Crack-front instability in a confinedelastic film

BY M. ADDA-BEDIA1

AND L. MAHADEVAN2,*

1Laboratoire de Physique Statistique de l’Ecole Normale Superieure,24 rue Lhomond, 75231 Paris, France

2Division of Engineering and Applied Sciences, Harvard University, Pierce Hall,29 Oxford Street, Cambridge, MA 02138, USA

We study the undulatory instability of a straight crack front generated by peeling aflexible elastic plate from a thin elastomeric adhesive film. We show that there is athreshold for the onset of the instability that is dependent on the ratio of two length-scales that arise naturally in the problem: the thickness of the film and an elastic lengthdefined by the stiffness of the plate and that of the film. A linear stability analysispredicts that the wavelength of the instability scales linearly with the film thickness. Ourresults are qualitatively and quantitatively consistent with recent experiments, and showhow crack fronts may lose stability due to a competition between bulk and surface effectsin the presence of multiple length scales.

Keywords: fracture; peeling; adhesion; instability

1. Introduction

The problem of peeling a thin plate from a soft confined adhesive film arisesrepeatedly in a number of scientific and technological applications, ranging fromthe ubiquitous band-aid to insect foot-pads. In these systems, peeling separatesthe joined solids and failure occurs via the loss of adhesion at one of the surfacesattached to the adhesive. When the intercalating adhesive material between theplate and the substrate is a fluid or viscoelastic solid, early studies on peeling(McEwan & Taylor 1966; Fields & Ashby 1976; Conley et al. 1992) drew ananalogy to the Saffman–Taylor problem (Saffman & Taylor 1958) for thedynamics of an interface between two fluids in a confined geometry. Recentexperiments with purely elastic intercalating materials (Ghatak et al. 2000;Monch & Herminghaus 2001; Ghatak & Chaudhury 2003) have shown that staticpatterns almost identical to those observed in the Saffman–Taylor problem ariseat the peeling interface. In figure 1a, we show the schematic for such anexperiment where a thin flexible cover-slip is peeled off from a soft, thin elasticadhesive film that is itself attached firmly to a rigid substrate. Examples ofthe resulting complex crack morphologies that arise are shown in figure 1b.An important qualitative difference is that the difference between the fingering in

Proc. R. Soc. Adoi:10.1098/rspa.2006.1708

Published online

*Author for correspondence (lm@deas.harvard.edu).

Received 7 June 2005Accepted 3 March 2006 1 q 2006 The Royal Society

elastomeric films shown in figure 1b and that in fluid or viscoelastic films is thatthe former does not involve the transport of matter and is a purely elasticinstability that is rate independent, unlike the Saffman–Taylor instability and itscousins which are strongly rate dependent. Recent attempts to understand thesepurely elastic interfacial patterns have focused on the case of two rigid solidsseparated by a thin adhesive layer (Ghatak et al. 2000; Shenoy & Sharma 2002),and account for the detailed short-range interfacial forces using a combination ofanalytical (energetic) and computational analyses. However, the simplestoccurrence of this instability in the context of an undulatory crack at peelingfront, documented by Ghatak et al. (2000) and Ghatak & Chaudhury (2003), hasnot been analysed theoretically (but see Ghatak (2005)). In this paper, wepresent a continuum elastic analysis of the stability of a linear crack front in aconfined elastic film. Our main result is a simple criterion for the instability ofthe peeling front or crack to planar undulations and follows from the effects ofelastic confinement due to the presence of lateral boundaries. This result isindependent of the detailed microscopic interfacial interactions, and is thus verygeneral and applicable to a variety of different systems.

For relatively thick films in contact with soft elastic plates, the peeling front orcrack is a straight line. However, for confined films in contact with stiff plates,the front loses the stability to an undulating geometry as can be seen in figure 1b.To understand this, consider a plate with bending stiffness D in contact with anelastomeric film of shear modulus m and infinite thickness. When the plate isdeformed by an amount d over a characteristic length-scale lp, the deformations

0.5

mm

0.5

mm

0.5

mm

0.5

mm

l

elastic film

flexible plate

crack front

spacer

edge of the filmaH

z

x

rigid glass slide

D

(a)

(b)

Figure 1. (a) Schematic of the problem. (b)The crack front loses stability to an undulatorymodewhenthe confinement parameter aZ!D=mH 3"1=3 is large enough. The various morphologies correspond toincreasing thicknesses of the adhesive film for a given flexural stiffness of the cover-slip. Typicalexperimental parameter values are 20/ a /100, and a film thickness Hw50 mm. See Ghatak &Chaudhury (2003) for further details. (Figure courtesy of A. Ghatak & M. Chaudhury.)

M. Adda-Bedia and L. Mahadevan2

Proc. R. Soc. A

decay exponentially into the bulk of the elastomer with a scale lp. To determinethis, we equate the bending energy per unit width in the plate Dd2=l3p with thatin the elastomer m!d=lp"2l2p so that lpw!D=m"1=3 defines the penetration depthof the surface deformations. For an adhesive layer of finite thickness H, thisallows us to define a ‘confinement’ parameter aZ!D=mH 3"1=3; the larger thisparameter, the more confined the system is. Experiments (Ghatak et al. 2000;Ghatak & Chaudhury 2003) show that the instability arises only if aOac,independent of the nature of the loading. The wavelength of the instability isfound to be independent of all parameters and scales linearly with the filmthickness H. To understand the qualitative nature of this phenomenon we notethat when a/1, the film is unconfined and can accommodate a straight peelingfront; indeed the related analysis of a crack front in an infinite three-dimensionalsolid shows that a straight crack is stable to small perturbations of its shape(Rice 1985). However, the introduction of a length-scale via confinement changesmatters qualitatively. When a[1, the penetration depth is no longer lp, but isinstead H; this geometrical confinement induces large stresses in theneighbourhood of the crack front which decay exponentially away from thisregion. This allows the crack to explore other configurations to relieve the storedelastic energy and possibly become unstable. The role of confinement is to favourshear deformations over normal deformations; to accommodate the localsqueezing of the elastic film at the crack-front, the crack prefers to undulate,since the excess cost of the undulation is more than made up by the energyreleased in the adhesive film, an effect that we will quantify in some detail.

