Moving punch on a two-layered viscoelastic medium

Int. J. Engng Sci. Vol. 34, No. 9, pp. 1047-1057. 1996

Pergamon

MOVING

PIi: S0020-7225(96)00004-3

PUNCH

ON

A TWO-LAYERED MEDIUM

Copyright© 1996ElsevierScience Ltd Printed in Great Britain. All rights reserved 0020-7225/96 $15.00+ 0.00 VISCOELASTIC

S. C. MANDAL~" Department of Mathematics, Indian Institute of Scient , Bangalore-12, India; and Jawaharial Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore-64, India A. C H A K R A B A R T I Department of Mathematics, Indian Institute of Science, Bangalore-12, India Abstract--The problem of a semi-infinite punch moving steadily with a constant velocity on the free surface of a viscoelastic strip resting in welded contact with a different semi-infinite viscoelastic medium (which can be taken as a model of a two-layered earth) and producing horizontal shear waves has been studied. The resulting mixed boundary value problem has been solved by using the Wiener-Hopf technique. Solutions have been derived in closed analytic form. Numerical results in certain special circumstances are presented graphically. Copyright © 1996 Elsevier Science Ltd

1. I N T R O D U C T I O N Problems involving the motion of obstacles on the surfaces of viscoelastic media have wide applications in road construction technology and geophysical research. While Galin [1], Gladwell [2], Suhubi [3], Tait and Moodie [4] and others have studied such problems in the classical theory of elasticity, Golden [5-7], Golden and G r a h a m [8, 9] and others have investigated the dynamic excitations of viscoelastic media under the influence of moving obstacles on the bounding surfaces. In a recent paper, Mandal and Ghosh [10] have tackled the problem of a moving punch on a viscoelastic semiL-infinite medium by utilizing the methods involving the use of the W i e n e r H o p f technique, As further generalizations and utility of the W i e n e r - H o p f technique (see [11]), we have undertaken, in the present paper, the study of the problem of a moving punch on the upper surface of the viscoelastic strip resting in welded contact with another semi-infinite viscoelastic medium of different properties as c o m p a r e d with those of the strip. Understanding the dynamic excitations of layered media is of great importance in geophysics, as has been described by Ewing et al. [12] through a wide range of boundary value problems of elasticity, since the Earth can be modelled as such a layered media. The final form of the solution of the problem is presented here in closed analytic form. The known particular case of a single viscoelastic half-plane as considered in [10] is observed to be solvable by using a limiting procedure of a standard type in the solutions of the present general problem. Numerical results depicting the distribution of the stress under the punch as well as the results concerning the stress intensity factor with specific values of the parameters involved are presented graphically.

2. F O R M U L A T I O N

OF T H E P R O B L E M

We consider an infinitely long viscoelastic strip of width h in welded contact with another semi-infinite viscoelastic medium which are set into motion by a semi-infinite rigid punch moving on the upper surface of the strip with a constant velocity v in the direction of the X-axis. The Y-axis is taken vertically downwards into the medium (Fig. 1). t Permanent address: Department of Mathematics, Jadavpur University, Calcutta-700 032, India. 1047

1048

S.C.

MANDAL

~ X

0

and A. C H A K R A B A R T I .--V

i///////////////////////////

L~ X

lh I : ~,n~,lJ~,

L

ll"

Y

Wt2~ 0 - I l l t~-tz) -13 • - 2 3

t~.2, ri2 , ~2,pz

x =X-vt

y=Y Y Fig. 1. G e o m e t r y of the problem.

Horizontal shear waves will then be generated with the displacements along the X and Y directions being zero and only the displacement W " ) = W ( ° ( X , Y, t), i - - 1 , 2 along the Z-direction will exist. The stresses under the punch are denoted by and

0-(o 13 --- o.(Ot¥, 13 ~,zx, y , t)

~ ( o = t.,23 ~(i)l'V tJ23 \~'l y , t)

where the superscript i = 1,2 refers to medium I and medium II (Fig. 1), respectively. The non-vanishing strains are Wl3"~(i)-

1 0 W (°

~(o~23

and

20X

1 0 W (° 20Y

i = 1, 2.

