Pull-off test for a cracked layer bonded to a rigid foundation

Int. J. Engng Sci. Vol. 25, No. 2, pp. 213-226, Printed in Great Britain

OC20-7225/87 Pergamon

1987

$3.00 + 0.00 Journals Ltd

PULL-OFF TEST FOR A CRACKED LAYER BONDED TO A RIGID FOUNDATION MEHMET Associate

Professor

RUSEN

GECIT

of Mechanics, Mechanical Engineering Department, & Minerals, Dhahran 31261, Saudi Arabia

University

of Petroleum

Abstract-This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semiinfinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.

1. INTRODUCTION

The structural strength of materials depends to some extent on the size and orientation of flaws existing in the medium. Even a normally ductile material may behave in a brittle manner if it contains cracks or other flaws which are sufficiently large. Bonded materials tend to have flaws due to the complexity of shape, chemical dissimilarities and assembly procedures. These flaws, under load, may develop into cracks with a resulting brittle failure. Irwin [l] claimed that the stress field in the vicinity of a crack tip can adequately be defined by a single parameter proportional to the stress intensity factor. When the intensity of the local tensile stresses at the crack tip attains a critical value, a previously stationary or slow-moving crack propagates rapidly. This critical value defines the “fracture toughness” and it is assumed to be constant for a particular material. Almost every engineering design requires connections between several components. A serious disadvantage of mechanical connectors, such as screws, rivets, or welding, is that they do not distribute the load uniformly and hence result in large local stresses. This problem can sometimes be disregarded by joining components adhesively. One of the major problems with adhesives is the determination of their strength characteristics. Many standard test methods have been developed on adhesives (see, for example, [a]). However, predicting the performance of the bond in actual applications is not straightforward using a strength of materials approach with these standard tests. The principles of fracture mechanics can be employed for this purpose. Often new tests or modifications of existing tests are required to obtain the related fracture mechanics properties. Consider a linearly elastic isotropic and incompressible layer bonded to a rigid foundation along its entire bottom surface (Fig. 1). The layer is under the action of an upward force applied through a rigid strip bonded to its top surface. Two different crack configurations are considered. The layer contains either a finite central crack or two semi-infinite external cracks. The problem under consideration may represent a typical pull-off test geometry for adhesives, such as Solithane 113 (a polyurethane elastomer with a Poisson’s ratio of v = 0.499) [2,3]. The untracked layer problem and the problem of a cracked layer tensioned between rigid cylinders have been treated recently by the author [4,5].

2. FORMULATION

OF

THE

FINITE

CRACK

PROBLEM

Consider the elastostatic plane strain problem for a layer shown in Fig. l(a). Material of the layer is assumed to be linearly elastic isotropic and incompressible. The body force is neglected. The layer of thickness h is perfectly bonded to a rigid foundation along its bottom surface and to a rigid strip of thickness 2a on its top surface through which the tensile force P = 2ap, is applied. There is a longitudinal crack of length 2c in the layer a distance of b away from the bottom surface. Due to symmetry about x = 0 plane, it is sufficient to consider the problem in x > 0 ES25:2-F

213

M. R. GECIT

214

P =20po

(a) t

k20

t : /,,/

-4

c

++

//////////,////////,///////////

-

x

P = 2upo

(b) t

i

/,/,/////// Fig. 1. Cracked

I-20-I

I h

AY 1

I--2c-l //////////,////////

i

layer loaded

through

\

c

x

b ////,

a rigid strip: (a) Finite crack, (b) Semi-infinite

only. Under these circumstances, the governing equations solved subject to the following boundary conditions

u(x,O) = 0,

u(x,O) = 0,

-

u(x, h) = 0,

v(x, h) =

a,@, h) = 0,

~&G h) = 0,

L&J,

where u and u are the x and y components u0 can be determined from the equilibrium

of the plane elasticity

(0 < x < m), (0 d x < a),

must be

(la, b)

(2a-d)

(a < x < CD),

~x~(x, b) = 0,

Q,(X, b) = 0,

cracks.

(0 d x -=c c),

of the displacement condition

Pa, W

vector, and the constant

s a

P a,(~, h)dx = 2 = up,.

