International Journal of Fracture 46: 237-256, 1990. ©1990 Kluwer Academic Publishers. Printed in the Netherlands. 237

Three-dimensional fracture analysis of thin-film debonding

H E R Z L CHAI Polymers Division, Institute Jor Materials Science and Engineering, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

Received 15 March 1989; accepted in revised form 8 October 1989

Abstract. A simplified mixed-mode fracture analysis combining nonlinear thin-plate stress solutions with crack-tip elasticity results has been developed to account for local variations of G~, GH and G~H in thin-film debond problems associated with large film deformations. Membrane and bending stresses from the plate analysis are matched with the crack-tip singularity solution over a small boundary region at the crack tip where the effect of geometric nonlinearity is small. Local variations in each of the individual components of the energy release rate are directly related to the "jump" in these stresses across the crack border.

Specific results are presented for 1-D and elliptical planeform cracks. Deformations were induced either by a pressure acting normal to the film surface or biaxial compression or tension stresses applied to the substrate in which the loading axes and debond axes coincide. The latter type of loading involves buckling of the delaminated film. The model predictions compare well with more rigorous solutions provided the film thickness is small compared to the debond dimensions. In all cases analyzed, G~I ~ was negligible. The ratio G~/GH typically decreases with increasing load or film deformation, the rate was moderate for pressure loading while generally sharp for compression loading. Film-substrate overlap may occur for certain debond geometry and loading conditions. Prevention of this by the substrate may critically increase the energy available for crack propagation.

1. Introduction

Debonding of a thin surface layer ("film") from the main load-bearing body ("substrate") is a failure mode that has been of long standing concern in a variety of industrial and technological applications. Film-substrate deformation gradients that develop upon load application give rise to intense stresses at the debond periphery which may enlarge the damaged area. Pressure loading ("blister") is a common cause of crack propagation. Such a stressing concept has been used extensively in the evaluation of adhesion strength [1-3]. Another effect which may induce debonding is temperature; mismatch in thermal expansion between the film and substrate may lead to buckling and growth of the coating layer [4]. In recent years the phenomenon of phase separation has also become of concern in composite material applications. Unwinding of individual plies in laminated tubes under external hydrostatic pressure [5] and collapse of compressively-loaded composite panels following impact by foreign objects [6] are only two examples. In both these cases, failure starts with local buckling of already delaminated ply groups and spreads to the rest of the laminate by further delamination and buckling.

The mechanisms by which a sub-surface debond may grow are complex, and numerous failure modes which depend on such factors as mechanical properties of film and substrate, strength of the interface and the details of stress distribution at the crack tip have been reported in the literature. Coplanar crack growth was reported for an axisymmetric blister consisting of a rubber film bonded to PMMA [2] and an epoxy bonded to glass [3]. For this debond configuration, both mode I and mode II stress intensity factors (k~, kH) exist [2, 7, 8]; the

238 H. Chai

nominal values vary with the ratio of film thickness, h, to crack radius, b. The adhesion fracture energy was evaluated based on linear elastic fracture mechanics (LEFM) concepts and was shown to increase with increasing ratio of shear stress to tensile stress at the crack tip, i.e. from 30N/m to 70N/m as k~/k~ increased from 0.23 to 0.69 [2]. The latter fracture energy was suggested as Glic [2], apparently from trends in Fig. 6-4 of [2] and the assumption that kll/k~ ~ ~ when h/b ~ O. However, the present analysis of the axisymmetric blister provides that kH/k~ --+ 0.78 as h/b ~ O. It is then quite possible that the true value of GHc will be considerably larger than that reported in [2].

Interlaminar crack propagation in laminated composites or adhesively-bonded joints, the subject that originally motivated this work, differs fundamentally from the case above in two ways. First, due to the adherends' rigidity, the growth of a crack is always coplanar, at least in a macroscopic sense. Second, plastic deformations at the crack tip are limited to the bondline region, and, in the case of shearing fracture modes, these deformations tend to be more and more homogeneous as the bondline thickness is decreased below the size of the natural crack-tip plastic zone of the adhesive [9]. These features make it easier to evaluate, and indeed define, engineering fracture toughness properties. The adhesive fracture energy was recently determined as a function of bondline thickness for a number of adhesive systems including brittle and ductile resins [9]. The results show that G,c and GH~c are always the same, with a nominal value that may exceed Gic by more than an order of magnitude. More recent tests [10] have shown that for each mode of loading, the interlaminar fracture toughness of laminated composites is identical to the corresponding adhesive fracture toughness if the comparison is based on equal resin thickness in the bondline or interlaminar region. Motivated by such findings, this work is concerned with evaluating energy release rates in mixed-mode debond problems so as to provide the necessary information needed for meaningful application of a fracture mechanics-based failure criteria. Although the fracture of adhesive bonds or composites may be accompanied by extensive plastic deformations in the length of the bond [3, 9-11], for simplicity reasons the present analysis will be limited to linearly-elastic material behaviour, with the bond thickness set to zero. Geometric nonlinear- ity associated with large film deformations will be accounted for in accordance with von Karman's kinematical assumptions.

