chapter 04: adhesion of solids: mechanical aspects · adhesion of solids: mechanical aspects 4.1...
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4Adhesion of Solids:
Mechanical Aspects
4.1 Introduction 4.2 Adhesion Forces, Energy of Adhesion, Threshold
Energy of Rupture4.3 Fracture Mechanics and Adhesion of Solids
Energy Aspects, Energy Release Rate • Example of the Flat Punch • Mechanical Aspects
4.4 Example: Contact and Adherence of SpheresHertz Solution • JKR Solution • DMT Solution • JKR–DMT Transition in a Dugdale Model • Application to a Liquid Meniscus
4.5 Liquid Bridges 4.6 Adhesion of Rough Elastic Solids — Application to
Friction4.7 Kinetics of Crack Propagation
Case of Viscoelastic Solids • Experiments of Fixed Cross-Head Velocity — Tackiness • Branch with Negative Resistance: Velocity Jump and Stick-Slip • Examples of the Additivity of Dissipations • Short Historical Background
4.8 Adhesion of Metals Adhesion of Microcontacts • Cold Welding • Adhesion of Metals at High Temperature
4.9 Conclusion
Abstract
Adhesion and adherence of solids are reviewed in the framework of fracture mechanics. The mainconcepts are recalled: surface energy, thermodynamic work of adhesion, energy release rate, stable andunstable equilibrium of cracks, controlled propagation, stress intensity factors, J integral, and Dugdalemodel. An example of application is given for the adherence of spheres (based on various models).Kinetics of crack propagation for viscoelastic solids is presented in terms of G(v) curves, tackiness, stick-slip, and velocity jumps. Examples of additivity of various losses are given, which include viscous drag,bulk dissipation, extraction of chains, and the problems of threshold values and hysteresis are discussed.Finally, the case of metals (cold welding and high-temperature adherence) is examined.
4.1 Introduction
When two solids approach each other, they undergo attraction forces (adhesion forces) which can be longor short ranged and of variable strength, corresponding to energies from a few mJ/m2 to some J/m2. We
Daniel MaugisLaboratoire des Matériaux et des Structures du Génie Civil
deal here with solid-state physics and surface physicochemistry. When we want to characterize the forceor the mechanical resistance of bonds so established between the two solids, we are faced with a morecomplicated problem. The separation never occurs spontaneously across the entire sample, but bypropagation of a crack. It is accompanied with deformation and energy loss, which is more importantthe stronger the adhesion force. The force needed to separate the two solids depends of course on thedensity of bonds, but in addition, to the geometry of the samples and to the velocity at which separationoccurs. The energy consumed can reach, in some cases, many hundreds of kJ/m2. It is thus necessary toseek analyses that apply to many geometries, how to employ fracture mechanics in various situations,and how to account for various energies that come into play.
If the surfaces are rough or if they are not clean, adhesion is generally not strong. If one introduces aliquid which wets the surfaces, it increases adhesion. In some cases, the liquid hardens afterward bycooling, evaporation, polymerization. If this liquid is a polymer one speaks of gluing, if it is a liquid alloybetween two metals one speaks of brazing (opposed to welding which is done without addition of matter).
4.2 Adhesion Forces, Energy of Adhesion, Threshold Energy of Rupture
Besides the Young’s modulus E and the Poisson’s ratio ν, an isotropic elastic solid has a surface γ. Young’smodulus and Poisson ratio reflect the behavior of intermolecular forces for small displacements of atomsaround their equilibrium points, and 2γ is the work per unit area necessary to break the bonds along animaginary plane and to separate reversibly two parts of a solid. Surface energy thus characterizes thenature of bonds which ensure the cohesion of the solid through this imaginary plane. So, metals andcovalently bonded solids have a high surface energy (1000 to 3000 mJ·m–2); the ionic crystals (100 to500 mJ·m–2) or molecular crystals (γ < 100 mJ·m–2) have a lower surface energy.
To calculate the surface energy of a solid, let us take the example of the Lennard–Jones potentialbetween two atoms
where C is the London constant. By a simple integration, one obtains the stress between two half-spaces(two plates) separated by a distance z:
(4.1)
where Z0 is the interatomic equilibrium distance. The constant A = π2n2C is called the Hamaker constant(n, the number of atoms per unit volume, is assumed to be constant, independent of the distance z, sothat we can deal with the interaction of two rigid solids not strained by the stress σ). This curve isdisplayed in Figure 4.1. At z = Zc it passes through a maximum termed the theoretical stress σth, whichis extremely high if compared to the elastic limit at 0.5%, for example. The part of the curve at the left-hand side of the maximum represents the elastic field, and the slope at z = Z0 is the Young’s modulus.The part at right of the maximum corresponds to two half-spaces in interaction; we term this portionof the curve the adhesion field. The area below the curve from z = Z0 up to infinity represents 2γ, andwe see that to create two new surfaces one must first supply a work 2γ ′ against the elastic restoring forces(going from Z0 to Zc) and then a work 2γ ″ against the adhesion forces (going from Zc to infinity)
UC
r
D
r= − +
6 12
σ
σ
zA
Z
Z
z
Z
z
Z
z
Z
zth
( ) =π
−
=
−
6
3 3
2
03
0
3
0
9
0
3
0
9
(4.2)
For two solids 1 and 2 in contact, the work necessary to reversibly separate them may be written as
W is the Dupré energy of adhesion and γ12 is the interfacial energy (which can be the grain boundaryenergy between two grains, or the twin boundary energy between two twins). The reversibility requiredin the definition of surface energy or adhesion energy is an important point which must be emphasized.
FIGURE 4.1 Lennard–Jones potential. Stress between two half-spaces. Jumps such as A to B and C to D are observedif a spring of insufficient stiffness is used to measure the force.
20 0
γ σ σ σ= ( ) = ( ) + ( )∞ ∞
∫ ∫ ∫z dz z dz z dzZ Z
Z
Z
c
c
w = + −γ γ γ1 2 12
In fact it is impossible to maintain the surfaces parallel down to the contact, and the symmetry isbroken. Only a part of the single crystals is in contact, and the separation instead, to occur on a whole,occurs along a line where bonds are broken one after another. We thus deal with a crack (Figure 4.2,where the trace of this line is the point E). The theoretical stress σth, instead of extending across the wholecrystal at the moment of separation, is localized at the crack tip, on the bond BC. Elastic forces appearahead of the crack tip, and adhesion (or cohesion) forces extend on some length, the cohesion zone,beyond which surfaces are interaction free. The stresses necessary to create a unit area of new surface aremuch less, and the crack can be seen as a stress transformer. The surface energy 2γ is thus the worknecessary to reversibly transfer an internal surface from a point where it is submitted to no stress (z =Z0) to the crack walls beyond the range of adhesion forces:
(4.3)
The problem of the reversibility is less crucial. The crack advances by a lattice mesh each time a bondsuch as BC is broken. This rupture is accompanied by a jump (the more important the shorter thecohesion zone), during which energy is lost (phonons emission, for example). Similar jumps occur duringhealing of the crack. The crack thus may be trapped (“lattice trapping”) by an energy barrier (as adislocation by Peierls–Nabarro forces) and may not be completely reversible. We will term G+
0 and G–0
these threshold energies to advance or recede the crack with G –0 < w < G+
0. This theory of “lattice trapping,”in which a lattice of point masses is linked by spring resistant to both traction and flexion, has beendeveloped by Thomson et al. (1971), and Fuller and Thomson (1978). It must be noted that due to theelastic “fretting” around this bond, the amplitude of jumps is less, but is very sensitive to the shape ofthe crack. This effect is probably small or negligible in inorganic materials, as experimentally shown onglass or mica (Wan et al., 1990) where hysteresis between advance and recession is not observed. On theother hand, it can be important for macromolecular solids, if we replace the word bond by the wordchain. We recover the argument proposed by Lake and Thomas (1967) for the rupture of elastomers, i.e.,
FIGURE 4.2 Schematic view of a crack tip. The last nonbroken bond is the bond CB sustaining the theoreticalstress. In the mechanics of continuous media the contour of the J integral just around the “cohesive zone” is the lineAED with a cusp. This contour should be ABCD in a discrete model.
21
2
γ σ= ( )[ ]∫ z xdz
dxdx
x
x
to advance, the adhesion has to break C–C bonds, but the work is much higher than the dissociationenergy of that bond because it is necessary to stretch all the bonds of the chain (molecular mass Mc)between two cross-linking points, and all the stored energy is irreversibly dissipated at the instant ofrupture. The calculation leads to:
(4.4)
where N is the number of chains cutting the fracture plane per unit area (proportional to 1/ ), n isthe number of bonds between two cross-link points (proportional to Mc), and U is the dissociation energyof a C–C bond. G0 thus is not an adhesion energy but a threshold rupture energy of with representativevalues of the order of 50 to 100 J/m2. Such a proportionality between G0 and was effectively observedby Genet and Tobias (1982) or in an equivalent manner between G0 and 1/ (Bhowmick et al., 1983)where E is the Young’s modulus. This argument is also valid for elastomers adhering to a solid by strongbonds: G0 must increase with the density N of bonds up to a value G*0 corresponding to the rupture ofthe bulk elastomer, and this effect was also observed by Ahagon and Gent (1975) and Chang and Gent(1981), G*0 is weaker the more cross-linked the elastomer.
4.3 Fracture Mechanics and Adhesion of Solids
4.3.1 Energy Aspects, Energy Release Rate
The work needed to extend the area of a crack by dA is 2γdA. This work is taken to the elastic and/orpotential energy of the system. Irwin and Kies, (1952) introduced the strain energy release rate, notedG by Irwin (1957) (initial of Griffith); elastic energy returned when the area of the crack increases by dA,
(4.5)
where δ is an imposed displacement, and Up = –Pδ represents the potential energy in the case where Pis constant.
Equilibrium at constant temperature and fixed load (dT = 0, dP = 0) corresponds to an extremum ofGibbs free energy �, whereas equilibrium at constant temperature and fixed grips (dT = 0, dδ = 0)corresponds to an extremum of Helmholtz free energy F. In both cases equilibrium is given by
(4.6)
which we will term Griffith’s criterion of equilibrium and which links two of the three variables δ, P, A ofthe state equations, so that equilibrium curves δ(A), A(P), P(δ) are functions of w.
If G ≠ w, the area of contact spontaneously changes to decrease the thermodynamic potential. If G <w, A must increase for the potential to decrease: the crack recedes. If G > w, the area of contact mustdecrease to have d� < 0 or dF < 0, and the crack advances. GdA is the mechanical energy released whenthe crack extends by dA. As the rupture of bonds requires the work wdA, the excess (G – w)dA is changedinto kinetic energy if there is no dissipation. G – w is the crack extension force, per unit length of crack;it is zero at equilibrium.
The equilibrium given by G = w can be stable, unstable, or neutral. A thermodynamic system undera given “constraint” is stable if the corresponding potential is minimum, i.e., if its second derivative ispositive. Stability at constant temperature (dT = 0) and fixed grips (dδ = 0) is given by:
G NnU
k Mc
0 =
=
Mc
Mc
E
GU
A
U
A
U
AE E P
P
= ∂∂
= ∂∂
+ ∂∂
δ
G w=
Similarly, at fixed load, stability will be defined by:
An important point is that the stability of equilibrium is generally not the same at fixed load and atfixed grips, and at fixed ∆ it depends on the stiffness km of the measuring apparatus. Its stiffness km isthat of the spring. A displacement ∆ can be imposed by turning the screw (which is thermodynamicallyequivalent to apply a load P to the spring, and then to clamp it without external work), and can bedivided into a spring elongation δm and a displacement δ of the two solids in contact. The spring exertsa force P = kmδm on the two solids, and we have:
(4.7)
Equilibrium at fixed ∆ is still given by G = w, and the stability by:
(4.8)
but this stability depends on the stiffness km. Equilibrium can be unstable at fixed load, and stable atfixed grips.