(4)

0

Due to the presence of the longitudinal crack, the elastic layer can be treated as being composed of two parts, namely 0 < y < b and b < y < h. Solution of the governing equations for the plane strain problem can then be written in the form [4,6]

pi = i

oz [(Ai + ryBi)e-‘Y + (Ci + ryDi)erY]sin(rx)dr, s

t+ = z om {[A, + (1 + ry)Bi]e-‘Y - [Ci - (1 - ry)Di]erY)cos(rx)dr, s

(h

b)

Pull-off

test for a cracked

4P m tf-(Ai zXYl.=lr s 0

+

layer bonded

rJJBJewrY

215

to a rigid foundation

(Ci + ryIIi)ery]sin(rx)dr,

+

4P cnY{- [Ai + (1 + rJJ)Bi]ePrY OYi= -n s 0 - [Ci - (1 - ry)Di]ery}cos(rx)dr,

@a, b)

where the subscript i assumes values 1 and 2 for the lower and the upper parts of the layer, respectively, and p is the shear modulus. The unknowns Ai - Di, (i = 1,2), are determined from the boundary and continuity conditions. At y = b, solutions for the lower and the upper parts must be matched to satisfy the following continuity conditions oylk

(2) and (8) may be replaced

(c < x < co).

0, b)

@a, W

by

&u(x, h) = 0,

&u(x,h) = 0,

4 = Plc4

&k 4 = PzW7

+,

(0 < x < co),

u,(x,b) = u,(x, b),

u,(x, b) = u,(x, Q The conditions

b) = gyz(x,b),

(0 d x < a),

Pa, b) Wa, b)

(0 < x < co),

such that Plb)

= Pz(4 =

07

Ula,W

(a < x < co),

and by

&Cuzk4 -

u,k @I = G,(x), Wa, b)

(0 d x < co), ;

Cuz(x, 4 - u,(x, WI = Wx),

such that G,(x) = G,(x) = 0,

s

(c < x < co),

U%W

(i = 1,2),

(14a, b)

c

Gi(x)dx = 0,

--c

respectively, by introducing new unknown functions pl, pz, G, and G,. These four unknowns are determined by using the conditions (3) and (9) which result in the following system of four singular integral equations

(-c
WAW

+

; S[ -(I

6 mn+2+/( t--x

Cm= 1,2), mn + 2(x,

t)

1 I POW

(-a


(15a-d)

=

0,


(m = 3,4),

216

M. R. GECIT

where 6,” is the Kronecker

delta and the Fredholm

s m

k,t,,(x,4 =

kernels

k,,, (m, n = l-4), are given by

K,,(x, t, r)e-‘(h-b)dr

0 1 + 2(1 + 2r2h2)e-2rh + e-4rh’

(16)

(m, n = l-4)

where

K,,

= [-

- 2rh + 2r2h2 - 4a,r*hb)

1 + 2a,a3 - a,(1 + 21% + 2r2b2) - 2a,(l

- 2a,u,

+ a:(1 - 2rb + 2r2b2)/u4]u8u1,,,

+ (1 + 2u,u,)u~

K12 = 2[af + u4r2b2 + 2u,r*h(h - 2b) + ufu: + u$*b2/u4]u9u1,,, K13 = a,(1 + a,)(1 + us)us + 2(u4 - u&,r*hb, Kl4

=

(a3

f

u4”6

+

a5a7

+

a2a4a5b9,

K,,

= [- 1 - 2u,u2 - a,(1 - 2rb + 2r2b2) - 2u,(l

K21

=

--Kl,,

+ 2rh + 2r2h2 + 4u,r*hb)

- 2u,u, + (1 - 2u,u,)u~ + ~$1 + 2rb + 2r*b*)/a,]u,u,,,

K2, = (a2+ K24 = K,, K32

=

u4u7 + a54

=

K,,

2K23,

a, =r(h-b),

K41

=

K,,

= K,,

u2=

1 +a,,

u7 = u6 - 2rh - 2rb,

2Kl4,

K42

=

-2K24,

a3 = 1 -a,, u6 = 1 + rh + rb + 2r*hb,

(18)

us = sin(t - x)r, a,,

The system of singular (14) and

+ 2rh + 2r2h2 + a,),

+ 8u5u,uIorh,

u5 = emZrb,

a, = cos(t - x)r,

K3i = -2K,,,

= -2u,u,u,,(l

r*h* )