Existing mixed-mode fracture analyses of surface debonds are mostly limited to the axisymmetric blister case discussed earlier. Recently, the more complex problem of an elliptically-shaped sub-surface flaw subject to surface shear traction was analyzed using a three-dimensional elasticity approach [12]. For this problem, all three fracture modes are present along the debond periphery. For debond problems associated with film buckling, geometric nonlinearity is present. A 1-D ("one-dimensional") debond configuration similar to that shown in Fig. la was analyzed using a geometrically-nonlinear finite-element pro- cedure combined with Irwin's crack closure approach [13]. Although both k~ and k H stress intensity factors were generally present, the problem was dominated by the mode II com- ponent, with kl being significant only in the early stages of buckling. When the applied stress greatly exceeded the film's buckling stress, film-substrate overlap associated with kl < 0 occurred. More recently, that analysis was extended to circular and elliptical sub-surface cracks which were either fully or partially embedded in the substrate [14]. Interesting fracture characteristics were found, some of which will be referred to later in this work.

Three-dimensional (3-D) fracture analyses such as in [12, 14] are complex, and typically provide results on a dimensional basis. In many debond problems of interest, the delaminatin~

Thin-film &bonding 239

layer is negligibly thin compared to all other dimensions of the structure so that simple nonlinear plate theories may accurately describe the elastic behaviour in regions removed from the crack border. In this work such solutions will serve as input data to available crack-tip elasticity solutions to produce the final results,

2. Analysis

2.1. Problem .formulation

The problems considered are shown in Fig. 1. Either 1-D or elliptical debonds will be analyzed. The following "thin-film" assumptions [15, 16] are made:

(a) Both the film and the substrate are assumed isotropic, with identical Young's modulus, E, and Poisson's ratio, v. The analysis of the more practical case of bi-material interface crack is mathematically more involved, and is deferred for future work.

(b) Loading may include internal pressure, q, and a biaxial strain field ("load") defined by 0 and 0 8 x By.

(c) The in-plane dimensions of the film are large compared to the film thickness, but small compared to all substrate dimensions.

(d) In calculating the stresses in the film in regions removed from the crack border, the film is treated as a thin plate with a clamp type boundary support. The in-plane deformations of the film's boundary are fully dictated by the substrate; the latter is unaffected by the debond.

The following conditions thus hold on the debond boundary defined by x2/a 2 ~- y2/b2 = 1:

U ~- X8 0.,

v = y~ , (la, b,c)

Ow Ow

~x @

_Oo (_~o) I

It

J

J

(a) 1-D Debond (b) Elliptic Debond

Fig. 1. Thin-f i lm d e b o n d models , sect ional view.

240 H. Chai

where u, v and w denote film displacement in the x, y and z direction, in that order, and "a" 0 and o and "b" are the ellipse semi-axes. Note the strains ex ey are related to applied stresses

through

0 0- x

E( o +

(1 - v 2) ' (2a, b)

o E( + ay = (1 - v 2)

Two in-plane loading cases will be considered in detail, namely "uniform" and "uniaxial". For the first,

0 0 8 x ~ B), ~ - 8 0,

E 8 ° GO 0 Go

(1 - v )

(3a, b)

For the uniaxial load case:

o O, o G x = ~y

0 - -Vg~ .

_- Ee 0, (4a, b)

Note the problem of debonding due to film-substrate thermal expansion mismatch is identical to the "uniform" compression case if

e ° - - ( 5 )

where AT is the temperature gradient, ~r and cq are the coefficients of thermal expansion for film and substrate, respectively, and c 9- > eL,..

2.2. Calculation of total energy release rate

It has been long recognized, albeit with an apparent lack of a formal proof that the total energy release rate, G, in plate crack problems can be accurately predicted from membrane stresses and bending moments at the crack border. Formal derivation of such correspond- ence was recently given for the relatively complex case where the debond boundary is curved and the film's deformation is nonlinear [17, 18]. These results will now be derived in a rather direct and straightforward way. Consider first the 1-D configuration shown in Fig. 2. As shown, the substrate is subjected to a uniaxial compressive stress while the film is acted upon at its end by a moment, a membrane axial force and a transverse shear force. Plane-strain conditions (8/8x = 0) are assumed. Referring to Fig. 2a for notation, stage II differs from stage I in that a small crack extension, Ab, has occurred. Consequently, point A is displaced by the amounts u2 and u3 ( - w ) as well as rotates by 0y. The energy released during this transition (on a per unit width basis) is made up of the following contributions: moment 34,.