Another important concept is that of controlled rupture. Let us consider a stable equilibrium definedby G = w and (∂G/∂A) > 0. If a fluctuation decreases the area of contact (dA < 0), G decreases and wehave G < w; the crack recedes to return to its equilibrium position. We have the same return to equilibriumif the fluctuation increases the area of contact (the system of loading remaining the same). The crackwill move only if the load P or the displacement δ is modified, and it will move such as to bring back Gto its value w. We deal thus with a controlled rupture, but for the crack to be always in equilibrium withthe loading, it is necessary that the variations of P or δ are very slow, so, we deal with a quasistatictransformation. The fact that G remains constantly equal to w during this controlled propagation iswritten:
(4.9)
Starting from a stable equilibrium with the two solids compressed (P > 0, δ > 0), let us decreaseprogressively the imposed displacement ∆. One encounters generally a progressive reduction of the areaof contact A, i.e., a controlled rupture with G = w and (∂G/∂A)∆ > 0 until a critical displacement ∆c
where (∂G/∂A)∆ = 0. Beyond this point the equilibrium becomes unstable, and the crack spontaneouslypropagates at ∆c = const. until the complete separation. It propagates with a crack extension force G – wwhich increases in proportion as A decreases (since (∂G/∂A)∆ < 0). This instability point on the equilib-rium curve P(δ) corresponds to the point where d∆/dδ = 0, i.e., after Equation 4.7 to:
(4.10)
∂∂
= ∂
∂
>2
20
F
A
G
ATδ δ,
∂∂
= ∂
∂
>2
20
�
A
G
AP T P,
∆ = + = +δ δ δmm
P
k
∂∂
>G
A∆
0
dGG
AdA
Gd
A
= ∂∂
+ ∂∂
=∆
∆∆ 0
kdP
dm + =δ
0
On the equilibrium curve P(δ) the stability ends with a horizontal tangent at fixed load P (km = 0), andwith a vertical tangent at fixed displacement δ (km = ∞).
The critical load Pc corresponding to the limit of stability of the system:
defines the pull-off force of the two solids in contact, force which generally depends on the stiffness km
of the measuring apparatus. In some geometries and in some loading conditions, the equilibrium isalways unstable; as soon it is reached the crack propagates spontaneously (at constant loading conditions),and the criterion defining adherence is simply reduced to G = w. It is the case, as we will see, for thedouble cantilever beam at fixed load, or the flat punch on an elastic half-space. It is also the case for theclassical Griffith crack in an infinite solid, as seen above.
4.3.2 Example of the Flat Punch
The problem of the adherence of a flat punch was solved for the first time by Kendall (1971) by a balanceof energy. Let us consider, Figure 4.3, an axisymmetric punch with a flat end, of radius a, pressing anelastic half-space. The penetration δ of the punch and the stress distribution in the area of contact werecomputed by Boussinesq in 1885, and are:
(4.11)
(4.12)
where
(4.13)
(E and ν are the Young’s modulus and Poisson’s ratio of the elastic half-space). Note that the compressivestress is infinite on the edge of the contact. The same formulae can be applied when adhesion forcesmaintain the contact over a circle of radius a, while one pulls with a force P (Figure 4.3b). Compressivestresses become tensile stresses, infinite on the edge. To compute the energy release rate G, one mustcompute the mechanical energy (elastic plus potential) and its variation when the area of contact variesby dA. To vary the radius of the punch to study the variations of mechanical energy may not be theobvious way to proceed, but examination of Figure 4.3c shows that Equation 4.11 applies to the causewhere the radius of contact a is smaller than the radius of the punch. One can thus study variations ofelastic and potential energies for fluctuations of areas of contact at constant load P. For a load P and aradius of contact a, the energy release rate is:
(4.14)
Note that Equations 4.11 and 4.14 are the two equations of state of the system.
G w
G
A
=
∂∂
≤ 0
δ = 2
3
P
aK
σz
P
a r a= −
π −2
1
12 2 2
1 3
4
1 112
1
22
2K E E= − + −
ν ν
GP
a K
K
a=
π=
π
2
3
2
6
3
8
δ
The equilibrium defined by G = w is always unstable, for we have:
Let us consider the unstable equilibrium of Figure 4.3c, under the load P. If a fluctuation temporarilyincreases the area of contact, G decreases, and as G < w, the crack recedes and the area of contact increases,reducing G still more. One deviates more and more from equilibrium, and the process stops when onereaches the edge of the punch. If, on the contrary, a fluctuation decreases the area of contact, G increases,and as G > w, the crack propagates and the area of contact decreases, increasing G more. The processstops only when a complete separation is obtained under this load P.
Let us return to the adherence of the punch of Figure 4.3b. For P = 0, we have G = 0, but the area ofcontact cannot increase further. In proportion, the traction force P is increased, G decreases at constanta, until the equilibrium (unstable) G = w is reached under a critical load Pc. At this moment, a fluctuationcan only decrease the area of contact, and this decrease continues until complete rupture under this load,Pc defined by G = w, i.e.,
(4.15)
FIGURE 4.3 Analogy between the adherence of a circular flat punch to a half-space and the rupture of a deeplynotched cylindrical bar.
r
PP
Pc
P1P2
[a] [b]
[c]
[uz]
[d]
a
b
uz
uz
δ1
δc
δ2
aδ δ
a
v1
v2
a
δ
∂∂
= ∂∂
= −π
= −π
<G
A
G
A
P
a K
K
aP
34
9
160
2
2 5
2
2 3δ
δ
P a Kwc = π6 3
which is, by definition, the adherence force of a punch of radius a. Similarly, at fixed displacement,instability will appear at the critical displacement:
(4.16)
Note that this adherence force is not proportional to the area of contact, nor to the perimeter, nor to theenergy of adhesion.
4.3.3 Mechanical Aspects
4.3.3.1 Stress Intensity Factors
From the beginning, a crack was considered as a flattened cavity with stress-free walls. In this case thestress on the edge becomes infinite. The singularity in 1/ of the stresses at a crack tip was pointed outby Sneddon (1946), and Irwin (1958) introduced the stress intensity factors KI, KII, KIII characterizing theforce of the singularity for the three components of displacement at the crack tip* (Figure 4.4): mode Ior opening mode, mode II or sliding mode, mode III or screw mode.
FIGURE 4.4 The three modes of displacement at a crack tip.
*One must beware of the fact that thee modes are defined by the local state at the crack tip, and cannot be deduceddirectly from the external loading. For example, when an axisymmetric punch is pressed onto the surface of a brittlesolid, a vertical crack (Hertzian fracture) appears on the edge of the contact which is opened in mode I and not inmode II, as might be suggested by the geometry.
Mode I
Mode II
Mode III
δc
aw
K= π8
3
r
Let us consider the Inglis elliptical cavity, with semi-axes a and b, and traversing a plate submitted atinfinity to biaxial stresses: σ along Oy ′ and kσ along Ox ′. The ellipse centered at O has its major axisalong Ox, which makes an angle β with Oy ′. One knows the exact solution as well in stresses as indisplacements (see Maugis, 1992b). When b tends toward zero, one obtains the Griffith crack, and thestresses around the crack tip, at
can be written
(4.17)
with
(4.18)
(4.19)
and m = 1 + k, n = 1 – k. These angular variations of stresses in mode I and II were given by Sneddon(1946) and Irwin (1958), and the stress intensity factors by Eftis and Subramonian (1978), generalizingthe results of Sih et al. (1962) for uniaxial loading.
The discontinuities of displacements ux, and uy, between the upper wall (θ = +π) and the lower wall(θ = –π) at a point of the crack are given by:
(4.20)
(4.21)
where µ is the shear modulus, ν is Poisson’s ratio and
x a r
y r
= +
=
cos
sin
θ
θ
σ θ θ θ θ θ θ σ β
σ θ θ θ θ θ
x I II
y I II
rK K n r
rK K
=π
−
− +
+ ( ) + ( )
=π
+
+
1
2 21
2
3
2 22
2
3
22 0
1
2 21
2
3
2 2
3
2
1 2cos sin sin sin cos cos cos
cos sin sin sin cos ++ ( )
=π
+ +
+ ( )
0
1
2 2 2
3
21
2
3
20
1 2
1 2
r
rK K rxy I IIτ θ θ θ θ θ
cos sin cos sin sin
K m n aI = −( ) π1
22cos β σ
K nII = 1
22sin β
u Kr
ry I[ ] = +µ π
+ ( )κ 1
20 3 2
u Kr
rx II[ ] = +µ π
+ ( )κ 1
20 3 2
κ νν
κ ν
= −+
= −
3
1
3 4
in plane stress
in plane strain
In mode I, a crack opens with a parabolic shape.Equations 4.18 and 4.19 show that stress intensity factors are expressed in Pa · . They must not be
confused with stress concentration factors which are nondimensional. For a crack in mode I (β = π/2),we have simply
(4.22)
(but σx depends on the biaxiality of the constant term, also named the T-stress). It can be shown thatterms in 1/ for the stresses and terms in for the displacements depend on the geometry of the systemonly by the KI and KII. The other terms of the power expansion are specific to the studied problem. Thesestress intensity factors play an important role in fracture mechanics, an interesting property arising fromthe superposition principle: for a given mode stress intensity factors are additive. They can be obtaineddirectly, without knowing the exact solution, and one can compute them for a large number of geometriesand loadings.
Let us consider another example, useful for the following, that of an exterior circular crack (case of adeeply notched bar). Paris and Sih (1965) gave the stress intensity factor for a force P or a stress σ appliedat infinity:
(4.23)
Note that this is the result obtained by the power expansion of Equation 4.12 in function of t = a – r,with a sign plus since it is a traction, giving
Let us also examine the case where a uniaxial and constant pressure p is applied inside the crack ona length d = c – a such that it opens and propagates it. This internal loading leads to a stress intensityfactor (Maugis, 1992a)
(4.24)
with m = c/a > 1. The stress in the area of contact (ρ = r/a < 1) and the elastic displacement of the crackwalls (ρ = r/a > 1) are given by
(4.25)
We will use these results for the contact of spheres.
m
K aI = πσ
r r
K aP
aa
P
a aI = π =
ππ =
π1
2 2 22
σ
σzIP
a a t
K
t=
π π=
π2
1
2 2
Kpa
am m mI =
π−
+ −−2 2 1 21 1tan
σρ ρz
IrK
a
p m, tan0
1
1
2 1
12
12
2( ) =π −
+π
−−
−
uE
K a pa m mt
t m tdtz I
m
=−( )π
π − − − −
−−
−
− −
( )
∫2 1 1
2 1 11
2
1 2 2 1 22 2
2 2 21
ν
ρρ
ρρ
ρ
cos cos
min ,
(4.26)
4.3.3.2 Irwin Formula
The work GdA to make the crack advance or recede corresponds to the work of stresses at the crack tipto open or close that crack. Irwin (1957) has shown the equivalence between the thermodynamic aspectand the mechanical aspect (or more precisely between G and K, since the notion of surface energy doesnot appear in the mechanical aspect), by computing the work of the singular stress, Equation 4.17, toclose on da a crack of parabolic shape, Equation 4.20 and found
(4.27)
This result is obtained with the following hypotheses: (1) propagation of the crack in its plane, (2) linearrelationship between stress and strain, and (3) no interaction between crack walls.
It must be emphasized that Equations 4.27 are not a definition of G, but an estimation of G within theabove hypotheses. Their domain of validity is limited; it is that of the linear elastic fracture mechanics,LEFM. In a number of cases, stress intensity factors can be written in the form KI ≈ σ . If, in theabsence of a crack, the sample is submitted to a constant stress σ, the density of elastic energy at anypoint is W = σ2/2E, and one can write G under the form G = kaW. We will see below how the Rice J-integral will generalize the use of that elastic energy density.
4.3.3.3 Dugdale and Barenblatt Models
The origin of these infinite stresses, physically unacceptable, arises from the purely elastic analysis of theproblem, analysis in which the adhesion forces between neighboring surfaces (at right of the peak inFigure 4.1) are not taken into account. In fact they cannot be taken into account directly since they arenot restoring forces: the stress decreases when the distance increases. One must use a stratagem. Dugdale(1960) and Barenblatt (1962) have shown how to eliminate these singularities: the adhesion or cohesionstresses acting between the walls of the crack can be seen as an inner loading, leading to a stress intensityfactor Km compensating the stress intensity factor KI due to external loading. The terms in 1/ for thestresses and in for the opening of the crack disappear. Under loading, the crack takes the shape of acusp (terms in r 3/2) and is no longer parabolic. The relation KI + Km = 0 ensures the continuity of stressesand determines the length d of the cohesion zone, but an equation is missing to calculate G since Equation4.27 is no longer valid. This last equation was given by Rice (1968a,b).
4.3.3.4 Rice J-Integral
Rice has introduced the following integral:
(4.28)
Where Γ is an open contour starting from the lower crack face and extending counterclockwise aroundthe crack tip to a point on the upper surface (see Figure 4.5), starting point and arrival point being onfree surfaces.