Considering a "standard linear solid" as the viscoelastic model, the stress-strain relations are o~(i) /~a(i) ~ a(i) ~ t"13+t3~(i)=2l£il~+t.tic. 13), 0t

/'~i°13

i=1,2 (2.1)

o~(i)

/.,~(i)

~ ~(i)~

u 2Ot " - - - - ~/ 3J i-u~ 23 -~(i)=2[£il~+t'ti~23)'

i=1,2

where ot~1,/31 a r e positive constants and/xl is the instantaneous elastic modulus of rigidity of the material I. c~2,/32 and/x2 represent similar quantities for material II. The equations of motion are o~(i) u13

~~'t~' ( i 23 )

OX

OY

--+

02w(i)

=p~--

Ot2 '

i=1,2

(2.2)

where pi(i = 1,2) are the densities of materials I and II, respectively. The boundary conditions of the problem are w ( i ) ( x , O, t) = WO, or(I)( " V\2. 3x ,

O, t) = O,

W ( ' ) ( X , h, t) (t)tX, 0"23 ~,

X - vt < 0 X-vt>O

= w(Z)(s,

h, t)

h, t) = o_2( 32 )~..,'x t v ~ h, t)

W(2~(X, oo, t) = O,

- oo < X < =.

(2.3) (2.4) (2.5) (2.6) (2.7)

Since we are interested in the steady-state motion of a punch, it is convenient to define a moving coordinate system (x, y) whose origin coincides with the tip of the punch and whose axes are parallel to the fixed (X, Y)-axes, respectively (Fig. 1).

Moving punch

1049

Hence setting x = X - vt, y = Y, the stresses, the strains and equations (2.1) and (2.2) become become ~r(1) ~(') ,3 = ¢~(x, y), "23 = o'~9(x, y) e(i) _ 13

1 0 W (° - -

..~__(i) - 1 1 t~Vl3 q- /'~ ~ ( i ) __

OX

-v

~(i) _ t=23

(X, y ) ,

20x

[

1 0 W (° - -

oW(i)~

02w(i)

IJitll3--]'gi~--l)7"FOli

OX

0~(;)

(

02W (;)

Ox

\

3x Oy

v 2 3 "F [3i0"(2i~ = ]'gi --13

(X, y )

20),

-F Oli

i=1,2

]'

0W(;)~

i=1,2

Oy ]'

}

(2.8)

and 0n(i) ~(i) 2 32W(i) t........_23 ,-, + z~,~, 23 0x 0y =P;v ax 2 ,

i=1,2.

(2.9)

Also the boundary conditions (2.3)-(2.7) become X< 0

W ( I ) ( x , 0 ) = W0,

o'~)(x, O) = O,

x> 0

(2.11)

W(')(x, h) = W(2)(x, h),

(2.12)

o-O)/~ ~(2)r~ 23 rt.~.~ h ) = t.~23 \.,.., h),

(2.13)

W(2)(x, oo) = 0,

3. R E D U C T I O N

(2.10)

TO W l E N E R - H O P F

- ~ < x < ~.

(2.14)

P R O B L E M AND ITS S O L U T I O N

Taking a Fourier transform of (2.8) and (2.9) we have, (i~v + aI~j)tl ~(J) l3 = ]~](~2U-- i~ai)ff'U), ( i & + /aJ j ) o~' 2. 3v . u ) ---

&(i~v + aj)

j=l, 2}

dl,~,o)

j=l,

dy

(3.1)

2

and i#~

d.7.(/) ~23 -)+ - -

dy

Djv2~2W(J),

(3.2)

j=1,2.

From equations (3.1) and (3.2) we have two equations d2~r(J)

..2 t.ly

_ 2i]-7(j) Y,' ~' =0,

(3.3)

j=l, 2

where j=l,

2

(3.4)

and ci = m / & (J = 1,2). The branches of 39 (J = 1,2) are so chosen that Re(T))>0,

for

-a i
where (v[3,_ ai=\c2 ~)/(1

v2

- ~2j2)"

(3.5)

1050

S . C . M A N D A L and A. C H A K R A B A R T I

For region I, the solution of equation (3.3) can be written as 0
l,V(I)(¢~, y ) = A(~)e-~"Y + B(~)e ~'~y,

(3.6)

where A and B are arbitrary functions of ~. Also for region II, the solution of equation (3.3) bounded as y --) o0 is ff,(2)(~, y) = C(~)e-~,,_(y-h),

y >h

(3.7)

where C is another arbitrary function of ~. Taking Fourier transforms of equations (2.12) and (2.13) we obtain ~r(I)(~, h ) = ~r<2)(~, h )

(3.8)

and

(')+c 23 I ~ h)=

,,~.(2)1~: tJ23 I,~

h).

