Indentation modulus of elastically anisotropic half spaces

The unloading process in an indentation experiment is often modelled as a contact problem of a rigid punch on an elastically isotropic half space. This allows one to derive simple formulae to determine the indentation modulus from experimental data. We have studied the contact problem of a flat circular punch and a paraboloid on an elastically anisotropic half space and have shown that the formulae used for isotropic materials can be used for anisotropic materials as long as the half space has three or fourfold rotational symmetry. In the case of lower symmetry, the indentation modulus depends on the shape of the indenter. We have calculated the indentation modulus of {100}, {111} and {110} surfaces of cubic crystals for a wide range of elastic constants. The {110} indentation modulus was calculated for the case of a flat circular punch. The single-crystal indentation moduli differ substantially from the isotropic polycrystalline indentation moduli and the differences increase with increasing anisotropy factor and decreasing (100) Poisson's ratio. The indentation modulus of a polycrystalline cubic material with a (111) fibre texture in the direction of the indentation has also been calculated and is much larger than the {111} modulus of the corresponding single crystal.