Daniel Maugis

Adhesion of spheres: The JKR-DMT transition using a dugdale model

Abstract In the Johnson-Kendall-Roberts (JKR) approximation, adhesion forces outside the area of contact are neglected and elastic stresses at the edge of the contact are infinite, as in linear elastic fracture mechanics. On the other hand, in the Derjaguin-Muller-Toporov (DMT) approximation, the adhesion forces are taken into account, but the profile is assumed to be Hertzian, as if adhesion forces Could not deform the surfaces. To avoid self consistent numerical calculations based on a specific interaction model (Lennard-Jones potential for example) we have used a Dugdale model, which allows analytical solutions. The adhesion forces are assumed to have a constant value σO, the theoretical stress, over a length d at the crack tip. This internal loading acting in the air gap (the external crack) leads to a stress intensity factor Km, which is cancelled with the stress intensity factor KI due to the external loading. This cancellation suppresses the stress singularities, ensures the continuity of stresses, and fixes the radius c and the crack opening displacement δt. The energy release rate G is computed by the J-integral and the equilibrium is given by G = w. The equilibrium curves a(P), a(δ), and P(σ), the adherence forces at fixed load or fixed grips, the profiles, and the stress distributions can therefore be drawn as a function of a single parameter λ. When λ increases from zero to infinity there is a continuous transition from the DMT approximation to the JKR approximation. Furthermore the value of G for the DMT approximation is derived. It is shown that it is not physically consistent to have tensile stresses in the area of contact and no adhesion forces outside or no tensile stresses in the area of contact and adhesion forces outside. In the JKR approximation the distribution of adhesion forces is reduced to a singular stress at r = a+. The total attraction force outside the contact being zero, the integral of stresses in the contact is equal to the applied load P and negative applied loads are supported by the elastic restoring forces. In the DMT approximation the adhesion stresses tend toward zero to have a continuity with the stress at r = a−, but their integral is finite and the total attraction force outside the contact is 2πwR. In the area of contact the distribution of stresses is Hertzian, and their integral is P + 27πwR. Negative applied loads are sustained by adhesion forces outside the contact.

Fracture mechanics and the adherence of viscoelastic bodies

The strain energy release rate G and its derivative delta G/ delta A are used to provide a general picture of the adherence of viscoelastic bodies. Two bodies in contact on an area A are in equilibrium if G=w, where w is the thermodynamic (or Dupres) work of adhesion. The quasistatic force of adherence is the load corresponding to delta G/ delta A=0. When G>w, the separation of the two bodies starts, and can be seen as the propagation of a crack in mode I. Three geometries are investigated: adherence of spheres, adherence of punches, and peeling. The variation of energies with the area of contact is given, and the kinetics of crack propagation are studied. The theory is supported by experiments on the adherence of polyurethane to glass.

Adhesive contact of sectionally smooth-ended punches on elastic half-spaces: theory and experiment

Attractive molecular forces give rise to singular tensile stresses and to a discontinuity of displacement at the edge of contacts. These boundary conditions are those of fracture mechanics and correspond to adding a rigid body displacement to the classical elastic solution. The adhesive contact of sectionally smooth-ended punches (sphere or cone) is studied. Stress distribution, shape of the deformed surface, and relation between load and penetration are given either for equilibrium or kinetic conditions. For a flat-ended sphere, the adherence of a flat punch and the JKR theory for sphere are obtained as particular cases; for zero surface energy systems, the solution reduces to that of Ejike. The case of a sphere with a spherical cap of different radius and that of a rounded cone are also examined. Experimental verification has been performed with a glass ball (R=2.19 mm and flats between 100 and 240 mu m radius) on polyurethane. Radii of contact, penetration and kinetics of detachment agree with the theory within 2%.

Stick-Slip and Peeling of Adhesive Tapes

The everyday experience shows that in a given range of temperatures and velocities, the peeling of an adhesive tape is jerky with emission of a characteristic noise. This phenomenon of self sustained oscillations (stick-slip) often described for peeling [1–10] is also observed in other fracture mechanics geometries such as tearing [11–13], wedge loaded double cantilever beams (DCB) [14–22], double torsion [23–29] and blister pressurized by incompressible liquid [30]. All these tests are characterized by the fact that the strain energy release rate G does not increase with the crack length, otherwise a single velocity jumps would occur [31]. The main experimental results are the following: 1/ When the imposed cross-head velocity increases, stick-slip appears abruptly with a large amplitude (defined by the difference between Gi for crack initiation and Ga for crack arrest) at a given velocity (depending on temperature), and its amplitude decreases as the cross-head velocity is further increased, until stable propagation is observed anew [1,7,24,26,27]. Generally it is the peak value Gi which decreases, Ga remaining more or less constant or increasing slowly.

Adherence and Fracture Mechanics

Cohesion of solids is insured by a number of well-known forces (van der Waals, ionic, metallic, covalent) and their rupture occurs by propagation of a crack during which the bonds are broken one by one, like a zip fastener. The energy needed to break these bonds is 2γ per surface area (where γ is the surface energy of the material) and is taken from the potential energy of an applied load and/or the elastic energy stored in the body. Surface energy reflects the strength of bonds and rarely exceeds 2 J/m2.

Sub-Critical Crack Growth, Surface Energy and Fracture Toughness of Brittle Materials

A tentative attempt is proposed to interpret subcritical crack growth in brittle materials without any mention to stress corrosion. Based on theoretical and experimental results on adherence of viscoelastic solids on glass, where the true Griffith criterion (G = 2 γ) was observed, with crack healing for G 2γ, it is proposed that subcritical crack growth is a normal mode of propagation in any material. The crack velocity v results from a balance between crack extension force G — 2γ and losses at the crack tip, fonction of the crack speed and the intrinsic surface energy γ:

Rupture and Adherence of Elastic Solids

G - 2\gamma = 2\gamma {\Phi _T}\left( v \right)

Elements of Surface Physics

Crack propagation is stable only on branches of the ϕ(v) curve with positive slope. At a critical crack speed vc, dϕ/dv becomes negative, and the velocity jumps on a second positive branch. This criterion for crack speed unstability which defines the fracture toughness Kc must not be confused with the Griffith criterion.