Weibull effective area for Hertzian ring crack initiation

Spherical or Hertzian indentation is used to characterize and guide the development of engineered ceramics under consideration for diverse applications involving contact, wear, rolling fatigue, and impact. Ring crack initiation can be one important damage mechanism of Hertzian indentation. It is caused by surface-located, radial tensile stresses in an annular ring located adjacent to and outside the Hertzian contact circle. While the maximum radial tensile stress is known to be dependent on the elastic properties of the sphere and target, diameter of the sphere, applied compressive force, and coefficient of friction, the Weibull effective area too will be affected by these parameters. However, estimations of a maximum radial tensile stress and Weibull effective area are difficult to obtain because the coefficient of friction during indentation is not known a priori. Circumventing this, the Weibull effective area expressions are derived here for the two extremes that bracket all coefficients of friction; namely (1) the classical, pure-slip frictionless case and (2) the case of an infinite coefficient of friction or pure stick.