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Adherence of elastomers: Fracture mechanics aspectsD. Maugis aa Equipe de Micanique des Surfaces du CNRS, LCPC, 58, Bd Lefebvre, 75732 Paris Cédex15, FrancePublished online: 02 Apr 2012.

To cite this article: D. Maugis (1987) Adherence of elastomers: Fracture mechanics aspects, Journal of Adhesion Scienceand Technology, 1:1, 105-134, DOI: 10.1163/156856187X00120

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Adherence of elastomers: Fracture mechanics aspects

D. MAUGIS

Equipe de Micanique des Surfaces du CNRS, LCPC, 58, Bd Lefebvre, 75732 Paris Cédex 15, France

Revised version received 17 February 1987

Abstract-Application of fracture mechanics concepts to adherence of elastic and viscoelastic solids is examined. It is shown that the Dugdale-Barenblatt model could be applied both for explaining the transition from Johnson-Kendall-Roberts to Deryagin-Muller-Toporov solution in adherence of

spheres, and for liquid bridges. The problem of the threshold energy release rate G0 for elastomers is considered. It is proposed to add to the viscoelastic losses a supplementary friction term when

polymeric chains strongly held at the interface by primary bonds are pulled out.

Keywords: Fracture mechanics; adherence; elastic solids; viscoelastic solids.

1. INTRODUCTION

The pull out force (or adherence force) needed to separate two solids in contact

not only depends on the adhesion forces which link the solids together, but also

on a number of additional parameters such as geometry, stiffness of the

measuring apparatus, withdrawal speed, residual stresses and rheological

properties of the solids. The only knowledge of the thermodynamic work of

adhesion (Dupre energy of adhesion), w = Y 1 + Y 2 - Y 12 2 (where y 1 and y are the

surface energies of the solids, and Y12 their interfacial energy),* and even the

precise knowledge of exact attraction force laws, are insufficient to predict adherence forces.

A way to rationalize the influence of the various parameters is fracture

mechanics. It tells us that separation never occurs as a whole, but by propagation of a crack when the mechanical energy GdA which can be released when the

area of contact decreases by dA overcomes the work wdA to break the bonds.

This strain energy release rate G, introduced by Irwin [1] takes into account the

geometrical factor, the elastic properties of the solids, the existence of prestresses or residual stresses, and allows for the prediction of adherence forces of elastic

solids when w is known. As pointed out by Sneddon [2] the elastic analysis of a

crack tip leads to discontinuities of displacement and to local stresses varying

respectively as p112 and p-ll2 (where p is the distance to the crack tip), the

magnitude of which is described by the stress intensity factors KI, Ku, KII, introduced by Irwin [3, 4] for opening, shearing and antiplane modes. In this

analysis the work GdA is the work done by these singular stresses, a part of

which, wdA, being used to break the bonds, and the remaining dissipated in

*The symbol My is sometimes used in place of w This can be confusing since adsorption reduces the surface energy y by Ay, and decreases or increases the Dupr6 energy of adhesion w by 0 w.

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irreversible processes. For an equilibrium crack these works (virtual works) are

just balanced and one has G = w which is the Griffith equation. However this mathematical description is insufficient to understand the role of

molecular forces in adhesion and the action of surrounding medium. In such a

description only stresses due to external loading are considered, giving rise to a stress intensity factor K1 which tends to open the crack, and there is no inter- action between the lips of the crack. This energy of interaction

(where g(y) is the intermolecular force, and Zo the equilibrium interatomic

distance) appears abruptly* beyond the crack. In reality, as shown by Barenblatt

[5], these cohesive forces extend to a length d, and as would do any other external

forces, they elastically deform the crack and give rise to a stress intensity factor

which is a material constant, and which if unbalanced tends to close the crack. The crack is in equilibrium when Ktot= K1 + Km=O. This condition of cancel- lation for equilibrium is identical to the Griffith criterion when the cohesive zone is sufficiently short [6, 7]. Any singularity is thus removed, and the crack has a

cusp-shaped tip, varying as p112 irrespective of the laws of molecular forces (Fig. 1). In this more physical model, the work GdA is the work done by these cohesive forces. For a constant cohesive stress p in the Barenblatt tip (yielding stress in the Dugdale model [8, 9] of small scale yielding, or capillary pressure in

liquid bridges) this work is simply where 6, is the crack opening displace- ment (COD) at the end of the cohesive zone. It is in this cohesive zone that the various attractive forces observed in air (10, 111 and attractive or repulsive forces

(disjoining pressures) observed in liquids [12-15] are operative. As the Dupr6 energy of adhesion w is the integral of these forces from the equilibrium interatomic distance Zo to a distance where they are negligible, it is easily understood that the surrounding medium must reach this cohesive zone to screen or enhance these forces and modify w If the crack is running too fast for the medium to follow it, there is no variation in w, although the crack lips at some distance have their surface energy reduced by adsorption. Similarly, as the rate of

transport of the active species to the crack tip is limited, either by the arrival rate of molecules at a given vapor pressure or by diffusion through a viscous liquid, the reduction of w can be less for a moving crack than for an equilibrium crack

[16]. To emphasize the importance of this cohesive zone it is sufficient to point out that in n/2 peeling a hung load P is sustained only by the forces in the Barenblatt tip since the adherence is independent of the contact area.

However, there are some differences between fracture mechanics of homo-

geneous and isotropic solids and adherence of solids. In a homogeneous and

*In fact, just beyond the crack tip one has Z > Zo since the solid is stressed, but here Z-Zo is considered as an elastic displacement.

