Using the connection between depth-sensing indentation by spherical indenters and mechanics of adhesive contact, a new method for non-direct determination of adhesive and elastic properties of contacting materials is proposed. At low loads, the forcedisplacement curves reflect not only elastic properties but also adhesive properties of the contact, and therefore one can try to extract from experiments both the elastic characteristics of contacting materials (such as the reduced elastic modulus) and characteristics of molecular adhesion (such as the work of adhesion and the pull-off force) using a non-direct approach. The direct methods of estimations of the adhesive characteristics of materials currently used in experiments are rather complicated due to the instability of the experimental force-displacement diagrams for ultra-low tensile forces. The proposed method is based on the use of the stable experimental data for the elastic stage of the force-displacement curve and the mechanics of adhesive contact for spherical indenters. Since the experimental data always have some measurement errors, mathematical techniques for solving ill-posed problems are employed.
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.
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