Thin-film debonding 241

z

x b ,-]

O y ~

My

f2 ~ , / Crack Tip

h

(-e °)

(a)

O)

(11) My

Oy (111) f3 ' ~

W (IV) f2

(V) -~y U2

(b)

Fig. 2. Thin-film model for calculating total energy release rate.

acting through a rotation 0 3, (iI-III) , transverse force f3 acting through a displacement u3 (III-IV) and an in-plane force f2 acting through a displacement u2 (IV-V). Similarly, the work associated with deformations in the x - y plane is that of a shear forcefx acting through a displacement Ul and a twisting moment Mx,. acting through a rotation 0,.,.. Thus

G - 2Ab f ui + M,, 0.~, + M~,, 0.yy v=b A~," (6)

Taylor expansion around y = b for the variables of interest, together with (lc), gives

w(b - Ab) = w(b) aw(g)

~y - - A b + O(Ab ~) = O(6b~),

O~ (b - Ab) -

Ow(b - Ab) 0w(b) (?y (?y

O2w(b) , , 02w(b) - - + " o + o(Ab2) - Ab + O(Ab ),

O~, (b - Ab) = 0 2w(b - Ab) ~2w(b)

OxOy Ox@ + o ( 6 b ) = O(Ab),

(7a-f)

, £ ( b - A b ) = . / ; ( b ) + O ( A b ) , i = 1 - 3 ,

ui(b - Ab) = ui(b ) + O(Ab), i = 1 - 2,

.3(b - 6b) = w(b) + O(Ab) = O(Ab).

242 H. Chai

Combining first order contributions in Ab, and noting the vanishing of M.~y on the plate boundary, one has:

a = + (8)

0 The displacement 112 is produced by a stress difference G , - G, where G is the film's membrane stress at the crack border. Thus,

Ab u2 = - (a~ ° - G)(1 - v 2) --if, (9)

similarly

2(z~), - zv,)(1 + v)Ab ul = E (10)

As shown in Fig. 2b, the work associated with f2 corresponds to increasing the membrane 0 stress from -O-y to - G ' over a displacement u2. With a similar argument for f l , one has

(11)

.f~ = (~° v - %.)h. (12)

Combining terms and noting that for the coordinate system in Fig. 2a My = Dc32w/Oy 2, one has

I 2 h(o-0 _ %)2 (1 + v)h ] = - - " T 2 G 2 ~ + (1 v 2) 2E + E ( ' d y - xy) , (13)

A y = b

where the bending stiffness, D, is given by

Eh 3 D -

12(1 - y 2 ) "

2.3. Effect o f nonlinearity

Within the scope of von Karman's plate theory, geometrical nonlinearity is affected through the term (#w/@) 2 in the following relationship which pertains to the midplane of the plate:

o< 1/ow 2 - 8y + 2 [ a y /

(14)

Because in the present thin-film formulation dw/@ vanishes at y = b, there will be a boundary region where linearity is expected to hold reasonably well. The extent of this region

Thin-film &bonding 243

1.0

I

0.5 c5

c5

I I I I [ I I I I [ 0 5 10

b/h (a)

--h

7 b

~ M

Y n

t ~ ~ x

(b) (e)

Fig. 3. Basic sub-surface crack configuration (b) and associated normalized mode I or mode i1 energy release rates (a) as deduced from [22]. (c) Curvilinear coordinate representation.

can be assessed from:

_ _ _ 0 2 w ( b ) Ow(b - Ab) Ow(b) Ew(b) Ab + . . . . . Ab -t- O ( A b 2 ) . ( 1 5 ) OY @ Oy 2 6qy 2

Assuming Ab = 6h, 6 is a constant, then

2

e l ( b - Ab) - Ou2(b@- Ab) + 21 k ~ ~°y- 6h . (16)

For the 1-D problem (Fig. la), one has [15]:

2b w(y) = - - ( 1 + cos 7w/b)x/-(1 - v2)(e ° - M),

7~

ec,. = 12(1--- v 2) '

(17a, b)

Because ~U2/~y = 8or = constant, where M is the buckling strain, one has

8/ (b Ab) 80 - - 1 _ {2rc2(1 _ v2)(~o _ 1)62} ~ , ~:o - . (18)

8cr 8~r

Equation (18) shows that the smaller the load ratio, g0, or the square of the slenderness ratio, h/b, the larger is the boundary region in which the effect of geometrical nonlinearity could be neglected. (For example, i fg ° = 3 and h/b = 50, the correction term in (18) amounts to less than 10 percent if 6 < 2.6.) Assuming h/b to be sufficiently small, the effect of the film section in Fig. 1 a may thus be uniquely expressed by a moment M and a midplane force P, as indicated in Fig. 3b, operating at a distance b ~ h. Note no transverse force is super- imposed in Fig. 3b because its effect was shown to be secondary.

2.4. Effect of debond curvature

Analogously to Irwin's treatment of an elliptically-shaped crack in an infinite solid [19], it is assumed here that, provided the radius of curvature of the debond boundary is sufficiently

244 H. Chai

large, the 1-D results are directly applicable to the more general debond case (Fig. lb) if y and x are replaced by n and t (Fig. 3c), the coordinates normal and tangent to the curved boundary. Field quantities in the new coordiante system can be expressed in terms of x and y [20]:

O" n = O- n C 0 S 2 ~ 71- O'y sin2~ + v~y s i n 2~,

sin 2c~ %, = r.,,~, cos 2c~ + (ay - Ox) 2 ' (19a, b,c)

M.. = M xcos2c~ + M v s i n 2 e - M~ysin2c~.

where

c o s e - sinc~ = ../22 + p21r12' ~1~22 + y21,~'

tl =-- b/a, 2 =- x/a, f; =- y/b. (20)

The midplane stresses at the plate boundary and the boundary moments can be expressed in terms of displacements [20] as:

O" x = ~xx + v (1 - v2) '

( v ~ + ( l - v2) ' ~ , =

% - 2(1 + v) ~y + ~xx ' (21a-e)

M+ (~2w O2w) = - - D \ ~ x 2 + v--@2 ,

(o w O w) - D \ o y 2 + V~x2 j .

