363
Internat. J. Math. & Math. Sci.
VOL. 13 NO 2. (1990) 363-372
ON LAMB’S PLANE PROBLEM IN MICROPOLAR VISCOELASTIC
HALF-SPACE WITH STRETCH
RAJNEESH KUMAR
Department of Mathematlcs,
Doaba College
Jalandhar, India
M.L. GOGNA
Department of Mathematics
Kurukshet ra University
Kurukshetra, India
LOKENATH DEBNATH
Department of Mathematics
University of Central Florida
Orlando, Florida 32816, U.S.A.
(Received November i, 1988 and in revised form December 30, 1988)
ABSTRACT.
space
A study is made of Lamb’s plane problem in mlcropolar viscoelastic half-
with
stretch.
The
viscoelasticity
is
characterized
by
the
rate-dependent
theories of mlcro-viscoelastlclty generalizing the classlcal Kelvln-Voigt theory.
displacement
components,
force
stress,
The
couple stress and vector first moment are
obtained for a half-space subjected to an arbitrary normal load.
Two particular cases
of a horizontal force and a torque which are time harmonic have been considered.
Several limitlng cases are obtained as special cases of the present analysls.
KEYWORDS AND PHRASES.
Lamb’s problem, Micropolar Viscoelastic Medium and Fourier
Transforms.
1980 AMS SUBJECT CLASSIFICATION CODE.
73D30.
INTRODUCTION.
Lamb’s problem [I] has received considerable attention by several researchers in
different elastic media with various kinds of loading.
In particular, Nowackl and
Chadha [3] has
Nowackl [2] have studied the Lamb problem in mlcropolar elastic media.
investigated
the
Lamb
problem
in
mlcropolar
elastic
media
and
discussed
the
propagation of waves in a seml-lnflnlte mlcropolar elastlc solid due to loading at the
plane boundary of a seml-half space.
Acharya and Sengupta [4] have recently studied
the same problem in a thermo-elastlc medium under the influence of temperture.
They
have examined the longltudlnal and transverse thermo-elastlc waves in a mlcropolar
R. KUMAR, M.L. GOGNA AND L. DEBNATH
364
semi-infinlte space bounded by a plane in which a normal loading is applied.
recent paper by Ray and
Sengupta [5], a study
micropolar
is made of a two-dimensional waves in a
More recently,
medium.
thermo-viscoelastic
In a
Kumar
and
Chadha
[6]
and
Chadha, Kumar and Debnath [7] have made an investigation of the Lamb problem in a
microelastic half-space with stretch,
and in a thermoelastic micropolar medium with
stretch.
Based upon the linear theory of micropolar viscoelastic waves due to Eringen [8-
9], a study
is made of Lamb’s plane problem in micropolar viscoelastic half-space with
the effect of stretch.
The displacement components, force stress, couple stress, and
vector first moment are determined for a half-space subjected to an arbitrary normal
Two particular cases of a horizontal force and a torque which are time harmonic
load.
are cited as examples of the general theory.
2.
BASIC CONSTITUTIVE AND FIELD EQUATIONS.
Making reference to Eringen [8-10] and Nowackl [11], the constitutive and field
equations for mlcropolar viscoelastic solids with stretch in the absence of body
forces, body moments and stretch forces are
+ 2 (a + a
(mjl
8o ekji,k
+ (8 +
81 -) ink, kSji
+ {(v- ) + (v-
+ [( + ) + ( +
lJ ao
+
#,j
[( + a) +
80 ekJl
(I
+
k,i’
2
V u+ [( +
al)]
+ 2(a + a
-
(2.1)
kjik ),
) (ui, j
)o:} j’,i
(2.2)
)ow:l ’,i’
(2.3)
a) + (
-)
+
al --]
I
grad dlv u
2u
rot
p
8t
(2.4)
2
2
-4(
a-n
o
where
tji,mji
and
j
J
2
o
are
and I,
,
is
the
a, 8, Y,
,
t
J
rot
t
2
(2.5)
(2.6)
2’
the components
80
no
of
force
stress,
couple stress and vector
J is the rotational inertia, u is the displacement
microrotation
ao’
1)
2
first moment, p is the density,
vector,
+ 2 ( +
)
* 1
vector
and
is
are material constants.
the
scalar
microstretch,
LAMB’S PLANE PROBLEM IN MICROPOLAR VISCOELASTIC HALFSPACE WITH STRETCH
3.