In §2, we formulate the problem for the equilibrium of the flexible plate inpartial contact with the elastomeric adhesive film. We assume that the film is anincompressible linearly elastic solid, and that the plate is bent weakly so that thecurvature of the plate is small (and primarily in one direction). Then, we set-upthe problem for the linear stability analysis of an initially straight crack front tosmall in-plane sinusoidal perturbations and determine the stress field in the filmin terms of the plate-induced boundary displacement field. In §3, we determinethe solution corresponding to the straight crack front, and in §4 we determine theperturbed local stress field due to a wavy perturbation of the crack front. Whenexpressed in terms of stress intensity factors, our solution leads to a criterion forthe stability of the crack front and yields both a critical confinement thresholdand the wavelength of the perturbation. In §5, we summarize our results and putthem in the larger perspective of crack-front stability in confined systems.

2. Formulation of boundary-value problem

The geometry of peeling pertinent to our problem shown in figure 1a shows thetwo length-scales in this problem: the thickness of the adhesive film H and acharacteristic dimension determined by the balance between the energy ofbending the cover-slip and the energy of stretching/shearing the adhesivefilm, given by lpZ!D=m"1=3, where m is the elastomer shear modulus and D is theflexural rigidity of the plate. The first length-scale characterizes variations in thevertical or z-direction while the second characterizes variations in the plane ofthe crack, i.e. along the !x; y"-directions. The experiments of Ghatak &Chaudhury (2003) show that the ratio of these length-scales determines the

3Crack-front instability

Proc. R. Soc. A

onset of the instability which occurs when lpR18H . Although the vast disparityof these two length-scales at the onset of the instability !H/ lp" suggests that anelastic lubrication approximation similar to that used in Ghatak et al. (2004)should be sufficient for analysing the present problem, we shall see that someaspects of the problem are closely related to the behaviour of the stress field inthe vicinity of the crack front, where the lubrication approximation clearlybreaks down. Thus, we will not use the lubrication approximation here.

To make the problem dimensionless, we scale the coordinates xi h!x; y; z"using H, and scale the displacements in the adhesive film ui h!u; v;w" and thedeflection of the flexible cover-slip h!x; y" using D0, the separation between thefilm and the plate at the spacer. We assume that the adhesive layer may bemodelled as a linear incompressible elastic solid, and scale the stresses in it bymD0=H . Then, we may write the scaled constitutive equation for the adhesiveelastic layer as

sij ZKPdij Cvuivxj

Cvujvxi

; !2:1"

and the equilibrium equations in the film and the condition of incompressibility are

VP ZDu; !2:2"V$u Z 0: !2:3"

Combining equations (2.2) and (2.3) leads to a Laplace equation for the pressurein the layer,

DP Z 0: !2:4"To complete the formulation of the problem, we need to specify the boundaryconditions. Defining the position of the crack front to be xZ f !y", with f !y"Z0corresponding to a straight front, we may write the boundary conditions as

u!x; y; 0"Z v!x; y; 0"Zw!x; y; 0"Z 0; !2:5"sxz!x; y; 1"Zsyz!x; y; 1"Z 0; !2:6"

w!x; y; 1"Z h!x; y"; !2:7"

szz!x; y; 1"Za3D2h!x; y"; x! f !y";

0; xO f !y":

(

!2:8"

The conditions (2.5) reflect the attachment of the adhesive layer to the substratezZ0. The rest of the conditions are a consequence of the continuity ofdisplacement and traction at the adhesive interface between the layer and theflexible cover-slip. In particular, conditions (2.6) follow from the assumption of ashear-free interface between the flexible plate and the adhesive layer. From aphysical point of view, we argue that the separation between the plate and thefilm of order of a ‘van der Waals’ distance (approx. 5 nm) allows the plate to‘slip’ relative to the layer so that the shear stresses are minimal; in reality, thereis some evidence of slip, but the exact boundary condition is still a matter ofsome debate (Ghatak et al. 2005). As we will see later, this choice may beresponsible for the small discrepancy between the theoretical prediction for thecritical wavelength and that observed experimentally. Equations (2.7) and (2.8)state that the layer is in contact with the flexible plate, i.e. the cover-slip only in

M. Adda-Bedia and L. Mahadevan4

Proc. R. Soc. A

the domain x! f !y". We note that aZ!D=mH 3"1=3, the ratio of the twocharacteristic length-scales is the only control parameter in the problem.

Far from the crack, the deflection of the elastic plate h!x; y" and the stress atthe interface szz!x; y; 1" must also satisfy the asymptotic relations

h!x; y"Z 0 as x/KN; !2:9"

szz!x; y; 1"Zh!y"!!!!!!!!!!!!!!!!!!!!!!!

p!f !y"Kx"p CO

!!!!!!!!!!!!!!!!f !y"Kx

p" #as x/ f !y"K: !2:10"

The condition (2.9) ensures the decay of the traction and displacements far fromthe crack front, while (2.10) enforces the inverse square-root singularity of thestress field at the crack front, with h!y" being the local stress intensity factor.Finally, we note that the vertical displacement of the cover-slip hg!x; y" in theregion where it is free !xO f !y"" satisfies the equation

D2hg!x; y"Z 0; !2:11"with the boundary conditions

hg!a; y"Z 1; !2:12"Dhg!x; y"jxZa Z 0; !2:13"

where a is the dimensionless position of the spacer that lifts the cover-slip (seefigure 1a). At the crack front, xZ f !y", the following continuity conditions hold

h!x; y"jxZf !y"K Z hg!x; y"jxZf !y"; !2:14"Vh!x; y"jxZf !y"K ZVhg!x; y"jxZf !y"; !2:15"Dh!x; y"jxZf !y"K ZDhg!x; y"jxZf !y"; !2:16"

which correspond to the physical requirements that the displacement of thecover-slip (elastic plate), its gradient and its curvature are all continuous at thislocation. However, the derivative of the curvature, which is proportional to thevertical shear force will be discontinuous at the crack front.

