞ with respect to t, we
obtain the transformed equations:
( )σ µ
σ σ
ρ
=
+ =
= −















2
2
1
2
2
p r p e
d
dr r
p u
e
du
dr
u
r
, ,
,
,
(2.4)
where use has been made of the quienscent
initial conditions
( )u r o
u
t
o
t o
, .=






=
=
∂
∂ (2.5)
Research Paper -Mathematics
Aug- Oct , 2013
Wave Propagation In Non Homnogeneous
Viscoelastic Plates.
* Dr. Hardeep Singh Teja
* Associate Professor Govt. Mohindra College, Patala.
( ) ( )σ µ τ
∂
∂τ
τr t r t
e
d
O
t
, , ,= −∫2
Eliminating e and u from the set of equations (2.4),
we get
( ) ( )
d
dr
P r
d
dr
Q r p
2
2
0
σ
σ
σ+ =, (2.6)
where
( ) ( )
( )
p r and Q r p
r
p
r p
= = − −
1
2
4
2
,
,
ρ
µ (2.7)
To remove the first order derivative in
equation (2.6) we write
( ) ( )σ φr p r r p, ,/
= −1 2
(2.8)
and obtain
( )φ φ, , ,rr g r p o+ = (2.9)

2. 16
International Indexed & Refereed Research Journal, ISSN 0974-2832, (Print), E- ISSN- 2320-5474, Aug-Oct, 2013 ( Combind ) VOL –V * ISSUE – 55-57
R E F E R E N C E
1. Bland, D.R. Theory of linear viscoelasticity Perganon Press, Ine. New York, 1960.
2. Christaneous, R.M.Theory of viscoelasticity, an introduction, academic press: New York and London 1971.
3. Kolexy, M. Stress waves in solids (Dower, Mxroyon 1963).
4. Moodie T.R. On the propagation, reflection and transmission transient cylindrical shear waves in non-homogeneous
viscoelastic media. Bill Austral, Math Soc - 8
where
g (r,p) = Q − −
1
2
1
4
2dP
dr
P (2.10)
Nowthestressfieldhastobedeterminedfrom
the equations (2.8) and (2.9) under the prescribed
boundaryconditionandfortheparticularformof µ (r,t).
Solution By Wkbj - Method
We observe that equation (2.9) is in a form to
which the WKBJ - method is applicable. Using this
method we see that an approximate solution is given
by
φ = ±








−
∫c g Exp g dr
r
r
o
1
1 4 1 2/ /
(3.1)
g (r,p) > o and | ( )d
dr
g−
<<1 2
1/
| . (3.2)
However, the solution is
( ) ( )φ = − ± −








−
∫c g Exp g dr
r
r
O
i
2
1 4 1 2/ /
, (3.3)
If g (r,p) < o. (3.4)
In these equations c1
, c2
and r O
i
are arbitrary constants.
Conclusion
The mnethods of integral transforms and the
WKBJ method were applied to find an approximate
solution of the problem. A quantitative comparison of
the approximate solutions for the displacements and
stresses between the non-homogeneous and
homogeneous viscoelastic plates at various times have
been discussed.