Elastic contact to a coated half-space - Effective elastic modulus and   real penetration
  A new approach to the contact to coated elastic materials is presented. A relatively simple numerical algorithm based on an exact integral formulation of the elastic contact of an axisymmetric indenter to a coated substrate is detailed. It provides contact force and penetration as a function of the contact radius. Computations were carried out for substrate to layer moduli ratios ranging from 0.01 to 100 and various indenter shapes. Computed equivalent moduli showed good agreement with the Gao model for mismatch ratios ranging from 0.5 to 2. Beyond this range, substantial effects of inhomogeneous strain istribution are evidenced. An empirical function is proposed to fit the equivalent modulus. More importantly, if the indenter is not flat-ended, the simple relation between contact radius and penetration valid for homogeneous substrates breaks down. If neglected, this phenomenon leads to significant errors in the evaluation of the contact radius in depth-sensing indentation on coated substrates with large elastic modulus mismatch. 
Etienne Barthel, Antoine Perriot