The use of a long wavelength plate theory for the cover-slip, embodied in (2.8)needs to be justified, especially since we treat the adhesive film as a bulk solid,including the effect of the singular stress distribution at the crack tip. Two factshelp us: (i) both faces of the cover-slip are free of shear stresses and thus allow itto deform by bending, while the adhesive film, which is stuck to the substrateand confined in the normal direction by the cover-slip can only deformisochorically by shearing and (ii) the three-dimensional effects of the singularstress on the cover-slip in the vicinity of the crack tip are relevant only in aboundary layer whose size scales as the thickness of the cover-slip (Keer & Silva1972). Experimental observations show that the critical value of the confinementparameter for the onset of the undulatory instability awacz18 so that thecritical adhesive layer thickness Hczh!Ep=m"1=3=40, where we have used the factthat the bending stiffness of the cover-slip DZEph

3=12!1Kn2", where Ep and nare the Young modulus and Poisson ratio of the glass cover-slip of thickness h.Since Epz100 GPa; mz1 MPa; nz0:25, this yields Hczh. Thus, over scalesmuch larger than the thickness of the cover-slip, we may use a simple platetheory to characterize its deformations, while we must account for the completeelastic field in the adhesive film.

5Crack-front instability

Proc. R. Soc. A

The piecewise boundary condition (2.8) suggests that we change thecoordinate system from !x; y; z" to !X hxKf !y"; y; z". With respect to thissystem, we may then write

v

vxA!x; y; z"Z v

vXA!X ; y; z"; !2:17"

v

vyA!x; y; z"Z v

vyKf 0!y" v

vX

$ %A!X ; y; z"; !2:18"

DA!x;y; z"Z !1Cf 02!y"" v2

vX2Kf 00!y" v

vXK2f 0!y" v2

vy vXC

v2

vy2C

v2

vz2

$ %A!X ;y; z":

!2:19"Furthermore, we assume that the perturbed crack has the form f !y"Ze sin uy,with jej/1. Then, anticipating a perturbation expansion in e to account for theundulations of the crack, we introduce the following forms

A!X ; y; z"ZA0!X ; z"Ce A1!X ; z"C v

vXA0!X ; z"

$ %sin uyCO!e2"; !2:20"

B!X ; y; z"Z eB1!X ; z"cos uyCO!e2": !2:21"Here Ahu, w, h, hg, P, sxx , szz , sxz , syy, while Bhv, sxy, syz . The advantage ofthe form of these perturbation expansions is that it makes the equations for thezeroth- and first-order problem in e similar; in particular, by changing thesubscripts from 1 to 0, and taking the limit uZ0 allows us to deduce the zeroth-order system from the first-order one and provides some notational andcomputational efficiency in our algebraic manipulations. Substituting the forms(2.20) and (2.21) into the equilibrium equations in the film (2.2)–(2.4) yields

vP1

vXZ

v2u1vX2Ku2u1C

v2u1vz2

; !2:22"

uP1 Zv2v1vX2Ku2v1 C

v2v1vz2

; !2:23"

vP1

vzZ

v2w1

vX2 Ku2w1Cv2w1

vz2; !2:24"

vu1vX

Kuv1Cvw1

vzZ 0; !2:25"

v2P1

vX2 Ku2P1 Cv2P1

vz2Z 0; !2:26"

subject to the boundary conditions (2.5)–(2.8) which may be written asu1!X ; 0"Z v1!X ; 0"Zw1!X ; 0"Z 0; !2:27"

s1Xz!X ; 1"Zs1yz!X ; 1"Z 0; !2:28"w1!X ; 1"Z h1!X"; !2:29"

s1zz!X ; 1"Z a3 v2

vX2Ku2

" #2h1!X"; X!0;

0; XO0:

8><

>:!2:30"

M. Adda-Bedia and L. Mahadevan6

Proc. R. Soc. A

The above linear problem for P1; u1; v1;w1 may be solved in terms of Fouriertransforms. In terms of the definitions

~A!k; z"Z&CN

KNA!X ; z"eikXdX ; A!X ; z"Z

&CN

KN

~A!k; z"eKikX dk

2p; !2:31"

we may write the solution in the Fourier domain as~P1!k; z"Z a!k"cosh!Kz"Cb!k"sinh!Kz"; !2:32"

~u1!k; z"ZKik

2K!b!k"z cosh!Kz"C!c!k"Ca!k"z"sinh!Kz""; !2:33"

~v1!k; z"Zu

2Kb!k"z cosh!Kz"C Kb!k"Kk2c!k"

u2 Ca!k"z' (

sinh!Kz"' (

; !2:34"

~w1!k; z"Z1

2a!k"z cosh!Kz"K a!k"

KKb!k"z

' (sinh!Kz"

' (; !2:35"

where

K Z!!!!!!!!!!!!!!!!k2Cu2

p: !2:36"

These forms satisfy the equilibrium equations (2.22)–(2.26) and the boundaryconditions at zZ0 given by (2.27). Applying the boundary conditions (2.28) and(2.29) at zZ1 leads to a relationship between ~s1zz!k; 1" and ~w1!k; 1"Z ~h1!k"given by

~h1!k"ZF!K"~s1zz!k; 1"; !2:37"where

F!K"Z K2K Csinh 2K

2K!1C2K2 Ccosh 2K": !2:38"

The piecewise boundary condition (2.30) on s1zz!x; 1" suggests the use of theWiener–Hopf method of factorization and solution. For this purpose we write