→T and
→u are stress traction vectors and displacements on the contour Γ, oriented along
the outward normal. W is the density of elastic energy in any point of the solid (linear elasticity or not).This integral is independent of the contour Γ. Taking for Γ the external contour of the solid, one canshow that J is identified with G. In the case where the walls of the crack are assumed to be withoutinteraction (LEFM), it suffices to take for the contour a circular contour or radius r around the cracktip; when r tends toward zero only singular terms give a contribution and one recovers the Irwin result,Equation 4.29.
GE
K
GK
E
= −
=
1 2
12
12
νplane strain
plane stress
πa
r
r
J Wdyx
ds
r
= − ⋅ ∂∂∫ T
u
If one now introduces a cohesion zone of length d such that singular terms disappear (Dugdale andBarenblatt models), and in which the stresses σ(u) are tensile stresses along Oy and function of theopening u, it suffices to take as contour Γ a line enclosing this cohesion zone. With dy = 0 (in undeformedstate), and T = δ(u) on Γ, the J-integral reduces to:
(4.29)
with δ = u+ – u– the crack displacement. Let δt be the opening of the crack beyond which cohesion forcesno longer act, termed the COD (crack opening displacement). We get
(4.30a)
In the case where σ(u) = σ0 is constant in the cohesion zone (Dugdale model), we get:
(4.30b)
4.4 Example: Contact and Adherence of Spheres
Let us examine now how all the notions introduced in section 2 apply in the case of the contact and adherenceof spheres. This problem is central in contact mechanics and tribology, and must be studied with great care.
4.4.1 Hertz Solution
Assuming an elliptical distribution of pressure in the area of contact, Hertz showed in 1882 that the areaof contact aH and the approach δH of two spheres of radius R1 and R2 under a load P just as the stressdistribution in the area of contact were given by:
(4.31)
FIGURE 4.5 Contours of Γ for the J integral.
B C
E D
hx
y
AF
Γ
J G ux
dxd
≡ = ( ) ∂∂∫ σ δ
0
J G dt
≡ = ( )∫ σ δ δδ
0
J G t≡ = σ δ0
aPR
KH3 =
(4.32)
(4.33)
with K given by Equation 4.13 and
We limit ourselves to the case of a rigid sphere on an elastic half-space. Note that: (a) the penetrationδH of the sphere in the half-space is twice the thickness of the portion of sphere of chord 2aH; (b) thatthe radius of contact is zero under zero load; and (c) that the stresses cancel on the edge of the contact,which imposes a tangential connection between the sphere and the half-space. Note also (d) that the force-displacement relation is not linear and that the stored elastic energy under the load P is:
(4.34)
Boussinesq showed in 1885 that, in the general case of a frictionless convex punch indenting an elastic-half space, the area of contact is generally unknown and that the problem can be solved by adding thecondition that the normal stress σ2 cancels on the edge of the contact, which ensures a tangentialconnection between the punch and the half-space and determines the penetration δ of the punch. Anydifferent value of the penetration leads to infinite stresses and to a discontinuity of displacements at theedge. Compressive singular stresses are impossible since they lead to interpenetration, whereas tensilesingular stresses need the solids to be endowed with adhesion. This remark was apparently forgotten foralmost a century.
4.4.2 JKR Solution
It was only in 1971 that Johnson, Kendall, and Roberts (1971) published their theory of adherence ofspheres (or sphere onto a plane), by making a balance of elastic energy UE, potential energy UP andsurface energy US = –πa2w of the system. The calculation of elastic energy is rendered difficult by thefact that the surfaces are not conformal: adhesion forces distort the neighboring surfaces, and elasticenergy is stored even under zero load. To evaluate it, JKR used a clever strategy.
Let P1 be the apparent Hertzian load which would give the same radius of contact a as the one observedin the presence of surface energy. By definition, we have:
(4.35)
To determine the penetration δ and the elastic energy UE stored under the load P, JKR used the loadingpath of Figure 4.6. Assuming no adhesion, apply the apparent Hertzian load P1 (point L). The radius ofcontact is then a, the penetration δ1 = a2/R, and the stored elastic energy the area OLMO. Reduce theload from P1 to P, making the radius of contact a constant and increasing the energy of adhesion up tow (point B). This unloading curve is a straight line (LB) since one works at constant area. Although wedeal with a sphere, we have a flat punch displacement since the motion is made as a whole. The variationof penetration is thus given by Equation 4.11:
δHHa
R=
2
σH
H H
P
a
r
a= −
π−3
21
2
2
2
1 1 1
1 2R R R= +
U PE H= 2
5δ
Pa K
R1
3
=
and the variation of elastic energy is given by the area of the trapezium LBJM. So, at point B, under theload P and in the presence of adhesion we get the elastic energy UE and the penetration
(4.36)
Derivating UE at constant displacement δ, or introducing the potential energy Up = –Pδ, the energyrelease rate is:
(4.37)
FIGURE 4.6 JKR loading path. G = 0 from O to L, then da = 0 from L to B. The elastic energy stored at point Bis the area OLBJO.
∆δ =−( )2
31P P
aK
δ = +a
R
P
aK
2
3
2
3
GP P
a K
a K
RP
a K
K
a
a
R=
−( )π
=−
π=
π−
1
2
3
32
3
22
6 6
3
8δ
Equations 4.36 and 4.37 are the equations of state of the system. At this stage the radius of contact aunder the load P is still unknown. It will be determined by Griffith equilibrium condition G = w, whichis the supplementary condition of Boussinesq, necessary to solve the problem. For w = 0 one recovers,as expected, Hertz results. Note that the loading path is the only one that allows us to obtain the elasticenergy from a diagram (P, δ).
Here also we can note that G could be obtained directly from the knowledge of the stresses on theedge of the contact. The distribution of normal stresses in the contact area results from the superpositionon the same area of Hertzian stresses under the load P1, and the tensile stresses due to the flat punchmovement, Equation 4.12, under the unloading P1 – P:
(4.38)
The stresses are compressive in the center and tensile on the edge, becoming infinite at r = a. The principalterm of the power expansion in function of ρ = a – r is a function only of the punch unloading, and wehave:
(4.39)
hence Equation 4.37 by using the Irwin formula, Equation 4.27, which can be written for two differentmaterials in the form:
(4.40)
We are correctly in the field of linear fracture mechanics. Stresses and displacement discontinuities inthe JKR theory can be written:
(4.41)
(4.42)
The connection between the sphere and the elastic half-space is now a vertical connection (opening ofthe crack in )
The equilibrium relation G = w links two of the three variables, a, P, and δ, of the equation of state(36), and draws equilibrium curves a(P), P(δ), δ(a). In Figure 4.6 the equilibrium curve is the curveBCDO. One notes that, if starting from L, one regularly decreases the load, one cannot go beyond pointC, whereas if one regularly decreases the penetration one cannot go beyond point D. These two instabilitypoints are defined by (∂G/∂A)P = 0 and (∂G/∂A)δ = 0. We get, for the adherence force:
(4.43a)
(4.43b)
σρ
ρ ρz rP P
a
P
a,0
2
1
1
3
21 11
2 2
12
2( ) = −π −
−π
− <
KP P
a aI = −
π1
2
GE E
K I= − + −
1
2
1 112
1
22
2
2ν ν
σ ρρ
ρ ρ, 01
1
3
21 1
2
2( ) =π −
−π
− <K
a
aK
RI
uv
EK a
a
Rz I[ ] =−( )
ππ +
π− + −( )
>− −
2 1 11 2
11
2
12
2 2 1cos cos ρ
ρ ρρ
ρ
r
P wRc = − π3
2 at fixed load
P wRc = − π5
6 at fixed grips
If the measuring apparatus has a stiffness km the limit of stability will be placed between C and D at apoint where the tangent of the curve is –km, in agreement with Equation 4.10.
Figure 4.7 displays the equilibrium relation δ(a), function of w, on which are superimposed the curvesδ(a, P) independent of w, Equation 4.36. If, starting from the point L (P = 10), one decreases very slowlythe load down to P = 0 (point N), one makes a controlled rupture at G = w following the equilibriumcurve LN. If, on the other hand, one abruptly imposes an unloading at P = 0, one will observe aninstantaneous elastic displacement at constant radius a from L to M, a point where G > w, then a crackpropagation at P = 0 with a progressive return to equilibrium at point N. The curve marked “slowingdown-acceleration” represents the locus of the points where (∂G/∂A)P = 0. If from point L one imposesan abrupt unloading at P = –5, one will observe a slowing down of the crack, followed by an accelerationuntil complete separation. Note that if one imposes an abrupt loading from N to L, one does not followa path similar to LMN, since one cannot have G < 0. One will observe an instantaneous penetration NSat constant a, then the path ST at G = 0, along the Hertz curve, and finally a crack recession at P =constant, up to the point L (Maugis and Barquins, 1978a, 1980). We will return later to these crackpropagation kinetics at fixed load.
The JKR theory has been verified a number of times. The variation of contact area at zero load withthe concentration of surfactant in a liquid allows us to draw the Gibbs adsorption isotherms (Haidaraet al., 1995).
4.4.3 DMT Solution
However, the JKR theory presents a difficulty: the adherence force Pc = – πwR, independent of theYoung’s modulus, was incompatible with an old calculation by Bradley in 1932 giving an adherence forcePc = –2πwR between a rigid sphere and a rigid plane in point contact. In 1975 Derjaguin et al. (1975)gave a completely different theory, the DMT theory, in which adhesion forces act in an annular zonearound the contact but do not deform the profile. In this theory, there are no singular stresses, the
FIGURE 4.7 Equilibrium relations between penetration δ and radius of contact a, in reduced coordinates. Super-imposed are the curves at constant P, independent of w.
32
connection is tangent, and separation occurs when the contact is reduced to a point (thus without abruptinstability) under an adherence force Pc = –2πwR.
A long controversy occurred when Tabor (1977), comparing both theories, emphasized that the maindefect of the DMT theory was to neglect the deformations due to adhesion forces outside of the contact,whereas that of the JKR theory was to neglect these adhesion forces. Considering the height h of thediscontinuity of displacements in the JKR theory, he proposed that one must pass continuously fromone theory to the other in function of a parameter w2R/E2Z 3
0 where Z0 is the equilibrium distance betweenatoms. After an exchange of notes with Tabor, Muller et al. (1980) gave a self-consistent numericalcalculation, using the Lennard–Jones potential and abandoning the hypothesis that adhesion forces donot deform the Hertzian profile. The result was a continuous transition from the DMT theory to theJKR theory when a parameter µ, proportional to the parameter introduced by Tabor, increases. The DMTtheory applies when µ � 1 (hard solids, small radii, weak adhesion) and that of JKR for µ � 1 (softsolids, large radii, strong adhesion). As shown later (Muller et al., 1980), neither limit theory dependson the exact form of the interaction potential between the surfaces.
4.4.4 JKR–DMT Transition in a Dugdale Model
In 1982, Barquins and Maugis (1982), used the theory of Sneddon (1965) for the contact of axisymmetricpunches, in which a rigid body displacement χ(1) was cancelled to have zero stresses on the edge of thecontact. They showed that, if χ(1) ≠ 0, the singularities of stresses and the discontinuities of displacementwhich appear are those of the fracture mechanics, with a stress intensity factor KI proportional to χ(1).Writing the classical relation between G and K 2
I , Equation 4.41, for plane deformation,* the JKR resultsappear immediately, not those of DMT. In fact, as recalled above, this relation is valid only in linearelastic fracture mechanics, when the interaction zone is negligible compared to a characteristic dimensionof the system, here the radius of contact. Applying a Dugdale model (Barenblatt model where adhesionforces are assumed constant and equal to σ0 on an annulus of width d around the contact), one recoversthe JKR–DMT transition, but this time with the advantages of analytic formulae (Maugis, 1992a).
The method consists in calculating two stress intensity factors, the one KI due only to external loading(Equation 4.39) is that of the JKR theory; the other corresponds to a constant pressure acting on a lengthd = c – a in an external circular crack (Equation 4.24). Comparison of Equations 4.25 and 4.26 withEquations 4.41 and 4.42 shows that terms in KI are identical. If we replace the pressure p by –σ0 oneobtains a negative stress intensity factor Km. One then adds the two loadings making
(4.44)
which cancels the singular stresses, ensures the continuity of stresses, and fixes the length d. One canthus calculate the profile of the crack, and in particular its opening δt at the end of the cohesion zone atr = c (i.e., ρ = m)
(4.45)
The energy release rate is computed by the J-integral, which reduces here to σ0δt, and the problem issolved by writing the equilibrium equation, G = w. In the calculation it appears as a parameter
*Deformations are plane in the vicinity of an axisymmetric crack.
