Stress concentration due to a finite rigid ribbon in the presence of a finite crack—II. SH case

Int. J. Engng Sci. Vol. 30, No. 4, pp. 419-429, 1992 Printed in Great Britain. All rights reserved

002~7225/92 $5.00+ 0.00 Copyright @ 1992Pergamon Press plc

STRESS CONCENTRATION DUE TO A FINITE RIGID RIBBON IN THE PRESENCE OF A FINITE CRACK-II. SH CASE K. VISWANATHAN Defence Science Centre, Metcalfe House, Delhi-110 054, India

U. SHARMA Lyailpur Khaka College for Women, Jalandher City-144 001, India Abstract-The stress intensity factors in the regions close to rigid ribbon like inclusions caused by the presence of a crack are studied. Here we treat the SH-Wave motion taking the crack as well as rigid ribbon to be finite. Both loading over the crack face by prescribed tractions as well as excitation due to a plane wave incidence are considered. Several numerical results are computed.

1. INTRODUCTION

The interaction of elastic fields with a crack or a rigid inclusion forms an important branch of study for application to material behaviour under given loads and also to many geophysical applications, en~ronment~ and non-destructive evaluation studies (see, e.g. [l-5]). While many authors have studied a single or several crack cases [6-81, the role played by rigid inclusions even in the form of thin ribbons have scarcely been studied in detail. Recent studies [9, lo], on source models in Seismology and the need for noise reduction in environmental protection have regenerated interest on this front. Also, the interaction of cracks (or soft materials) and rigid ribbons can occur in such applications. In recent papers 111,121, the authors have studied the interaction between a semi-infinite crack and a finite rigid ribbon under several loading conditions. A more practical model, however, will be that of a crack and ribbon both of finite dimensions. Therefore, we consider in this paper a model of a finite crack and a finite rigid ribbon under the conditions of loading over the crack faces considered alon~ith (or without) a plane wave incidence. The general case of an inclined crack or a rigid ribbon will be reported in our future communications. Numerical results, particularly for the stress intensity factors associated with the inside edges of the rigid ribbon are evaluated. These are expected to help in forecasting the possible debounding of such rigid inclusions for applications both in material behaviour areas as well as in the earth-quake analysis.

2. FORMULATION

OF THE

PROBLEM

We consider a uniformly loaded finite crack and a finite rigid ribbon in an isotropic, linear, homogeneous, and perfectly elastic medium with Lame’s instants (d, p) and density p. A Cartesian co-ordinate system is introduced such that the x-axis lies along the line of the crack and the rigid ribbon as shown in Fig. 1. The time-harmonic SH-wave i&“) = A expf-ik(x

cos 0, + y sin 0,) + iot]

Of

is incident obliquely on the crack and the rigid ribbon. Let the scattered field be given by w(sc)

2:

I

-

--m

[p(f) f QGi)lexp(-ib- v Ivl) d& 419

(2)

420

K. VISWANATHAN

and

U. SHARMA

ttttt_o

A

kIIk4

B 2-x

d_ -d+l---+

c-c-

Fig. 1. Geometry

The f sign in (2) correspond

of the problem

to y S 0, and y = (5” - kZ)“2, k = o/c,,

Re( v) 2 0, C.Y= (p/p)1’2,

(k = k, + ik2).

The total displacement

(3)

field is than given by W = WW + W(W

(4)

The P(c) term in equation (2) corresponds to symmetric antisymmetic part of the solution about the line y = 0. The boundary conditions are taken as (i) p$

part and the Q(c)

= FPOeiw, (y = 0, -c
term to the

(5)

(ii) w = 0, (y = 0, d d + 1). (7) Here & is a constant and d denotes the distance between the crack tip and the nearer edge of the ribbon. The boundary condition (5) yields,

Here the relevant part of equation (7) is also considered with S:(c) denoting the unknown contribution due to the possible shear stess discontinuity across the ribbon and T(c) denoting the contribution due to the unknown (but continuous) shear stress from the entire region along y = 0. These functions are to be solved as part of the solution of equation (8). Comparing the like terms on the both sides of equation (8), we get (9)

vvQ(C) = -

2ng!floor,,) (1 - e-‘(C-k ‘OS‘O)‘)-

From the boundary condition (6), we obtain n [P(g) f Q(I;)]e-‘@ d< = -AeCikxcosC)o, I -m

d
(10)

T(g).

y =o.