2.5. Hybrid analysis

A fracture mechanics solution for the basic 1-D crack problem shown in Fig. 3b is given in [21]. A similar solution was also derived in [22]. As shown in [21] or Fig. 3a which was constructed from data given in [22], the stress intensity factors or energy release rates depend on the ratio h/b. In the limit case h/b ~ 0, which is of interest in ths work, the solution reduces to [21, 22]:

k~ = (C~P + CzM/h)/h 1/2,

k l l =-- (C3P + c4m/h)/h U2.

(22a, b)

Thin-film debonding 245

Figure 3a shows that this asymptotic solution can be applied to a finite thickness film with a reasonable accuracy if h/b < 0.1; the error in G l and G n then becomes less than 5 and 12 percent, respectively.

The constants Ci in (22) differ slightly between [21] and [22]. Adopting the values from [22], one has:

C 1 = 0.434, C 2 = 1.934, C 3 = 0.558, C 4 - 1.503. (23)

From earlier discussions, these results should be directly applicable to the present 1-D thin-film model (Fig. la) provided the film's bending and membrane stresses are applied at a distance b ~> h. The error introduced by representing these stresses by the boundary stresses instead can be assessed from:

My(b - ~h) = My(b) ~M, 62h 2 ~2M, Oy 6h + 2 @2 + . . . . etc. (24)

For the 1-D model in Fig. la, Eqns. (17a), (21e), and (24) give

'( My(b - g~h)/My(b) = 1 - ~ no5 (25)

Thus, the correction term in (25) again diminishes in proportion to (h/b) 2. It should be noted that a relationship analogous to (22) was proposed in connection with a finite-element analysis of the 1-D compression problem [13, 23]. That relationship was not derived from first principal but it became apparent from superposition of data obtained from a variety of debond configurations.

Replacing P and M in Fig. 3b with h (a°n - o-n) and - Mn, respectively, and invoking our earlier argument concerning debond curvature, one has for the general curved-boundary, thin-film debond problem:

ki = h~/2[O.434(a°n - an) - 1.934Mn/h2],

kn = h1/2[0.558(a°n - an) + 1.503Mn/h2], (26a-c)

ki l l 1/2 o = h (Tn t - - T, n t ) ,

where kin was deduced from (13) and (27c). The plane-strain energy release rates can be found from

a , = (1 - v2)k~/E,

Gill = (1 + v)k~ll/E.

i = I, II, (27a-c)

Several interesting characteristics of this solution are readily apparent:

246 H. Chai

(1) ki and kn are negative when

a~ - a n <~ 4.46 M,/h 2,

ao _ an <~ _ 2.69 Mn/h 2,

k~ ~<0

klj ~< 0

(28a, b)

Note negative k~ corresponds to the physically unacceptable situation of a film-substrate overlap. A contact analysis must be developed to account for this effect.

(2) When a 1-D film debond has a free edge (i.e., P = M = 0 in Fig. 3b), and there is a far-field stress a° (> 0), one has

k l / k n = 0 . 7 8 or GI/GII = 0 . 6 0 . (29)

Such a configuration is generally denoted as "cracked-lap-shear". A solution for the closely related double-crack lap-shear configuration (i.e., one film on each side of the substrate) was produced using a geometrically-nonlinear finite-element approach [24]. Results were given (Fig. 6 of that source) as a function of film thickness, with substrate thickness, H, and crack length being fixed at 6.4 mm and 50.8 ram, respectively. GI/GH was found to decrease from 0.8 when h = 6 m m (or h/b = 0.12, h/H = 0.94) to 0.59 when h = 0.7ram (h/b = 0.014, h/H = 0.11). The latter result compares well with the present prediction of 0.6 (Eqn. (29)) pertaining to h/H ~ 1, h/b ~ 1.

(3) When the applied load or film deformation is relatively small (as is the case during the O a n d % t , ~ o Thus, k m = 0 a n d initial stages of buckling), a n ~ an "on,.

k~l/k~ = - 0 . 7 8 or G~I/G~ = 0.60 or Gn/(G~ + G H ) = 0.375. (30)

L E F M analysis of the axisymmetric blister [8] shows that kH/k ~ increases monotonically with decreasing h/b. Specifically, for h/b = 1, 0.2 and 0.1, kH/k ~ was - 0 . 0 1 , - 0 . 5 7 and -0 .76 , in that order. (No data were given for thinner films.) The latter result is consistent with (30), and it seems to provide a good indication of the range of applicability of the present asymptotic model.