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I

Figure 1. Variation of the energy of interaction with the distance p to the crack tip (a). In the Griffith model, there is no interaction between the crack tips, and the energy w=2y appears as a step function giving a Dirac force. In the Barenblatt model, adhesion forces act in a 'cohesive zone' of finite length and he energy of interaction increases progressively. These adhesion forces elastically deform the crack (b), and there are no singular stresses (c). The maximum stress at the crack tip is the theoretical strength The forces are recognized as adhesion forces for p < 0, and as elastic forces for p > 0. The hatched area is not exacly equal to w since the interatomic distance at p=0 does not have the equilibrium value Zo of the unstressed solid.

isotropic body (where y is not direction dependent) the crack freely follows the

principal stresses and deflects to always have Kn = 0 at its crack tip, and to maximize G [17] thus reducing the thermodynamic potential of the system at the

highest rate. If the surface energy is markedly anisoptropic, it can be energetically favourable for the crack to follow these cleavage planes where y has the lowest

value, even in a mixed mode, since what is to be maximized is now the

generalized force G-2y [18]. Similarly, when interfaces are present (poly- crystals, composites, or for any adherence problem), the crack will follow the interface even with a mixed mode if the value w is sufficiently below the values

2y of the adjoining phases (so, the fracture can to transgranular in pure poly- crystalline metals, where the anisotropy of y is a few percent, and can become

intergranular in an active medium or after segregation of impurities, since

adsorption markedly increases the anisotropy). Interfacial cracks have a special behaviour when an elastic mismatch (described by the Dundurs a and P

parameters [19]) exists: no direct transition from adhesion to separation is

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possible, and an extremely small slip zone (mode II) must precede the open crack

(20J, even under pure normal loading, otherwise unrealistic oscillatory singulari- ties appear at the crack tip. Furthermore, at the transition from slip to separation, the normal stresses must vanish [21]. If a tangential load is added, not only the

slip zone increases considerably, but the crack cannot be fully closed, even under

compressive load [22, 23]. These effects can be seen as due to a kind of butt

stressing of the crack which cannot veer into an adjacent phase. So under

compressive and tangential loadings, interfacial cracks behave as bubbles moving along the interface, as observed in fibre pull out [24]. Similar problems have been found in contact mechanics, with unrealistic oscillatory singularities [25], removed by a slip zone [26], extended slip zone under tangential loading [27], and propagation of bubbles [28-30], showing the complete analogy between fracture mechanics and contact mechanics.

2. ADHERENCE OF ELASTIC SOLIDS

The problem can be solved in two different ways. In the first one (the thermo-

dynamic approach) the energy available to break the bonds is evaluated from the external forces applied to the system, and a mechanical energy release rate is evaluated. In the second, the exact distribution of stresses at the interface is

analyzed, and the stress intensity factors are computed. Of course the two methods must give the same result.

We will focus on the thermodynamic approach (Griffith equation) and will give a comparison with the elasticity approach for the case of adherence of punches.

2.1. The Griffith equation

z

Let us consider the system made of two elastic solids in contact over an area A. This system is allowed to exchange work and heat, but not matter, with the exterior. A force P (compressive or tensile) can be applied to either of the two elastic bodies, either by a dead load as in Fig. 2a, or by imposing a displacement A to a spring of stiffness km as in Fig. 2b.

To avoid the problems of elastic mismatch, it is assumed that the two solids have the same elastic properties, or that one is rigid and the other has a Poisson ratio v=1/2, so that the Dundurs [3 parameter [19] is zero. The energy of the

system U= U(S, 3, A) is a function of the extensive variables S (entropy), d, and A. It can be decomposed into elastic energy UE and the interface energy US = - wA, where W=Y1+YZ-Y12 is the Dupr6 energy of adhesion, a material

property independent of crack velocity which replaces the intrinsic surface

energy 2 y at an interface. The first differential of the energy can be written in the form

with

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G, which describes the variation of elastic energy with A, at constant 6, is the strain energy release rate. Note that wA is considered here as a kind of potential energy which can be recovered by crack healing. The three relationships,

expressing intensive parameters in terms of the independent extensive para- meters, are the equations of state of the system. Knowledge of_these three

equations of state is equivalent to knowledge of the fundamental equation

U=U(S, 6, A), and gives a thermodynamically complete description of the

system. A system is in equilibrium if virtual perturbations of the extensive variables

leave its energy constant. However, equilibrium is often studied in the presence

Figure 2. Equilibrium contact of two elastic solids: (a) dead load, (b) testing machine with stiffness km.

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of constraints such as constant pressure, constant volume or constant tempera- ture. In this case, the Legendre transformation of the energy U is used to

exchange any variable Xj with its derivative Pj =(a UlaXj). The equilibrium of the

system at constant Pj corresponds to the extremum of the function T = which is the Legendre transform of the energy U, and is called the thermodynamic potential. In the present case we wish to study equilibrium at constant temperature, by allowing perturbations of the area of contact at constant load P, or at constant displacement 6 (fixed load or fixed grips conditions). Of interest are thus the Helmholtz free energy F=U-TS, and the Gibbs free energy g= U- whose differentials are

Noting that -P3 is the potential energy UP of the load, these expressions show that

I I I I

Note that the Maxwell relations (obtained from the equality of the mixed partial derivatives of g or F) give

with

Equilibrium at fixed temperature and fixed load conditions (d T=0, dP= 0) corresponds to an extremum of g, and equilibrium at fixed temperature and fixed

grips (d T=0, d d= 0) to an extremum of F. In either case equilibrium is given by

(The same equilibrium condition would be obtained at fixed A with a spring of stiffness km [3 1]. Equation (9) is the Griffith criterion which connects two of the three variables d, P, A of the equations of state (equation (2)), so that the

equilibrium curves 3 (A ), A (P), P( 3 ), are functions of w If G = w, the area of contact will spontaneously change so as to decrease the

thermodynamic potential. If G < w, equations (3) and (4) show that A must

increase, and the crack recedes. Conversely, if G > w the area of contact must decrease to give dg < 0 or dF < 0, and the crack extends. GdA is the mechanical

energy released when the crack extends by dA. The breaking of interfacial bonds

requires an amount of energy wdA, and the excess ( G - w)dA is changed in

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kinetic energy if there is no dissipative factor. G-w is the crack extension force, which is zero at equilibrium.

The equilibrium given by G=w can be stable, unstable or neutral. A thermo-

dynamic system under a given constraint is stable if the corresponding thermodynamic potential is minimum, i.e. if its second derivative is positive. Thus, from equations (3) and (4) stability is defined by

I I

or more generally by (,OGIaA), > 0 if the machine has a finite stiffness km. It can be shown that the stability range monotonically increases with the stiffness, from the fixed load case (km =0) to the fixed grips case (km C1;)) [31, 32]. If, under a stable equilibrium, a fluctuation decreases A(dA < 0), G incrementally decreases and one has G < w: the crack recedes to its equilibrium position. It can only advance if the load P or the displacement 6 is slowly varied, bringing back G to the value w: one is dealing with controlled rupture of an adhesive joint. In this case one has

or

Starting from a stable equilibrium with the two bodies compressed (P > 0, 3 > 0), let us quasistatically decrease the load: one generally encounters a

progressive reduction of area of contact, i.e. a controlled rupture at d G= 0, until a negative load P, is reached where (aGlaA)p < 0; the equilibrium becomes unstable and the crack spontaneously extends toward complete separation at constant Pc, with G-w increasing as A decreases. The load Pc, corresponding to the limit of stability, is the adherence force in an experiment at fixed load. At fixed grips the adherence force could be different.