(11)

Stressconcentration Comparing obtain

the symmetric

1

J-m

parts on the both sides of equation

and antisymmetric

P(f‘)e-‘”

421

d
dc = -AewikrCOSeo,

y = 0,

(11), we

(12)

00

I

-m

Q(l;)e+@

d
dc = 0,

Finally, the continuity of w for x < -c, O
I

m _-mQ(5)e-‘@ df‘ = 0,

y=O.

x>d+l;y=O,

xc-c,

gives x>d+l;

O
03)

y =o.

04)

From equations (13) and (14), we obtain

I

x >o;

XC-C,

_: Q(S;)e-i’x d5 = 0,

y =o.

(15)

Therefore, Q(c) = u*(5),

(say),

(16) with U(x)==0

for

x>O

3. SOLUTION

and

xc-c.

FOR Q(c)

Prom equations (10) and (16), we get rm

J.-coW*(l;)e-‘@

dg = -iAk sin 90e-ikxmseo,

Using the following integral repre~ntation

-c
(17)

for U*(c)

U*(g) = 1-1 eir*’ U(x’) dx’,

(18)

we can show that the equation for finding the function U(x) is given by U(~)H~2~(k Ix - ~1) drj = f x(x) Jd

(19)

where Me-h

x(x) =

co5

k sin 8, 0,

B. 9

f%*o0),

(20)

(&J=O).

(The derivation of the above result is straightforward and hence the details are omitted.) The numerical solution of equation (19) finally leading to the function Q(c) in equation (16) will be similar to the procedure adopted for the function P(c) and f( q) given later in Section 4 [equation (30)]. Since our main aim is the discussion of stress singularities at the tips of the ribbon, we do not further discuss the solution due to Q(t) here.

K. VISWANATHAN

422

and U. SHARMA

4. SOLUTION

FOR

P(t)

From equations (9) and (12), we see that

= Aepikx cOsall, (d
(21)

Let S:(c)

=2

i’+‘f(s)ei’n

dn

(22)

in accordance with the definition of ST(c) as arising from a distribution d
defined

over

e-i5r f(rl)e”‘drl

= Ae-ikx

>

d5 cos

en

_ PO

+2nip

~0

I

e-X.yl

_

(d
Equation (23) can be written as (assuming the validity of the interchange in the usual sense; see, e.g. Carslaw [13], pp. 198-200) -

1

of integration

(23) orders

d+l

2J’t I d

f(q)Hk2)(k

(x

-

~1)

dn =2iAe-‘*xc~so+p,Z(n) nC1

(24)

where (25) = I,(x) - 1,(x + c),

(say),

(26)

with (27) Evaluating the integral Z,(x) along the contour as shown in Fig. 2, we get a

Z,(x) = 2ie’”

I

o G(k

+ iFy--j

dt

(28)

=

Substituting

(29) in equation (24), we get ik ~0s%WW+O+d) + $[

z,(x),

(_ 1 < x < 1)

(30)

423

Stress concentration

k

I

Fig. 2. Contour for the evaluation of the branch integral.

where (31)

=--

4i

k [

e ik(lt2(X+l)+d)

v

;(X

+ 1) + kd)

_ eiW+Vf2)(~+l)+4 v kc + t(x

+ 1) + kd

>I

(32)

and (33)

5. EVALUATION

OF f(q)

AND

#(q’)

The Hankel function ZYZh2)(z) can be expressed as follows [13] H$2)(z) = Jo(z) - iY,(z) =J,(z)[l-5

(y + ln(z/2)] + zS(z)

(34)

where y = 0.5772 - - - (Euler’s constant), S(z)=$~(;)yl+;+;+...+‘)

(35) m

(36)

and (37) ES 30:4-B

K. VISWANATHAN

424

and U. SHARMA

Let us define a new function @?‘(z) = E@(z) + 2i In(z) Jr that is regular at z = 0. Then, the equation (30) becomes

4iA z-e 1

-ikxcos8”(1/2(X+l)+d)

+&?!I,(X)

(-1


<

1)

wl

(39) where

(40) We next solve equation (39) for $(q ‘) by a well known expansion technique (see [ll]). As in the last reference cited, the solution of equation (39) is accomplished by the expansion of @(r]‘) in Chebyshev polynomials of the first kind [14]

where T,(x)

= cos(m

cos-’ x).

(42)

The solution of equation (39) is then given by (Gj)

=

(43)

(Wj)-‘(gi)

where -2 2 gm - (M + 1) i=o

(44)

g(xi)Tm(xi).