-K34,

a4 = emzrh,

a3a4a5)a9,

- 4(~4 - u,)u,r*hb,

K 34 = 4u 5910 a a K43

+

(1 7)

= e

-r(h-b)

integral

equations

(15) must be solved subject to the conditions

s

(4),

(I

-0

For convenience

in the numerical

z,Jx, h)dx = 0.

scheme, introduce

the following

x = uw,

t = us,

C-a -c lx, 4 < al,

x = cw,

t = cs,

c-c

< (x,t) < c].

(19)

dimensionless

variables

(20a-d)

Pull-off

test for a cracked

layer bonded

217

to a rigid foundation

Then, eqns (4), (14), (15) and (19) may be rewritten as

+ cIc,,(cw, cs)]pG,(cs)

&

(-1

k&w,

4G(4


6 z

+

+ ak,,+ Jew, as)p,(as))ds

(m = L2),

1I

+ akmn+2(aw,as) p,(m) ds = 0

(-l
s -1 1

= 0,

s

(m = 3,4),

(21a-h)

-1 1

G,(cw)dw

=

0,

p,(aw)dw

= 306 1my

(m = 1,2).

One may observe that some of the kernels have Cauchy-type singularity at w = s due to the term (s - w))‘. Therefore, solution will be sought in terms of sectionally holomorphic functions [7]. The unknown functions G, and pm, (m = 1,2), will be singular at w = k 1 [S-lo]. This singular behavior may be determined by writing

< Re(l)

G,(cw)= g,W(l - w’)- 'PO/P,

[0

P*h4

CO< Wy) < 11,

= &+*W(l

- w2)-yPo,

< 11,

(m = L2),

(22a, b)

(m = 1,2)

(23a, b)

where g,, (m = l-4), are Holder-continuous functions in [ - 1, 11, and by examining the singular integral equations (21a-d) near the end points w = + 1. Following the complex function technique outlined in [7] and using the procedure described in [l 11, one can obtain the result that 1 = y = 0.5

(24a, b)

which is in agreement with previous results (see, for example, [9,12]).t Now one can make use of the Gauss integration formula given in [ 131 and replace eqns (21) by

+ c,,kmn(cWj, c,Si)]g,(si) = O,(m = 1,2; j = 1,. . . , N - I), c,k,,(awj,

+

c,sJ]gJsJ

5 diglAsJ = ida,,

(25)

= O,(m = 3,4;j = 1,. . . , N - I),

(m = l-4),

i=l

tIf the layer material is compressible (i.e. if v < 0.5), y = 0.5 f (i/2x)ln(3-4v), v being the Poisson’s ratio. In this case, the scheme for numerical solution becomes entirely different [14,15] which is beyond the scope of this paper.

218

M. R. GECIT

where d, = d, =

di = 1 N-

’

2(N - 1)’

(i = 2,..., N - l),

1’

(i = 1,.

)

(j = l,...,N

)

c3

4. EXTERNAL

, N),

=

c4

=

(26)

- l),

a.

CRACKS

Now,

consider the problem shown in Fig. l(b). There are two semi-infinite cracks at y = b in the layer such that the net ligament has a width of 2c. In this case, the boundary conditions (3) and (8) must be replaced by

(0 < x < c),

u,(x,b) = 4x, bh

u,(x, b) = u,(x, b),

6% b)

(c < x < a),

a,(~,b) = 0,

G’ga,b)

respectively. Defining

, (0 6 x < a),

P&I = ~yk b),

PAX)= g,(x,b),

(2% b)

instead of G,(x) and G2(x) defined in (12), and following a procedure similar to the one used in the previous sections, one can reduce the formulation to the following singular integral equations

