International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 1
ISSN 2229-5518
Scattering of thermo elastic waves at wavy boundary of a micropolar semi-space.
Dr. Ayaz Ahmad
Abstract -In this work we discusses the scattering of plane thermo-elastic waves at wavy boundary of a micropolar semi-space . Method of small perturbations has been used. The analyses shows that surface wave breaks into three parts. Rayleigh wave with velocity C scattered waves with
velocity of propagation cw
w rc
and cw
w rc
.It is also seen that scattered wavevelocity depends on the wave length and also on the wavy
nature of the boundary.
[1,25,86] in classical theory of elasticity.Considerations of
(ii) The surface under consideration dissipates according to Newton,s law of cooling
T
thermal heating and the resulting thermoelastic field has been a subject of interest for many years.
HT 0
n
( 1.2)
Literature survey shows that the corresponding analysis for micropolar elastic solid has not been dicussed probably
Nowacki (1969) showed that when displacement u
because om much mathematical complexities. In the present analysis we discuss scattering of plane waves in a
= u1 , u2 , u3
and rotation w =
w1 , w2 , w3
depend
micropolar elastic half space bounded by sinu-soidal
on the variables
x1 , x2
and t ,we face two mutually
surface under the following assumption ;
independent systems of equations ;
..
(i) Semi-space is homogeneous, free from any heat source
(ii) The surface is slightly rough i.e. the amplitude and
curvature of the roughness are sufficiently small. The sinu-
2u
2
1e1 223 u1 1T
..
2
soidal model of roughness has been considered. The
u2
2e2 213 u
..
2T
method of small perturbation is used to investigate the
wave propagation.
Mathematical model;
And
2 4 2
1u2 2u1
j w3
(1.3)
We consider a micropolar elastic half space-
..
2
x1 , x3
, x2 hf x1
bounded by a surface
1
11
41 22u3 j w1
..
1
2
2 1
42 21u3 j w2
x2 hf x1 ,
where
f x1
r sin
rx1
and h
2
(1.4)
..
represent a small perturbation parameter such that h2
2u 2 ( ) j
and its higher order terms are neglected (i.e the surface is slightly wavy ) and we assume that
Where
, , , , , are the elastic constants of the
The wavy boundary has a normal traction of the concentrated type , zero shear and zero couple stress.
micropolar material, is the density, J IS The rotational
inertia and dots denote the time derivative. The following
notation have been used in the equations (1.3) and (1.4)
nn p s , ns 0, sp 0
………………(1.1)
for
x2 hf x1
1 2
, e1 1u1 2u2
, 1 11 22
Where nn
is the normal stress component for the wavy
3 2 t
and
t is the coefficient of linear
boundary in the direction of normal to the curve ,
ns is
thermal expansion.
T = Temperature distribution in the material satisfying
the shear stress component for the wavy boundary along
coupled heat equation in the absence of any heat source .
the curve ,
sp
is the couple stress component in the
(2 1 )
div u = 0 (1.5)
direction of binormal to the curve. t
(i) The temperature and deformation fields do
not depend on the variable x3 .
IJSER © 2012
International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 2
ISSN 2229-5518
Where
k , k
denoting the heat conducting
Where
, m
coefficient and
Ce the specific heat at constant
two other wave equations.
deformation. = 0
k
,0 being the absolute temperature
1
2 2
2
2 2
1 2
2 t
4
2
0
of the natural state.
The latter system (1.4) is unperturbed by the thermal field,
As such we shall consider types of waves governed by the
former system (1.3), with boundary condition given by
2
(1.11)
1 2 2 2
1 2 2 0
(1.1).
We introduce the elastic potentials
and connected
2
2
t
(1.12)
4 2 t
4
2 4 3
with the displacements u1
and
u2 by
It may be noticed that equation (1.10) represent
longitutudinal wave where as equations (1.11) and (1.12)
represent transverse and tortional waves respectively.
u1 =
u2 =
1 2
2 1
(1.6)
Conclusion ; After solving the above problem, the fourier transformation of components of displacements and
Inserting (1.6) in (1.3) ,we get the following
microrotation are given by
2 1 2
u1 =
u1 + u1
K 2 t
mT
(1.7) 0 ,
2
1
1 2
0
(1.8)
u2 =
u2 + u2
2 t 2 3
2
3 =
3 3
(2 2
1 )
2 0
(1.9)
4 2 3 4
4
Where ( 0 , 0
3 ) represent transformed
Where k 2 , k 2 , k 2
displacement vector for plane boundary
x2 0
and
k 2 , k 2 , k 2 2 , ,
( u1 ,
u , ,
, ) arises due to the wavy
1 2 4
j
boundary. It is seen that surface waves breaks into three
, 2
1 1
parts
(a) Rayleigh wave with velocity c
m = 3 2
(b) scattered waves with velocity of propagation
cw
w rc
2 t cw
Eliminating Tfrom the equations (1.5) and (1.7).
and
w rc
We get the wave equation
,
(c) The scattered waves depend on the wave length as well
2
1 2 2 1
2 0
(1.10)
as on the wavy nature of the boundary.
k 2
t
t t
1
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International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October-2012 3
ISSN 2229-5518
7. Moon , F .J .Appl . Mech . 37, 153 (1976)
8. Flecher ,R. C. ,J . G eophys . Res .78 , 7661 , 19739.
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Department of mathematics
NIT ,Patna ,Bihar
India 800005
Email; ayaz1970@gmail.com
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