3. Results

3.1. 1-D problem

For this relatively simple yet instructive case, one has from (4) and the technical theory of beams/plates:

G O = E~3 0,

~o = "c = 0, (31a-c)

M = --D 02w @2 .v~h"

%

~'- 1.0

E" w L9

d

re 0.5

u J - a

N

E o 7

Axisymmetric (~B = -0,941)

5 2 ~o __- eo/eg =_ eo g=/XB

I I I F t I I

2 3 4 5 6 7 8

Normalized Strain, ~o

(a) Compression Loading

d r e

g~

w

E o Z

Thin-film debonding 247

200

3~ _ x21J=

100

a n \ a l - E

o

0 1 2 3 4

Normalized Central Deflection, W(0,0) / h

(b) Pressure Loading

Fig. 4. Variation of energy release rates with normalized load or normalized plate deflection for 1-D and axisymmetric debond configurations, v = 0.3.

The membrane normal stress at the film boundary , a,, remains the same as the buckling stress for all load levels. Thus

GI/G

GII/ G

G i l l

a./E = ecr = e °, (32)

where the subscript " B " denotes buckling. Using (31), (32), (17) and (26), one easily finds

= 2.511 - 0.389x/(g ° - 1)12/(~ ° + 3),

= 1.511 + 0.643x/(~ o - 1)]2/(go + 3),

= 0,

Eh (1 - v 2) G = - G 1 ~- Gll -~- Gli I - - 2 ~r(E° - 1)(~° + 3), E ° ~ 1.

(33a-d)

Note the expression for G in (33d) coincides with that derived in [15] based on a direct differentiation of the strain energy.

The variat ion of G with g0 is shown in Fig. 4a. Following film buckling (~0 = 1), the energy release rate rises sharply with increasing load, attaining a maximum when the load becomes three times the buckling load (i.e., ~0 _- 3). Thereafter G descends towards a plateau which is only slightly smaller than its peak value. As discussed in detail [15], for the special case where Gic = G l i c , the propagat ion o f a crack will be unstable if ~0 < 1.5 while stable if ~0 > 3. In the intermediate range, the growth will be unstable and followed by crack arrest. The relative contr ibut ion of mode I and mode I! energy release rates is shown in Fig. 5 as a function of load ratio (~0) or normalized central deflection, w (0, O)/h. GI/G seems to decline

248 H. Chai

1.0

G__I G

0.5

0.0

Normal i zed Centra l Def lect ion, W(0,0) / h 1 2 3

I I I I i I I ~ I I I I I

x isymmetr ic /B l is te r

~ / ~ x i s y m m e t r i c / C o m p r e s s i o n

e n t (Whitcomb)

I i v ~ , . . . . -O J I 3 5 7

Normal i zed Strain, ~o

1.0

G_~I G

0.5

0.0

Fig. 5. Variation of energy release rates with load ratio or normalized deflection for I-D and axisymmetric debonds, v = 0.3. Open symbols represent data from [23]. Dash-dot curve is best fit to open symbols.

sharply with load, from a value of 0.625 at buckling to zero at i 0 = 7.6 (or w (0, O)/h = 3.0). For larger loads, film-substrate overlap occurs, and the present solution becomes invalid. This interesting aspect of the 1-D compression problem has been previously established by Whitcomb (e.g., [13, 23]). GI/G data were constructed from Fig. 4 of [23] and are represented in Fig. 5 by open circles. The dash-dot curve in this figure is a best fit to the open symbols. (Because of differing material properties, comparison for G could not be easily made.) Results are given only for i ° > 2; no accurate data could be extracted from [23] for smaller load ratios. The data from the present model seem to agree reasonably well with that of the rigorous finite element analysis.

3.2.3-D problem

The elastic deformation problem of a thin elliptical plate subjected to the boundary con- ditions in (1) was solved numerically [25] using a Rayleigh-Ritz analysis based on the following polynomial series solution:

a2h 2 N N-m m + l , n + 1 Y , u = ° m=0E .-0E a

b~h 2 ~ N-m V = bf2t3°y ~- - 7 (O E bm+l,n+l ~2mf)2n, ( 3 4 a - c )

m = 0 n = 0

M M m

rn+ 1,n+l Y m = 0 n=0

Thin-film debonding 249

where

q~ - 2 2 + 9 2 - 1, (35)

2 =- x/a, f~ - y/b. (36)

The displacement coefficients (a . . . . bm, n and c,~,n) were determined from the requirement that the potential energy be minimized with respect to a variation with any one of them. The summation parameters M and N were systematically increased until sufficient convergence of the desired field quantities was achieved. As many as 77 displacement terms (i.e., M = 5, N = 6) were employed in order to ensure an accuracy better than a few percent for all quantities to be reported here. Utilizing (34), (26), (27) and (19-21), one has:

Di Eh5 22m f;2n 22 _2 b ci - (1 - 7 ) b 4 - o,=0 n = 0 E a +l,n+l + Y m+l,n+l

M M m ~ 7 2 '

+ + Z = Y m+,,~+, i I, II, (37) m~O n=O

GIII -- ( 1 - ~ - V ) m=0 n=0E "~2my2n bm+l,n+l qeam+l,n+l

where

0 ~< 2 ~ < 1 ,

p = x/1 - 22,

D~ -- 0.750, D H = 1.245, (38)

E~ = 1.485, Eli - 0.898.