+ Rij(w, s)]pAccjs)ds= O,(- 1 <

~

W <

l),

(i = l-4),

(30aad)

where ‘(h-b) + $ Tje-zrb

Rijw, S) = Ctj

S14 = -2a,(l

+ aJato)a8, S22

=

SII

s,,

=

-s,,,

S24

=

-

s33

=

S,,P,

s34

=

-s,2/2,

s42

=

s24/2,

s4,

=

-s,,,

2h

-

(i,j = l-4)

dr,

S,, = -4a:a,aIo,

S,, = 2(af + a: - a:Ja8alo,

S13 = 2(-a,

1

-

s23

4aIa8aIo, %I

a2dda8,

- a:&,,

=

&2

s,,/z s4, s44

= =

=

s14, =

-s,,/z

-s,4/2,

s22/2,

T,, = T,2 = T13 = T14 = T21 = T,, = T23 = T24 = T31 = TX2 = T41 = T42 = 0, Tj3 = -(l T43

=

-

+ 2rb + 2r2b2 -I a5)a8, T34,

T44 = T33 + 4rba,,

A = 1 - 2(1 + 2af)af, + a:,, aI 7 a2_= a,

Ts4 = 2r2b2a,a,,

aJ = a4 = c.

V = 1 + 2(1 + 2r2b2)a, + a$,

(31)

Pull-off test for a cracked

layer bonded

to a rigid foundation

219

Note that the consistency conditions (14) are, in this case, replaced by equilibrium conditions

s

s c

c

,WYx

p&)dx -C

= Zap,,

--c

= 0.

(32a, b)

It can be shown that the singular behaviors of p3 and p4 are similar to those of p1 and pz. Hence, one can write Pm(X)= P&W)

=

where g,, g, are Holder-continuous

g,(W - w21"2Po,

(m = 3,4),

(33a, b)

functions in [ - 1, 11.

5. NUMERICAL

RESULTS

Some of the calculated results for the finite crack problem are shown in Figs 2-9 and Table 1. Figures 2 and 3 show the variations of the normalized stresses pl/po and p2/p0 along the rigid strip-layer interface for b = O.Sh, c = h. Here one may note that the case

-21 0

I 0.25

I 0.5

I 075

1.0

x/a

Fig. 2. Normal

stress between

the layer and the rigid strip for b = 0.8h, c = h (finite crack).

x/a

Fig. 3. Shear stress between

the layer and the rigid strip for b = 0.8h, c = h (finite crack).

220

M. R. GECIT

Fig. 4. Normal

stressbetween the layer and the rigid strip for a = h, b =

-0

0.25

0.5

0.75

0.7h (finite crack).

1.0

x/a

Fig. 5. Shear stress between

the layer and the rigid strip for a = h, b = 0.7h(finite crack).

a/h = 0.01 approximates that of a concentrated tensile force directly applied to the layer. Note also that, since the stresses are normalized using po, the resultant force P = 2ap, is proportionai to u/h ratio for a fixed layer thickness h. As can be observed from Figs 2 and 3, variations of p1 and pZ over the normalized interval lx/al < 1 depend heavily on a/h ratio. As the end point x = a is reached, both stresses tend to infinity. p2 tends to - 00 when a/h ratio is less than 1.07415 and to + co when it assumes greater values for b = 0.9h and c = h. For a/h = 1.07415 the shear stress vanishes at x = a. Figures 4 and 5 show p1 and p2 for a = h and b = 0.7/t. These figures indicate typically the dependence of interface stress distributions on the crack length. With increasing crack length, distributions become more disturbed. As the crack length exceeds the width of the rigid strips, due to large stresses induced by the crack tips, the resultant tensile force is accumulated near the corners and p, becomes even compressive over a wide interior portion.

Pull-off test for a cracked layer bonded to a rigid foundation

b/h

Fig. 6. Stress intensity factors at the corners of the rigid strip when a = 0.31 (finite crack).

0

05

1.0

1.5

c/h

Fig. 7. Stress intensity factors at the tips of the finite crack for

a = h.

E b/h *

0.9

4

2 I*?

0 1r

i

4

C

0.5

1.0

1.5

a/h

Fig. 8. Stress intensity factors at the tips of the finite crack for c = h.

221

222

M. R. GECIT

0

0.5

1.0

15

C/h

Fig. 9. Probable

cleavage

angle at the tips of the finite crack when a = h.

