The JKR-type adhesive contact problems for power-law shaped axisymmetric punches

Abstract

The JKR (Johnson, Kendall, and Roberts) and Boussinesq-Kendall models describe adhesive frictionless contact between two isotropic elastic spheres, and between a flat-ended axisymmetric punch and an elastic half-space respectively. However, the shapes of contacting solids may be more general than spherical or flat ones. In addition, the derivation of the main formulae of these models is based on the assumption that the material points within the contact region can move along the punch surface without any friction. However, it is more natural to assume that a material point that came to contact with the punch sticks to its surface, i.e. to assume that the non-slipping boundary conditions are valid. It is shown that the frictionless JKR model may be generalized to arbitrary convex, blunt axisymmetric body, in particular to the case of the punch shape being described by monomial (power-law) punches of an arbitrary degree d ≥ 1. The JKR and Boussinesq-Kendall models are particular cases of the problems for monomial punches, when the degree of the punch d is equal to two or it goes to infinity respectively. The generalized problems for monomial punches are studied under both frictionless and non-slipping (or no-slip) boundary conditions. It is shown that regardless of the boundary conditions, the solution to the problems is reduced to the same dimensionless relations between the actual force, displacements and contact radius. The explicit expressions are derived for the values of the pull-off force and for the corresponding critical contact radius. Connections of the results obtained to problems of nanoindentation in the case of the indenter shape near the tip has some deviation from its nominal shape and the shape function can be approximated by a monomial function of radius, are discussed.

Contact probing of stretched membranes and adhesive interactions: graphene and other two-dimensional materials

“…Although the approach was applied to a sphere described as a paraboloid of revolution z = r 2 /(2 R ), very far generalizations of the approach are possible. Recently, the approach has been extended to non-slipping boundary conditions [47], transversely isotropic solids [48] and the punches of arbitrary axisymmetric blunt shapes contacting elastic materials with rotational symmetry of their elastic properties [44]. It is attempted here to follow the original JKR approach as closely as possible, hence we denote as a 1 , P 0 and δ 2 the true values of the radius of adhesive contact, the external load and the displacement, respectively.…”

“…For example, as the characteristic scale for a 3D adhesive contact problem, Borodich et al [47] took the radius a 1 of the contact region at P 0 =0. For 2D membranes, the dimensionless variables could be also specified using various characteristic scales, in particular, via the radius of the membrane.…”

Nanoindentation of compliant materials using Berkovich tips and flat tips

“…35 In the same year, Borodich provided another derivation of Galanov's solution; however, it was published much later. 36 Some recent results related to the generalization of JKR theory for axisymmetric adhesive contact problems have been reviewed by Borodich. 37 For the contact problems involving non-axisymmetric shapes, the absence of radial symmetry presents significant difficulty in the numerical simulation, because the number of nonlinear equations increases from N to N 2 , as discussed Sec.…”

Evaluation of adhesive and elastic properties of materials by depth-sensing indentation of spheres

“…These models include the JKR (JohnsonKendall-Roberts), DMT (Derjaguin-Muller-Toporov), and Maugis models [1,16]. The extension of the JKR approach to axisymmetric solids of arbitrary monomial shape f (r) = B d r d [17][18][19] shows that the shape of the probe can considerably affect the value of the force of adhesion. Here f is the shape of the probe, r the radial coordinate, d is the degree of the monom (parabola) and B d is a constant.…”