Results for several crack problems of interest will be presented based on (37) and the solution for the displacement coefficients [25]. In all cases considered, v = 0.3.

3.3. Ax isymmetr ic case

Although this is effectively a 1-D crack problem, the associated deformation is that of a plate 0 0) or as opposed to a column. G, G~ and G n data for the case of pure pressure (eo = ey =

pure compression (q = 0) loadings are shown in Figs. 4 and 5. For both cases, Gin = 0. It should be noted that the total energy release rate for pure compression has been determined for small [26] and for large [27] film deformations. Figures 4a and 5 show that for this case, the trends in G and GI/G a r e similar to the 1-D model except these quantities substantially exceed their 1-D counterparts when the load ratio becomes large. Because the buckling loads for a column and an axisymmetric plate are nearly the same (i.e., e°b2/h 2 = - 0 . 9 0 4 (17b)

250 H. Chai

A e-

l -6 t~ O

_1 "O

.N m

O z

6

~- - - - - - " - -~ w(x,y) > 0

4 ~ k I > O i

2 Buckling ~ ' t ~ - - - -

I I I I 0 2 4

a/b (a) "Uniform" Compression, G ° = 6~ - G °

e -

g

I

.6 tll O

,.,,1 "O

s O

Z

j / /

J / / i / / I / i / • / /

/ / w(x,y) > 0

" ~ ki>O

Buckling Curve ] I I

0 2 a / b

1-D I I 4

(b) "Uniaxial" Compression, 6°/G~ = -P = - 0 . 3

Fig. 6. Buckling curves and film-substrate overlap conditions as predicted from nonlinear thin-plate theory [25] and present crack-tip elasticity analysis. "Uniform" (a) and "uniaxial" (b) compression load cases. "Plate" type overlap occurs in those domains where w(x, y) < 0 while "local" type overlap occurs when k~ < 0. v = 0.3.

vs - 0.941 (Fig. 6)), the circular crack will grow at a lesser load than the 1-D crack. Figure 5 shows that no film-substrate overlap may now occur (k I is always positive), a departure f rom the 1-D case which is due to differing post-buckling responses; while the membrane normal stress at the crack border is fixed for a column, for a plate it increases with end

displacements. This tends to reduce the stress difference la ° - ~rnl, and thus (see (26a)) increase k I. Fo r a "blister", o -° = 0 and o-, > 0. Consequently, the ratio Gn/G (Fig. 5), while starting again at 0.375, now increases only moderately with increasing load or deformat ion

level. It can be seen f rom Fig. 4b that both G~ and Gn increase monotonical ly with debond size. Thus, under constant pressure, the growth of a blister must be catastrophic.

3.4. General debond case

For all representative debond configurations to be discussed, Gi11 was negligible so that the combinat ion G and GI1/G (or G1/G ) fully defines the mixed-mode fracture problem. Vari- ations of these quantities along the boundary o f an elliptical blister are shown in Fig. 7. Results are given for loadings associated with small, medium and large plate deformations. For small deformations (w/h < 1), an exact plate solution is available [20]:

(1 - v 2)7 (1 - - 2 2 - 352)2 w/h - 2 (1 + ~t/2 + I/4) ' u = v = 0, (39)

q b 4 7 = E h 4'

Thin-film debonding 251

2.0

T Z L5 = 1.0

(5

T

0

y/b

x/a 0 1.0 G qh or GU / G

Fig. 7. Variation of normalized total and mode II energy release rates along the debond boundary for an elliptical blister, b/a = 0.5, v = 0.3.

From this and (37) (subject to vanishing of all displacement coefficients except c~,~), one has

G 2 (1 - v2)~[1 - 22(1 - q2)]2 qh 3 (1 ..~ ~_~2 2 ~_ /,]4)2 0 ~ f f ~ 1, 0 ~ ~ 1,

Gn/G --- 0.375 (or Glt/G1 = 0.60), (40a-c)

Gi11 ~ 0.

Comparison with the nonlinear plate solution (37) showed that this result is satisfactory if w/h <<. 0.4. Figure 7 shows that G is always maximized at the end of the ellipse minor axis. Because the relative contributions of G~ and Gll do not change appreciably along the debond boundary, for all practical mixed-mode fracture toughness conditions the crack is expected to grow towards a circular planeform. For a given 7 and b, it is easily seen that the values of G~ and Gn at the end of the ellipse minor axis are always substantially larger than their axisymmetric counterparts (Fig. 4b), which strongly suggests that the growth of an elliptical crack will be catastrophic; the crack will first grow into a circle and then continue to grow in that shape indefinitely. Similarly to the axisymmetric blister, G./G starts at a value of 0.375 everywhere along the debond boundary, and then increases moderately from that value as ~ is increased.