Table I. Strain energy release rate W at the tips of the finite crack for o=h Wh

clh 0.03 0.25 0.40 0.75 1.00 1.25 1.50

0.1

0.3

0.5

0.7

0.8

0.02712 0.12389 0.17643 0.33928 0.55160 0.85611 1.2668 1

0.03287 0.15150 0.19820 0.42316 0.78467 1.34731 2.07 158

0.03393 0.16541 0.20588 0.51341 1.18580 2.40147 4.04461

0.03254 0.14980 0.19907 0.57528 2.10980 5.97918 11.87402

0.03076 0.13643 0.18956 0.55882 3.26719 13.13478 30.24916

The stress state near the corners factors defined by

of the rigid strip can be described

k,, = Fz [2(a - ~)]l’~a,(x,

by the stress intensity

h),

(3% b) k,, = ?‘r?,[2(a - x)]~‘~z~~(x, h).

Figure

6 shows the variations

of the normalized

Ei, = ki,/podi2,

stress intensity (i = 1,2),

factors Wa, b)

with bJh for a = 0.5h and for various crack lengths specified by c/h. As b/h increases, the crack gets closer to the top surface. As indicated in Fig. 6, the effect of the crack on EI, and E2. is magnified greatly with increasing b/h and/or c/h. The stress intensity factors at the crack tips may be defined as k,, = lim [2(x - c)] li2a,(x, b), x-+c kZE = &

and normalized

[2(x - c)]

li2Txy(X,

Wa, b)

b),

as

kit = kic/pOc1'2,

(i = 1,2).

W’a,b)

PulLotT test for a cracked

layer bonded

223

to a rigid foundation

Figures 7 and 8 show these factors for a = h and for c = h, respectively. As can be seen in Fig. 7, for a = h, I;,, and EzCbecome significant when c > -h. Figure 8 shows significant variations in Ei,, and EzCwith a/h for a fixed crack length 2c = 2h. El, and I&~become maximum when a/h is less than c/h = 1.0. It may be worth to note that maximum values of &, and I;,, do not occur simultaneously. If the material of the layer is brittle, the crack propagation may be expected to take place, as suggested by Erdogan and Sih [6], in a direction perpendicular to maximum cleavage stress which is defined by k,,(l - 3cos0,) - k,,sin8, = 0, 3k,,sint?, - k,,costl,

W, b)

< 0.

Figure 9 shows the variation of the probable cleavage angle 8, at the crack tips with c/h for LI= h. A positive value for 0, indicates that the crack is expected to propagate towards the top surface of the layer. When the crack is closer to the bottom surface (for example, consider the case when b = O.lh), the crack tends to escape from the rigid foundation. A similar behavior had been reported by Erdogan and Gupta [8]. When the crack is closer to the top surface, smaller cracks tend to escape from the rigid strip whereas larger cracks tend to reach the free top surface. If an energy balance type criterion is used for estimation of the crack propagation load, one has to calculate the strain energy release rate given by [16,17]

au = ac

-

$(kt

+ k:,).

(39)

Table 1 gives the dimensionless strain energy release rate

w=--2P au nhp; ac

(40)

for a = h. As can be seen in Table 1, W increases significantly with increasing crack length. It can be said that W increases also with increasing b/h ratio except for relatively small crack lengths. Then, an interpretation may be given by saying that larger cracks and cracks that are closer to the top surface have greater tendency to propagate. Some sample results for the problem of external cracks are shown in Figs 10-15. Figures

c/h

0

0.25

= 0.25

0.5

0.75

1.0

x/a

Fig. 10. Normal

stress between

the layer and the rigid strip for a = h, b = 0.7h (semi-infinite

cracks).

224

M. R. GECIT

x/a

Fig. II. Shear stress between the layer and the rigid strip for a = h, 6 = 0.7h(semi-infinite cracks)

x/c Fig. 12. Normal stress in the plane of semi-infinite cracks for a =Lh, h = 0.7A.

