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.

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

Since the early 1990s, adhesive contact mechanics has emerged as an area of considerable interest in nano- and bio-sciences. Here we review the methods which have been developed in the past 75 years to account for adhesive interactions in elastic contact problems. Emphasis is given to the connection between the local, physical mechanism of adhesion and the macroscopic, mechanical loading. The discussion centres on the contact equations. In an attempt to provide a broad view of the field, we outline the key concepts and their progressive developments, starting from the approximate calculation by Derjaguin in 1934 and ending with recent results for coated systems and time dependent materials, through the well established DMT and JKR models.

https://hal.archives-ouvertes.fr/hal-00256886/document

Adhesive elastic contacts: JKR and more

The two-dimensional problem of an arc shaped crack lying along the interface of a circular elastic inclusion embedded in an infinite matrix with different elastic constants is considered. Based on the complex variable method of Muskhelishvili, closed-form solutions for the stresses and the displacements around the crack are obtained when general biaxial loads are applied at infinity. These solutions are then combined with A.A. Griffith's virtual work argument to give a criterion of crack extension, namely the de-bonding of the interface. The critical applied loads are expressed explicitly in terms of a function of the inclusion radius and the central angle subtended by the crack arc. In the case of simple tension the critical load is inversely proportional to the square-root of the inclusion radius. By analyzing the variation of the cleavage stress near the crack tip, the deviation of the crack into the matrix is discussed. The case of uniaxial tension is worked out in detail.

A crack along the interface of a circular inclusion embedded in an infinite solid

https://ntrs.nasa.gov/api/citations/19910022378/downloads/19910022378.pdf

An equivalent domain integral method for three-dimensional mixed-mode fracture problems

We have formulated the restricted self-consistent model for the adhesive contact of linear viscoelastic spheres. This model is a generalization of both the Ting (J. Appl. Mech. 33 (1966) 845) approach to the viscoelastic contact of adhesionless spheres and the restricted self-consistent model for adhesive axisymmetric bodies. We also show how the model can be used in practice by giving a few examples of numerical solutions.

The adhesive contact of viscoelastic spheres

The distribution of stress in the vicinity of an external crack in an infinite elastic solid

The effect of shear on a penny-shaped crack at the interface of an elastic half-space and a rigid foundation

A pair of coplanar cracks at the interface of two bonded dissimilar elastic half planes

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The distribution of stress in the vicinity of an external crack in an infinite elastic solid

The effect of shear on a penny-shaped crack at the interface of an elastic half-space and a rigid foundation

A pair of coplanar cracks at the interface of two bonded dissimilar elastic half planes