2.2. Adherence of punches

The calculation of G and KI for a number of geometries such as peeling, double-

cantilever, double torsion, blister test can be found in text books. We will concentrate on the case of adherence of punches (and specially of spheres) which is conceptually an important topic to understand the connection between

adherence, mechanics of contact and fracture mechanics, or more simply what is an area of contact.

The case of an axisymmetric flat punch of radius a on an elastic half-space was solved by Kendall [33] by evaluating the elastic and potential energies from the elastic displacement under the load P:

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where

(E and v are the Young modulus and Poisson ratio). The result is

The equilibrium corresponding to G=w is unstable (as for a crack of radius a in an infinite body) and the load P given by G=w is the adherence force at both fixed load and fixed grips. Equation (14) could also be deduced from the value of

KI given by Paris and Sih [34] for a deeply notched bar. Stress and displacement at the edge of the contact are, of course, those of fracture mechanics in mode I.

The case of a sphere of radius R is more subtle, since the contact is no longer conformal and elastic energy is stored under the action of molecular forces. To evaluate it, Johnson, Kendall and Roberts [35] (JKR theory) first apply the Hertzian load .

and then reduce the load from P, to P at fixed radius of contact. In this flat punch displacement under the load Pi -P:

tensile stresses appear in an annulus inside the area of contact, with singularities and discontinuity of displacement at the edge. Minimization of elastic energy leads to

the equilibrium given by G=w is stable now over a range of loads. The limit of

stability (adherence force), given by (aGlaA)=0, is reached at fixed load for

and at fixed grips for

Compared to the radius of contact ao under zero load, these minimum

equilibrium radii of contact are, respectively, ao and am;n=0.30 a 0. On the other hand, Derjaguin, Muller and Toporov [36] (DMT theory) arrived

at a completely different picture, by taking into account the molecular attraction in a ring-shaped zone outside the contact area which is only under Hertzian

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compression. Considering the reverse problem of a deformable sphere on a rigid plane, they assumed that under the influence of surface forces, the sphere is deformed near the contact region according to Hertz solution ('hemispherical' compression stress distribution in the area of contact and tangential connection to the plane), but with an enlarged radius of contact. They found an adherence force

with separation at point contact, a result similar to that of Bradley [37] for a rigid sphere on a rigid plane. Their paper was criticized by Tabor [38] and a long dispute followed [39-48] on the relative merits of the two theories, on the shape of the deformed zone (vertical tangent or smooth profile) and on the action of molecular forces outside the contact area. Finally, it was agreed that, as suggested by Tabor [38], the JKR analysis fails when the 'neck' at the edge of the contact due to the flat punch displacement, (equation (16)), is of the order of magnitude of the equilibrium interatomic distance Zo. In this case, intermolecular forces outside the contact area can no longer be neglected.

Assuming a Lennard-Jones potential between the deformable sphere and the

plane, Muller et al. [43] have numerically studied the transition from the DMT solution to the JKR solution as a function of a single dimensionless parameter

(which is approximately the ratio of the 'neck' 6. at zero load, to Zo [38, 43]. For

,u < 1 (small or/and rather rigid particles) the profile is close to the Hertzian one, and the adherence force is 2?wR. For ,u < 1 (large or/and soft spheres) the

profile is close to the JKR one, and the adherence force 3/2JrwR occurs at a finite contact radius. In this model the radius of contact corresponds to the circle where the Lennard-Jones forces are extremum, and the external force is simply the sum of the compressive Lennard-Jones forces in the central part of the

contact, and the tensile Lennard-Jones forces acting both in an annulus of the contact area, and outside this contact area. So, when all the tensile stresses are

inside the area of contact one has the JKR model; whereas when all the tensile stresses are outside the area of contact one has the DMT model. It must be

emphasized that the break in the JKR theory is not due to a break in the

hypothesis a/R < 1, for the same assumption exists in the DMT theory, nor to any immersion of the whole sphere in the field of surface forces.

Before going further in the discussion, it is noteworthy that the adherence of

rigid punches can be derived by a direct route [49-51] from the first principles of

elasticity. In the general problem of a convex frictionless punch indenting an

elastic half-space under a normal load P, the area of contact is generally unknown. Since the work of Boussinesq [52] the problem has been solved by

adding the condition that the normal component of stress must be zero at the

limit of the contact area; this condition ensuring a tangential connection between

the punch and the elastic half-space, and prescribing the value do of the depth of

penetration of the punch. As noted by Boussinesq, a difference in the depth of

penetration corresponds to the superposition of a rigid displacement ao-a of

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the punch and gives rise to singular stresses and to the displacements dis- continuities typical of a flat punch; singular compressive stresses are to be ruled out because they lead to a negative discontinuity of displacement and, therefore, to a penetration of the half space into the punch. On the other hand, singular tensile stresses seemed to be impossible, unless the punch and half-space are allowed to sustain normal tensile forces; they were also ruled out, so that the

necessity of zero stress at the edge seemed well-established for a century. Sneddon derived a rigorous solution of the frictionless axisymmetric Boussinesq problem from which he deduced simple formulae for the depth of

penetration 3 of the tip of a punch of arbitrary profile given by f(r), for the total load P which must be applied to the punch to achieve this penetration, for distribution of pressure a1 under the punch, and for displacement u, of the free surface. In his solution an arbitrary rigid body displacement

appears, which must vanish in order for the stresses at the edge of the contact area to be finite. By developing in power expansion the Sneddon formulae, Barquins and Maugis [50] showed that if 0 the stress singularities and

discontinuity of displacement are those of fracture mechanics in mode I

with a stress intensity factor

which can be written

where P1 is the load that would give the same radius of contact a for x ( 1 ) = 0, as that obtained with X (1) 0 0. (For a sphere for a flat punch Pi==0, for a cone of semi angle n/2 - f3, P, =

3/8nKaZtgfJ). The strain energy release rate is thus given by equation (17) whatever the profile of the punch,* and the supple- mentary equation needed to determine the radius of contact is thus the Griffith criterion G=w Note also that the rigid body displacement Jr/2 x(1 ) for 0 is

always given by equation (16) for any punch. When w= 0, one has P=Pl and all the classical results for the Boussinesq problem are obtained.