~h1!k"Z&CN

KNh1!X"eikXdX Z ~h

K1 !k"C ~h

C1 !k"; !2:39"

where

~hK1 !k"Z

&0

KNh1!X"eikXdX ; ~h

C1 !k"Z

&CN

0h1!X"eikXdX ; !2:40"

and

~s1zz!k; 1"h ~sK1zz!k; 1"Z&0

KNs1zz!X ; 1"eikXdX ; !2:41"

where the condition s1zz!XO0; 1"Z0 has been used. We pause here to notethat if k is complex !khkRC ikI", the integrals in (2.40), (2.41) and theirderivatives with respect to k are bounded at infinity only when !kIX"O0.Therefore, in the complex k-plane, ~h

C1 !k" is analytic when ImkO0, while ~h

K1 !k"

and ~sK1zz!k; 1" are analytic when Imk!0. Then, in the Fourier domain, theboundary condition (2.30) may be written as

~sK1zz!k; 1"Za3G1!k"Ca3K4~hK1 !k"; !2:42"

with

G1!k"Z h 0001 !0K"Kikh 00

1 !0K"K!k2 C2u2"h 01!0K"C ik!k2 C2u2"h1!0K": !2:43"

7Crack-front instability

Proc. R. Soc. A

Substituting the above expression for ~hK1 !k" in (2.37) leads to

a3K4~hC1 !k"Ka3G1!k"ZC!K"~sK1zz!k; 1"; !2:44"

where

C!K"ZK1Ca3K4F!K"; !2:45"is a real function with the following properties: (i) it is an even function of K;(ii) as K/0, C!K"/K1Ca3K6=3, and (iii) as jK j/N, C!K"/a3jK j3=2.Therefore, C!K"Z0 always admits two real roots KZGb!a" (see figure 2),which fact will be important for subsequent analysis. Thus, we may rewriteC!K" as

C!K"Z !K2Kb2"D!K"; !2:46"where D!K" is an even function that has no real roots.

We now need to account for the asymptotic boundary conditions both far fromand in the vicinity of the crack front. Since the front can undulate and thus is afree boundary, the similarity between the first-order problem and the zeroth-order problem stops here. For later reference, we note that the importantequations for subsequent analysis are (2.43), (2.44) and (2.46), where the zeroth-order problems in e are deduced by changing the subscripts from 1 to 0, andtaking uZ0.

The far-field condition (2.9) can be expressed in the Fourier domain as

~hG0 !k"Z cte as k/0; !2:47"

which is equivalent to the statement that ~hC0 !k" and ~h

K0 !k" have no poles at kZ0;

the same condition holds for ~hG1 !k". The normal stress component at the interface

between the plate and the adhesive film szz!X ; y; 1" may be written out as aperturbation expansion as

szz!X ; y; z"hs0zz!X ; z"Ce s1zz!X ; z"C v

vXs0zz!X ; z"

$ %sin!uy": !2:48"

Additionally, the perturbation of the stress intensity factor h!y" due to the shapeperturbation of the crack may be written as

h!y"Z h0 Ceh1!u"sin!uy"CO!e2": !2:49"

5 10 15 200.2

0.3

0.4

0.5

0.6

a

b

Figure 2. The solution b!a" of C!K"Z0 as defined by (2.45).

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Proc. R. Soc. A

The stability of the straight crack front follows from the sign of the perturbedstress intensity factor (Rice 1985); if h1!u"O0, the straight front is stable, whileif h1!u"!0 it is unstable.

In light of the crack-tip singularity as given by (2.10), the behaviour ofs0zz!X ; 1" and s1zz!X ; 1" in the vicinity of the crack front XZ0 are given by

s0zz!X ; 1"Z h0!!!!!!!!!!KpX

p CO!!!!!!!!KX

p" as X/0K; !2:50"

s1zz!X ; 1"C v

vXs0zz!X ; 1"Z h1!u"!!!!!!!!!!

KpXp CO!

!!!!!!!KX

p" as X/0K: !2:51"

Equivalently in the Fourier domain, the behaviour of ~sK0zz!k; 1" and ~sK1zz!k; 1" forjkj/N are given by

~sK0zz!k; 1"Zh0!!!!ik

p CO1

!ik"3=2

!as jkj/N; !2:52"

~sK1zz!k; 1"Zh0!!!!ik

pCO

1!!!!ik

p' (

as jkj/N; !2:53"

so that the first-order perturbation of the stress intensity factor given by (2.51) is

h1!u"Z limjkj/N

!!!!ik

p!~sK1zz!k; 1"Kik~sK0zz!k; 1"": !2:54"

Finally, the vertical displacement of the elastic plate hg!X ; y"Zhg!X" inthe region where it is free (XO0) can be determined by integrating (2.11) withthe boundary conditions (2.12) and (2.13). In terms of the notation introducedat the beginning of this section, the zeroth-order solution hg0!X" of the deflectionof the plate is given by

hg0!X"Z c1!XKa"3 Cc2!XKa"C1; !2:55"where c1 and c2 are real constants. At the crack front, XZ0, the continuityconditions (2.14)–(2.16) impose

h0!0K"ZKc1a3Kc2aC1; !2:56"

h 00!0K"Z 3c1a

2 Cc2; !2:57"h 000!0K"ZK6c1a: !2:58"

For the first-order problem that accounts for the undulatory perturbations of thecrack front, the deflection of the glass plate hg1!X" is given by

hg1!X"Z d1!XKa"cosh#u!XKa"$Cd2 sinh#u!XKa"$; !2:59"where d1 and d2 are real constants. At the crack front, XZ0, the continuityconditions (2.14)–(2.16) yield

h1!0K"ZKd1a cosh#ua$Kd2 sinh#ua$; !2:60"h 01!0K"Z !d1Cd2u"cosh#ua$Cd1ua sinh#ua$; !2:61"

h 001 !0K"Z dh 000

0 !0"Kd1u2a cosh#ua$K!2d1 Cd2u"u sinh#ua$: !2:62"

We note that dh 0000 !0"hh 000

g0!0"Kh 0000 !0K" due to the discontinuity of the derivative

of the plate curvature at the crack front.This completes the problem formulation. At each order of the perturbation,

there are three constants of integrations and the stress intensity factor which

9Crack-front instability

Proc. R. Soc. A

remain to be determined. For the zeroth-order problem, these quantities are c1; c2;h 0000 !0K" and h0, while for the first-order problem they are d1; d2; h

0001 !0K" and h1.

