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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.

Medium under Influence of Temperature,

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.

6.

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).