(3.9)

Equations (3.6)-(3.8) imply that C(¢) = A(¢)e -~'h + B(¢)e r'h.

(3.10)

Now, from the second relation of equation (3.1) the expressions for ,v.(l) .v.(2) can be written ,-23 and 023 as

~<,) , , (a, +i~v)dlg '(') ] 23 (¢, Y) = izl (fl----(+i~v----~ d----~ [ (3.11)

[

-,2" (a2 + i~v)dff '<2) / o'~3'(~, y ) = ~2(-~2 ~ ~ 'J From equations (3.6), (3.7), (3.9) and (3.11) we obtain B(~)e~,, h _ A(~)e_.V, h = _ i.z2 y2(a2 + i~v)(/3~ + i~v) C(~).

(3.12)

tz, y,(/32 + i~v)(a, + i~v) Solving equations (3.10) and (3.12) the values of A(s¢) and B(~) can be obtained as A ( ¢ ) = ~1 C(¢)e r'h [ 11+ -/z2Y2(a2+isrv)(/3-1+i~:v)] -

}

/-~l Y~ (/32 + iCv)(a~ + isCv)J '

[

1

B(~) = ~ C(~)e-~'h

/U'23+2 (a2 _+ i~v)(/3, _+ iev)]

(3.13)

#~ y~ (/32+i~v)(a, + i~v)J "

Substituting the values of A(~:) and B(~) in equation (3.6) we obtain y) = C(~)[cosh{(yffy - h)} L

p(e)r2 sinh{y,(y Yl

h)}],

(3.14)

l

where p(~) =/z2 (a2 + iscv)(/3, + i&) /"~1 (/32 "j- i ~ ) ( a ,

+

(3.15)

i~v)"

We now assume that W°)(x, O) = Wo = Woe% x < O, e > 0 (e will be made to tend to zero finally) = Woq(x) (say),

x>0

(3.16)

Movingpunch

1051

and o-(,)/,. 23\a,) 0) = 0,

x > 0

= Wot(X) (say), x <0,

(3.17)

where q(x) and t(x) are unknown functions such that

q ( x ) - O ( e -k'x)

as x---~oo, k , > 0

and

t ( x ) - O ( e k~)

as x---~-oo, k2>0.

Taking a Fourier transform of relation (3.16), we obtain

Wo Wo +~X +(~), I,V(')(~, O) - V2-~ (e + iO

(3.18)

where X + ( 0 = Jo q(x)ei~x dx.

(3.19)

In equation (3.118) the first term on the right-hand side is analytic in the lower half plane Im(O = r < e and X+(~) is analytic in the upper half plane v > -k,(k~ < a, say). Again taking a Fourier transform of relation (3.17), we obtain

(~(')~ 23I$, O ) = ~Wo Y-(~),

(3.20)

Y-(O

(3.21)

where

t(x)e i~xdx.

Y_(£) represents an analytic function in the lower half plane r < k2. Therefore, ff'(')(£, 0) is analytic for - k l < r < e and ,,23~(l)tc~s,0) is analytic in the lower half plane r < k2. From the second relation of (3.1), we have [(i&+/31 ','T'( 1)1 - [ lZl(i~v + al) ~ ] /tJ23Jy=O--

y=o

from which we get Wo Y-(C) =/.t, X/~

-C(~)y, sinh(y,h) + P(~)72cosh(ylh) Y,

.

(3.22)

Again from relations (3.14) and (3.18) we obtain C(O[cosh(ylh ) + p(~:) y, yz sinh(ylh)] _ X / ~ ( Wo e + iO + ~Wo -~X+(O.

