THREE-DIMENSIONAL STRAIN/STRESS FIELDS IN BERKOVICH/VICKERS INDENTATION: COMPARISONS WITH THIN FILM DELAMINATION AND SHEAR BAND FORMATION

Abstract

Starting from the simple assumption that the displacement field underneath a pyramidal (Berkovich/Vickers) tip for very shallow indents is described by the Boussinesq solution, while for deeper ones is dictated by the tip's specific geometry, a combination of the two leads to a combined displacement field for the material at contact with the tip. Continuum mechanics is then utilized for the calculation of the strain tensor, while gradient elasticity theory is adopted for the determination of the stress tensor underneath the tip. Gradient elasticity was assumed in order for the gradient term to be able to model, in this case, the inhomogeneously applied load due to the specific tip geometry, which is different than a flat punch. The thus calculated stress component along the loading (vertical) direction is a parametric function of the material's elastic constants (Poisson's ratio, Lame constants), as well as of the maximum elastic deformation and the gradient coefficient. By appropriately modifying these parameters, the proposed formulation seems to be able to predict the mechanical response of the material underneath the indenter for the specific pyramidal tip geometry (Berkovich or Vickers), without any approximations or empirical relations. The validity of the proposed formulation's predictions was checked against thin film delamination and shear band formation AFM/SEM micrographs, which showed a very good qualitative as well as quantitative comparison with the theoretical predictions.