Indentation of an elastic half space with material properties varying with depth

We consider the effect of an elastic modulus that decreases with depth on the load-displacement relation for indentation of a graded half space by a rigid indenter. A closed-form approximation incorporating features of the plate on an elastic substrate and the Hertzian contact theory is compared with finite element results for the case of a uniform stiff layer on a homogeneous substrate. Some general results are presented for the case where the grading has inverse power-law form and the effects of truncation to a finite surface value are investigated numerically. Finally, a more practical error-function grading is considered. In all cases, the load-displacement relation is closer to linear than in the homogeneous case. We conclude that the experimental data can be used to determine parameters in a predetermined form of grading, but that comparative insensitivity to the exact form of the grading would make it difficult to distincguish experimentally between different models based on indentation experiments alone.