3. The straight front

We now turn to a solution of the straight front via the zeroth-order problem.Changing the subscripts from 1 to 0, and taking uZ0 in (2.43) and (2.44) leads to

a3k4~hC0 !k"Ka3G0!k"Z !k2Kb2"D!k"~sK0zz!k; 1"; !3:1"

G0!k"Z hK0000 !0"KikhK00

0 !0"Kk2hK00 !0"C ik3hK0 !0": !3:2"

Here ~sK0zz!k; 1" is analytic for Imk!0 and ~hC0 !k" is analytic for ImkO0 suggesting

the use of a Wiener–Hopf decomposition. Since the function D!k" can bedecomposed into the form (see appendix A)

D!k"ZDC0 !k"DK

0 !k"; !3:3"where DC

0 !k" (resp. DK0 !k") is analytic for ImkO0 (resp. Imk!0), (3.1) can be

written as

a3!k4~hC0 !k"KG0!k""!k2Kb2"DC

0 !k"ZDK

0 !k"~sK0zz!k; 1"; !3:4"

which is an equality on the real axis between an analytic function for ImkO0 (theleft-hand side of (3.4)) and an analytic function for Imk!0 (the right-hand side of(3.4)). The only analytic function in the complex plane that satisfies this property isa polynomial function of k (Muskhelishvili 1953). The asymptotic behaviour of~sK0zz!k; 1" given by (2.52), and the property that DK

0 !k"x!!!!ik

pfor jkj/N (see

appendix A) allows only a real constant as the general solution of (3.4), so that

~sK0zz!k; 1"Zh0

DK0 !k"

; !3:5"

~hC0 !k"Z

1

k4G0!k"C

h0a3 !k

2Kb2"DC0 !k"

h i; !3:6"

where h0 is the stress intensity factor of the straight crack front.Finally, substituting (3.6) into the Fourier-domain equivalents of

(2.56)–(2.58), and noting that ~hC0 !k" has no poles at kZ0 leads to the

determination of the constants c1, c2, h0000 !0K" and h0 as functions of a and a with

c1 ZKiDC00 !0" b

2h06aa3 ; !3:7"

c2 Z !2DC0 !0"Cb2aiDC0

0 !0"Kb2DC000 !0"" h0

2a3 ; !3:8"

dh 0000 !0"Z h 000

g0!0C"Kh 0000 !0K"ZK!aDC

0 !0"C iDC00 !0"" b

2h0aa3 ; !3:9"

h0 Z6a3

6aDC0 !0"C!6C2b2a2"iDC0

0 !0"K3b2aDC000 !0"Kb2iDC000

0 !0": !3:10"

In typical experiments where the degree of confinement is high, a[1. Therefore,the stress intensity factor h0 is well approximated by

h0x3a3

b2a2iDC00 !0"

: !3:11"

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Proc. R. Soc. A

We are now in a position to find the equilibrium configuration of thestraight front. The total dimensionless energy E0 of the system is the sum ofthe energy in the adhesive film, that of the elastic plate and the interface, andis given by

2H 3

Dd2U0 ZE0!a"Z

1

a3

&dSs0ij

vu0i

vxjC

&dx!h 00

g0!x""2 CGa: !3:12"

Here GZ2gH 4=Dd2 is the dimensionless line tension (or equivalently the surfaceenergy of the cohesive interface) with g the dimensional surface energy. By usingthe divergence theorem and the far-field boundary conditions, the previousexpression is simplified to

E0!a"ZK1

a3

&0

KNdxs0zzh0!x"C

&N

KNdx!h 00

g0!x""2CGa: !3:13"

Simplifying further, by integrating by parts and using the boundary conditions(2.8), (2.12) and (2.13) give

E0!a"ZKhg0!a"h 000g0!a"Chg0!0"dh 000

0 !0"CGa: !3:14"For the asymptotic case a[1, this yields

E0!a"Z3

a3CGa: !3:15"

The equilibrium position of the straight front with respect to the position of thespacer, ac, is determined by minimizing E0!a" with respect to a and yields

ac Z9

G

' (1=4

: !3:16"

We observe that the limiting case a[1 corresponds to G/1, a result goingback to the work of Obreimoff (1930) who used the peeling of a thin film of micato determine the value of the interfacial energy. In this limit, the dimensionalcrack length is given by acHZ!9Dd2=2g"1=4, which is independent of m and Hand thus of the properties of the adhesive elastic layer.

The above analysis also allows us to compute the crack opening stresss0zz!x; 1" given by

s0zz!x; 1"Zh0

&CN

KN

eKikX

DK0 !k"

dk

2p; !3:17"

which exhibits the usual square-root singularity at the crack tip. In figure 3, weplot szz!x; 1" for different values of a. When a!abx9, the stress decreasesmonotonically from the edge. However, when aOab, we see that szz!x; 1"becomes non-monotonic, with a new maximum at xZxb. An important pointworth emphasizing here is the appearance of a threshold in the confinementparameter ab that determines whether or not there is a secondary maximum inthe stress behind the crack tip. Under certain conditions (Ghatak et al. 2004)corresponding to the case when the contact line is pinned by a perpendicular edgeso that the singularity is weaker than that for a crack, crack nucleation proceedsvia cavitation behind the pinned edge followed by the coalescence and growth ofthese resulting bubbles. A simple lubrication theory that neglects the stresssingularity at the crack tip (Ghatak et al. 2004) suffices to capture thisphenomena and yields predictions that are consistent with experiments. Since

11Crack-front instability

Proc. R. Soc. A

the local energy release rate at a perpendicular edge vanishes and furthermore,the incipient crack is pinned at the edge by inhomogeneities and defects it isprevented from moving. Then, cavitation bubbles may arise where the stress ismaximally tensile behind the pinned crack tip. In the current situation, the cracktip (or contact line) is not pinned so that in general we do not expect cavitationbehind the crack tip since the stresses at the tip will typically always be largerthan that behind for two reasons: the contact line is a location where the energyrelease rate is finite (tantamount to the presence of a classical square-rootsingularity), and there are typically no pinning sites along the featurelessadhesive surface, unlike at an edge.