K KI m+ = 0
δ
σ
t
a
Rm m m
v
Ea m m m
=π
− + −( ) −
+−( )
π− − − −( )
−
−
22 2 1 2
2
02 1 2
1 2 1
4 11 1 1
tan
tan
(4.46)
When λ → ∞, (c/a → 1) one recovers the JKR theory, and when λ → 0, (c/a → ∞), the DMT theorywith an energy release rate
(4.47)
We have seen in Figure 4.1 that surface energy represents the work against two types of forces: firstelastic forces, then adhesion forces. It could seem physically incompatible to have tensile stresses in anarea of contact and no adhesion forces outside (JKR), or to have only compressive forces in an area ofcontact and adhesion forces outside (DMT). Figure 4.8, which displays the stress distribution in surface,allows us to understand the difficulties carried by these two limit cases.
The JKR theory which is an LEFM theory, corresponds to the limit λ → ∞, and we see that adhesionstresses outside the area of contact are reduced to a peak infinitely narrow of zero measure. (One couldsay that the joining point of stresses on the edge of the contact is reported at infinity.) The total adhesionforce outside the contact being zero, the integral of stresses in the area of contact is equal to the appliedload P, and negative loads are sustained by tensile elastic stresses. In the theory of DMT (λ → 0) thestress distribution is assumed to be Hertzian in the contact. The adhesion forces must tend toward zeroto have the continuity of stresses on the edge, but their integral is finite and the total attraction forceoutside the contact is 2πwR. The integral of normal stresses in surface is then P + 2πwR, and negativeloads are sustained by adhesion forces outside the area of contact.
Figure 4.9 displays the equilibrium curves a(P) as a function of the parameter λ. One observes acontinuous transition from the JKR theory to the DMT theory when λ decreases. The last equilibrium
FIGURE 4.8 JKR–DMT transition. Surface stress distribution for P = 3πwR and for various values of the parameter λ.
λ σ=π( )
2 0
21 3
wK R
G
a K
RP
R=
−
π
3
2
radii of contact at fixed load or fixed grips (defining the corresponding adherence forces) are depictedby dotted lines on the JKR curve. The Hertz curve is given for comparison. Figure 4.10 displays the crackprofile for P = 0 and the transition of the crack tip shape from the vertical to the tangential when λ decreases.
FIGURE 4.9 Equilibrium radius of contact vs. applied load, in reduced coordinates, for various values of theparameter λ. The Hertz curve is shown for comparison.
FIGURE 4.10 Profile of the air gap for P = 0 and for various values of the parameter λ. This is also the shape ofan elastic sphere on a rigid half space. The dashed line shows the increase of the crack opening displacement δt andof the ratio c/a when λ decreases. The parabolic approximation for the JKR profile is also given in the dashed line.
4.4.5 Application to a Liquid Meniscus
A liquid meniscus at the edge of a contact is a perfect example of a Dugdale zone where the constantpressure σ0 is the Laplace pressure, and, as a matter of fact, experiments with crossed mica cylinders(geometry similar to sphere/plane geometry) had shown that a liquid meniscus changes the shape of theprofile in the vicinity of the contact (Israelachvili, 1992; Christenson, 1985, 1988). These experimentshave been repeated with great precision by Maugis and Gauthier-Manuel (1994) and are in good agree-ment with the above theory.
4.5 Liquid Bridges
When the contact between the solids has disappeared, the meniscus is transformed into a liquid bridgewhich also give an adherence. The exact calculation of that force is difficult but is simplified for smallvolumes of liquids for which one can use a circular approximation of the meniscus. This attraction forceis due both to the Laplace pressure and to the surface tension of the liquid (Fischer, 1926; Heady andCahn, 1970), but as the second term is negligible for small volumes, one can evaluate the adherence forceby fracture mechanics (Maugis, 1987b, 1991), using a Dugdale model with a Laplace pressure σ0 = γ/ρ(where ρ is the radius of the meniscus) acting at the tip of an external crack with an opening of δt = 2ρcos θ. We get
(4.48)
Let us take a rigid punch with a profile f(r) and let b equal the wetted radius and z the separationbetween the punch and a rigid plane. One can compute the energy release rate G from the potentialenergy Up = –Pz:
As a first approximation, take a cylindrical liquid bridge of constant volume V:
With the condition dV/dz = 0, G becomes
(4.49)
Equating Equations 4.48 and 4.49 one obtains the force P as a function of the separation z.For a flat punch, f (r) = 0, we get the classical result:
(4.50)
For a sphere, f (r) = r2/2R:
(4.51)
a result given by Israelachvili (1992).
G t= =σ δ γ θ0 2 cos
GU
A
P
b
dz
dbP
P
= ∂∂
= −π2
V zb z rf r drb z
= π ( ) + π ( )( )∫2
0
2
GP
bz f b=
π+ ( )2
Pb
z= π2 2γ θcos
PR
zR
b
= π
+
4
12
2
γ θcos
For a cone of semi-angle β, f (r) = r/tan β:
(4.52)
This last equation is an approximation for large β of an expression given by Coughlin et al. (1982) forthe case z = 0.
For two spheres, evaluation of the liquid volume leads to
(4.53)
Since experiments are generally made at constant volume, let us introduce the volume V in the previousequations, with V = πb2z for two parallel plates, V = πb2z + πb4/4R for a sphere and a plane, V = πb2z +πb4/2R for two spheres. Equations 4.50, 4.51, and 4.53 become
(4.54)
(4.55)
(4.56)
which are decreasing functions of z. Figure 4.11 displays the equilibrium curves P(z) for a sphere and aplane.
The stability of equilibrium depends on the stiffness km of the measuring apparatus and can be studiedeither by (∂G/∂A)∆ or by Equation 4.10. The second method is more direct and shows that equilibriumis stable if km > –dP/dz. For a liquid bridge between a sphere and a plane, for example, the initial slope is
It is only if km > k0 that the curve can be entirely followed, eventually up to the point where dP/dz = –km
again, as for the curve drawn in the dashed line. This shape arises from the fact that, for a given volume,there are two solutions for the gorge radius (Erle et al., 1971; Fortes, 1982; Boucher et al., 1982; Lianet al., 1993). The criterion of stability at fixed grips is the point where the two solutions converge (Lianet al., 1993), the point with a vertical tangent on the curve P(z). If km < k0 the equilibrium at fixed ∆ isunstable for z = 0, and one jumps along the straight line
Pb
z
b
= π
+
1
1
γ ββ
tan
tan
PR
zR
b
= π
+
2
12
γ θcos
PV
z= 2
2
γ θcostwo plates
P RV
Rz
= π −+
π
4 11
12
γ θcos sphere plane
P RV
Rz
= π −+
π
2 11
12
2
γ θcos sphere sphere
dP
dzk
R
Vz
= − = − π π
=0
0
3
4 γ θcos
P R k zm= π −4 γ θcos
down to the point where it cuts the equilibrium curve, which can be followed until the rupture. If thisstraight line does not intersect the equilibrium curve, the adherence force is simply P = 4πγR cos θ. Someexperimenters have observed an instability at z = 0 (McFarlane and Tabor, 1950; O’Brien and Hermann,1973; Fisher and Israelachvili, 1980), others extended equilibrium curves (Mason and Clark, 1965; Hottaet al., 1974). When one studies the contact of a sphere with a meniscus, one can thus expect two jumps:the first at the limit of stability of the solid–solid contact, the second at the limit of the stability of theliquid bridge (depending on the volume of the liquid).
We must note that in this cylindrical approximation of the liquid bridge, the adherence force at z = 0is independent of the volume of the liquid bridge. In fact a circular approximation for the meniscus oran exact solution (Mazzone et al., 1986) shows that it decreases with the volume.
In such experiments one must take care to avoid viscous effects. For a liquid bridge between twoparallel plates, for example, the Stefan law, modified by Healey (1926) tells us that the time to separatetwo plates from z1 to z2 with a force F is
i.e., that the viscous force is:
an equation given by Dienes and Klemm (1946) (see also Trahan et al., 1987). For a liquid bridge betweena sphere and a plane, the viscous force was given by Matthewson (1988) under the form
FIGURE 4.11 Force-displacement equilibrium curve for liquid bridges between a sphere and a plane for variousvalues of the reduced volume V/πR3 of the liquid bridge. Stable equilibria can only be observed for stiffness km ofthe measuring apparatus larger than –dP/dz.
tb
F z z= π −
3
8
1 14
12
22
η
FV
z
dz
dt=
π3
2
2
5
η
The influence of the viscosity on the adherence due to liquid bridges was clearly shown by McFarlaneand Tabor (1950) and can be suspected in experiments by Mason and Clark (1965) on oil bridges.
4.6 Adhesion of Rough Elastic Solids — Application to Friction
The JKR theory is well verified on soft elastic materials, like rubber, but experiments on harder materialsonly revealed a very weak adherence. The explanation was given by Johnson (1975) and Fuller and Tabor(1976). On rough surfaces there is a competition between compressive forces exerted by the contact ofhigher asperities and the attraction force exerted by the contact of lower asperities, so that on hard elasticsolids the least roughness decreases the adherence force.
Fuller and Tabor (1976) used the model of Greenwood and Williamson (1966): a surface of nominalarea A0 having N asperities is in contact with a smooth rigid plane. Asperities have all the same radiusof curvature R and have a Gaussian distribution of height with a standard deviation σ
Where φ(z)dz is the probability for an asperity to have a height between z and z + dz above the planedefined by the mean height of asperities. If all the spheres were at the same level (σ = 0), the adherenceforce should be NPc = NπwR. If σ ≠ 0, the more stretched asperities pull off first, as soon as theirelongation reaches δc, critical elongation at fixed grips (roughnesses individually behave as if they aresubmitted to a fixed grips motion). The pull-off force between the two surfaces is given in the dottedline in Figure 4.12, as a function of the adhesion parameter σ/δc. The modified parameter
Where E* = E/(1 – ν2), is, except for a coefficient, the ratio of the force needed to deform the asperityby a quantity δ = σ to the adherence force of that asperity. For metals, even clean, the Young’s moduliare so large that standard deviations less than a few tens of Ångströms should be necessary to have anadhesion parameter less than one.
There is no difficulty in making a similar calculation for contacts of the DMT type. The result (Maugis,1996) is rather similar and is displayed in the solid line in Figure 4.12. Due to adhesion forces aroundthe contact, an overload Z appears which must be added to the load P applied by the experimenter. Thisoverload certainly plays an important role in the friction of solids. An analogy can be made with magneticforces. The friction coefficient f = T/P of steel on steel is about 0.2, but if the rider is magnetized, thefriction coefficient increases enormously. The mechanisms of dissipation have not changed, but the loadon asperities is now the applied load plus the overload due to magnetic forces. If T is the friction forceand P the applied load, the friction coefficient µ should be defined by
(4.57)
F RzR
b
z
z= π −
6 122
2
2
η˙
φσ σ
zz( ) =
π−
1
2 2
2
2exp
32
θ σ σδ
= = π
E R
wR c
* 1 2 3 23 2
9
8 3
T P Z= µ +( )
FIGURE 4.12 Elastic contact of a rough surface on a smooth rigid plane. Normalized adherence force F/NPc as afunction of the parameter σ/δc. In the dotted line, the theory of Fuller and Tabor for JKR contacts; in the solid line,the theory for DMT contacts.
FIGURE 4.13 DMT contacts. The total load Z + P as a function of the applied load P, for various values of theadhesion parameter θ. Normalized coordinates. Dashed line: the limit of stability (pull-off force).
and the shape of the friction curve T vs. P should be that of Figure 4.13. A similar equation has beenrepeatedly proposed in the past with Z a constant adherence force, so that the equation looks like thetwo-term Coulomb law
(4.58)
(see Tabor, 1975 and Derjaguin, 1988). Such equations account for friction under negative load. In thepresent theory Z is not the adherence force and it increases with the applied load P since the number ofcontacts increases, and thus the overload.
4.7 Kinetics of Crack Propagation
We have seen that a crack in an elastic solid is in equilibrium if G = w. If G > w the crack advances, ifG < w the crack recedes. G – w is the thermodynamic force which makes the crack advance. In a solidwithout dissipation, a crack subjected to a constant force G – w would constantly accelerate until theRayleigh waves velocity is reached. However, such nondissipative materials do not exist. First, a realmaterial can have a term of static friction ψ such that the crack only advances if G – w > ψ, giving theappearance of a large energy of adhesion G0 = w + ψ, and leading to hysteresis between crack advanceand crack recession. This is the case when there is plastic deformation or polymer chain extraction.Secondly, when G > G0, the crack assumes a velocity depending on dissipative processes, as a ball fallingin oil. (Of course, if (∂G/∂A) ≠ 0, G varies during propagation with the velocity.) So we are led to writein a very general manner
(4.59)
where Ud1, Ud2, … are the dissipation energies (depending or not on the velocity) and UK the kineticenergy.
