187
Internat. J. Math. & Math. Sci.
(1987) 187-198
Vol. i0 No.
LAMB’S PLANE PROBLEM IN A THERMO-ELASTIC
MICROPOLAR MEDIUM WITH STRETCH
T.K. CHADHA and RAJNEESH KUMAR
Department of Mathematics
Guru Nanak Dev University
Amritsar, India
and
LOKENATH DEBNATH
Department of Mathematics
University of Central Florida
Orlando, Florida 32816, USA
(Received May 23, 1984 and in revised form January 8, 1985)
ABSTRACT. A study is made of the Lamb plane problem in a thermooelastic micropolar
medium with the effect of stretch. The problem is solved for an arbitrary, normal
load distribution by using the double Fourier transform. The displacement components,
force stress, couple stress, vector first moment and the temperature field are determined for a half space subjected to an arbitrary normal load. Two special cases of a
horizontal force and a torque which are oscillating with a frequency m have been
investigated. It is shown that results of this analysis reduce to those without
stretch.
KEY WORDS AND PHRASES.
Lamb’s problem, Fourier transform, and micropolar medium.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
Eringen and Suhubi [1] have developed a general theory of linear and nonlinear
micro-elastic continua. This theory contains the Cosserat continuum and the intermediate couple stress theories as special cases. In a subsequent paper [2], Eringen
The
recapitulated his work and renamed his theory as micropolar elasticity.
micropolar theory essentially deals with such materials whose constituents are
dumbbell type molecules and are allowed to rotate independently without stretch.
Later on, Eringen [3-4] extended his work to include the effect of axial stretch
during the rotation of molecules and developed theories for both micropolar elastic
solids with stretch and micropolar fluids with stretch.
The mechanical model
underlying the theory of micropolar elastic solids with stretch can be envisioned as
an elastic medium composed of a large number of short springs. These springs possess
average inertia and can deform in axial directions.
T.K. CHADHA, R. KUMARand L. DEBNATH
188
Lamb’s problem [5] has been investigated extensively by several researchers in
different elastic media with various kinds of loading. In particular, Nowacki and
Nowacki [6] have studied the Lamb problem in micropolar elastic media. Recently,
Chadha [7] has investigated the same problem in micropolar elastic media, and discussed wave propagation in a semi-infinite micropolar elastic solid due to loading at
the plane boundary of semi-half space. Acharya and Sengupta [8] have recently studied
Lamb’s problem1 in a thermo-elastic medium under the influence of temperature. They
have examined the longitudinal and transverse thermo-elastic wave propagation in a
micropolar semi-infinite space bounded by a plane in which a normal loading is applied.
In spite of these studies, no attention is given to Lamb’s problem in
thermo-micropolar elastic half-space with stretch. The main purpose of this paper is
to investigate the problem with the assumption that the heat is radiated from the free
plane boundary surface of the semi-infinite space and the maximum temperature difference across the surface is always small. The displacement components, force stress,
couple stress, vector first moment and the temperature field are determined for the
half-space subjected to an arbitrary normal load. Two special cases of a horizontal
force and a torque which are harmonic in time have been discussed. The problem is
solved by the double Fourier transform method.
2.
THE FORMULATIOI OF THE PROBLEM AND THE BOUNDARY COIDITIONS
We consider a homogeneous micropolar elastic semi-infinite space with stretch
under the influence of temperature. We assume that there is a uniform stretch in the
O.