The behaviour under axial compression is more involved, and it furthermore critically depends on the problem parameters. For such loading, film-substrate overlap may occur over part of the plate surface. Overlap may develop as a result of °'sinusoidal" deformations (which may take place anywhere inside the plate) or if k I becomes negative. The first type

0 1.0

1 1.0

i v

0

252 H. Chai

Fig. 8. Variation of normalized total and mode II energy release rates along debond boundary for an elliptical debond, uniform compression load case (d = t°). b/a = 0.5, v = 0.3.

of overlap will be termed "plate overlap" while the second, which may be confined only to the vicinity of the crack border, will be termed "local". The conditions for a "plate" type overlap were determined in [25] while that for k~ < 0 from (26a) and the plate solution [25]. The results are given in Fig. 6 as functions of ellipse aspect ratio and load level. In this figure, overlap occurs in domains bounded by the buckling curve and the dash-dot ("plate") or dashed ("local") curves. For the "uniform" load case Fig. 6a shows that a "plate" type overlap (which is associated with w (x, y) < 0 over some part of the plate surface) may occur only if a/b > 2.2. Such an overlap develops immediately upon film buckling, but it dis- appears when the applied load becomes several times the buckling load. For the "uniaxial" load case, Fig. 6b shows that no "plate" overlap is possible unless a/b < 1.4. In that range, overlap may occur immediately upon buckling (a/b < 0.9) or, otherwise, once the load ratio becomes sufficiently large. Figure 6 shows that a consideration of "local" overlap further limits the range where no film-substrate interaction occurs. In the domains bounded by the dash-dot and dashed curves, the overlap is limited to the vicinity of the crack border.

Figure 8 shows results for an elliptical debond (1/ = 0.5) subject to a "uniform" far-field 0 to buckling compression load. Data are given for three load ratios (ratio of applied load, ey,

load, soB), i.e., ~0 = 1.5, 3 and 5. In this and subsequent illustrations, the following nor- malization applies:

O, = Gi/Ehe °2, i = I, II, III. (41)

As shown in Fig. 6a, for this debond "local" overlap occurs when -e°b2/h2 < 1.3 (or < 1.9). That overlap was found to concentrate near the end of the long axis of the ellipse.

In view of the smallness of G in that region, however, the effect of film-substrate contact on the dominant values of G can clearly be neglected. Initially, G seems to be localized over a narrow region around the end of the minor axis but as the load ratio is increased, the energy

Thin-film debonding 253

1.0

T

1(5 0~

~ I t 4 - t t - 6

(a)

\

1.0

I V

1,0 ~ I ~ I i I I I I

;\ GI] /G /i \ \

I \

;== Finite.Element 1; ~ 113 rn\ 0.5 (Whitcomb) GI]/G /

/ /l //[3 pJ~

( , / /

0.0 0.5 1.0 x/a (b)

Fig. 9. Variation of normalized total and mode II energy release rates along the boundary of a circular debond. The uniaxial compressive load is applied along the vertical coordinate: (a) present model; (b) comparison of present model predictions (solid and dashed curves) with finite-element results (open symbols) [14], v = 0.3. The load ratio in (b) is e~/e°~ = 2.84.

1.o-

o

\ ~ / ~ ' , B = ,

t I I I t I I I l ' o l s '

(a)

"-q, \ \

(5 " Finite.Element Io m G "EL,,,,

0.5 (Whitcomb) GI]/G

I I I I I I I I I 0.0 0.5

x / a

(b) Fig. 10. As in Fig. 9, b/a = 0.5. The load ratio in (b) is ey/e).B° 0 = 2.84.

1.0

release rate tends to spread out, with the locus of maximum shifting away from the minor axis. In this region of interest the ratio GI]/G increases rapidly from its initial value of 0.375 at buckling as the load ratio is increased, in much the same way as for the 1-D case (Fig. 5). Such correspondence is easily understood from the relatively large radius of curvature of the debond in the region around the end of the minor axis, which renders the problem there as essentially 1-D.

Results for some "uniaxial" load cases are shown in Figs. 9-11. For a circular debond (Fig. 9a), the total energy release rate tends to localize over the boundary region which is

254 H. Chai

IO

1.0-

0 0.5

(a) Overlap Allowed

10 ̧

o

IO 0

0 10

(b) Overlap Constrained

Fig. 11. Varia t ion of normal ized total and mode II energy release rates a long debond bounda ry for a uniaxially- loaded ellipse, b/a o o = 2, e)./eyB = 3: (a) f i lm-substrate overlap allowed; (b) results f rom contac t analysis; no overlap is allowed. Also shown are corresponding con tour plots; solid and dashed contours identify outward (w > 0) and inward (w < 0) out-of-plane deflections, respectively. Fr inge cons tan t = Aw/h = 0.2. v = 0.3.

normal to the loading axis, the concentration effect that seems to increase with increasing load. The ratio Gn/G seems to be little affected by the load level in that critical region, a departure from earlier examples which is due to the absence of a direct compression loading

0 0). As can be seen from Fig. 6b, "local" and "plate" type film-substrate there (i.e., o- x = overlap are present once the normalized load exceeds 3.1 and 4.0, respectively (or, alterna- tively, once ~ exceeds 1.25 and 1.6, respectively). The magnitude of G in the boundary region where ki < 0, identified in this and subsequent figures by a dotted line, seems to increase with load ratio, though it remains small compared to the dominant energy release rate at 2 = 1 for all load ratios shown. Whitcomb [14] has recently developed a finite-element contact analysis for this debond configuration to account for the redistribution of stresses when overlap between the film and substrate is prevented. His results show that the effect of contact becomes significant only when the load ratio is increased beyond about 10.