*As one solid is rigid, one has G VZ 2 *As one solid is rigid, one has G = - 2 1 1 _ E V2 K 2 ' '

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It is thus easy to have the adherence force for any punch whose profile f(r) is known. For a cone, for example [49], one has a fixed load

and at fixed grips

Compared to the radius ao under zero load, these minimum values are, respec- tively, amin = 0.56 ao and am;n=0.06 ao. The same calculations have been done for flat ended spheres and cones [511. About these various adherence forces it is

interesting to note that they are not always proportional to w They vary as J w for the flat punches, as w for the sphere, as W2 for the cone; but the mean stress at

rupture is always proportional to (wKIa) 1/2 The problem now is the following: what must be modified in this apparently

rigorous approach to find the DMT results, and how to introduce the parameter ,u (equation (23))? It seems that the solution can be gleaned from the Dugdale treatment of small scale yielding [8, 9]. In metals the stresses ahead of the crack

tip given by the linear elastic fracture mechanics (LEFM) must be limited to the

yield stress ay, giving rise to a plastic zone whose length dy is computed by cancelling the total stress intensity factor due to external load and to these

restraining stresses in the plastic zone which give singularities of opposite sign. However, such a stress limitation in the plastic zone does not satisfy equilibrium along the crack line with the initial crack length. Use is made of a 'notional' elastic crack length with an apparent crack tip at distance i.e. in the centre of the plastic zone (see e.g. Knott [54]) and this simple correction to LEFM can account for plasticity effects, as long as fracture stress oF is below Qy. Similarly, in adherence problems, the tensile stress in the contact zone, computed by the LEFM, cannot exceed the theoretical tensile strength Œth'

As JKR and DMT models are limiting cases which are not sensitive to the detail of molecular forces, let us assume a constant negative pressure -po acting at the circular crack tip on a small length d =b -a. From Lowengrub and Sneddon (55), or Fabrikant (56) the stress intensity factor is ,

As in the Dugdale-Barenblatt theory this length d can be computed by cancelling the total stress intensity factor KE=Km+KI, where KI is given by equation (28). Hence

a classical result in small scale yielding theories. The correction to LEFM would be to replace a by a -dl2 in equation (28), putting the crack tip at the centre of the cohesive zone, but the resultant equation is more cumbersome than for a

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linear crack in an infinite solid. A proper theory would be to compute the J

integral [57]

from the crack opening displacement d, at the end of the cohesive zone, as done

by Rice [57] for the penny-shaped crack, and to write J=w at equilibrium. For

spherical punches, an evaluation of this crack tip opening can by obtained from the discontinuity of displacement in the LEFM [50] approach:

where e = rla -1. An approximate value can be found by putting x ( 1 ) = 0 and 8=dla

with J = w and d = 3 nKwl 16 p o at equilibrium, the ratio ald is given by

If we assume Lennard-Jones forces as in ref. [41] the maximum stress is given by Q,h = wIZ.. Taking po = w/Zo it thus becomes

Thus the LEFM analysis (i.e. the JKR theory) holds as long 1, as found by Tabor (38] and Muller et al. [43] on the basis of slightly different arguments. Note that in the JKR theory the mean stress at rupture under fixed load is well above

alh when p < 1. If we return to the calculation of G by an energy balance we see that the

problem lies in the flat punch displacement under Pl-P which gives rise to infinite stresses. As the stresses at the edge of the contact cannot exceed there is a stress redistribution which could be taken into account by a small reduction in the radius of contact as in the small scale yielding theories but this correction fails when this reduction is of the order of magnitude of the radius itself.

2.3. Verification of the JKR theory

The equilibrium relationships between load P and radius of contact a (i.e. G = w with G given by equation (17)) was verified by JKR [35] in their original paper and by Maugis and Barquins (51] for flat-ended spheres with a flat of radius af (Fig. 3). In this case, one has

.... ,

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Figure 3. Equilibrium radii of contact for four spheres with a flat of radius af in contact with poly- urethane. Drawn lines are for theory. (From ref. (51].)

The relationship between the penetration 6 of the punch and the radius of contact a was verified by Maugis and Barquins for sphere, both at equilibrium or

during unloading [58] (Fig. 4) and during loading [31], and for flat ended spheres [51].

Adherence at fixed grips was not directly tested, but proofs of the influence of the stiffness of the measuring apparatus were given by studying the kinetics of crack propagation (59) (see below).

The shape of the deformed surface was indeed shown to be non Hertzian [38, 44, 48, 60, 61] except when adsorption considerably decreases w [35], when

repulsive forces act [48], or in loading stages during which Kj = 0 [31].

2.4. Influence of prestresses

One advantage of the use of strain energy release rate G is that the influence of

pre-stresses or residual stresses can easily be studied. These stresses increase the stored elastic energy, hence the available energy for crack propagation: they reduce the quasistatic force of adherence. This point has been shown by Kendall for peeling [62]: when an elastic strip (thickness h, width b) is stretched to e=dL/L before placed on a rigid plane, the elastic energy stored is UE=1/2 (eZEhA) and thus.the quantity 1/2 (e2Eh) released when the crack advances by dA must be included in G. For 7r/2 peeling, for example, one has

so that the elastic adherence force (for G=w) is reduced by 1/2 (e2Eh) and

disappears (spontaneous peeling) when the elastic energy stored in the strip is

equal to the total energy of adhesion wA at the interface. If such a deformation

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Figure 4. Displacement vs. radius of contact for adherence of glass ball on polyurethane, for various unloadings. Comparison between theory and experiments. The curve b(a) is given by G -w; the curves (3)p represent the equation of state, equation (2b). (From ref. (58J.)

appears by thermal dilatation, the elastic strip must be as thin as possible to avoid spontaneous peeling.

Such spontaneous peeling has been studied by Barquins [63] by placing a rubber strip on to a second rubber strip which is stretched. Spontaneous peeling and kinetics of crack propagation (see below) are in agreement with the theory within 3%.

A special case of residual stress effect is the adherence of rough surfaces:

higher asperities are compressed, more elastic energy than on a smooth surface is stored and the adherence decreases and may even disappear [64].