10 and 11 show the normal and the shear stress distributions, respectively, along the rigid strip-layer interface when a = h and b = 0.711.These distributions, as one may expect, depend on c/h ratio heavily. It seems that a very large portion of the resultant force P is transmitted through the central portion of the interface which has a width equal to that of the net section between the two crack tips at y = b. As c/h increases, stress distributions become smoother. Figures 12 and 13 show the normal and the shear stresses, respectively, in the net section between the two crack tips when a = h and b = 0.7h. These stresses are normalized using q0 = P/k. As c/h increases the load gets more accumulated around the center. Finally, Figs 14 and 15 show the stress intensity factors at the corners of the rigid strip for c = h and at the tips of the external cracks for a = h, respectively. The normalized stress intensity factors rt,,, (i = 1,2), are defined as El, = kic/q(p,

(i = 1,2).

(41a, b)

As can be observed in these figures, El,, decreases with increasing a/h ratio and similarly does RI, as c/h increases.

Pull-off

test for a cracked

layer bonded

to a rigid foundation

225

0.1

0.c

c 9”

-0.1

r

c/h

= 1.5

t

-02 .

Fig. 13. Shear stress in the plane of semi-infinite

cracks

for a = h, b = 0.7h.

0.E

I.2

0.4

D Is”

1s

I.1 0.2

0.01 I

C

0.25

cx,)

I

0.5

0.75

1

10

b/h

Fig. 14. Stress intensity

factors

at the corners

of the rigid strip when c = h (semi-infinite

cracks).

REFERENCES [1] G. IRWIN, Fracture Mechanics. In Structural Mechanics (Edited by J. N. Goodier and N. J. Hoff). Pergamon Press, Oxford (1960). [2] G. P. ANDERSON, S. J. BENNETT and K. L. DEVRIES, Analysis and Testing of Adhesive Bonds. Academic Press, New York (1977). [3] T. N. FARRIS and L. M. KEER, Williams’ blister test analyzed as an interface crack problem. Int. J. Fracture 27, 91-103 (1985). [4] M. R. GECIT, Bonded contact problem for an elastic layer under tension. Arabian J. Sci. Engng 12, To appear (1986). [S] M. R. GECIT, Analysis of tensile test for a cracked adhesive layer pulled by rigid cylinders. Int. J. Fracture, To appear. [6] M. R. GECIT, A tensionless contact without friction between an elastic layer and an elastic foundation. Inc. J. Solids Struct. 16, 387-396 (1980). [7] N. I. MUSKHELISHVILI, Singular Integral Equations. P. Noordhoff, Groningen, Holland (1953). [8] F. ERDOGAN and G. D. GUPTA, The stress analysis of multi-layered composites with a flaw. Int. J. Solids Struct. 7, 39-61 (1971).

226

M. R. GECIT

0 .6

0.0 I 0 .4 I

u

II*-

b

IIC

0.2t

-0.1

0

/

I 0.25

I 0.5

I 0.75

)02

b/h

Fig. 15. Stress intensity

factors

at the tips of the semi-infinite

cracks

when a = h.

[9] D. B. BOGY, Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. ASME J. Agpl. Me&. 38, 377-386 (1971). [lo] K. ARIN and F. ERDOGAN, Penny-shaped crack in an elastic layer bonded to dissimilar half spaces. Int. J. Engng Sci. 9, 213-232 (1971). [1 l] T. S. COOK and F. ERDOGAN, Stresses in bonded materials with a crack perpendicular to the interface. lnt. J. Engng Sci. 10, 677-697 (1972). [12] G. G. ADAMS and D. B. BOGY, The plane solution for the elastic contact problem of a semi-infinite strip and half space. ASME J. Appl. Me& 43, 603-607 (1976). [13] S. KRENK, Quadrature formulae of closed type for solution of singular integral equations. J. Inst. Math. Applic. 22, 99-107 (1978). [14] F. ERDOGAN, Approximate solutions of systems of singular integral equations. SIAM J. Appl. Math. 17, 104-1059 (1969). [15] F. ERDOGAN and G. D. GUPTA, Layered composites with an interface flaw. Int. J. Solids Strut. 7, 1089-l 107 (1971). [16] F. ERDOGAN and G. C. SIH, On the crack extension in plates under plane loading and transverse shear. ASME J. Basic Engng 85, 519-527 (1963). 1171 B. M. MALYSHEV and R. L. SALGANIK, The strength of adhesive joints using the theory of cracks. Ink J. Fract. Me& 1, 114-128 (1965). (Received

15 May 1986)