A first type of dissipation, very general, can be understood from the Barenblatt model. When a crackpropagates at a velocity ν, the peak of stress at its tip moves with it, and a volume element neighboringits trajectory undergoes a cycle stress when the crack tip moves near it,* cycle whose characteristicduration is about d/ν (where d is the width of the stress peak) and whose magnitude is proportional tothe theoretical stress, hence to w. Energy is lost in such a cycle,** and the crack undergoes a dragproportional to w (as proposed for the first time by McLean [1957] for plastic dissipation in intergranularfracture of metals), and function of the crack velocity.
Besides internal friction in the vicinity of the crack tip, other types of dissipation can be added. Whenthe crack propagates in a liquid medium, viscosity can impede the opening of the crack and cause aNewtonian drag which can be larger than the internal drag above a given velocity (see Michalske andFrechette [1980] for fracture of glass and Carré and Schultz [1985] for peeling). One can also havedissipation in the bulk of the material, at a distance from the crack. In the case of the peeling of a
*One can make the analogy with the drag of a cylinder (nonadhesive) rolling on a viscoelastic material. This dragarises from the energy dissipated in bulk during the passing of compressive stresses below the roller. This drag tendstoward zero at very low or very high velocities and is maximum for velocities around a/τ where a is half the contactwidth and τ the relaxation time. This maximum correlates with the maximum of tan δ taking as period T = 3.4 a/Vis the duration of passage of a volume element below the roller (see Hunter, 1961; Johnson, 1985).
**This internal friction can have various origins: thermal diffusion between compressed and stretched volumeelements (thermoelasticity), diffusion of interstitial atoms toward octahedric sites in face-centered crystals (Snoekeffect), stress-induced migration of alkali cations in glasses, oscillations of pinned dislocations in metals(Granato–Lucke effect), etc.
T k P= + µ
G wdU
dA
dU
dA
dU
dAd d K− = + + +1 2 L
viscoelastic band, for example, energy can be dissipated in the transitory flexion of the band when theinterfacial crack advances, and whose importance depends on the band thickness. In the case of a ballor cylinder rolling on an elastomer, losses by deformation of the bulk must be added to dissipation atthe edge of the contact (Roberts, 1979, Zaghzi et al., 1987). Finally, in the case of polymers the essentialpart of the dissipation can arise from the work of extraction of fibrils (crazes) or polymeric chains. Forsome years a great deal of attention has been focused on the role of these connectors (Raphaël and DeGennes, 1992; Creton et al., 1992, 1994; Brown, 1993; Brochard-Wyart et al., 1994; Deruelle et al., 1994).
4.7.1 Case of Viscoelastic Solids
Let us examine first the case of adherence of viscoelastic solids where the only dissipations are localizedat the crack tip. As said above, a crack submitted to a constant force G – w undergoes a drag proportionalto w and function of its velocity ν and one can write the phenomenologic equation (Mueller and Knauss1971; Andrews and Kinloch, 1973; Maugis and Barquins, 1978b).
(4.60)
which generalizes for any geometry the equation given by Gent and Schultz (1972) for the peeling invarious liquids. aT is the WLF translation factor (Williams et al., 1955) for the velocity–temperatureequivalence, given by
(4.61)
where TS = Tg + 50 (Tg is the temperature of viscous transition), used for the first time in fracturemechanics by Mullins (1959) for tearing of rubber, and by Kaelble (1960) for peeling.
So, in Equation 4.60 surface properties (w) and viscoelastic properties are completely decoupled fromelastic properties, geometry, and applied loadings which appear in G. The dimensionless function φ(aTν)is characteristic of crack propagation in mode I. Once this function is known, Equation 4.60 allows usto predict the kinetics of pull-off at fixed load, fixed grips, or fixed cross-head velocity. Equation 4.60has been verified for adherence of glass/polyurethane on various geometries where G could be computed(sphere, flat punch, flat-ended sphere, peeling) at various temperatures. Whatever the studied geometry,one obtains the same master curve, which shows the advantage of fracture mechanics.
Adsorption of water decreases the Dupré energy of adhesion w, and thus viscoelastic losses in agreementwith Equation 4.60. This point was verified by measuring the rolling resistance � of a glass cylinderrolling on an inclined plane as a function of the velocity for various room humidities (Roberts, 1979;Maugis and Barquins, 1980). In this geometry we have essentially a π/2 peeling at the rear of the cylinder(the points of the cylinder follow a cycloid). The result is a translation of the logG–logν curves withhumidity. As in these experiments we have G � w, this translation clearly arises from the multiplicativeterm w in the right-hand side of Equation 4.60, in agreement with earlier results for peeling in variousliquids (Gent and Schultz, 1972) or on various substrates (Andrews and Kinloch, 1973).
So, logG–logν curves are horizontally shifted to the right when temperature increases (due to the aT
factor) and shifted down when the Dupré energy of adhesion decreases.
4.7.2 Experiments of Fixed Cross-Head Velocity — Tackiness
Adherence of solids is more often studied with a testing machine at constant cross-head velocity than atfixed load or fixed grips, but the kinetics of detachment is more difficult to interpret since the testingmachine makes two parameters vary simultaneously: displacement ∆ and crack length. The variation ofG with time is given by:
G w w aT− = ( )φ ν
log.
.a
T T
T TT
S
S
= −−( )
+ −8 86
101 6
(4.62)
At the beginning G is low and the crack advances slowly, the first term prevails, and the recorded forceincreases. Then, as the crack accelerates, the second term prevails and the recorded force decreases. Themaximum which results from the competition between the two effects has no physical signification.
Let us examine the case of a glass ball on a rubber half-space, pulled with the velocity·∆ by a testing
machine of stiffness km (Barquins and Maugis, 1981). This experiment corresponds to the practicaldetermination of tackiness, a normalized test since it depends on a number parameters and is withouta theoretical model, and whose simplified version is the thumb test (one puts a thumb on a material andwithdraws it to see if it sticks). To solve the problem we have the two equations of state of the system,Equations 4.36 and 4.37, the equation of propagation Equation 4.60, and the relation between
·δ and
·∆extracted from Equation 4.7. A numerical computation allows us to have P as a function of t (i.e., of ∆).The comparison between theory and experiment is very good. Figure 4.14 (for km infinite) shows thatthe maximum force recorded essentially depends on the imposed velocity
·δ. The dotted curve corresponds
to quasistatic unloading (see Figure 4.6) and the point D is the adherence force at fixed grips (F = πwR).One can see that great influence of the viscoelastic dissipation on the adherence and how much care
FIGURE 4.14 Adherence of a glass ball on polyurethane measured with a traction machine. Influence of the cross-head velocity
·δ on the recorded force. The JKR equilibrium curve (for
·δ → 0) is shown in the dotted line. Points C
and D correspond to quasistatic adhesion force at fixed load and at fixed grips. (Modified from Barquins, M. andMaugis, D. (1981), Tackiness of elastomers, J. Adhesion, 13:53-65.)
equi
libriu
mcu
rve
dG
dt
G G
AA
A
= ∂∂
+ ∂∂
∆
∆∆
˙ ˙
56
must be taken when using a testing machine. Confusion between the curve at·δ = 1 µm/s with the
equilibrium curve would lead to a very large apparent adhesion energy.
4.7.3 Branch with Negative Resistance: Velocity Jump and Stick-Slip
The relation between the form of the function φ(aTν) and the viscoelastic properties of a material is aproblem still unsolved. Mullins (1959) had shown, for the tearing of various elastomers, that there is alinear relation between rupture energy G at given temperature and velocity with the imaginary part ofthe complex shear modulus measured at the same temperature and a given frequency. For the adherenceof glass on polyurethane a strong correlation was found by Maugis (1982) between variation of G withcrack velocity, G ∝ (aTν)0.6, and variation of the imaginary part of the Young’s modulus E″ ∝ ω0.6, at lowfrequency.
Let us study the implications of such a negative resistance branch in the G(ν) curve. This curve canbe seen as the superposition of an elastic solution with G increasing when ν approaches the velocity cR
of Rayleigh waves and a wide peak due to viscoelastic losses (Figure 4.15). Branches with positive slopescorrespond to stable propagation, i.e., if a small perturbation temporarily increases the velocity, thedissipation increases and the crack returns to its initial velocity. The branch AC, on the other hand,corresponds to unstable propagations and cannot be observed. In fact, the behavior depends on thestability of the studied geometry.
If geometry and loading are such that (∂G/∂A) < 0, G and thus ν increase with the crack length upto a value Gc, νc (point A) where the velocity abruptly jumps (with acoustic emission) on the secondbranch with positive slope. This value Gc is the critical energy release rate for catastrophic fracture, andis a characteristic of the material. We are thus faced with three characteristic values of G: w, G0, Gc. Thiscatastrophic fracture is thus always preceded by a slow propagation (subcritical) as clearly noted, perhapsfor the first time, by Rivlin and Thomas (1953) in the case of the rupture of rubber. This subcriticalpropagation can pass unnoticed if νc is low and extends a fraction of mm only. This criterion ofcatastrophic propagation for ν = νc can be written G = wc (with wc a thousand times w in the case ofrubber) but absolutely cannot be confused with the Griffith criterion of equilibrium. The value Gc mustnot be deduced from the maximum of the recorded force in a tensile machine at constant cross-headvelocity, since this maximum can be observed in the subcritical crack propagation regime (Margolis et al.,1976, Maugis, 1985). For many decades, velocity jumps have been observed by a number of experimenters(see Maugis, 1985), and a hysteresis between the velocity jump AB for an accelerating crack and thevelocity jump CD for a crack which decelerates was observed by Kobayashi and Dally (1977) in epoxyresins.
FIGURE 4.15 G(ν) curve seen as the superposition of an elastic solution with G increasing as the Rayleigh velocityis approached, and a broad peak due to viscoelastic losses.
If the geometry and the loading are such that (∂G/∂A) > 0, the crack propagation can be controlledand the crack velocity imposed. If, however, one tries to impose a velocity V between νc and ν2 in thenegative resistance branch, a phenomenon of stick-slip motion appears. According to the simple modelof relaxation oscillations proposed a number of times (Clark and Irwin 1966; Williams et al., 1968;Williams, 1984), and classical in friction, the first branch is followed up to Gc (point A, Figure 4.15)where the velocity νc is too low, so that the velocity jumps to ν1 (point B) where it is too fast; the crackslows down until point C where the velocity ν2 is still too fast. Then the velocity jumps to point D wherethe crack appears arrested. Therefore the velocity increases up νc, and so on. In this periodic stick-slipmotion, most of the time is spent on the branch DA of low velocities and the recorded force has asawtooth shape, with Pmax giving classically Gi or Ki for crack initiation and Ga and Ka for crack arrest.
Maugis and Barquins (1988) have studied the stick-slip in peeling using adhesive rollers and the resultswere in agreement with the above model. This model accounts for the frequencies observed in the rangeof velocities V, for the acoustic emission at each jump, for the marks let on the band at each cohe-sive–adhesive transition (allowing an independent period measurement), but does not explain why theamplitude of oscillations decreases when V increases, why oscillations are sinusoidal, or why the markson the band disappear at high velocity. For that purpose it is necessary to introduce the inertia of thesystem. The relaxation cycles are replaced by limit circling including point A or point C, or both, withthe possibility of chaotic motion when a third degree of freedom is added, like the peel angle (Maugis,1987a; Hong and Yue, 1995).
4.7.4 Examples of the Additivity of Dissipations
4.7.4.1 Viscous Drag
When a crack propagates in the presence of a viscous liquid, this liquid can decrease the energy of adhesionat the crack tip, giving a translation of G(ν) curves, as seen above. But beyond a given velocity, the dragdue to hydrodynamic phenomena inside the crack can be larger than the drag wϕ(ν) due to internalfriction. In this case a knee arises in the G(ν) curve and one follows the curve corresponding to viscousdrag until cavitation appears. The crack then propagates without contact with the liquid medium, andone follows the G(ν) curve for propagation in air, as shown by Carré and Schultz (1985) for peeling ofelastomers in oils of various viscosities. The curve for viscous drag can even cross the G(ν) curve forpropagation in air, with a velocity jump when cavitation occurs, as observed by Michalske and Frechette(1980) for the fracture of glass in water.