3. LIQUID BRIDGES

Adherence between elastic solids can be obscured by capillary forces which may have the same order of magnitude. Exact calculations of these forces are complex

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[65, 66] but are simplified for small liquid volumes and with the circle

approximation for the meniscus. For a rigid sphere of radius R on a plane, it becomes

as verified by a number of authors [67-701. This force due to the Laplace pressure appears when a meniscus with a negative radius is set up, as well in contact with another liquid [71] as when a nonwetting liquid recedes spontane- ously to form vapor-filled cavity [72].

This Laplace pressure can be seen as a restraining pressure Po= y/ p (where p is the meniscus radius) acting at the tip of an external crack, as in the Dugdale model, with a crack tip opening 2pcos 0. Equation (31) thus gives

If b is the wetted radius of an axisymmetric rigid punch with a shape f(r), and z is the separation, G can be computed from the potential energy Up = -Pz

As a rough approximation, let us take a cylindrical liquid bridge of constant volume V

With the condition d Vldz =0, it becomes

Equating equation (36) and (37), we have for a flat punch ( f(r)=0) the classical

result,

for a sphere ( f(r)=rzl2R)

a result given by Fisher and Israelachvili [70], and for a cone of semi-angle {3

(f(r)= r/tg{3):

This last expression is an approximation for large f3 of the expression given by Coughlin et al. [73], for the case z=0.

Equation (37) shows that only at the edge of the meniscus is the shape of the

punch taken into account, so that equation (35) is also valid for flattened sphere. For adherence of a deformable sphere with a radius of contact a and a wetting

menicus at r=b the problem is more difficult. Experiments [70] have shown that

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the contact radius decreases with increasing capillary condensation with vapor pressure, and that under tensile load it always falls before the surface finally jumps apart. The shape of the surfaces, which is of the JKR type with dry surfaces, becomes progressively more rounded as the meniscus increases and as a tensile load is applied to tend towards their original undeformed shape as the

separation force increases under high vapor pressure (large meniscus). These

experiments have also shown that equation (35) is still valid in presence of a finite radius of contact, which is understandable since the shape is surely spherical at r=b, but only as long as the surface energy of the liquid is

sufficiently high to exceed the adhesion force in air. Clearly this behavior mimics with a lower po the JKR-DMT transition. Here, the cohesive zone is considerably increased, (the size of the meniscus), and is imposed by the vapor pressure, with a known crack tip opening 2pcos 8; but true cohesive forces (although reduced

by adsorption) continue to act at the very crack tip so that the integral J is

The same analysis as above could be given with b/a obtained by equating equations (28) and (29), but the exact analysis remains to be done.

Note that the first application of fracture mechanics to a liquid bridge is

probably that of Burns and Lawn [741 in their experiment of a liquid between two cantilever arms.

4. ADHERENCE OF VISCOELASTIC SOLIDS

Elements of volume near the trajectory of a moving crack undergo a cycle of stress when the crack tip comes on and then comes off, whose characteristic time is of the order of d/v (where d is the length of the cohesion zone and v the crack

velocity) and the magnitude proportionnal to crh and hence to w In a dissipative material energy is lost during such a cycle, in proportion to w as pointed out by McLean [75], so that losses arise only if the interface itself is capable of with-

standing stress [76]. Thus, instead of continuously accelerating until the Rayleigh velocity, the crack takes a limit velocity v function of the generalized force G - w that drives it. In a viscoelastic solid the drag force on the crack will be a function of the frequency dependent loss modulus E", as pointed out by Mullins [77] and Mueller and Knauss [78], so that one can write [58, 76].

which generalizes for any geometry the equation given by Gent and Schultz [79] to account for peeling in various liquids.

In this equation kinetic energy is neglected (although it can be taken into account by adding a term d UxldA, where UK is the kinetic energy of the system), the viscoelastic effects are assumed to be limited at the crack tip zone, so that

gross displacements are elastic and G can still be evaluated by the elastic theory

(with relaxed modulus) during kinetic phenomena. Results at various temperatures can be shifted to a reference temperature

Ts = Tg+ 50 (where Tg is the glass transition temperature measured at zero

frequency) by using the WLF shift factor [80] aT, where

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so that a master curve is obtained when studying crack propagation as a function of the reduced parameter aTv, as first shown by Smith [81] for tearing, and Kaelble [82] for peeling.

Thus, in equation (40) surface properties (w) and viscoelastic properties 0(v) are completely decoupled from elastic properties, geometry and loading condi- tions included in G. The dimensionless function Ø(aTv) is a characteristic of crack propagation in Mode I in the material. Once Ø( aTv) is known, equation

(40) allows one to predict any feature such as kinetics of detachment at fixed

load, fixed grips or fixed cross-head velocity. Equation (40) was verified for glass on polyurethane for various geometries

where G can be computed (sphere, flat punch, flat ended spheres, peeling) by studying crack propagation at a fixed load [31, 51, 58) or at a fixed displacement [59, 63] at various temperatures. As shown in Fig. 5, 0 is independent of the

geometry and varies as (aT V)0.6 over about five powers of ten, a result often found for the peeling of rubber-like material [78, 83, 84]. The fact that equation (17)

'

gives the same results for crack propagation as flat punches and peeling is a proof of its correctness and a further verification of the JKR theory.

Water adsorption decreases the Dupr6 energy of adhesion, and hence the viscoelastic losses according to equation (40). This point was verified by measuring the rolling resistance t% of a glass cylinder rolling on an inclined

sample (G = R / 1) as a function of velocity for various humidiy conditions [31, 84]. The result is a translation of the G(v) curves with water adsorption. As G > w in these experiments, the shift clearly arises from the multiplicative term w on the right hand side of equation (40), as previously deduced from peeling in various liquids (79) or on various substrates [76]. ,

Figure 5. Reduced crack extension force against crack speed for glass/polyurethane systems. (From ref. [5 8].) Results for various radii of spheres (R), various initial loads (P) and final loads (P'), for two temperatures (using the WLF shift factor), fall on a master curve. The continuous line is for peeling. Other geometries tested (axisymmetric flat punches, flat-ended spheres) gave results falling on the same master curve.