(3.23)

Eliminating C(~) from equations (3.22) and (3.23), we then find that Y_(,) = - K ( , ) [ e + - - ~ + X + ( , ) ] ,

(3.24)

(a, + i~v)[yzP(~)+3fitanh(y,h)] K(~) =/~, y~ (/31 + i~v) I_y~ + yzP(~)tanh(y,h) J"

(3.25)

where

The functional equation (3.24) is the desired Wiener-Hopf equation and it can be handled for solution by using standard Wiener-Hopf arguments (see [11]).

1052

S.C.

MANDAL

and A. C H A K R A B A R T I

We factorize K(¢) as

K(~) = K+(~)K_(~)

(3.26)

where K + ( , ) = / x , [~¢(1- ~21z)+ i( vfl'

,/2

and

13 /

(3.27)

-

U /

with 1 f y z P ( ¢ ) + 3', tanh(y,h)] ] /~±(a) = e x p t + - - I !f~+id"z ogl'yl + y2P(~)tanh(ylh) ~d(~

L

2 n : i J--~+id,,2

where - k , < d l , de
iK+(ie) s¢ - ie

upper and lower half-plane r > - k ,

K+(~)X+(~) + ~ i

(3.28)

J

b~ -- a

and "r
[K +(~) - K+(ie)].

(3.29)

The functions on the right-hand side of equation (3.29) are analytic and non-zero in the upper half-plane r > - k 2 , and the functions on the left-hand side are analytic and non-zero in the lower half-plane r < e (e < k,). Since both the functions are analytic in the strip -k2 < r < e, the principle of analytic continuation states that each of them represents an entire function M(~) in the whole ~ plane. Now the left-hand side of the equation (3.29) approaches zero as I~:l~ ~. It may then be concluded by Liouville's theorem that M(~) = 0, and therefore we obtain Y_(~) = ~i K +(0)K_(~),

as

e----~O,

(3.30)

and i

X+ (~:) = - [K+(s~) - K+(0)], ~K+(~)

as e--~ 0.

(3.31)

Using the representation (3.31) for the function X+(~:) we obtain, from equation (3.18),

ff't')(s¢, 0) =

Wo i K + (0)

V~-~K+(~)'

which, on applying the Fourier inversion formula, gives

W~')(x, 0)

=

-

iWoK+(0)

2x

r~_i~ e-iCr d~

.k~,-i~£K÷------(£'-)' x > 0 (0< 8
(3.32)

Movingpunch

1053

Assuming al < a2, at
W(l)(x, 0)

1fix,

1))x~--l/2---/ri'4fx e-(.+.,)~g_(_i7 _

-

k

c2]

e

$1(7)~(//) 4- $2(7)$4(7) d-

x

ia2)

-Io (7 + a2)(7 + a2 - a,) 'n

s~(7----; ¥ ~(,7---5

(3.33)

7,

where ( ..../z2 JId'l 1

S,(7)

132\1/2 } C2, '

"~{/\7

O/i'~1/2/ OlX~ll2{ ~) +a2 q-T) ~ 7 -}-a2 -I- ? ] ~7+a2 -[- '

$2(7)=(1 -c-~l] v2~'/2{~7 + a 2 - at)|n(7 + a2 +~!)(7 +a2+-~)tan(~'lh), ,~3(7)=( 1-v2~1/2c 2] (7 +a2-aD''z(7+a2+~)(7+a2+~) $4(7)

.....

(

/a2 [d'l 1

' 112{

} C2/

7

\ 7 +az

+O/1) 1/2(

o,,,2:

7+az+?}

P/

~7+a2+

tan(~'lh),

( _ v2~ uz(7 +a2)(7 + a 2 - a l ) 1/2

-,

7+a2 +

k-(-in - ia2) = exp [2- ~(-2--,)/-

dp V

.

