Asymptotic analysis of the substrate effect for an arbitrary indenter
  A quasistatic unilateral frictionless contact problem for a rigid axisymmetric indenter pressed into a homogeneous, linearly elastic and transversely isotropic elastic layer bonded to a homogeneous, linearly elastic and transversely isotropic half-space is considered. Using the general solution to the governing integral equation of the axisymmetric contact problem for an isotropic elastic half-space, we derive exact equations for the contact force and the contact radius, which are then approximated under the assumption that the contact radius is sufficiently small compared to the thickness of the elastic layer. An asymptotic analysis of the resulting non-linear algebraic problem corresponding to the fourth-order asymptotic model is performed. A special case of the indentation problem for a blunt punch of power-law profile is studied in detail. Approximate force-displacement relations are obtained in explicit form, which is most suited for development of indentation tests. 
I. I. Argatov and F. J. Sabina
math.AP