365
BASIC EQUATIONS AND THE BOUNDARY CONDITIONS
We consider a seml-inflnlte homogeneous, isotropic, micropolar viscoelastic solid
We take the rectangular Cartesian coordinates x, y, and z with the
origin at the plane boundary of the half-space, z > o and the z-axls is normal to the
medium.
We assume that there is a uniform stretch in the x-dlrectlon and a given
with stretch.
_
loading function g(x,t) normal to the free surface z
We
the
consider
two-dlmensional
problem
microrotation are independent of the y coordinate.
(Ul,
u__
o, u
3)
and
(o,
2’
o).
o.
that
so
the
Hence, we may
displacement
We introduce displacement potentials
$(x,z,t) and $(x,z,t) defined by u
+
and u
3
(2.6) can be written in terms of $ and $ as
Sx Sz
Sz Sx"
Consequently,
equations (2.4)
(3.1)
(3.2)
(3.3)
(3.4)
where
2
3
v2 --+
,2
c2
2
2
3
-,
(I
+
c
2
2
2ao/J’ Pl
c4
or
m2
3
[{(c c 2 )
V2
Eliminating
/
c2= ( + 2)/, c2=
(T + e)/J, c
(TI + el )/J’
2
(+ 2 )/,
al)/P’ c3
2alp,
2
p 2allP
4a/J,
Y2
T
nolao"
4alia, 3
from (3.2) and (3.3), we obtain
3
2
2}
3t
{(c 23 +
c
2 3
2
-) v
a
+ (P + P )
and
write
(v2
+
3
2
+ 3 2)
Y
--at
a 1
)
(,2)
(2
+
o
(3.5ab)
The boundary conditions of the problem are
(3.6abcd)
where
(3.7)
(3.8)
366
R. KUMAR, M.L. GOGNA AND L. DEBNATH
[(T + e) +
m32
(R)
X3
-z +
Co
(’Y1
+
82
el) -’]
6o
}
ox
(3.
3
In addition, we assume that
, , m2
and
tend to zero as z
=.
SOLUTION OF THE PROBLEM.
4.
We apply the double Fourler transform
(k,z,s)of f(x,z,t)
defined by (Mylnt-U
and Debnath, [13]) as
=ff
(k,z,s)
exp{i(kx + st)} f(x,z,t) dxdt
(4.1)
and its inverse is given by
ff
f(x,z,t)
exp[-i(kx+st)]
(k,z,s)dkds
Applications of this transforms to equations (3.1)
d
2
d
;=o,
2
2
(3.4) gives
2
d
(4.2)
2
2
o,
d
2
(4.3)
2
(dz 2
(4.4)
0
4
where
s
2
2
2
k
c
s
2
+ s 23
2
(s22
2
s4=
k
k
2
2
2k
isc2
2
s
c-is c
--
2) (s 23-k 2
2
s
+ T3
c
s
+
2
s2
2
2
(c2-1s
2
-s z
B1
c ,2
2
2
(c3-1s
c ,2
3
-
c2
(4.6)
(4.7)
-s z
+ C e
-s z
+
2
(T2-1sy)
2
(c2-1s c 2 2) (c3-is
2
(4.4) with the boundary conditions at infinity are
-s z
B e
2
(2-is)]
(p l-iS p’
4
A e
e
2
(4.8)
The solutions of (4.2)
-s z
T2-t sT-s
2
2
c3-1s c 3
2
(4.5)
Cle
3
3
(4.9ab)
-s z
D e
4
(4.10ab)
LAMB’S PLANE PROBLEM IN MICROPOLAR VISCOELASTIC HALF-SPACE WITH STRETCH
where it is assumed that
B
a2B,
C
aj
Pl_i
sPl’
Re(sj)
J
0 for
I, 2, 3, 4 and
(4.11ab)
a3C,
[(c 22
sc2
i
2
(sj
k
2
+ s 2 ], J
(2.10) and combined with (4.gab)
A
A
(4.10ab), we
A
4
-)
{_!, E-’ E-’ E
2
3
(4.12ab)
2,3
(3.1) to the boundary conditions (3.6abcd)
Applying the Fourier transforms
(A, B, C, D)
obtain
(k,s)
4.13abcd
where
(r3t 4
A
A
m
A
m
3
(4.14abcd)
(r4t 2 r2t4)
s
k
.2,
Jffil
-2i*ksj
21* ksj,
J=2,3
m
j=
-qlm3 ),
l(r2t 3 r3t2).