x-direction only and a loading g(x,t) normal to the free boundary surface z
Further, we assume that the micropolar semi-space is free to exchange heat within the
region z > O; and prior to the appearance of any disturbance, both media are everywhere at the constant absolute temperature T
O
We consider the two-dimensional problem so that the displacement and rotation
are independent of the y coordinate. Thus we may write u (u 1, O, u
and
(0, m2’ 0). The displacements are related to the displacement potentials
(x, z, t) and (x, z, t) as follows:
_
3)
Ul=
x
e
v2@
+
B--z
u3=--z- B-9-X
(2. lab)
so that
where
B2
V 2 ----+
B2
x2
-
v2 Bul @u3
I)Z
Bu
e
and
(2 2ab)
)X
-+
Bu 3
(2 3ab)
z
We follow Eringen [4] and Nowacki [9] to write down the basic field equations in
a thermo-micropolar elastic solid medium with stretch and without body forces and body
moments. These field equations are
(p+)v 2 u + (+I-) grad div u + 2
(y+)v2
m
+ (y+8-) grad div
m
rot m__
4
+ 2
u
@2 u
grad o
rot u
p
J
B-tt-’
@t 2’
(2.4)
(2.5)
LAMB’S PLANE PROBLEM IN A THERMO-ELASTIC MICROPOLAR MEDIUM
,,,
where
_
’J
ao v2@ n =-t--
(2.6)
are material constants, p is the density of the
n
o
o
(3+2) a t, a t is the coefficient of
rotational inertia, u
o
absolute
T-T
temperature minus the initial
solid,
the
B, y, e,
material, J is the
linear expansion of
absolute temperature
Using the values
189
To
of u and
in equations
-- -
(2.4)-(2.6) we get
@2
Be
2
(+) v 2 u I + (},+p-a) T-
Bo
-+ 2 Bm2
(u+) v 2 u 3 + (+p-) Be
(Y+)
v2 m2 4m2
a
o
Bu
v
(Bz I B 3
no
J B2
2 Bt 2
B2Ul2
(2 7)
B2u32-
(2.8)
Bt
Bo
p
Bt
B2m 2
Bu
+ 2
v2
p
J
(2.9)
Bt 2
(2 I0)
The temperature field o(x,z,t) satisfies Fourier’s Law of heat conduction,
which in the present case can be written as
Kv2o
where
B2O
pC
+ T u B
O
(v2),
(2.11)
K
is thermal conductivity and C is the specific heat at constant strain.
Using (2.1ab)-(2.3ab) and (2.11) in equations (2.7)-(2.10), we obtain
B2
(v2
(v2
(v2
(v2
o,
I
c
Bt
2
1 B
c2
--)
where
.+2
p
c
-
P2
Y _1__
c
2
c
r-{ (v 2
--)o
+a c
T’
I___B 2
K
p--C’ c
(2.14)
0,
2
)m2
(2.13)
0,
+ s
v2*
(2.15)
0,
o,
24
Y+
--ZF-’
(2.16)
c 25
2
J
(2.17abcde)
T.K. CHADHA, R. KUMAR and L. DEBNATH
190
4
y+’
2
YI
Y2
no
--’o p
r
T’ s
2
2
’
To
v
q
(2.18abcd)
+-T2--’
2
(2 19ab)
y+.
e
from equations (2.12)-(2.13), and
We next eliminate @ or
(2.14)-(2.15) to obtain the following partial differential equations"
(v2
32
1
c--t-)(v
1
I__
(V2
2
32
1
T
---T
2
c3
2
32
1
-
v
](’e)
ps.
qr,
[4] and Nowacki [9] the stress tensor
Eringen
Following
are given by
stress tensor
where
m2
from
(2.20)
O,
2V21
2
2
2]
or
o..
and the couple
j
o..
(;u k,k -v0)a ij + (-) (ui ,j + uj ,i
ji
Bo
j
Bj
where jki
i,j,k-1,2,3.
So ’j
Bo
+
(Y-)j,i
+
(2.22)
kji k
(2.23)
(Y+)mi,j’
(2.24)
kji mk,i’
is the vector first moment and
unit antisymmetric tensor, Bj
These expressions in the present case reduce to the form
is
033
3.
+
’k + Bmk,k aij
kji
+ 2(u
,j
2
32
’32 B-z
Bz--324,
32,
u32
(Y+) @T-
Bo
B3
eo
+
v2
+
(2.25)
0
32,]
(V2,
2 2)
(2.26)
(2.27)
(2.28)
Bo 3---"
BOUNDARY CONDITIONS.