_11

_,{Bz(p)] 1 f~

tan-'(B4(p)~ \A4(p)/

]

dP"I-~-cJ0 ( £ T ; T a 2 )

'dp '

l)

---~l) p k p + a , + v ) B 4(p ) = l~l \(1- -

[,P-

1/2( [ tz ~1/2[ P ln[p -I or2 - a m. p + a2 + V]~" ~P + az ~ fl~)tan(~h),

1/2) I/2

1)

~W'~p,np( A3(p)=/'t2 ( 1_~22) I.~

A4(p)= 1-~l,] I'; =

(1

,,~-,,,)"~( p + a 2 + ~"~/ ) kp

kp+a'+a2v/ v2~'2[

)

fl,-a,. ,

v

v

P

v

a ~,/2[

p + a 2 - a l )1/2

P+ -.~

v ,

'

1054

S.C. MANDAL and A. CHAKRABARTI

and •

K+(O) = , . , ~

~i~/4{ 1J~l ~

\C

0~1\1/2 o

_~

~.) K+(O),

I

with [ 1 L ,an " - 1 {A~,B,---;T~) I ('r/)'~ I k+(0) = exp (77 + a2) dn+ 2

dr/

Io

1

tan-'( ~, ~ l

:

/0

(7"/+or,) ~ 2_a, (r/+ al)

Al(~)=l.t.....22(1_v2~ 1/2rll/2(rl+a2)(a2+fll+rl)(a2+°t2+'q) v 1)

r/j,

'

1/

( _ 11£~I/2 (a 2 -t- 'l~)(a 2 -- a I --t-n) 1/2 B,(rt) = 1 c~] tan(q;h), 13

/3z('O) =/z---Zz(1-v--~2'~'/2( r / - a2 + a,,l/2(r/+

(a,

a,)(a,+ ~l+v 71)(a'+ az+u 7/) '/2

+°'+.) 13

It

__ v~ ''~ (a, + ~)T] 1/2 ~/'= 1 - c 2 } (a, + °gl-~- "0)1/2' U

and

:, ( _ v__22~ '/2 (a2 + r/)(r/+ Y,=1c2 ] (

a2 - a,) '/z ),/2 a2+ a~+ r/ 13

Now from relations (3.20) and (3.30) we obtain -(')t~, 0) = ~Wo i K+(0)K_(f) 0-23,. which after Fourier inversion gives rise to the expression (')+x 0"23 I, ~0 ) =

W_2o.K+(0) (j_: = 1~K_(f)e- ,~xdr, Z/~'I

X <0.

Considering a branch cut along the positive imaginary axis starting from f = iaz/v and changing the path of integration from the real f-axis to the path around the branch cut it can be easily shown that If(i ( 1/2 p4 o'[~)(x, 0) = - W°K+(0)e 3~ri/4e a-.rio v p + a 2 _ / 3 , /~+ ip +i/3,

(

~)

UI

/ o _ O/ \1/2

\

~

I

]

X~,(P)~(P) +- ~2(P)J4(P). e(a,-:_.)
(3.34)

1055

Moving punch

-$“‘(p+T+ a,)“*(, +yq(p - _)tan(g,h),

g*(p)= (1

“,(p)=(l-~)“*(p+~+a])“*(p+~)(p-~), j&)

= ;’

(1 _ KJ2Pl/2(,

+ a2 + E?)“2(P

+ (y2

u

)

I’* p + 7-: (

>

tan(y,h),

v2),,2(P+~)(P+ul+y2

(

1-7

9,=

v

’

(Y*-ck!, I’*

cl (

Pfy-

>

and

& V2

II2

( )

q;=

V

+& I’*

a I

(

V

1

I_,

p,-a,

Cl

(

I’*

’

1

V

with

tan-’

1 = --In

?I ( P*(v) >

d77 ,

(3.35)

nz-a’ q+a,+p+& (

V 1

I

and the value of Z?+(ip,Iv) is obtained by putting p = 0 in relation (3.35). In the particular case when h = 0, i.e. when a semi-infinite rigid punch is allowed to move on a semi-infinite viscoelastic medium, we just put (Y]= CY* = LY, p, = p2 = p, y, = y2 = y, pI = p2 = p, cl I= c2 = c and a I = a2 = a in the above results. Hence, using relation (3.33), the displacement W(x, 0) for x > 0, for the special case h = 0, can be found to be in a form given by the relation

w (x,

0) =

F6

e-(Ix

$5

du

(X > 0).

(3.36)

where a=($-:)/(I-$).