(* + 2*)
qj
(r2t 4- r4t2) (q3ml
m2(r3t 4 r4t3),
ml(r3t 4 r4t3)
2
A4
*(k 2 +
s) +
(y + e*)
r
-q2ml +
r4t 3) (qlm2
m3(r2t 4 r4t 2)
A
, (sj2
a
k
2
2aj)
J-2,3
J=2,3
sjaj,
j=
J=4
iBok,
i
Bokaj
J=2,3
3aoSj
I -is
k*
-
*
B -is
a*=a-is a ,y*
V -isv
*
e-ise
Application of the inverse Fourier transformation to (4.9ab)
,
-s
f f
9"-- f f
367
Ae
lz-i (kx+s t
-s z
(Be
2
+ C e
-s z
3
(4.19)
dkds,
e
-i (kx+s t
(4.10ab) gives
dkds,
(4.20)
368
R. KUMAR, .L. GOGNA AND L. DEBNATH
-s z
f f
(R)
Using (4.19)
-s z
3
2
2= f
(2
+3 c
-s4z-i(kx+st) dkds.
D e
e
-i (kx+s t
(4.21)
dkds,
(4.22)
(4.22), we obtain the formal solutions for the displacement components,
mcrorotations, force stress, couple stress and vector first moment in the form
Ul
2--
ff Fl(Z’k’s) (k,s)
e
u3"
2--
ff F2(z’k’s) (k,s)
e
2
:-
t33
t31
m32
X3
-
Fl(Z,k,s
F2(z,k,s
k s)
+
(4.25)
St)dkds’
(4.26)
(4.27)
(4.28)
ff F7(z’k’s) (k’’s)e-i(kx + St)dkds’
(4.29)
-
[ikAle
-SlZ
+
s2A2
e
-s z
[SlAle
-s z
2
-s
_ik(A2
e
2
+ a 3 A3
[{(A* + 2U*)s 2
[2il*k
SlA
e
*
-SlZ +
=
F7(z,k,s) = [18ok(a2A
[(y* + e*)
e
2
+
(4.30)
3
)]
(4.31)
(4.32)
-s z
2
a*)s + (*
2
a*)s3+(l*-s z
2
a3A3
-s z
2
-211*k(s2A2e
Ale
a*) k 2
2*
e
e
a3s3a3
-s z
3
3
-s
+
s3A3e
3z )]
k2-2a* ct2}A2e
a*)
a3} A3e
-s z
+
-s z
2
-s3z ],
3z ),
{(I* +
(a2s2a2e
e
-s z
e
k
s3A3
+ A3 e
-s
2
A2 e
+
2z
-s z
(a
+{(I* +
F6(z,k,s)
(4.24)
dkds,
ff F6(z’k’s) (k’s)e-i(kx + St)dkds’
=
F4(z
=
F5(z,k,s) =
F3(z,k,s)
(4.23)
f Fs(z,k,s) ’(k,s)e -i(ks + St)dkds,
2--
where
-i(kx + st)
ff F4(z’k’s) (k’s)e-f(ks
2
6
)dkds,
ff F3(z’k’s) (k’s)e-t(kx + St)dkds
:!_
I__
-t(kx + st
-s z
3
],
(4.33)
-s z
2
(4.34)
-s z
-tBoka4e
4
],
(4.35)
-s z
+
aos4A4
e
4
(4.36)
LAMB’S PLANE PROBLEM IN MICROPOLAR VISCOELASTIC HALF-SPACE WITH STRETCH
5.
369
PARTICULAR SOLUTIONS OF INTEREST.
We first consider a time harmonic concentrated normal load in the form
g(x,t)
P (x) e
it
(5.1)
where P is a constant and (x) is the Dirac delta function and
is the frequency.