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4.1. Tackiness

The adherence of solids is more often studied with a tensile test machine at constant cross-head velocity than at constant load, but the kinetics of separation is less easier to interpret, due to the competition for increasing G with time between increasing 3 with time at constant a, and decreasing a at constant 6. If the machine has an infinite rigidity km (Fig. 2), one has A = <5 and the variation of G with time is given by

The recorded force first increases, then decreases. The maximum value, termed the tack force, is a measure of the adherence under this particular experimental condition, and has no clear physical significance. The area under the curve, termed tack energy, is equal to the work jGda of the cohesive stress at the crack

tip. 'Tackiness' refers to the ability of an elastomer to adhere instantaneously to a solid surface, or to itself, after a brief time of contact under low pressure. Probe tack testing can be analysed by equation (40), and tack curves obtained by computer integration closely coincide with experimental ones [85]. Figure 6 is for a spherical probe and shows that even at very low cross-head velocity the visco- elastic effects considerably increase the adherence force compared to the elastic

(or quasistatic) adherence force at fixed displacement (Point D).

4.2. Viscoelastic losses and negative resistance branch

Viscoelastic losses at the crack tip are related to the frequency dependence of E", the imaginary part of Young's modulus (loss modulus). As a matter of fact, E" varies as m°.6 for polyurethane at low frequency [86]. A viscoelastic model such as the parabolic Zener model

can be used to obtain E"( w) - (w-rf when cor- 0. Such a model corresponds to a continuous spectrum of relaxation times, and leads to symmetrical Cole-Cole

diagram with E'(co) -(to T) -11 when (or - OJ . Assuming the viscoelastic effects to be limited to the Barenblatt tip, Coussy [87] has shown that equation (42) leads to 0(v) in equation (40), varying as v" at low speeds, with a maximum at a value

Vo confirming thus the model proposed in refs. [86, 88]. Beyond the curve

G( v) has a negative resistance branch which cannot be observed and the velocity jumps on a second positive branch (Fig. 7) at a value

which is the critical strain energy release rate for catastrophic crack propagation. It is the lowering of Gc by adsorption or segregation that can explain embrittle- ment effects [89].

Such a correlation between 0(v) - v, and E" - (cuz)n is not limited to poly- urethane. It can be found for PMMA for which 0(v) - V°.17 is observed for crack .

propagation [90] and E" - (W-r)0.17 at low frequency (91J.

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Figure 6. Influence of the withdrawal speed on the tack force between a spherical probe and a plane. (From ref. [85].) The dashed curve is for vanishing cross-head velocity, when viscoelastic losses can be neglected (elastic analysis). At finite cross-head velocity, the recorded force first increases because 6 increases faster than the area of contact decreases, and then decreases when the area of contact decreases faster than 3 increases. The maximum recorded force (tack force) follows from this competition and has no simple physical meaning.

Figure 7. Schematic diagram of a log G vs. log v plot showing viscoelastic effects. The curve results from the superposition of a peak of viscoelastic losses to the classical G(v) curve (elastic analysis) showing the increase of G near the Rayleigh velocity. As the branch AC with a negative slope cannot be observed, at G, the velocity jumps from A to B (catastrophic failure) in specimens where G increases with the crack length. Trying to impose a mean velocity V, between v, and v2, leads to stick- slip motion of the relaxation type (ABCD) when inertial effects can be neglected; otherwise limited cycle can be observed, including inside the point A, or the point C, or both.

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4.3. Velocityjump and stick-slip

If the geometry and the loading are such that aGlaA < 0, the crack velocity cannot be monitored and velocity jumps from A to B, or C to D should be observed. On the other hand, if aGlaA > 0 a stable crack velocity can be

imposed, except in the negative resistance branch where trying to impose a mean

velocity V gives rise to stick-slip following the cycle ABCD [86, 92] in absence of inertia. Velocity jumps and stick-slip motion have been discussed in ref. [89]. It should be added that for rubbers the slow crack growth preceding catastrophic rupture was clearly described by Rivlin and Thomas [93], and the range of stick-

slip associated to a branch with negative resistance by Greensmith and Thomas

[94]. Furthermore the hysteresis between the velocity jump AB for accelerating 'cracks, and the velocity jump CD for decelerating cracks was shown by Kobayashi and Dally [95] for epoxy resins.

Stick-slip in peeling of adhesive tapes (thickness h, width b) has been studied

by Barquins et al. [96]. Figure 8 displays the two positive branches observed. The three parameters of control are pull rate V, peeled length L, and inertia I of the roller (radius R). Three first order differential equations describe the motion, which reduce to two when neglecting the variation of peel angle on the roller:

with <I>(v)= w[1 + Ø(v)], m = 1/R2 and k =EbhlL. This system is equivalent to

which is the classical Lienard equation for self-sustained oscillations. The behaviour is governed by the parameter S =,u d 0/d v (with u = b/jkm). For large ?-values, oscillations follow the relaxation cycle ABCD, with a period T

essentially due to the time spent on the branch DA

Figure 8. Log G vs. log v curve for peeling of an adhesive tape, showing the two stable branches.

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When p decreases, the limit cycles in the phase space (G, v) become more and more distinct from the 0(v) curve, including in their interior either both points A and C, or only one of them. Point A and C are Hopf bifurcation points which can be either subcritical or supercritical depending on the shape of the <I>(v) curve in their vicinity, i.e. oscillations can appear abruptly or progressively near these

points. It must be noted that the variation of peel angle gives a third degree of freedom and thus a road to the chaos [97], the limit cycles being changed into

strange attractions.

4.4. Dwell time

It is well known that increasing dwell time increases the tackiness of elastomers. The results given above for adherence of glass on polyurethane [31, 51, 58, 59] were all for a constant dwell time tA=10 min. This value was chosen because the radius of contact of a glass ball under constant load ceases to increase after that time. Thus fitting the curve of radius of contact versus load with the JKR theory gave values of w ranging from 50 to 100 mJ/m2 in function of the room humidity content, but independent of the dwell time. However Barquins [98] made the

surprising observation that although radius of contact under a load P does not

change with tA > 10 min, the time tR to rupture from P to P' continues to increase

regularly with tA as t - tJ2. Analysing these unloading curves as before, gives parallel G(v) curves for various tA which can be superimposed on the curve for

tA=10 min by assigning to each curve a value w This value increases with tA as

w - tJ,l (Fig. 9), ranging from w=63 mJ/m2 at tA=1 min to w=75 mJ/mz at

tA=10 min and w=121 mJ/m2 at tA=15 h. So apparently the Dupr6 energy of adhesion continuously increases inside the contact area, without changing the radius of this contact area, above tA=10 min.

Increase of the tackiness of elastomers with increasing dwell times is well known [91, 100] and is attributed to a flattening of asperities increasing the real area of contact, followed by interdiffusion of chains in case of contact between

Figure 9. Variation of the work of adhesion as a function of dwell time. Open circles: values obtained by fitting the JKR equation for equilibrium contacts. Full triangles: values obtained by shifting the G(v) curves. (From ref. [98].)