Also, from relation (3.34), the shearing stress is given by !$i ~23(X,

0)

=

($

_

;)“2e~~/~

$

“;Iadu

@

-

<

0).

(3.37)

u-V

The above two results agree fully with the results obtained expected.

by Mandal and Ghosh [lo], as

1056

S. C. MANDAL and A. CHAKRABARTI

15

IJ~ =1. 0 Pt

Stomdord L~neor Solid

fl'-~l= 5.0 v-. = 0.1 %

10

-2 ~',','.' " 100

Z

5

0

h"= ,

0.2

I

~

I

0.4

0.6

- - h ' = ~o --- h" = 20

¢= o.6

~'=o.,, *b,-/*

~=S.O

s /i/

~._.2: 10.0

//

131 =0.005

J~

~ =0.'1 ¢1

/ ~

I 0.8

-8

1.0

I

0

v ] C2---~

,

I

0.4

,

I

~

0.6

I

0.8

1.0

X I----.~

Fig. 2. Stress intensity factor N vs v/c2.

4. N U M E R I C A L

I

0.2

Fig. 3. Stress r* vs distance x~.

RESULTS

AND

DISCUSSION

The non-dimensional stress given by

o)

17*-

/*,Wo#i/c, just below the punch (x < 0) and the stress intensity factor (SIF) at the tip of the punch defined by N = l i m (r*), x-~0+

have been computed

numerically and plotted against dimensionless distance xl(=-x~l/cl)

a n d v/c2, r e s p e c t i v e l y , f o r d i f f e r e n t v a l u e s o f t h e p a r a m e t e r s

~ j h / c ~ , / ~ 2 / / ~ , /32/~1, a 2 / a l ,

a~/#, and v/c,. The values of N have been plotted against v/c2 for different values of the dimensionless strip width h*(=[3~h/c~) and for/x2//x~ = 1, fl2/fll = 5, v/c~ = 0.1. It is found from the graph (Fig. 2) that SIF increases with the increase in the values of h* (=10, 50, 100). In all cases the SIF increases with the increase in the value of v/c2, attains a maximum value and then decreases gradually. The stress r* has been plotted against x~ (Fig. 3) for different values of v*(=v/c2). It is interesting to note that the value of stress r* first decreases and then increases with the increase in the values in v*. Also, the value of r* is higher for lower values of strip width h*. Finally it is found that the value of r* increases with the increase in the value of x~ and, after attaining a maximum value, decreases and tends to zero. Acknowledgement--The first author is grateful to Prof. M. L. Ghosh for valuable discussions on this problem. We are also grateful to the referee for his invaluable suggestions for improving the presentation of the paper.

REFERENCES [1] A. I. GALIN, Contact Problems in the Theory of Elasticity (Edited by I. N. SNEDDON). North Carolina State University, Raleigh (1961). [2] G. M. L. GLADWELL, Contact Problems in the Classical Theory of Elasticity. Sijthoff and Noordhoff, Alphen and den Rijn (1980).

Moving punch [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

1057

E. S. SUHUBL Bull. Tech. Univ. Istanbul 25, 71 (1972). R. J. TAIT and T. B. MOODIE, Int. J. Engng Sci. 19, 224 (1981). J. M. GOLDEN, Q. Jl Mech. Appl. Math. 32, 25 (1979). J. M. GOLDEN, J. Aust. Math. Soc. 21B, 198 (1979). J. M. GOLDEN, Acta Mech. 43, 201 (1982). J. M. GOLDEN and G. A. C. GRAHAM, Boundary Value Problems in Linear Viscoelasticity. Springer, Berlin (1988). J. M. GOLDEN and G. A. C. GRAHAM, Q. Jl Mech. AppL Math. 49, 107 (1996). S, C. MANDAL and M. L. GHOSH, Indian J. Pure Appl. Math. 21, 847 (1990). B. NOBLE, Met~ods Based on the Wiener-Hopf Technique. Pergamon Press, New York (1958). W. M. EWlNG, W. S, JARDETZKY and F. PRESS, Elastic Waves in Layered Media. McGraw-Hill, New York (1957). (Received 30 December 1994; accepted 3 December 1995)