The double Fourier transform of g(x,t) is
(k,s)
P (s-o)
Consequently the solutlons assume the form
Ul
P27 f [Fl(z,k,S)]s. exp[1(kx + t)]dk
u3
2-
2
f [F2(z’k’s)]s=exp[-1(kx
= f [F2(z’k’s)]s=exp[-1(kx
P
t33
t31
f [F4(z’k’S)]s =mexp [-i (kx
+ t
27
f [Fs(Z’k’S)]s =mexp
+ t) dk
-
)’3
67
-i (kx
f [F6(z’k’S)]s =toeXp[-1(kx
f [FT(z’k’S)ls =exp
(5.3)
(5.4)
+ t)]dk
2--
P
2
m32
+ mt)]dk
(5.2)
dk
+ tot)]dk
(5.5)
5.6
(5.7)
+ t dk
-i (kx
Neglecting the viscous effects, we obtain the corresponding results for the mcropolar
elastic half space with stretch
[Gl(z
k s)]
+ t)]dk
u
P
u3
2-- f [G2(z’k’S)]sffimexp[-1(kx
2
P
s
=exp[-i(kx
f [G3(z’k’s)]s =mexp [-I (kx
+ mt)]dk
+ t) dk
(5.9)
(5.10)
(5.
t33
2--
f [G4(z’k’S)]s=exp[-1(kx
+ t)]dk,
(5.12)
t31
2-- f [Gs(z’k’S)]sfexp[-1(kx
+ mt)]dk,
(5.3)
m32
3
2
P-2,
f [G6(z’k’s)]s =mexp[-1(kx
f [G7(z’k’s)[]s=exp[-1(kx
+ mt)]dk,
+ t)]dk
(5.14)
(5.15)
R. KUMAR, M.L. GOGNA AND L. DEBNATH
370
where
G
l(z,k,s)
o
G2(z’kis)--
G3(z’k’s)
G4(z’k’s)
---
-s
[ikAll
-s
[Sll A 11 e
o
-8
(a21 A2
o
11
11
21
z
+
z
-tk(a21 e
-s
z
+
-s
[2iUkSl 1A 11
[( +
o
z
11
II
s21
s
e)(a21s21A21e
2
A
o
(5.18)
),
-s
e
11
z
-2ik(s
21A21 e
-s21 z
-s31z )],
-s
21
-s21 z
-8
+a31s31e
-s
z
+
a31A31
e
31
31
z
-i6okA41 e
-s
(5.19)
21
(5.20)
-s4z )],
(5.21)
-s z
z
+
3aoS4A41
e
4
],
(5.22)
(5.23)
+
c2
2
(s31
k
s
2
2
2
j
z
2’
2
31
_k
(5.17)
2
c
s21
2
2
-s31z )],
2
+ {(U+a)s 21 +
o
k
31
A31e
(5.16)
(-a)k2-2aa21 }A21e
-s31 z
2
+{ (+a)s31 +(-a)k2-2aa31 }A31e
e
=G7(z,k,s) =- [lok(a21A21e
2
-s21 z +
-s31z ],
s31A31e
31A31 e
o
G6z,k,s)
s21A21e
-s21 z +
a31A31e
2
2U)Sll -,k2} A
e
[{(% +
=’-o
G5(z’k’s)
e
=Cj+c
2
c3
P
c3
2c 2
(5.24)
(s2-T2)
2 2
c c3
2
2
k ),
J
(5.25)
(5.26)
2,3
2
r4t 31 )(q llm21 -q2 ml
r31t4
Y2
+
(r2 t4-r4t2 )(q3
m
-ql Im31
(5.27abcd)
All m31(r21t4-r4t21) m21(r31t4-r4t31)
&21 mll(r31t4-r4t31 )’ A31ffimll(r4t21-r21t4)’
A41fmll)r21t31-r31t21),
( +
2.)si
Xk
2
J=l
(5.28ab)
qJl
-2i
ksj
J=2,3
LAMB’S PLANE PROBLEM IN MICROPOLAR VISCOELASTIC HALF-SPACE WITH STRETCH
371
21.ksj
2
+
(U+a)sj
rjl
(" +
tjl
iSokajl
(5.29ab)
(-a)k2-2aajl
J--2
g)ajl Sjl
3
J=2,3
(5.30)
j=2,3.