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two elastomers. A crude estimation of the elastic energy stored in the rough- nesses of the polyurethane [98] gives a supplementary energy release rate G'=140 mJ/m2 analogous to the effect of a prestress, which was not taken into account in equation (40), giving thus a value of w too low. It was thus proposed [98] that as the stresses in roughnesses relax, G' decreases, so that the apparent w increases to reach the true value w = 200 mJ/m2 when G'=0. However, an additive term such as G' cannot explain parallel curves on a log-log plot. Furthermore, the w values are rather high for Van der Waals forces and seem to still increase for longer dwell times. These points will be discussed later.

Dwell time effects play a role in cyclic loading-unloading of a glass ball on

polyurethane studied by Barquins et al. [101, 102]. After a dwell time tA=10 min

(w=46.4 mJ/m2) a load P=30 mN for 1 s and a load P'=-30 mN for 5 s were

cyclically applied. During loading part of the cycle the crack heals towards a radius of contact giving w= 9 mJ/m2; during unloading the crack advances, first with kinetics given by equations (17) and (40) with w= 9 mJ/m2, and when

reaching the central zone, with a kinetics given by w= 46.4 mJ/m2, and so on. The annular zone with low w increases at each cycle. Theoretical curves giving the evolutions of the radius of contact and of the elastic displacement with time are in complete agreement with experiments. The main results are: (1) cyclic loading-unloading increases the life-time before rupture, i.e. there is no fatigue effect, and (2) the area of contact in an adhesive contact depends only on the value of w at the edge of the contact, independently of the value of w in the central part, showing again the local character of both fracture mechanics and balance of energy. The tensile load P' is essentially sustained by cohesive forces in the Barenblatt tip which reach theoretical strength; whereas inside the contact,

negligible stresses are transmitted since interatomic displacement corresponding to elasticity are small. It is evident, once again, that mean stresses at rupture are of little interest in the field of adherence and fracture mechanics.

5. THE PROBLEM OF Go FOR POLYMERS

So far, we have assumed that the threshold value Go for vanishing crack speed is the Dupr6 energy of adhesion w for adherence, or twice the surface energy y for fracture of homogeneous bodies. Indeed, for adherence of glass on polyurethane, crack propagation was observed for G > w, and crack healing for G < w [31, 101,

102]. However, examination of the literature shows that the threshold values Go are

generally considerably higher than the thermodynamic values w or 2y. It is only for interfacial cracks, and in absence of strong chemical bonds, that values lower than 2 J/m2 can be obtained. In case of strong interfacial bonds, values up to 100 J/m2 for Go are easily reached, which is of the same order of magnitude as the values Go obtained for bulk polymers (e.g. by tearing).

To explain these high threshold energy rates, the theory of Lake and Thomas

[103] is generally used. The argument is the following. When a crack propagates it has to break the C-C bonds, but the work required is higher than the dissociation energy of these bonds, because it is necessary to stretch all the bonds of the chain (of molecular weight Me) comprised between two adjacent

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crosslinks, the stored energy being irreversibly dissipated at the instant of the break. The calculation gives

where N is the number of chains crossing the facture plane, n is the number of C-C bonds between two adjacent crosslinks, and U is the dissociation energy of a C-C bond. This proportionality between Go and M::/2 was effectively observed

by Gent and Tobias [104]. Following this line Carre and Schultz [105] proposed that the observed increase in the peeling force of cross-linked elastomers on treated aluminium was due to decrease of cross-linking near the interface and wrote

with Go = w J Me for the kinetic of crack propagation. However, the arguments of Lake and Thomas can hardly be followed. Surface

energy is a reversible work which does not take into account any dissipation, nor elastic energy stored in the bonds. A single crystal is a kind of macromolecule, and the elastic energy stored in its bonds before a bond is broken is taken into account in G, but not in y. Furthermore, it must be pointed out that equation (46) leads to a mere translation of G(v) curves, which is not always the case.

Ahagon and Gent [106] increased the density of covalent bonds (primary bonds) between the glass and elastomer by varying the proportion of vinylsilane in ethylsilane. Figure 10 shows that G is considerably more increased at low

Figure 10. Adhesion energy W (i.e. G) for polybutadiene crosslinked with 0.2% dicumyl peroxide against various glass surfaces. Treated with 0% vinylsilane (100% ethylsilane), (0); 50/50 mixture of ethyl and vinylsilanes, (8); 100% vinylsilane, (0); clean glass, (+). Dotted curve: cohesive strength vs. rate of tear. (From ref. (106J.) -

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crack velocities than at high velocities, and that curves do not remain parallel. Go seems to be proportional to the concentration of vinyl groups at the surface of the treated glass. Other experiments by Chang and Gent [107] indicate that Go is also a function of cross-linking. Figure 11 from ref. [108] shows that in adhesive

rupture Go increases linearly with the density of chemical bonds between two elastomeric plates, to reach a limiting value G* characteristic of the bulk elastomer. It also shows that the less the elastomer is crosslinked (low amount of

dicumyl peroxide) the higher the Go. Similar experiments have been done by Lake and Stevenson [109] and Lake [110], pointing out that G( v) curves do not remain parallel when interfacial cross-linking is increased at constant bulk

properties.

Figure 11. Threshold energy release rate Go for adhering polybutadiene sheets vs. the degree of interlinking Av. Two levels of crosslinking were employed, using different amounts of dicumyl peroxide. (From ref. [108].)

