The extended Hertzian theory and its uses in analyzing indentation experiments

Abstract

Since the presentation of the complete elastic field of an extended Hertzian approach by the author (treating the problem of mechanical contact of two bodies with shape functions of symmetry of revolution of the type z(r)=r 2 /d 0 + r 4 /d 2 + r 6 /d 4 + r 8 /d 6 + …) quite a few publications have appeared showing some useful applications of this new theory. They are mainly dedicated to the problem of parameter identification in the case of nanoindentation experiments. In this paper these new analysing techniques are summarised and further applications concerning indentation testing are proposed.http://mc.manuscriptcentral.com/pm-pml

Completely Analytical Tools for the Next Generation of Surface and Coating Optimization

“…During a small conference in Italy in 1999, the author learned of that concept and decided to work out a theory solving not only the problem for the mechanical contact of an indenter with general shape of symmetry of revolution, but also to extend this solution to layered structures [24]. Later on, he also presented a variety of applications together with Pharr, Chudoba, Richter and others [5,6,8,9,[25][26][27]. Soon, it became clear, however, that the "effective indenter theory", even though powerful, was not something one could easily give to the engineer or an indenter experimentalist and expect her or him to use it as a tool for the analysis of indentation data.…”

“…In [6] and [24], it is shown how these general contact approaches can be applied and how the complete elastic fields have to be evaluated while the extension to the case of layered materials is been elaborated in [22]. Extension to tilting and lateral loads is given in [6] and [24].…”

“…This, of course, requires reevaluation of the Integral in Equation (33) but caused by the distribution structure (properties of the Dirac delta function) in the integrand any such additional time-dependency would be very easy to handle. In addition one could go for extended Hertzian load dots instead of Hertzian ones (e.g., [6]), incorporate tilting, twisting and lateral loads, defects, and intrinsic stresses as done in [12,39,40]. Then the method is also applicable to asymmetric scratch-or tribo-like contact situations.…”

“…Here, t and r are denoting time and space coordinates, respectively. Instead of a phenomenological SLS with the well known bulk modulus of the kind: (5) (N gives the number of particles and c defines a parameter connected with the volume of an individual particle) and a subsequent Young's modulus time dependency like we have a resulting phenomenological Young's modulus structure of the form: (6) With a known principle structure for the temperature distribution function deduced from the shear stress distribution (adiabatic approach) such an approach seems to be manageable by the means of iterations. With a typical dependency for the temperature dependent relaxation time like [37,38]: (7) (with a constant W A and the T = T 0 -viscosity η 0 and the Boltzmann-constant k B ) one can evaluate the mechanical contact situations applying the techniques elaborated in [2] (theory part II) under the assumption of completely adiabatic temperature fields resulting from shear and completely following the second stress invariant distribution.…”

“… the effective indenter concept [3][4][5][6], made time dependent;  the extension of the Oliver and Pharr method to analyze nanoindentation data to layered materials and time dependent mechanical behavior [7][8][9][10];…”