  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.