These results clearly show that neither equation (40) nor equation (46) can

explain observed phenomena. As a multiplicative factor such as wJ Me in

equation (46) give only parallel G(v) curves, one must probably add to equation

(40) a term which takes into account both the number of primary bonds and the

length of chains in the neighbourhood of the interface. A way to approach this

problem is to study the healing of interfaces. It is well known that if after rupture of a polymer, the two lips of the crack are

brought in contact under a light pressure at T > T9, a welding occurs by pro- gressive interdiffusion of chains. (This interdiffusion is the mechanism of solvent

welding of plastics, where a 'good' solvent is used to swell surface layers and increase chain mobilities [111].) Jud, Kausch and Williams [112] have shown that the critical energy release rate G, to break again the partially welded crack increased with contact time t as Ge(t) - tll2 to finally reach the value of the bulk

polymer, after some interdiffusion length (Fig. 12). For PMMA at 385 K, the maximum of G, is reached within 10 min after an interdiffusion of 2.5 nm. It is these interdiffused chains which must be pulled out at the moment of re-fracture. _ Using his reptation theory, de Gennes (113-115) proposed that G(t) varies as

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Figure 12. Plot of fracture toughness Kii against tp (penetration time, tp). Curves 1-4: healing of broken PMMA specimens immediately after fracture. Curves 5: surfaces welded after vacuum drying and polishing. (From ref. [112].) .

where N is the number of chains having diffused, f the force to extract them from their tube, and

the length of the chains penetrated. D is the diffusion coefficient along the tube, and is inversely proportional to the molecular weight M (or to the total chain

length L). He thus found the variation proportional to tl/2 as observed in

experiments. If one assumes that dissipation during re-fracture occurs by pulling out of

chains as well as by viscoelastic losses at the crack tip, one is tempted to write

The second term on the RHS represents Go- w For a healing crack, N is of the same order of magnitude as the number of chains broken by the crack, and thus varies as M-ll2. The maximum of l(t) being L/2, one has, assuming f to be

independent of L

Equation (49) corresponds to Gent and Tobias [104] results relating Go to 4M., and equation (50) was given by de Gennes [l 1 3] and Prager and Tirrel [116].

However, the situation is not yet very clear, firstly due to some confusions between Go and G, which probably do not have the same variation with the molecular weight, and secondly because different models may be proposed

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according to the emphasis is placed on the number of bridges, on the curvilinear

length of the diffused chain, or on mean interdiffusion distance. For example, Kim and Wool [1171 proposed a model leading to Go(t) ~ tl/2/Ml/2 and

GO(oo) - M. One can assume that in case of strong chemical bonding at an interface, chains

or bundles of chains are pulled out of the polymer. If they are short, they are

easily pulled out; if long they can be broken at a critical extension. An analogy could be the adherence of a dish of entangled spaghetti, or the rupture of

composites where fibres are pulled out if they are not too long and broken when the friction exerted on a long chain is too high. A layer of oriented polymer is thus left on the surface as in the Bikerman weak layer theory. The work done in this extraction has to be added to viscoelastic losses due to the moving stress,

tentatively as

where N is the number of primary bonds firmly holding the chains to the substrate. Such an equation could account for the results of Ahagon and Gent

[106] (Fig. 10) and Chang and Gent [107] (Fig. 11) on the proportionality of Go to the concentration of vinyl groups and on the influence of crosslinking. However, examination of Fig. 10 shows that f probably varies with the crack

speed (possibly by a mechanism of biased random walk). In the experiments on adherence of glass on polyurethane by Maugis and Barquins with 10 min dwell time there was probably no primary bond nor chain pull out so that equation (40) was well verified. On the other hand, Gent and Vondracek [118] have shown that adherence of a silicone rubber to glass measured at a velocity v=100 ,um/s increases linearly with dwell time from G=25 J/m2 up to G=600 J/m2 a value characteristic of the bulk rubber at this velocity (Fig. 14), whereas the values measured at 1 ,um/s (near Go) increased from 1 J/m2 to 13 J/m2 (for bulk rubber the threshold value being Gg=42 J/m2). They explain their results in terms of

hydrolytic decomposition of the polymer leading to the formation of reactive

groups which combine with surface hydroxyls of the glass. The main drawback of equation (51) is to mix fracture mechanics and

thermodynamic considerations to microscopic ones. Here also, the introduction of the Dugdale model would be particularly useful, as first proposed by Marshall et al. [119] in the closely related problem of crazing in bulk polymers. (For a recent review on crazes, see Kramer [120].) As the extracted fibrils in the craze sustain a nearly constant stress ao, the craze zone is the most perfect example of the Dugdale plastic zone at a crack tip. Ward and coworkers [121-124] have shown that the craze profiles are in agreement with the theory, from which they deduce fracture toughness values in good agreement with those obtained directly. The crack opening displacement was found to vary considerable with molecular weight. Since Berry [125], the influence of molecular weight and chain

length, and the role of entanglements on the fracture behavior of polymers has been the subject of considerably interest [126-131]. Recently Prentice [131] proposed a model for fracture of glassy polymers in which the force to pull out a chain is proportional to its length, and is a function of the pull out velocity. He obtained G - M2 as long as the chain length was not too high. Above a critical

length, i.e. a critical molecular weight, the chains break and G becomes

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Figure 13. The critical energy release rate Go of PMMA as a function of molecular weight M,. M, is the critical molecular weight for entanglement. (From ref. [131 ].)

Figure 14. Development of adhesion with time of contact between a lightly cross-linked layer of PDMS and a Pyrex surface, at various storage temperatures. Peeling at v=100 pm/s and at 25°C. (From ref. [1181.)

independent of M. Accordingly, his experimental results (Fig. 13) on brittle

rupture of PMMA show that G, becomes independent of M above a critical molecular weight.

All the studies done on fracture of bulk polymers are certainly relevant to the adherence of polymers, the mechanisms of losses at a crack tip being the same: viscoelastic losses due to moving stresses, work to extract chains or fibrils, viscous drag in presence of a liquid. It is probable that the various theories of

adhesion, namely theory of wetting, theory of the rheological factor, theory of the chemical bond, theory of the weak boundary layer, theory of interdiffusion, are all valid, each of them corresponding to a particular emphasis on a dominant mechanism.

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6. CONCLUSION

Adherence of two elastic solids seems to be well understood on the basis of fracture mechanics, although complications arise when they are elastically dissimilar, or when the extent of the cohesive zone, where the adhesion forces

act, is no longer negligible compared to the area of contact. In the latter case, for which the linear elastic fracture mechanics fails, the Dugdale-Barenblatt model could probably be used, in particular to study the transition from JKR to DMT

solution. The same model could also be used for capillary bridges, if the meniscus is seen as the tip of an external crack.

For viscoelastic solids, kinetic effects in crack propagation can be understood

by assuming a nonlinear viscous drag at the crack tip proportional to w and

opposed to the driving force G - w This viscous drag depends on the viscoelastic

properties of the material, and if it decreases above some critical velocity, velocity jumps and stick-slip phenomenon appears. This simple model can be

applied to the adherence of elastomers as long as strong chemical bonds do not exist at the interface. Otherwise, polymeric chains strongly held at the interface are pulled out (crazing) giving rise to a supplementary friction term, which is a

function of the chain lengths and probably of the crack opening velocity.

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