4.7.4.2 Bulk Dissipations
Let us take the example of a cylinder of mass M and length b rolling on an elastomer inclined by anangle β on the horizontal. The energy (potential) release rate is G = Mg sin β/b (Roberts and Thomas1975, Kendall 1975). (One can view the rolling cylinder as an adhesive roller which unglues by rollingdown: we have a peeling at π/2 under the load P = Mg sin β). Two types of dissipation exist. The first,in the bulk, as discussed in note 4 and which is independent of surface properties, the second in thevicinity of the crack tip, as in peeling. By decreasing the mass of the cylinder toward zero, the bulk termdisappears, whereas, by spreading talc powder onto the surface, adhesion disappears. Zaghzi et al. (1987)have clearly shown that the two types of dissipation must be added, with the same factor aT, to have
(4.63)
Figure 4.16 illustrates rolling on EPDM. For a mass of 15 g, bulk losses are negligible, and that curve,which can be represented by the equation
G G aT= ( )0Φ ν
G w= +( )1 0 6αν ,
with w = 0,25 J/m2, corresponds to peeling alone. We must note that in Equation 4.63 G0 is not a truemultiplicative factor since in a log–log diagram curves are not shifted in parallel when one changes G0.It is more explicit to retain the form
4.7.4.3 Chains or Fibrils Extraction
When one has strong bonds between a substrate and a polymer, separation can occur with chains orfibrils extraction as in the fracture of bulk polymers. Energy dissipated in this process is to be added tothe energy dissipated by internal friction by moving stresses (Maugis, 1995). Let us consider, as anexample, the results of Ahagon and Gent (1975) concerning the peeling of polybutadiene on treated glass(Figure 4.17). These authors increased the density of covalent bond (primary bonds) by increasing theproportion of vinylsilane in ethylsilane during the glass treatment, so that the density of vinyl groupsused for the bonding to glass could vary from 0 to 100%. They observed that curves are not shiftedparallel to themselves in log–log coordinates, when one changes the density of primary bonds, but theyconverge at a high velocity of peeling, and that the upper curve is the same as for tearing of the elastomer.(Recall that Chang et Gent [1981] have shown that G0 linearly increases with the density of primarybonds up to a limit value G*0 characteristic of the bulk elastomer, and that the less the elastomer is cross-linked the higher G*0 , in agreement with the model of Lake and Thomas [1967].) Assume that twodissipative mechanisms coexist with the same factor aT for the velocity–temperature correspondence, butwith different sensitivity to velocity, as for rolling seen above, and that the lower curve for 0% of vinylsilane(and thus no chemical bonds) corresponds to internal friction. Let us subtract from the three upper
FIGURE 4.16 LogG–logν curves for cylinders of various masses rolling on an elastomer (EPDM). For 15g the lossesin the bulk are almost negligible, and this curve corresponds to peeling. (Modified from Zaghzi, N., Carré, A.,Shanahan, M.E.R., Papirer, E., and Schultz, J. (1987), A study of spontaneous rubber/metal adhesion. I. The rollingcylinder test, J. Polym. Sci. Polym. Phys., 25:2393-2402.)
G w a g aT T= + ( )
+ ( )10 6
2 2α ν ϕ ν,
curves the values of the lower curve: one obtains three curves (dotted lines) which are vertically shiftedwhen one increases the density of chemical bonds. We are thus tempted to write schematically
(4.64)
where the Nfl term is that proposed by de Gennes (1982) in his theory of chain suction (l is the chainlength, f the force to extract it, and N the number of chains crossing the fracture plane unit area). Noticethat this nonparallelism of logG–logν curves is observed only if the two dissipative phenomena are ofthe same order of magnitude. If dissipation by internal friction is negligible compared to the work ofchain or fibril extraction, the curves would appear parallel and one would be tempted to write
(4.65)
FIGURE 4.17 LogG–logν curves for the peeling of an elastomer on glass surfaces treated with a mixture of ethylsilaneand vinylsilane: (0% vinylsilane [O], 50% vinylsilane [∆], 100% vinylsilane [�] or nontreated [+]). (Data fromAhagon, A. and Gent, A.N. (1975), Effect of interfacial bonding on the strength of adhesion, J. Polym. Sci. Polym.Phys. Ed., 13:1285-1300.)
G w w a Nf a lT T− = ( ) + ( )ϕ ν ν
G G aT= ( )0 Φ ν
where G0 is, this time, a multiplicative factor and where Φ → 1 when ν → 0. At zero velocity Equation4.64 is written
(4.66)
Note that the term Nf0l is identical to the σ0σt term of the Dugdale model (Equation 4.30b).In summary, it seems that two regimes must be distinguished. The first is where adherence is low,
there are only losses by internal friction at the crack tip, and where equation
(4.67)
is well verified, giving parallel log–log curves when surfaces energy is modified. The second is whereadherence is high, due principally to chain or fibril extraction and where Equation 4.65 is verified, givingparallel log–log curves when the measure of primary bonds is modified, but with ∆(aTν) ≠ F(aTν).
4.7.4.4 Remarks on Hysteresis
It is clear that the dissipations at crack tip or in the bulk in a viscoelastic solid are of hysteresis origin; avolume element submitted to cycles of stresses dissipates more or less energy according to the imposedfrequency.
Another type of hysteresis, often described, that we will name static hysteresis, concerns the thresholdfracture energy G0. We have seen above that hysteresis between advancing or receding cracks was negligiblein glass or mica. In the case of JKR contacts we will say that a static hysteresis occurs if the contact areaunder load P is larger when obtained by unloading (crack advance) than by loading (crack receding).Very low crack velocity or very long times must be reached to ascertain whether hysteresis occurs or not.However, there are a number of cases where there is no doubt (see Kendall [1974] where hysteresis isvery neat.) One generally admits that values of G obtained after increase of the contact area (recedingcrack) correspond to Dupré energy of adhesion, w, whereas the higher values G0 obtained by decreaseof the crack area arise from a mechanism of chain extraction. Shanahan and Michel (1991) have shown,for glass-elastomer contacts, that the more the elastomer is cross-linked the lower G0, the value of wbeing independent of the degree of cross-linking, and that hysteresis disappears at high degrees of cross-linking. When loading and unloading are carried out in a quasistatic manner one obtains two curvesa(P) both obeying the JKR relation (Chaudhury and Whiteside, 1991). Thus, when one speaks ofadherence energy during advancing or receding, one must understand that equilibrium or pseudoequi-librium has been reached.
Note that there always exists a dissymmetry between advancing and receding cracks. The thermody-namic “motive” G – w for crack advancing can be as high as demanded. On the other hand, as G cannotbe negative, the motive w – G for crack receding cannot exceed w (so that the velocity of crack closingremains low). We have seen in Figure 4.7 that if one starts from an equilibrium point on the JKR curveand applies an overload, most of the path follows the Hertz curve (G = 0). The joining is tangent; thereis no dissipation on the edge of the contact; and this path is almost instantaneous (Maugis and Barquins,1978a). It is only after this quasi-instantaneous increase of the area of contact that a true crack slowlyrecedes. This dissymmetry explains the difference of kinetics between coming on and coming off exper-iments, but has no relation to the static hysteresis which could be observed.
4.7.5 Short Historical Background
After Griffith, fracture mechanics expanded in a rather anarchic manner in various fields (glass, elas-tomers, metals) and the history of appearance of different concepts is rather involved. The first geometriesstudied were unstable. The Griffith criterion of equilibrium was confused for a long time with a criterionof catastrophic fracture, particularly for glasses where the critical fracture energy Gc for brutal fractureis slightly above the 2γ0 of glass in a vacuum. For other materials Gc can be higher by many orders of
G w Nf l0 0= +
G wF a w aT T= ( ) = + ( )[ ]ν ϕ ν1
magnitude. For the catastrophic rupture of metals, Irwin (1948) and Orowan (1949) have generalizedthe Griffith relation by replacing surface energy with a fracture energy γp. Similarly but independently,the criterion of catastrophic rupture of rubber was written as
by Rivlin and Thomas (1953), where Tc is the tear energy characteristic of the material and considerablylarger than surface energy. However, they noted that catastrophic fracture was preceded by a slowpropagation above a characteristic threshold T0 well below Tc (They obtained T0 = 3700 J/m2, comparedwith Tc = 13,000 J/m2.) Later, Greensmith and Thomas (1965) observed that T has not an single valuebut depends on the tear velocity, and gave the first G(ν) curves of the literature. In the field of glass, dueto the confusion of the criterion for brutal rupture with the Griffith criterion, the slow propagationpreceding brittle fracture (studied for the first time by Grenet, 1899), remained mysterious, as reflectedby its names: static fatigue, retarded fractured, stress corrosion. The time to rupture in a static fatigueexperiment was seen as the time for a defect to increase from subcritical size to the Griffith critical size(subcritical propagation). However the problem was that a crack of this size should close instead ofpropagate. Shand (1961) was probably the first, in the field of glasses, to draw the crack velocity as afunction of the energy release rate (so, eliminating the crack length and the geometry) followed byWiederhorn and Bolz (1870). Shand (1961) was also the first to affirm that this was the threshold valueG0, and not Gc, which was to compare with surface energy. (Nowadays the idea of a drag by internalfriction to explain the subcritical growth, and despite the analogy with problems of adhesion [Maugis1985, 1986], is not yet accepted by the scientific community working on glasses and ceramics, whichprefers to keep the stress corrosion.)
Concerning the kinetics of crack propagation in rupture and adherence of elastomers, it was remarkedearly that the peeling or tearing force strongly depends on velocity (Rivlin, 1944; Busse et al., 1946). ThenMullins (1959) for tearing and Kaelble (1960) for peeling used the WLF translation factor aT to havemaster curves. Introducing the energy release rate (named rupture energy θ), Gent and Kinloch (1971)showed that G(aTν) curves were independent of the tested geometry.* Gent and Schultz (1972), workingwith peeling elastomer bands at various velocities in various liquids, observed that log–log curves givingthe peeling force as a function of velocity were parallel shifted by a quantity predicted by the reductionof Dupré energy of adhesion w, i.e., that the work of detachment W was proportional to w:
(4.68)
Similarly Andrews and Kinloch (1973) studied adherence of rubber to various polymer substrates.They obtained, in log–log coordinates G(aTν) curves, all parallel to the tearing curve of rubber. Fromthe threshold tear energy, they deduced the threshold adherence θ0, compatible either with a Dupréenergy of adhesion when secondary bonds are present at the interface, or higher in the presence ofcovalent bonds, and proposed to write
(4.69a)
or
(4.69b)
*Following Rivlin and Thomas (1953), the interfacial or cohesive rupture energy, for a nonlinear elastic solid wasevaluated from the density of stored elastic energy in the sample at the moment of propagation, a method notapplicable to all geometries, particularly the adherence of a flat punch.
∂∂
≥U
ATE
c
δ
W wF w k n= ( ) = +( )ν ν1
G w aT= ( )Φ ν
G G aT= ( )0Φ ν
An equation similar to Equation 4.69b was also given by Mueller and Knauss (1971) for the tearing of anelastomer. Maugis and Barquins have rewritten Equations 4.68 and 4.69a in the form of Equation 4.60, inorder to separate clearly the thermodynamic driving force from the drag term. We have seen that one of theproblems was the identity of the mechanisms and the dissipative functions in Equations 4.68 and 4.69. Itremains to predict the form of functions F(aTν) and Φ(aTν) from the rheological properties of the materials.
4.8 Adhesion of Metals
Metals have an elastoplastic behavior which complicates further adherence problems. Furthermore, cleanmetallic surfaces have high surface energies and make strong metallic bonds when in contact. Separationgenerally occurs by ductile rupture of the softer material, and adherence depends on rheological prop-erties. If impurities are adsorbed onto the surfaces, the metallic bonds are screened, adhesion decreasesstrongly, and rupture will be adhesive rather than cohesive. Contact of clean surfaces must be avoidedin friction, but is a necessary condition in cold welding.
4.8.1 Adhesion of Microcontacts
When a hard sphere is progressively pressed on an elastoplastic half-space, it begins to elastically deformthe half-space; then plastic deformations appear at the Hertz point (0, a/2) for a mean pressure pm =1.1σ0 (σ0 is the elastic limit). The mean pressure continues to increase as the plastic zone develops, thenremains constant and approximately equal to pm = 3σ0 = H when plastification reaches the surface (H isthe hardness). In the presence of adhesion forces these deformations are enhanced. One can even haveplastic deformation under zero load. After elastoplastic or full plastic contact (radius of contact af) andelastic recovery, the adherence force F depends on the type of observed rupture. This rupture can becohesive if the energy of adhesion is high, or interfacial, occurring after reduction or not of the radiusof contact. These various points have been discussed in Maugis and Pollock (1984) and play an importantrole in the adhesion of particles (Rimai et al., 1994).