In view of the normal loading of magnitude
g(x,t)
applied on
z
O,
the
boundary conditions are given by
g(x,t),
33
31
O,
32
O, B 3
O, at z
O.
(3.1abcd)
In view of the assumption that the temperature difference across the free surface
is always small, the linearized form of the radiation condition is valid on the
boundary z 0 so that
30
+ he
0
(3 2)
on z 0
3Z
on
bounded
and
finite
is
g(x,t)
Further, if we assume that the loading function
e,
and
vanish at infinity.
z O, then @,
,
m2
LAMB’S PLANE PROBLEM IN A THERMO-ELASTIC MICROPOLAR MEDIUM
191
SOLUTION OF THE PROBLEM.
We solve the above equations (2.16), (2.20) and (2.21) by using the double
Fourier transform defined as follows:
i(kx+qt) dxdt,
e
(4.1)
4.
f(k,z,n)
-#
_(f_f
f(x,z,t)
where the inverse transform is given by
2"--=f-=I(R) (k,z,n) e-
f(x,z,t)
(kx+nt) dkdn.
Thus the equations reduce to the form
2
d2
d
o,
az
,
( ) ( )(-)
2
d
(z-Z- 1
d
(4.2)
2
o
( )
o,
(.1
where
2
(4.6)
c3
cI
(4.7)
c4
c2
2
2
2
2 n
n
)(-(c
c
2
(4.9)
4
2
k2 +
)
n2
Yz2
(4.10)
c5
In view of the boundary conditions at infinity, the bounded solutions of (4.3)-
(4.5) assume the fom
$
Ae
->,1 z
+ Be
-Xl z
Ale
Ce -X3z
+
+
->,2 z
(4.11)
-2 z
Ble
De -X4z
(4.12)
(4.13)
-X3z
2 Cle + Dle -X4z
(4.14)
E e -X5z
(4.15)
where
A1
i A,
B1
2 B,
C1
3 C,
D1
4 D,
(4.16abcd)
T.K. CHADHA, R. KUMAR and L. DEBNATH
192
and
( + n____
k2),
for j-I 2
(4.17ab)
J:
2
(}, + n____ k2),
for j=3 4
It is assumed that Re(},j) > O, j=1,2,3,4,5.
Applying the Fourier transform (4.1) to (3.1abcd)-(3.2) and using (2.25)-(2.28),
it turns out that
2{
2_
d2
d---
-
k2
+ x( --T
dz
+ ik
dz
u{k2 + d2
d___}
2ik
+
dz
dz
-
(Y+)
So
de
dm
o-
(k,n),
(4.18)
0
(4.19)
,d2
td- k2-22)
2
(4.20)
O,
z + iBok
1
-g
(4.21)
O,
iBok m2
(4.22)
+hO=O,
where
(k,n) is the double Fourier transform of
(4.15) into (4.18)
Substitution of (4.11)
g(x,t).
(4.22) yields
ql A + q2 B + q3 c + q4 D
(4.23)
(k,n),
(4.24)
O,
pl A + p2 B + p3 C + p4 D
r3C + r4D + r5E
O,
(4.25)
t3C + t4D + t5E
O,
(4.26)
S
lA + s2B
(4.27)
O,
where
qj
[},j2(Xukxj,
+ 2)
k2>, -j),
j
L-2i
pj
I-2i
j
k},j,
,(k 2 +
2.)J + (},2._
J
j
k2
2j)
j
1’21
1’21
(4.28ab)
3,4
3,4
(4.29ab)
LAMB’S PLANE PROBLEM IN A THERMO-ELASTIC MICROPOLAR MEDIUM
193
(4.304b)
Solving equations
-h) j,
sj
(Xj
tj
li6ok
[3oXj
j,
A
1
ml(P3m 5
A
q3m4
j
3,4,]
(4.32ab)
for A,B,C,D and E we obtain
a4
a2
(k,n), C
a5
(k,n), E
a3
(k,n).