[g(X) being defined in (39)] (45)

(46)

v, =

--JGln(2),

n = 0,

-nln,

n >O.

1

0.5,

m=O

ym= 1 1.0.

m>O.

This completes the solution of $(n’) and hence f(q).

(47)

(48)

Stressconcentration 6. CALCULATION

OF THE

STRESS-INTENSITY OF THE RIBBON

425

FACTORS

AT THE

EDGES

The stress in the region of the ribbon is given by 5

=

P4w Pm) = qqy3j

Therefore, the dynamic stress-intensity found from the formula K = lim [a(lx X--+X0 = lim

*+x0

d
7

y=O.

factors at the inside tips of the rigid ribbon can be

-x,~)~‘%&,

0)]

(.x0= d, d + I)

J/%( Ix - xo])‘n

6 = lim 2 x+.Q

(Ix - ~oV

(X = *I)

d

44(X) (x-d)(d+Z-x)

I*

(50)

Therefore, K,,=S.I.F.

atx=d (51)

and Ki=S.I.F.

atx=d+l (52)

7. NUMERICAL

RESULTS

AND

DISCUSSION

The stress-intensity factors (S.I.F.) at the tips of the rigid ribbon due to the shear stresses at both the tips of the ribbon as given by equations (51) and (52) are numerically evaluated for the plane wave incidence, crack loading and for the combined loading (due to plane wave loading alongwith crack loading). Graphs are drawn for the Poisson’s ratio v = 0.25 and for the various combinations of the different parameters; namely, length c of the crack (length 1 = 1 of the rigid ribbon), angle of incidence of the plane wave (&), d, the distance between the crack and the nearer tip of the rigid ribbon and the frequency parameter (kl). We have used the notations S.I.F.l and S.I.F.2 respectively for the S.I.F. at the nearer and the farther tips of the ribbon. For the various loading conditions, we have obtained a few results which are summarized as follows. (i) Plane wave loading Figure 3 shows the values of the S.I.F. computed for different values of the angle of incidence (~9,). We observe that as the angle of incidence & increases, the S.I.F. at the left tip of the ribbon decreases and that at the right tip of the ribbon increases. Also we observe that the S.I.F. is not affected by the distance (d/l) between the crack and the ribbon, because for the plane wave case, the ribbon does not influence the crack as long as they are along the same straight line.

426

and

K. VISWANATHAN

U. SHARMA

L

I

I

I

I

- -1 30”

0

0.2

0.4

0.6

0.8

1.0

kl Fig. 3. S.I.F. at the inside edges of the ribbon for the plane wave insidence vs kl and for fixed B,,.

0.5 -

-7 I

0.4 -

i;i Ii L w ,:

0.3 -

4

0.2 -

i 0.1 -. rJY 0

0.1

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

kl Fig. 4. S.I.F. at the inside edges of the ribbon for the case of crack loading vs kl and for c/l = 0.8, d/l =O.O, 1.0, 2.0 and 4.0.

1.0 -

1.6

-----__ v)

---0 0.2

-

---

I

I

0.4

0.6

---

0.4 .0.4

-

8::

I

I

0.6

1.0

kl Fig. 5. S.I.F. at the inside edges of the ribbon for the case of crack loading only vs kl and for c/1=0.2, 0.4, 0.8, 1.6andd/l=O.O.

Stress concentration

427

1.6

0.6

0.6

kl Fig. 6. S.I.F. at the inside edges of the ribbon for the crack loading only vs kl and for c/l = 0.2, 0.8, 1.6 with d/f = 2.0 (- left edge, ----- right edge) and d/l = 4.0 (-x-xleft edge, ---x---x--- right edge).

(ii) Crack loading Figures 4-6 show the values of the S.I.F. computed for different wave numbers. Compared to the plane wave case, the S.I.F. values are now are much lower but the shape of the curves are more complex with fluctuations. We observe that as the crack to ribbon distance (d/l) increases, the S.I.F. decreases generally, as expected. From Figs 4 and 5, we notice that when there is no gap between the crack and the ribbon (d/l = O.O),the S.I.F. at the nearer tip of the ribbon increases while that at the farther edge decreases rapidally. This is true even as the wave number kl increases. 4

823 0.6

Ii

uj

1 0.6 0.4 0.2 01

0"

I

I

I

I

I

I

I

I

I

I

15"

30"

45"

60'

75'

90'

105'

120"

135"

150"

I

I

165* 180'

BO

Fig. 7. S.I.F. at the inside edges of the ribbon for the combined loading vs &, and for d/l = 1.0, c/l = 1.0, kl=O.2, 0.4 and 0.8.