4.8.2 Cold Welding
Milner and colleagues from 1959 to 1969 made a major contribution in explaining the mechanisms ofcold welding (Milner and Rowe, 1962; Tylecote, 1968). It is the fragmentation of the oxide film and theextrusion of underlying clean metal through the cracks which explains cold-welding by co-rolling (Vaid-yanath et al., 1959), punching, or torsion–compression. This extrusion is obtained above a deformationthreshold varying from 10 to 90% depending on the metals. This threshold is lowered by preliminaryscratch brushing with a metallic brush, by working in a vacuum (Sherwood and Milner, 1969), or by thereduction of oxide thickness (Cantalejos and Cuminski, 1972). Cave and Williams (1973) have shown inscanning microscopy how the metal was progressively extruded through fractured scratch-brushed layersin proportion as deformation increases, in cold welding by rolling. Adherence is more important thecleaner the metal in contact. A large proportion of virgin surface is immediately contaminated byimpurities trapped at the interface and adheres poorly. It is necessary to have large deformation to dispersecontaminants or to work in a vacuum (Sherwood and Milner, 1969). An important point is that if thefriction coefficient between oxide films is high, these films break up coherently, extruding clean metal(Cantalejos and Cuminski, 1972; Osias and Tripp, 1966). This point was shown in an elegant mannerby Osias and Tripp (1966) on plasticine models covered with brittle varnish.
4.8.3 Adhesion of Metals at High Temperature (Maugis, 1980)
Experiments show that above a threshold temperature, adherence of metals strongly increases withtemperature and contact time. This threshold decreases when the load increases, but a representativevalue is about 0.3 Tf where Tf is the absolute melting temperature. So, room temperature experimentsare high-temperature experiments for indium, tin, and lead.
When two metals come into contact under a given load, asperities compress one another and undergoplastic deformation and creep; the area of contact increases with temperature and contact time, as in ahot hardness experiment. Then, when the stress has sufficiently decreased, the area of contact increasesby surface diffusion or diffusion along grain boundaries as in a sintering experiment. As these variousmechanisms are thermally activated, the increase of the area of contact with temperature can be written:
(4.70)
where the activation energy QA can be deduced from the activation energies of these various mechanisms.In high-temperature contacts, strong metallic bonds are easily developed and separation generally
occurs by creep rupture (cohesive rupture), so that the adherence force can be written, as for cold welding:
(4.71)
where σu is the ultimate tensile strength and k a factor linearly decreasing from 2 to 1 when the real areaof contact increases. The proportionality between tensile strength and hardness (H ≅ 3σu) is maintainedat high temperature, so that σu has the same activation energy t QH as the hot hardness. In creep rupture,as in hot hardness, there is an equivalence between time and temperature, and master curves can beobtained by using the Sherby–Dorn parameter θ = t exp(–Q/RT) where Q is the activation energy forself-diffusion. Thus, creep rupture depends on time to rupture t, and the adherence force decreases whenthe rupture time increases.
The adherence at high temperature is thus characterized by a competition between the increase of thearea of contact by creep and sintering with time and temperature, and the decrease of mechanicalproperties with those parameters. These effects can be separated by measuring adherence at a constanttemperature lower than that of contact, as in experiments by Polke (1969) or Gras (1989). In fact, theproblem is rendered more complex by phenomena such as oxidation, oxide dissolution, segregation ofimpurities, interdiffusion, and phase transformation. Mutual solubility is not a condition of high adher-ence as believed some years ago, and insoluble pairs like Ag–Fe, Ag–Ni, and Pb–Au can have betteradherence than soluble pairs like Ag–Au. Interdiffusion changes only the mechanical properties in thevicinity of the junction so formed; if they are improved, adherence increases, but the formation of brittleintermetallic compounds or the development of porosities by Kirkendall effect can reduce that adherence.
4.9 Conclusion
This overview of mechanical aspects of adhesion leads to a rather coherent image. But in the details anumber of points remain to be elucidated. One should know how to predict the threshold energy G0 andmore generally the G(ν) curves from the mechanical properties of materials. Phenomena such as cavi-tation and crazes at crack tip remain to be clarified, and stick-slip is more complex that initially thought(Ryschenkoff and Arribart, 1996). Crack propagation in mixed mode is a subject still debated. Finally,despite recent progress (Kim and Kim, 1988; Williams, 1993; Moidu et al., 1995), we are still unable totake accurately into account elastoplastic deformations in the adherence of solids.
References
Ahagon, A. and Gent, A.N. (1975), Effect of interfacial bonding on the strength of adhesion, J. Polym.Sci. Polym. Phys. Ed., 13:1285-1300.
Andrews, E.H. and Kinloch, A.J. (1973), Mechanics of adhesive failure. I, Proc. London Ser. Soc. A.,332:385-399.
Barenblatt, G.I. (1962), The mathematical theory of equilibrium cracks in brittle fracture, Adv. Appl.Mech., 7:55-129.
A A e Q RTA= −0
F kA u= σ
Barquins, M. and Maugis, D. (1981), Tackiness of elastomers, J. Adhesion, 13:53-65.Barquins, M. and Maugis, D. (1982), Adhesive contact of axisymmetric punches on an elastic half-space:
the modified Hertz–Huber stress tensor, J. Méc. Théor. Appl., 1:331-357.Bhowmick, A.K., Gent, A.N., and Pulford, C.T.R. (1983), Tear strength of elastomers under threshold
conditions, Rubber Chem. Technol., 56:226-282.Boucher, E.A., Evans, M.J.B., and McGarry, S. (1982), Capillary phenomena. XX. Fluid bridges between
horizontal solid plates in a gravitational field, J. Colloid Interface Sci., 89:154-165.Brochard-Wyart, F., De Gennes, P.G., Léger, L., Marciano, Y., and Raphaël, E. (1994), Adhesion promoters,
J. Phys. Chem., 98:9405-9410.Brown, H.R. (1993), Effects of chain pull-out on adhesion of elastomers, Macromolecules, 26:1666-1670.Busse, W.F., Lambert, J.M., and Verdery, R.B. (1946), Tackiness of GR-S and other elastomers, J. Appl.
Phys., 17:376-385.Cantalejos, N.A. and Cuminski, G. (1972), Morphology of the interface of roll-bonded aluminium, J. Inst.
Metals, 100:20-28.Carré, A. and Schultz, J. (1985), Polymer–aluminium adhesion. IV. Kinetics aspects of the effect of a
liquid environment, J. Adhesion, 18:207-216.Cave, J.A. and Williams, J.D. (1973), The mechanisms of cold pressure welding by rolling, J. Inst. Metals,
101:203-207.Chang, R.J. and Gent, A.N. (1981), Effect of interfacial bonding on the strength of adhesion of elastomers.
I. Self adhesion, J. Polym. Sci. Polym. Phys. Ed., 19:1619-1633.Chaudhury, M.K. and Whiteside, G.M. (1991). Direct measurement of interfacial interactions between
semispherical lenses and flat sheets of poly(dimethylsiloxane) and their chemical derivatives, Lang-muir, 7:1013-1025.
Christenson, H.K. (1985), Capillary condensation in systems of immiscible liquids, J. Colloid InterfaceSci., 104:234-249.
Christenson, H.K. (1988), Adhesion between surfaces in undersaturated vapors. A reexamination of theinfluence of meniscus curvature and surface forces, J. Colloid Interface Sci., 121:170-178.
Clark, A.B.J. and Irwin, G.R. (1966), Crack propagation behaviors, Exp. Mech., 6:321-330.Coughlin, R.W., Elbirli, B., and Vergara-Edwards, L. (1982), Interparticle force conferred by capillary-
condensed liquid at contact points. I. Theoretical considerations, J. Colloid Interface Sci., 87:18-30.Creton, C., Brown, H.R., and Shull, K.R. (1994), Molecular weight effects in chain pullout, Macromole-
cules, 27:3174-3183.Creton, C., Kramer, E.J., Hui, C.Y., and Brown, H.R. (1992), Failure mechanisms of polymer interfaces
with block copolymers, Macromolecules, 25:3075-3088.De Gennes, P.G. (1982), The formation of polymer/polymer junctions, in Microscopic Aspects of Adhesion
and Lubrication, George, J.M. (Ed.), Elsevier, Amsterdam, 355.Derjaguin, B.V. (1988), Mechanical properties of the boundary lubrication layer, Wear 128:19-27.Derjaguin, B.V., Muller, V.M., and Toporov, Yu. P. (1975), Effect of contact deformation on the adhesion
of particles, J. Colloid Interface Sci., 53: 314-326.Deruelle, M., Tirrell, M., Marciano, Y., Hervet, H., and Léger, L. (1994), Adhesion energy between polymer
networks and solid surfaces modified by polymer attachment, Faraday Disc., 98:55-65.Dienes, G.J. and Klemm, W.H. (1946), Theory and application of the parallel plate plastometer, J. Appl.
Phys., 17:458-471.Dugdale, D.S. (1960), Yielding of sheets containing slits, J. Mech. Phys. Solids, 8:100-104.Eftis, J. and Subramonian, N. (1978), The inclined crack under biaxial loading, Engng. Fracture Mech.,
10:43-67.Erle, M.A., Dyson, D.C., and Morrow, N.R. (1971), Liquid bridges between cylinders, in a torus and
between spheres, A.I.Ch.E.J., 17:115-121.Fischer, R.A. (1926), On the capillary force in an ideal soil: corrections of formulae given by W.B. Haines,
J. Agric. Sci., 16:492-505.
Fisher, L.R. and Israelachvili, J.N. (1980), Determination of the capillary pressure in menisci of moleculardimensions, Chem. Phys. Lett., 76:325-328.
Fortes, M.A. (1982), Axisymmetric liquid bridges between parallel plates, J. Colloid Interface Sci., 88:338-352.Fuller, B.W. and Thomson, R.M. (1978) Lattice theories of fracture, in Fracture Mechanics of Ceramics,
Vol. 4, Bradt, R.C., Hasselman, D.P.H., and Lange, F.F. (Eds.), Plenum Press, 507.Fuller, K.N.G. and Tabor, D. (1976), The effect of surface roughness on the adhesion of elastic solids,
Proc. R. Soc. A, 345:327-342.Gent, A.N. and Kinloch, A.J. (1971), Adhesion of viscoelastic materials to rigid substrates. III. Energy
criterion for failure, J. Polym. Sci., A-2. 9:659-668.Gent, A.N. and Schultz, J. (1972), Effect of wetting liquids on the strength of adhesion of viscoelastic
materials, J. Adhesion, 3:281-294.Gent, A.N. and Tobias, R.H. (1982), Threshold tear strength of elastomers, J. Polym. Sci. Polym. Phys.
Ed., 20:2051-2058.Gras, R. (1989). Etude de l’adhérence et du frottement à haute température du Nickel. Thèse d’Etat, Paris.Greensmith, H.W. and Thomas, A.G. (1955), Rupture of rubber. III. Determination of tear properties,
J. Polym. Sci., 18:189-200.Greenwood, J.A. and Williamson, J.B. (1966), Contact of nominally flat surfaces, Proc. R. Soc. A, 295:300-319.Grenet, M. (1899), Recherches sur la résistance mécanique des verres, Bull. Soc. Encour. Ind. Nat., 4:838-848.Haidara, H., Chaudhury, M.K., and Owen, M.J. (1995), A direct method of studying adsorption of a
surfactant at solid–liquid interface, J. Phys. Chem., 99:8681-8683.Heady, R.B. and Cahn, J.W. (1970), An analysis of the capillary forces in liquid-phase sintering of spherical
particles, Metall. Trans., 1:185-189.Healey, A. (1926), Trans. Inst. Rubber Ind. 1-334.Hong, D.C. and Yue, S. (1995), Deterministic chaos in failure dynamics: dynamic of peeling of adhesive
tape, Phys. Rev. Lett., 74:254-257.Hotta, K., Takeda, K., and Ilnoya, K. (1974), The capillary binding force of a liquid bridge, Powder
Technol., 10:231-242.Hunter, S.C. (1961), The rolling contact of a rigid cylinder with a viscoelastic half space, Trans. ASME,
J. Appl. Mech., 28:611-617.Irwin, G.R. (1948), Fracture dynamics, in Fracturing of Metals, Trans. ASM, Vol. 40 A, 147-166.Irwin, G.R. (1957), Analysis of stresses and strains near the end of a crack traversing a plate, J. Appl.