(k,n),
(4.33abc)
(4.344b)
(4.35)
q4m4 ),
m2(P4m 5
AI s2(P3ml P4m2 )’ A2 sl(P4m2- P3ml )’
3 mlm4’ A4
(4.31)
5.
j
(k,n), B
D
where
1,2,
-- ---
(4.23)-(4.27)
A
J
(4.36ab)
(4.37abc)
-m2m4’ A5 m3m4’
and
mI
(r4t 5 r5t4),
m2
(r3t 5 r5t3),
m4
:(slP 2 s2Pl),
m5
(slq 2 s2ql).
m3
_I _f (Ae
_f
O
(4.2) in (4.11)
+ Be
+ D e
_ooI=_I (a3C e -x3z
=-
-x2 z )e- i(kx + nt)dkdn
-; (IAe -XlZ + 2Be -x2z )e -i(kx + nt)dkdn,
-x3z
_I _I (C e
27
2
-x 1 z
+
i= I= E e -x5z e -i(kx
=’2-
_
-X4Z)e-i(kx + nt)dkdn
44 D
+
(4.38abc)
(4.39ab)
Using the Fourier inverse transformation
1
27
(r3t 4 r4t3),
e -x4z )e- i(kx
+
nt) dkdn,
nt)dkdn
(4.15) we obtain
(4.40)
(4.41)
(4 42)
(4.43)
(4.44)
(4.40)-(4.44) we can obtain the displacement components, force stress,
couple stress tensor, vector first moments, and the temperature field in the integral
Thus, using
form
uI
u
_I_(R).r
1
3
m2
_,I _(R)I
2-- _I
*
Ul(Z,k,n)
*
u3(z,k,n)
*
m2(z,k,n)
nt)dkdn,
+ nt )dkdn,
(k,n)e -i(kx +
(4.45)
(k,n)e- i(kx
(4.46)
(k,n)e- i(kx + nt) dkdn,
(4.47)
194
033
2--,
31
2--2
___1
3
27
c)
where
*
_=.r o33
u 3 (z,k,n)
*
o31 (z,k,n)
_I _I
32(z,k,n)
Lik
=-
"
*
(Ale
-
z
+
+
(4.48)
(4.49)
dkdn,
nt)dkdn
(4.50)
nt )dkdn,
(4.51)
(k,n)e- i(kx + nt )dkdn,
o (z,k,n)
A2 e
-2 z
+
3A3 e
ik(A3e
-3 z
-}’3 z
(4.52)
+
4A4 e
+
A4e
-4z
-4 z
],
]
[3A3 e -X3 z + 4A4 e -X4 z ]
=- [
{(>, + 2),
2)
{( +
-2i uk
(,3A3 e
k2,
l}Ale
+
},4A4 e
(4.54)
(4.55)
-XlZ
k2 2}a2e -2
-,3 z
(4.53)
z
(4.56)
-},4 z )],
[2ik(XlA1e -Xl z + X2A2 e -2 z
o31(z,k,n)
+ {,(k 2 +
{(k2
+
*
32(z,k,n)
(z,k,n)
(k,n)e- i(kx
[,IAI e -I z + ,2A2 e -2 z
+
B 3 (z,k,n)
(k,n)e i(kx +
*
f
+ nt)
(k,n)e -i(kx
B3 (z’k’n)
-I
’2 (z,k,n)
o33 (z,k,n)
(k,n)e- i(kx + nt) dkdn,
(z,k,n)
_I
2---=I 7_
(z,k,n)
u
-
T.K. CHADHA, R. KUMAR and L. DEBNATH
I
+
,) + (x(,
k2
,)+ (x(,- k2
2(x3)}a3e
-},3 z
2c4)}A4e -x4z
(4.57)
],
[(y+)(X33A3e -3 z + 4X4A4 e -x4 z -iBokA5e -x5 z ]
[io k (3A3e -x3z + 4A4e -x4z + 3oX5A5e -X5z ]
F
L Ii e
-i z
22 e
-X2z l]
(4 58)
(4.59)
(4 60)
PARTICULAR CASES:
(i) We consider a time periodic concentrated force acting at the origin in the
direction of x-axis so that the loading function assumes the fom
5.
g(x,t)
F(x) e -it,
(5.1)
where F is the magnitude of the force, 6(x) is the Dirac function of distribution and
m is the frequency.