In figure 4, we plot the dimensionless location of maximum tensile stress xb asa function of the confinement parameter a. We see that xb varies quasi-linearlywith a for aT10, with the dimensionless location of maximum tensile stressKxbz0:1a. Then, the actual location of bubble nucleation is given byKxbHz0:1!D=m"1=3 which is independent of the thickness of the film. For comparison, wealso show the theoretically determined and experimentally observed dimension-less location of the maximum tensile stress for the case when the crack front ispinned at an edge (Ghatak et al. 2004). The main differences between theseresults can be traced back to the conditions at the crack front, embodied in (i)the effect of the crack-tip singularity and (ii) the implicit assumption in Ghataket al. (2004) of a pinned crack tip that is not allowed to move.

4. The wavy front

We now consider the first-order problem that accounts for the undulatoryperturbations of the crack front. To facilitate the application of the Wiener–Hopfmethod here, we write the function G1!k" given by (2.43) as

G1!k"Z !a0C ia1k"!kKiu"2C!b0C ib1k"!k2Cu2Kb2"; !4:1"

0.1 0.2 0.5 1 2 5 100

0.5

1.0

1.5

2.0

2.5

–x

s zz(

x,1)

h 0

Figure 3. szz!x; 1"=h0 for different values of a (from bottom to top aZ8; 10; 22). We see thatszz!x; 1"=h0 displays a local maximum behind the crack tip when aOabx9.

M. Adda-Bedia and L. Mahadevan12

Proc. R. Soc. A

where a0, a1, b0 and b1 are real constants related to h1!0K", h 01!0K", h 00

1 !0K" andh 0001 !0K" by

h1!0K"Z a1Cb1; !4:2"

h 01!0K"ZKa0K2ua1Kb0; !4:3"

h 001 !0K"Z 2ua0 C3u2a1 C!u2Cb2"b1; !4:4"

h 0001 !0K"ZK3u2a0K4u3a1K!u2 Cb2"b0: !4:5"

The boundary condition expressing traction continuity at the adhesiveinterface (2.44) may now be written as

!kC iu"2~hC1 !k"Ka0Kia1k

k2 Cu2Kb2K

D!K"~sK1zz!k; 1"a3!kKiu"2

Zb0C ib1k

!kKiu"2; !4:6"

where the function ~sK1zz!k; 1" is analytic for Imk!0 and the function ~hC1 !k" is

analytic for ImkO0. Furthermore, since the function D!K" can be decomposedinto the form (see appendix A)

D!K"ZDC1 !k"DK

1 !k"; !4:7"where DC

1 !k" (resp. DK1 !k") is analytic for ImkO0 (resp. Imk!0), (4.6) can be

written as

!kC iu"2~hC1 !k"Ka0Kia1k

!k2Cu2Kb2"DC1 !k"

KDK

1 !k"~sK1zz!k; 1"a3!kKiu"2

Zb0 C ib1k

!kKiu"2DC1 !k"

: !4:8"

The left-hand side of (4.8) is the real-valued difference between a function that isanalytic when ImkO0 and another function that is analytic when Imk!0. This

–x b

5 10 15 200

0.5

1.0

1.5

2.0

2.5

3.0

(a)

(b)

(c)experimental range

a

Figure 4. A comparison of the experiments and predictions for the location xb of the secondarymaximum in the normal stress szz!x; 1" in the case treated here when the contact line (crack front)is free, and the case when it is pinned (Ghatak et al. 2004), as a function of the confinementparameter a. Curve (a) corresponds to the present work and the dashed line is given byKxbz0:1a.Curve (b) corresponds to the theoretical results based on lubrication theory in Ghatak et al. (2004)and is given byKxbZ0:49

!!!a

p. Curve (c) corresponds to the experimental results in Ghatak et al.

(2004) and is given by KxbZ0:73!!!a

pK0:006a. We note that the present work shows that the

secondary maximum appears only when aRab, and is a direct consequence of accounting for thecrack-tip singularity.

13Crack-front instability

Proc. R. Soc. A

allows us to write the solutions for ~sK1zz!k; 1" and ~hC1 !k" as (Muskhelishvili 1953)

~sK1zz!k; 1"Za3!kKiu"2

DK1 !k"

&CN

KN

F!k 0;u"k 0KkC ie

dk 0

2ip; !4:9"

~hC1 !k"Z

a0 C ia1k

!kC iu"2C

!k2 Cu2Kb2"DC1 !k"

!kC iu"2

&CN

KN

F!k 0;u"k 0KkKie

dk 0

2ip; !4:10"

where

F!k 0;u"Z b0C ib1k0

!k 0Kiu"2DC1 !k 0"

: !4:11"

The function DC1 !k 0" is analytic when Imk 0O0 so that F!k 0;u" is analytic when

Imk 0O0 except at k 0Z iu where it exhibits a pole of multiplicity 2. This propertyallows us to evaluate the integrals in (4.9) and (4.10) leading to

~sK1zz!k; 1"ZKa3

DK1 !k"

v

vk 0b0 C ib1k

0

DC1 !k 0"

$ %

k 0Ziu!kKiu"C b0Kb1u

DC1 !iu"

$ %: !4:12"

To complete the solution of the wavy front problem, we need to fix the realconstants d1, d2 and hK000

1 !0". The other constants introduced during the analysisare determined a posteriori through (2.60)–(2.62) and (4.2)–(4.5). First, we notethat (2.42) yields

~hK1 !k"Z

~sK1zz!k; 1"Ka3G1!k"a3!k2 Cu2"2

: !4:13"