Mech., 24:361-364.Irwin, G.R. (1958), Fracture, in Encyclopedia of Physics, Vol. IV: Elasticity and Plasticity, Flugge, S. (Ed.),
Springer-Verlag, Berlin, 551.Irwin, G.R. and Kies, J.A. (1952), Fracturing and fracture dynamics, Welding Journal, 31:95-100.Israelachvili, J.N. (1992), Intermolecular and Surface Forces, 2nd ed., Academic Press, New York.Johnson, K.L. (1975), Non-Hertzian contact of elastic spheres, in The Mechanics of the Contact Between
Deformable Bodies, ed. de Pater, A.D. and Kalker, J.J. (Eds.), Delft University Press, 26.Johnson, K.L. (1985), Contact Mechanics, Cambridge University Press, U.K.Johnson, K.L., Kendall, K., and Roberts, A.D. (1971), Surface energy and the contact of elastic solids,
Proc. R. Soc. London, Vol. A, 324:301-313.Kaelble, D.H. (1960), Peel adhesion, Adhesive Age, 3:37-42.Kendall, K. (1971), The adhesion and surface energy of elastic solids, J. Phys. D: Appl. Phys., 4:1186-1195.Kendall, K. (1974), Kinetics of contact between smooth solids, J. Adhesion, 7:55-72.Kendall, K. (1975), Rolling friction and adhesion between smooth solids, Wear, 33:351-358.Kim, K.S., and Kim, J. (1988). Elasto-plastic analysis of the peel test for thin film adhesion, Trans. ASME,
110:266-273.Kobayashi, T. and Dally, J.W. (1977), A system of modified epoxies for dynamic photoelastic studies of
fracture, Exp. Mech., 17:367-374.Lake, G.J. and Thomas, A.G. (1967), The strength of highly elastic materials. Proc. R. Soc. Ser. A, 300:108-119.
Lian, G., Thornton, C., and Adams, M.J. (1993), A theoretical study of the liquid bridge force betweentwo rigid spherical bodies, J. Colloid Interface Sci., 161:138-147.
Margolis, R.D., Dunlap, R.W., and Markovitz, H. (1976), Fracture toughness testing of glassy plastics, inCracks and Fracture, ASTM STP 601, Philadelphia, 391.
Mason, C. and Clark, W.C. (1965), Liquid bridges between spheres, Chem. Eng. Sci., 20:859-866.Matthewson, M.J. (1988), Adhesion of spheres by thin liquid films, Philos. Mag., 57:207-216.Maugis, D. (1980), Creep, hot hardness and sintering in the adhesion of metals at high temperature,
Wear, 62:349-386.Maugis, D. (1985), Subcritical crack growth, surface energy, fracture toughness, stick-slip and embrittle-
ment, J. Mater. Sci., 20:3041-3073.Maugis, D. (1986), Subcritical crack growth, surface energy and fracture toughness of brittle materials,
in Fracture Mechanics of Ceramics, Vol. 8, Bradt, R.C., Evans, A.G., Hasselman, D.P.H., and Lange,F.F. (Eds.), Plenum Publishing, New York, 255.
Maugis, D. (1987a), Propagation saccadée de fissure en pelage, rôle de l’inertie, C. R. Acad. Sci. Paris,Série II, 304:775-778.
Maugis, D. (1987b), Adherence of elastomers: fracture mechanics aspects, J. Adhesion Sci. Technol., 1:105-134.Maugis, D. (1991), Adherence and fracture mechanics, in Adhesive Bonding, Lee, L.H. (Ed.), Plenum
Publishing, New York, 303.Maugis, D. (1992a), Adhesion of spheres: the JKR–DMT transition using a Dugdale model, J. Colloid
Interface Sci., 150:243-269.Maugis, D. (1992b), Stresses and displacements around cracks and elliptical cavities: exact solutions,
Engng Fracture Mech., 43:217-255.Maugis, D. (1995), On the additivity of losses of propagating cracks in polymer adhesion and tearing,
J. Adhesion Sci. Technol., 9:1005-1008.Maugis, D. (1996), On the contact and adhesion of rough surfaces, J. Adhesion Sci. Technol., 10:161-175.Maugis, D. (May 1997), Adherence des solides. Aspects mécaniques, La Revue de Métallurgie–CIT Sciences
et Génie des Matériaux, 655-690.Maugis, D. and Barquins, M. (1978a), Adhérence d’une bille de verre sur un massif viscoélastique: étude
du recollement, C. R. Acad. Sci. Paris, Sér. B, 287:49-52.Maugis, D. and Barquins, M. (1978b), Fracture mechanics and the adherence of viscoelastic bodies,
J. Phys. D: Applied Phys., 11:1989-2023.Maugis, D. and Barquins, M. (1980), Fracture mechanics and adherence of viscoelastic solids, in Adhesion
and Adsorption of Polymers, Part A, Lee, L.H. (Ed.), Plenum Press, New York, 203.Maugis, D. and Barquins, M. (1988). Stick-slip and peeling of adhesive tapes, in Adhesion 12, Allen, K.W.
(Ed.), Elsevier, London, 205.Maugis, D. and Gauthier-Manuel, B. (1994), JKR–DMT transition in the presence of a liquid meniscus,
J. Adhesion Sci. Technol., 8:1311-1322.Maugis, D. and Pollock, H.M. (1984), Surface forces, deformation and adherence at metal microcontacts,
Acta Metall., 32:1323-1334.Mazzone, D.N., Tardos, G.I., and Pfeffer, R. (1986), The effect of gravity on the shape and strength of a
liquid bridge between two spheres, J. Colloid Interface Sci., 113:544-556.McFarlane, J.S. and Tabor, D. (1950), Adhesion of solids and the effect of surface films, Proc. R. Soc.
London. A, 202:224-243.McLean, D. (1957), Grain Boundaries in Metals, Clarendon Press, Oxford.Michalske, T.A. and Frechette, V.D. (1980), Dynamic effect of liquids on crack growth leading to cata-
strophic failure of glass, J. Am. Ceram. Soc., 63:603-609.Milner, D.R. and Rowe, G.W. (1962), Fundamentals of solid phase welding, Metall. Rev., 7:433-480.Moidu, A.K., Sinclair, A.N., and Spelt, J.K. (1995), Analysis of the peel test: prediction of adherent plastic
dissipation and extraction of fracture energy in metal-to-metal adhesive joints, J. Testing Eval.,23:241-253.
Mueller, H.K. and Knauss, W.G. (1971a), The fracture energy and some mechanical properties of apolyurethane elastomer, Trans. Soc. Rheol., 15:217-233.
Mueller, H.K. and Knauss, W.G. (1971b), Crack propagation in a linearly viscoelastic strip, J. Appl. Mech.,38:483-488.
Muller, V.M., Yushenko, V.S., and Derjaguin, B.V. (1980), On the influence of molecular forces on thedeformation of an elastic sphere and its sticking to a rigid plane, J. Colloid Interface Sci., 77:91-111.
Mullins, L. (1959), Rupture of rubber, part IX: role of hysteresis in the tearing of rubber, Trans. Inst.Rubber Ind., 35:213-222.
O’Brien, W.J. and Hermann, J.J. (1973), The strength of liquid bridges between dissimilar materials,J. Adhesion, 5:91-103.
Orowan, E. (1949), Fracture and strength of solids, Rep. Prog. Phys., 120:185-232.Osias, J.R. and Tripp, J.H. (1966), Mechanical disruption of surface films on metals, Wear, 9:388-397.Paris, P.C. and Sih, G.C. (1965), Stress analysis of cracks, in Fracture Toughness and Testing Applications,
ASTM STP 381, Philadelphia, 30.Polke, R. (1969), Adhesion of solids at elevated temperatures, in Adhésion et Physico-Chimie des Surfaces
Solides, Colloque CNRS/DGRST, Soc. Chim. de France, p. 51-54.Raphaël, E. and De Gennes, P.G. (1992), Rubber-rubber adhesion with connector molecules, J. Phys.
Chem., 96:4002-4007.Rice, J.R. (1968a), A path independent integral and the approximate analysis of strain concentration by
notches and cracks, J. Appl. Mech., 90:379-386.Rice, J.R. (1968b), Mathematical analysis in the mechanics of fracture, in Fracture, Vol. 2, H. Liebowitz
(Ed.), Academic Press, 191.Rimai, D.S., Demejo, L.P., and Bowen, R.C. (1994), Mechanics of particle adhesion, J. Adhesion Sci.
Technol., 8:1333-1355.Rivlin, R.S. (1944), The effective work of adhesion, Paint Technol., 9:215-216.Rivlin, R.S. and Thomas, A.G. (1953), Rupture of rubber. I. Characteristic energy of tearing, J. Polymer
Sci., 10:291-318.Roberts, A.D. (1979), Looking at rubber adhesion, Rubber Chem. Technol., 52:23-42.Roberts, A.D. and Thomas, A.G. (1975), The adhesion and friction of smooth rubber surfaces, Wear,
33:45-64.Ryschenkow, G. and Arribart, H. (1996), Adhesion failure in the stick-slip regime: optical and AFM
characterizations and mechanical analysis, J. Adhesion, 58:143-161.Shanahan, M.E.R. and Michel, F. (1991), Physical adhesion of rubber to glass: cross-link density effects
near equilibrium, Int. J. Adhesion Adhesives, 11:170-176.Shand, E.B. (1961), Fracture velocity and fracture energy of glass in the fatigue range, J. Am. Ceram. Soc.,
44:21-26.Sherwood, W.C. and Milner, D.R. (1969), The effect of vacuum machining on the cold welding of some
metals, J. Inst. Metals, 97:1-5.Sih, G.C., Paris, P.C., and Erdogan, F. (1962), Crack-tip stress intensity factors for plane extension and
plate bending problems, J. Appl. Mech., 29:306-312.Sneddon, I.N. (1946), The distribution of stress in the neighbourhood of a crack in an elastic solid, Proc.
R. Soc. A, 187:229-260.Sneddon, I.N. (1965), The relation between load and penetration in the axisymmetric Boussinesq prob-
lem for a punch of arbitrary profile, Int. J. Engng. Sci., 150:243-269.Tabor, D. (1975), Interactions between surfaces: adhesion and friction, in Surface Physics of Materials,
Vol. II, Blakely, J.M. (Ed.), Academic Press, New York, 475.Tabor, D. (1977), Surface forces and surface interactions, J. Colloid Interface Sci., 58:2-13.Thomson, R., Hsieh, C., and Rana, V. (1971), Lattice trapping of fracture cracks, J. Appl. Phys.,
42:3154-3160.Trahan, J.F., Wehbeh, E.G., and Hussey, R.G. (1987), The limits of lubrication theory for a disk approach-
ing a parallel plane wall, Phys. Fluids, 30:939-943.
Tylecote, R.F. (1968), The Solid Phase Welding of Metals, Arnold, London.Vaidyanath, L.R., Nicholas, M.G., and Milner, D.R. (1959), Pressure welding by rolling, Brit. Weld. J.,
6:13-28.Wan, K.T., Lathabai, S., and Lawn, B.R. (1990), Crack velocity functions and the thresholds in brittle
solids, J. Eur. Ceram. Soc., 6:259-268.Wiederhorn, S.M. and Bolz, L.H. (1970), Stress corrosion and static fatigue of glass, J. Am. Ceram. Soc.,
53:543-548.Williams, J.G. (1984), Fracture Mechanics of Polymers, Ellis Horwood, New York.Williams, J.G. (1993), A review of the determination of energy release rates for strips in tension and
bending. Part I — Static solutions, J. Strain Analysis, 28:237-246.Williams, J.G., Radon, J.C., and Turner, C.E. (1968), Designing against fracture in brittle solids, Polym.
Eng. Sci., 4:130-141.Williams, M.L., Landel, R.F., and Ferry, J.D. (1955), The temperature dependence of relaxation mecha-
nisms in amorphous polymers and other glass-forming liquids, J. Am. Chem. Soc., 77:3701-3707.Zaghzi, N., Carré, A., Shanahan, M.E.R., Papirer, E., and Schultz, J. (1987), A study of spontaneous
rubber/metal adhesion. I. The rolling cylinder test, J. Polym. Sci. Polym. Phys., 25:2393-2402.