The double Fourier transform of g(x,t) is
LAMB’S PLANE PROBLEM IN A THERMO-ELASTIC MICROPOLAR MEDIUM
(k,)
_F6(x)e- imtei(kx+nt )dxdt,
_/
F
ei(n-m)tdt
I
(5.2)
F6(n-m).
(4.45)-(4.60) with (5.2), we obtain
Thus from
Ul
u3
2
c33
F e it I
(2)
F
TTT
F
(2)
I
e -ikx dk,
*
e- ikx
[u3(z,k,n)]n=
*
e- imt f [m2(z
k,n)] n=
e
(5.3)
(5.4)
dk,
-ikXdk
(5.5)
*
e-ikXdk,
e-imt_.i- [o33(z,k,n)]n=
m
v
F
e-imt
I
F e it i
"32
B3
e it
[u *l(z,k,n)]n=
F
31
195
F e
I
e
(2’’)
F
e
e-it
I
(5.6)
*
e-ikXdk
[o31(z,k,n)]
n= m
(5.7)
*
e- ik Xdk,
[,32(z,k,n)]n=m
(5.8)
imt_l-
[B(z,k,n)]n=m e- kXdk
(5 9
[0"( z,k,n)] n= e- ikXd k.
(5.10)
If we neglect the stretch effect, we recover the corresponding expressions for the
displacements, stresses, and the temperature field in the
F e -it
u
(5 11)
_=I [M
m
for
l(z,k,n)]n= e-ikXdk
u3
F e -imt
/TTT
o31
32
0
(5.12)
F e -it
I
[M3(z,k,n)]n:m e-ikXdk,
(5.13)
F e -imt
I
[M3(z,k,n)]n=m e-ikXdk,
(5.14)
i
[M3(z,k,n)]n=m e-ikXdk,
(5.15)
m2
o3 3
_(R)i= [M2(z,k,n)]n= e-ikXdk,
F e -it
F
e-
mt
(--)-F e -imt
i
f [M z(z,k
n)]n=m e-i kXdk
[M3(z, k n)]n=m e-ikXdk
(5.16)
(5.17)
T.K. CHADHA, R. KUMARand L. DEBNATH
196
where
.
[ik(Ale -i
M l(z,k,n)
, z
, -2
, -3 z
+ ),4A4 e -4 3,
A2e z) + >,3A3 e
, -,2 z
-3 z + A e -X4 z )],
A e
k
3
22 e
4
z
+
I-T
[ial
M3(z,k,n)
1._
,* e -X4Z]
[3A’3 e -X3Z + aa
M4(z,k,n)
I-T
[{+2), 2
+
e
.
k2
k2
. ->,3
z
2ik(},3A3 e
M5(z,k,n)
M6(z,k,n
+
va
.
4A4
(5.19)
(5.2o)
e
,al}al
2)},
+ {(, +
.
.
M2(z,k,n)
-X2 z
2}
e
A
2 e
-4 z )],
(5.21)
z
z
,
[2ik(XlA e -1 + >,2Z2 e -2
I-T
I__
+
{(k2+
1 a(},-
+
{p(k2+
x
[(y
+
A
+
)(,33
k 2-
2m3)}A3e
z
* -X4
],
24)}4e
, -L4 z
+ (X- k 2
.
z
a 3 e -3 +
-3 z
4 a4A4e
)],
z
, -2 z
,
],
Ml(Z,k,n) I--T [ sia e i + 2A2 e
(5.22)
(5.23)
(5.24)
and
A*= (P3r4 P4r3)(s2ql slq2) (q3r4 q4r3)(s2Pl slP2),
(5.25)
A
s2(P3r4- P4r3 ),
2
s 1(p3r4
3
r4 (slp2
s2Pl )’
(5.28)
4
r3(slP 2
s2Pl)
(5.29)
P4r3 ),
(5.27)
These results agree with those obtained by Acharya and Sengupta [8].