Since ~hK1 !k" is analytic for Imk!0, there cannot be any poles at kZKiu, leading

to the conditions

~sK1zz!Kiu; 1"Ka3G1!Kiu"Z 0; !4:14"v

vk#~sK1zz!k; 1"Ka3G1!k"$kZKiu Z 0: !4:15"

Using our knowledge of the singularity at the crack tip which implies that ~sK1zz!k; 1"wh0

!!!!!!!!!ik"

pfor jkj/N (see (2.53)) and the asymptotic behaviour of DK

1 !k"x!!!!ik

pfor jkj/N (see appendix A) in (4.12) leads to

iv

vk 0b0 C ib1k

0

DC1 !k 0"

$ %

k 0ZiuZ

h0a3 : !4:16"

The conditions (4.14)–(4.16) determine the constants d1, d2 and hK0001 !0" as

functions of a, a, u and h0, but we will not write them out explicitly due to theirlength. Instead, to understand the onset of the wavy instability of the crackfront, we will focus on the determination of the perturbation to the stressintensity factor h1!u" induced by the perturbation of the crack front given by(2.49) and (2.54). For this, we first simplify the expression for ~sK1zz!k; 1", by usingthe condition (4.16) and write (4.12) as

~sK1zz!k; 1"Zh0

DK1 !k"

!ikCuCk!u""; !4:17"

where

k!u"Z a3!ub1Kb0"h0D

C1 !iu"

: !4:18"

M. Adda-Bedia and L. Mahadevan14

Proc. R. Soc. A

When a[1, some algebra allows us to show that k!u" is simply given by

k!u"Z DC1 !iu"KDC

0 !0"iDC0

1 !iu"C 2ub2DC

1 !iu"; !4:19"

where the parameter b!a" is plotted in figure 2. Finally, (2.54), (3.5) and (4.17)allow us to determine the perturbation to the stress intensity factor

h1!u"Z !uCk!u"Cq!0"Kq!u""h0; !4:20"where q!u" is defined by the asymptotic expansion

DK1 !k"Z

!!!!ik

p1C

q!u"ik

' (as jkj/N: !4:21"

To understand the criterion for instability in terms of the perturbed stressintensity factor (Rice 1985), we note that if h1O0, regions which are furtheraway from the spacer !sin!uy"!0" have a stress intensity factor h!y" which issmaller than regions which are closer !sin!uy"O0". Thus, as the crack advancesquasi-statically the undulations will diminish and the crack front will tend tostraighten and stabilize. On the other hand, if h1!0 regions which are furtheraway from the spacer !sin!uy"!0" have a stress intensity factor h!y" that islarger than regions which are closer !sin!uy"O0" leading to a destabilizing effect.Evaluating h1!u" asymptotically using the expression (4.20) and the asymptoticexpansions of k!u" and q!u" yields

h1!u"=h0x2u; u/1;

h1!u"=h0xu=2; u[1:

Therefore, the straight front is always stable to both small and large wavelengthperturbations for all values of the confinement parameter a. When u/1,corresponding to long wavelength perturbations (relative to the film thickness),there is no energetic gain in having a curved crack, while for very shortwavelength perturbations !u[1", confinement plays no role, and in the absenceof any intrinsic short length-scale these perturbations are also stable. Thus, whena is small, h1!u" increases monotonically. However, as a increases, this relationbecomes non-monotonic owing to the effects of boundary-induced elasticshielding. Eventually, when aZac the perturbation in the stress intensity factorh1 changes sign for finite non-zero u, yielding an unstable wavelength lZ2p=u.

To quantify this, we compute h1!u" as given by (4.20) using the definitions(4.19) and (4.21) of k!u" and q!u". These quantities are computed numerically interms of the decomposition of D!K" into DC

1 !k"DK1 !k" (see appendix A), yielding

the results shown in figure 5, where we plot h1!u" for different values of a. We seethat there is a critical value aZacx21 for which h1!uc"Z0 with thewavenumber of the instability ucx1:85, which corresponds to a wavelengthlcx3:4H in reasonable agreement with the corresponding experimental values ofaexpx18, uexpx1:57 and lexpw4H (Ghatak & Chaudhury 2003). The smalldiscrepancy (of order 15%) for both the wavelength and the threshold is probablydue to the uncertainty in our knowledge of detailed nature of the adhesivecontact. In particular, it is not clear that the condition of zero tangential slip(2.6) between the glass plate and the adhesive film is accurate, owing tothe relatively weak nature of the van der Waals interactions present (Ghataket al. 2004).

15Crack-front instability

Proc. R. Soc. A

5. Discussion

Our study illuminates the crucial role of geometric confinement in quasi-staticfracture problems. In particular, we show how a straight crack front may losestability in the plane of the interface due to a competition between bulk andsurface effects in the presence of multiple length-scales. This must be contrastedwith the instability of the crack plane itself to transverse out-of-planeundulations in such examples as the sinusoidal cracking of a thin glass plate(Yuse & Sano 1993; Adda-Bedia & Pomeau 1995).

In the context of adhesion of flexible plates to soft confined films, our analysisallowed us to determine the equilibrium configuration of the straight front, aswell as the conditions for and the mechanism of the instability of a straightcrack to in-plane perturbations. In particular, we find that the equilibriumposition of the straight front does not depend on either the material propertiesor the thickness of the adhesive film. Our analysis in this case generalizes theclassical analysis of Obreimoff (1930) for peeling a solid (mica) film from thebulk material via cleavage by considering the case when the confinementparameter a is not large. As the degree of confinement is increased, the normaltraction at the adhesive interface displays a secondary maximum behind thecrack front when aOabx9. The location of this secondary stress maximum xbis essentially independent of the thickness of the film, withKxbHZ0:1!D=m"1=3.However, sinceKxbH/a, in the present case where the crack front is free tomove to its equilibrium position unhindered, the singularity at the crack tipdominates the stress field and bubble nucleation does not occur. Whenconfinement is increased still further, we see the onset of an undulatoryinstability of the crack occurring when aOacZ21, in agreement withexperiments (Ghatak et al. 2000; Ghatak & Chaudhury 2003). We find thatthe wavelength of the instability scales linearly with the thickness H, consistentwith experiments, although there is a quantitative discrepancy with the actualprefactor that shows a difference of about 15%, which may be due to the

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5w

0

0.1

0.2

0.3

0.4

h 1(w

)/h0

Figure 5. h1!u"=h0 for different values of a (from top to bottom aZ15; 20; 25). We see that h1=h0becomes negative for aOawx21 and indicates the threshold for the onset of the undulatoryinstability of the crack front. The corresponding value of the dimensionless wavenumber ucz1:85.