The case of an elliptical debond loaded in compression along the minor axis, Fig. 10a, is yet different from those discussed. Here G varies gradually along the debond periphery, attaining its extremes at the end of the ellipse axes. For small load ratios, the maximum occurs at the end of the minor axis while for large ones it occurs at the end of the major axis. As in previous examples, the ratio GI~/G increases rapidly with load in the boundary region under direct compression stress while it is quite insensitive to load ratio in the transverse direction where the far-field stress vanishes.

When the ellipse is compressively-loaded along its long axis, overlap occurs right at buckling(Fig. 6b), the effect that becomes more and more pronounced with increasing load. The contour plot in Fig. 1 la (~ = 3) shows that the inward plate deflection at (2, 35) = (0, 0.5) is approximately the same as the plate thickness. To gain an insight into how the stresses redistribute when inward plate deformations are prevented, a simplified contact analysis was developed [25] which is based on the premise that film-substrate contact is restricted to two isolated points, located at (x, y) = (0, Yc) and (0, -Yc)- The condition w(0, Yc) = 0 was

Thin-film debonding 255

then imposed on the equations for displacement coefficients in [25] to produce the modified solution. The parameter Yc was chosen from the requirement that the slope ~w/@ is mini- mized at the contact points. Figure 1 lb shows the new contour plot obtained from this analysis as well as the corresponding variation of G and Gn/G. As shown, inward plate deformations are completely suppressed in favour of much larger outward deformations. G is seen to increase monotonically as one moves from a point slightly away from the end of the long axis of the ellipse to the end of the short axis, its maximum being about five times greater than the largest value of G in Fig. 1 la. It should be noted that convergence of the contact solution with increasing number of displacement terms was not very satisfactory, and the values of G presented are believed accurate only to within _+ 40 percent.

The problem of elliptical debond under far-field uniaxial compression stress was recently analyzed [14] using a rigorous, 3-D geometrically-nonlinear finite-element approach. Figures 9b and 10b compare G and Gn/G data for a circle and an ellipse (t/ = 0.5) between the present model (dashed and solid curves) and [14] (open symbols). In both analyses G I I 1 ~ 0. The debond dimensions in [14] are: circle: h x b = 0.4 x 15mm; ellipse h x b x a = 0.4 x 15 x 30mm. This gives a slenderness ratio (h/b) of 0.027. The corresponding buck-

0 is found from Fig. 6b to be -0.0018 and -0.00088, respectively. In both ling load, eyS, cases, the results pertain to a load ratio of 2.84. Because of some differences in material parameters between the present analysis and [14], dimensional comparison for G could not be easily made. Thus, in both Figs. 9b and 10b, G from [14] was normalized so as to match with the present result at x = 0. The agreement for G and Gn/G is generally satisfactory, which tends to validate the numerous simplifying assumptions made in the present "thin film" model as well as in the derivation of energy release rates. A more detailed examination of Figs. 9b and 10b shows that the present GH/G ratio exceeds that of [14] in the region around 2 = 1 by as much as 30 percent. It is not clear as to whether this departure is due to the difference in material properties or a more profound consideration.

4. Summary and conclusion

A simplified hybrid fracture analysis was developed to evaluate local variations of crack driving forces in thin-film debond problems associated with curved debond boundary and large film deformations. Nonlinear thin-plate solutions for boundary membrane stresses and bending moments were used as input parameters in available crack-tip elasticity solutions. Comparison with more rigorous analyses indicates that the present model is quite effective if the film thickness is at least an order of magnitude smaller than the in-plane dimensions of the debond. For thicker films, the detailed variation of the energy release rates with h/b from Fig. 3a as well as a higher-order plate theory need to be considered in order to ensure accurate results.

Results were presented for a number of basic debond problems. A wide range of mix-mode behaviour was found depending on debond size and shape, load biaxiality, and load level. Many of these fracture characteristics could be qualitatively deduced from the results of the relatively simple axisymmetric blister and the 1-D compression models. The following general findings are noted:

(a) Gm is negligibly small. This may not be the case, however, when far-field shear stresses are applied or when material anisotropy exists.

256 H. Chai

(b) The ratio GH/G~ gradually increases from its initial value of 0.60 as the load or film deformation increases. The rate of that increase is much greater for compression loading than pressure loading.

(c) Film-substrate overlap may occur under compression loading. Overlap prevention by the substrate may drastically increase the energy available for crack propagation, particularly for narrow debonds loaded along their long axis.

For simplicity reasons the analysis was limited to isotropic materials, with the properties of film and substrate being identical. Anisotropy and mismatch in elastic constants between the film and substrate may cause overlap and inability to separate the individual components of the energy release rate. In applications to adhesive joints or composites, consideration must also be given to the interlayer or the binding phase. These complicating factors must be investigated before a more complete understanding of the debonding problem can be achieved.

References

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