(ii) In this case we consider a torque with its axis parallel to the z-axis so that
g(x,t) car. be written as
-it
g(x,t)
G[(x-a)-(x+a)]e
where G is the magnitude of the force.
The double Fourier transformation of
(k,q)
G
2i
(5.30) gives
I I’[ 6 (x-a)-6 (x+a) ]ei (kx+nt)e
,/ G
sin
(5.30)
(ka)6(n-m).
itd xdt
(5.31)
LAMB’S PLANE PROBLEM IN A THERMO-ELASTIC MICROPOLAR MEDIUM
Then from (4.45)
197
(4,60) with (5.31), we obtain
tof
u
iG
2)e -i
u3
ig
/(-2-7-)e -it I
*
-ikx
[Ul(Z,k,n
]n=mSin(ka)e d k,
(5.32)
*
[u3(z,k
]n= sin(ka)e-ikXdk
(5 33)
]n:Sin(ka)e-ikXdk,
(5.34)
n)
.
m
iG
v)e -it_.I
[m2(z,k,n)
033
iG
vT7-)e -it I
*
] n=msin(ka)e-ikXdk,
[o33(z,k,n)
(5.35)
o31
iG
2)e -imt_.I
*
[o31(z,k,n)
(5.36)
u32
iG
2v)e-imtl
83
iG
3
WT7-)e-imt
E)
iG
/(-e
imt
]n:msin(ka)e-ikXdk,
[2(z k n) ]n:msin(ka)e-ikXdk
(5.37)
i
[83* (z,k,n)]n:m sin(ka)e-ikXdk
(5.38)
i
[0" (z,k,n)]n=m sin(ka)e-ikXdk
(5 39)
In the absence of the stretch effect, we obtain the corresponding expressions for the
displacements, stresses and the temperature field in the form
[Ml(Z,k,n)]n=Sin(ka)e-ikXdk,
(5.40)
I" [M2(z,k,n ]n=msin(ka)e- ik Xdk,
I [M3(z,k n) ]n:m sin(ka)e-ikXdk
(5.41)
(5.43)
u
iG
/-)e-it_l
u3
iG
/T7-)e -imt
m2
/(-2--)e -imt
iG
o33
iG
(2/)e-it_.l [M4(z,k,n) ]n=mSin(ka)e-ikXdk,
o31
iG
2)e -imt
32
0
,I" [M5(z,k
n)
In=
m
si
n(ka)e-i kXdk
[M6(z,k,n)]n:msin(ka)e-ikXdk,
iG (2/)e -imt_-I [M7(z k,n) ] n:msin(ka)e-ikXdk
iG
)e-imt_,I"
(5.42)
(5.44)
(5.45)
(5 .46)
These results also agree with the corresponding results without stretch.
CONCLUS ION.
6.
The displacement field, force stress, couple stress, temperature field and vector
It is noted that the displacement field, force
first moment have been obtained.
stress, couple stress and temperature field involve the parameters s o, (o and n o of
the micropolar elastic media with stretch. In addition to the displacements, force
stress, couple stress, and temperature field, vector first moment 8j has been determined which vanishes in the case of thermo-micropolar elasticity. Some numerical
calculation for specific models of physical interest will be carried out and will be
communicated in a subsequent paper.
T.K. CHADHA, R. KUMAR and L. DEBNATH
198
Acknowledcement-
The second author expresses his grateful thanks to Guru Nanak Dev
University, Amritsar for providing financial assistance during the preparation of the
paper.
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Eringen, A. C.
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3.
Eringen, A. C.
Micropolar Fluids with Stretch, Internal Jour Eng. Sci 7
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Eringen, A. C.
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Lamb, H.
6.
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