428

K. VISWANATHAN

and U. SHARMA

4

T-3 I I

ci

I 0.2 . 0.4

\’

q\ \\ \\

- - - -

0”

I

I

I

15”

30”

45’

I 60’

1 I 75” 90”

I 105”

1 120"

I 135’

I 150’

1 165’

I 180”

Fig. 8. S.I.F. at the inside edges of the ribbon for the combined loading vs 8, and for d/l = 1.0, c/1=2.0, kl=O.2, 0.4and0.8.

Figure 4 further confirms that for larger values of d/l (1.0, 2.0 and 4.0), the orders of S.I.F. are much lower as compared with d/l = 0.0. From Fig. 5, we also observe that the increase in the crack length results in the increase in the S.I.F. at both the edges. (iii) Combined loading (d ue to plane wave incidence alongwith crack loading) For the combined loading, we observe that the preceding general observations are still true. However, an interesting feature emerges when we plot the variations as functions of &,. 4

T-3

- 1.6,

I 0.6,

I

si

. \

\

'\

1.6

0"

15O

30'

45'

60"

75"

90'

105'

120'

135"

150" 165' 160'

00

Fig. 9. S.I.F. at the inside edges of the ribbon for the combined loading vs 6,, and for d/l = 1.0, kl=O.2, c/l =0.2, 0.4, 0.8 and 1.6.

Stress concentration

429

Figures 7-9 show the values of the S.I.F. computed for different 0,. We observe that the S.I.F. attains its maximum value at the near grazing incidence, while all the graphs show a minimum value of the S.I.F. around 90” - 105” range. Also, we see that the S.I.F. at the right edge falls steeply with 13~in the range 0” s &, G 75” - 90”, while the S.I.F. at the left edge decreases less rapidally in the range 0” =Ze0 s 75” - 90”, and is considerably small for e0 5 75” - 90“. We further observe that the S.I.F. at the left edge of the ribbon is always less than the S.I.F. at the right edge for almost all angles and for all values of the wave number (kl), crack length (c/l) and distance (d/l) between the crack and the ribbon. From Figs 7 and 9, we observe that as the crack length increases, the S.I.F. at both the edges increases for all wave numbers (kl) and for e0 6 90“.

8. CONCLUSIONS

The results obtained in this model obviously suggest that there are significant influences of the crack face stresses on the S.I.F. at the rigid ribbon particularly for the smaller and larger values of the angle of incidence &, where we observe the drastic affect of the crack face stresses on the rigid ribbon.

REFERENCES Y. H. PA0 and H. F. TIERSTEN (Eds), Applications of Elastic Waves in Efectrical Devices, NDT and Seismology. Report of NSF Workshop, Northwestern University (1976). A. D. RAWLINS, J. Sound. Vib. 45(l), 53 (1976). D. A. HILLS and M. COMNINOU, Int. J. Solids. Struct. 21(4), 399 (1985). K. VISWANATHAN and J. P. SHARMA, Proc. Ind. Nat. Sci. Acad. 51,802(1985). K. VISWANATHAN and J. P. SHARMA, in IUTAM Proc. on Modem Problems in Wave Propagation (Edited

[l] J. D. ACHENBACH, [2] [3] [4]

[5]

by J. MIKLOWITZ and J. D. ACHENBACH), p. 171. Wiley, New York (1978). [6] J. H. M. T. VAN DER HIJDEN AND F. L. NEERHOFF, ASME J. Appl. Mech. 51,646 (1984). [7] L. M. KEER, W. LIN and J. D. ACHENBACH, ASME J. Appt. Mech. 51,65 (1984). [8] H. SO and J. Y. HUANG, Int. 1. Engng Sci. 26, 111 (1988). [9] B. V. KOSTROV and S. DAS, Geophys. J. Astr. Sot. 78, 19 (1984). lo] S. DAS and C. H. SCHOLTZ, BSSA 71, 1669 (1981). 111 K. VISWANATHAN and U. SHARMA, Inf. 1. Engng Sci. 25,295 (1987). 121 K. VISWANATHAN and U. SHARMA, Inf. J. Engng Sci. 26,267 (1988). 131 H. S. CARSLAW, introduction to the theory of Fourier’s Series and Integral. Dover, New York (1930). 141 L. FOX and I. B. PARKER, Chebyshev Polynomials in Numerical Analysis. Oxford University Press (1968). (Revision received and accepted 23 August 1991)