M. Adda-Bedia and L. Mahadevan16

Proc. R. Soc. A

experimental difficulty of controlling the tangential slippage of the flexible plateon the adhesive film.

On the other hand, when the crack tip is pinned at an edge of adhesive film(e.g. Ghatak et al. 2004), the elastic layer spans the region X!0 only, and thetwo effects conspire to change the qualitative nature of the stress field: thecrack tip is blunted and pinned strongly to the edge. This is consistent withother experiments reported in Ghatak et al. (2004) where the effect of a changein the angle of the edge that pins a crack was to change the relative importanceof pinning and the crack-tip stress singularity, leading to a suppression of thetendency for bubble nucleation. This is also why the simple elastic lubricationtheory in Ghatak et al. (2004) that ignores the stress singularity at the crack tipcompletely, but implicitly assumes that the crack is pinned at the edge of theadhesive film yields criteria for bubble nucleation that compare well withexperiments (see figure 4), but cannot provide a critical value of theconfinement parameter ab for bubble nucleation.

Taken together, these results show that by controlling confinement and crackpinning, it is possible to quantitatively understand and thus tailor the toughnessof interfaces using simple geometric attributes. Much still remains to beaccomplished in theoretical, experimental and technological arenas.

We thank John Hinch for his many comments that helped to sharpen and clarify our arguments.This work was supported by the Schlumberger Chair Fund at University of Cambridge (L.M.).Laboratoire de Physique Statistique de l’Ecole Normale Superieure is associated with the CNRS(UMR 8550) and Universities Paris VI and Paris VII.

Appendix A. The Wiener–Hopf decomposition of D(K)

Here we present the method of decomposition of the function D!K" defined in(2.46) for the zeroth- and first-order problem. Using the properties of thefunction C!K" defined in (2.45), we define a new function E!K" such that

E!K"Z 2D!K"a3

!!!!!!!!!!!!!!!!!g2 CK2

p Z2C!K"

a3!K2Kb2"!!!!!!!!!!!!!!!!!g2 CK2

p ; !A 1"

where

gZ2

a3b2:

The function E!K" is an even real function of K that satisfies E!0"Z1 withE!K"/1 as jK j/N. These properties allow to approximate E!K" using anumerical scheme introduced by Liu & Marder (1991). We first define a newvariable l such that

K Zbl!!!!!!!!!!!1Kl2

p : !A 2"

Now the function E!l" is a bounded non-oscillatory function on #K1; 1$. Theconstant b is chosen such that E!l" is well behaved. The function E!l" isapproximated to any desired accuracy as a sum of Chebyshev polynomials. Thecomplex roots ln of E!l" are then found numerically. Using these results and the

17Crack-front instability

Proc. R. Soc. A

properties of the function E!l", we write E!l" in the following approximate form

E!l"xYN

nZ1

!l2Kl2n"0E!K"xYN

nZ1

K2Cb2nK2 Cb2

; !A 3"

where

bn Zbln!!!!!!!!!!!l2nK1

p : !A 4"

We now use the property D!k"ZD!Kk" so that it follows that

0DC0 !k"DK

0 !k"ZDC0 !Kk"DK

0 !Kk"0 DC0 !k"

DK0 !Kk"

ZDC

0 !Kk"DK

0 !k"Z const:;

where the constant is arbitrary. The same property holds for the first-order decom-position. Therefore, the Wiener–Hopf decomposition of the zeroth- and first-orderproblem can nowbe carried outwithout any difficulty. For the zeroth order, one has

DK0 !k"Z

!!!!!!!!!!!!!gC ik

pEK0 !k"; !A 5"

DC0 !k"Z

a3

2

!!!!!!!!!!!!gKik

pEC0 !k"; !A 6"

where we have chosen the constant such that DK0 !k"w

!!!!ik

pfor jkj/N, and

EH0 !k"Z

YN

nZ1

bnGik

bGik: !A 7"

For the first order, one has

DK1 !k"Z

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!g2 Cu2

pC ik

qEK1 !k"; !A 8"

DC1 !k"Z

a3

2

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!g2 Cu2

pKik

qEC1 !k"; !A 9"

where

EH1 !k"Z

YN

nZ1

!!!!!!!!!!!!!!!!!b2n Cu2

pGik!!!!!!!!!!!!!!!!

b2Cu2p

Gik: !A 10"

Finally, the quantities needed for computing the first-order calculations are given by

q!u"Z 1

2

!!!!!!!!!!!!!!!!!g2 Cu2

pCXN

nZ1

!!!!!!!!!!!!!!!!!b2n Cu2

qK

!!!!!!!!!!!!!!!!b2 Cu2

p$ %; !A 11"

DC1 !iu"Z

a3

2

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!g2 Cu2

pCu

q YN

nZ1

!!!!!!!!!!!!!!!!!b2n Cu2

pCu!!!!!!!!!!!!!!!!

b2Cu2p

Cu; !A 12"

iDC01 !iu"

DC1 !iu"

Z1

2!!!!!!!!!!!!!!!!!!g2Cu2

pCu"

CXN

nZ1

1!!!!!!!!!!!!!!!!!b2n Cu2

pCu

K1!!!!!!!!!!!!!!!!

b2Cu2p

Cu

" #

: !A 13"

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19Crack-front instability

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