(5.31)
If we neglect both the viscous and stretch effects, we obtain the corresponding
displacements, microrotation and stresses for the micorpolar elastic half space.
For
the sake of brevity, we avoid writing down these results.
We next consider the effects of the harmonic torque with its axis parallel to the
z-axis in the form
M[(x-a)
g(x,t)
(x+a)]e
-imt
where M is the magnitude of the torque.
(k,s)
The double Fourier transform of (5.32) gives
to
2
(4.29), we obtain
f [Fl(Z,k,S)]s= msln(ka)e-ikXdk
(5.34)
Q
f [F2(z,k,S)]sftosin(ka) e-tkXdk,
(5.35)
Q
f [F3(z,k,S)]s=tosin(ka) e-ikxdk,
(5.36)
Q
u
3
(5.33)
2iM sin ka (s-m)
Using this expression in the equations (4.23)
u
(5.32)
t33
Q
f [F4(z,k,S)]s= sin(ka)e-tkXdk
(5.3z)
t31
Q
f [Fs(z’k’s)]s=sin(ka)e-ikXdk
(5.38)
Q
)-k
3
where
Q
3
[F(z,k
s)]
stn(ka)e-ikxdk,
f (FT(z,k,S)]s=msin(ka)e-lkXdk’
Q
iM
e
-it
(5.39)
(5.40)
(5.41)
These results are in excellent agreement with those for the cases without viscous
and/or stretch effects which have been discussed by several authors including McCarthy
and Eringen [9], Kumar and Chadha [6] Nowacki and Nowaki [2], and Acharya and Sengupta
[4].
R. K, M.L. G OGNA AND L. DEBNATH
3?2
ACKNOWLEDGEMENT.
This work is partially supported by the University of Central Florida.
REFERENCES
I.
LAMB, H., On the Propagation of Tremors Over the Surface of Elastic Solid,
Phil. Trans. Roy. Soc. London, A203 (1904), 1-42.
2.
NOWACKI, W.
3.
CHADHA, T.K.,
4.
ACHARYA, D. and Sengupta, P.R., Lamb’s Plane Problem
and NOWACKI, W.K., The Plane Lamb’s Problem in a
Micropolar Elastic Body, Arch. Mech. Stos. 21 (1969), 241-251.
Infinite
Semi
Plane Lamb’s Problem in a Semi-lnflnite Micropolar
Elasticity, Gerlands Beitr. Geophyslk. Leipzig, 88 (1979), 407-414.
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in Micropolar Elastic
Bull Polish Acad. Scl. Series Sci.
Tech. 27(1979), 414-427.
5.
ROY, S.K. and SENGUPTA, P.R., Two Dimensional Wave Propagation in a
Micropolar Thermo Viscoelastic Medium,
Techn. 34 (1986), 27-46.
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Bull.
KUMAR, R. and CHADHA, T.K., Lamb’s Plane Problem
Acad.
Scl.
Set.
Sci.
HalfTechn 34(1986), 59-72.
in Micropolar Elastic
space with Stretch, Bull. Acad. Polon. Scl. Sei.
7.
Polon.
$cl.
CHADHA, T.K., KUMAR, R., and DEBNATH, L., Lamb’s Plane Problem
in a ThermoElastic Mlcropolar Medium with Stretch, Internat. J. Math. and Math. Sci. I0
(1987), 187-198.
8.
ERINGIN, A.C., Linear Theory of Micropolar Visco-elasticity, Int. J. Eng.
Sci. 5 (1967), 191-204.
9.
McCARTHY, M.F. and ERINGIN, A.C., Micropolar Viscoelastic Waves, Int. J.
En. Sci. 7 (1969), 447-458.
I0. ERINGIN, A.C., Micropolar Elastic Solids with Stretch, Ari Kitabevi Math.
24 (1971).
II. NOWACKI, W., Theory of Micropolar Elasticity, International Centre for
Mechanical Sciences, Courses and Lectures No. 25, Springer Verlag, Berlin
(1970).