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 distinguish experimentally between different models based on indentation experiments alone.

我们考虑了弹性模量随深度减小对刚性压头对渐变半空间的压痕的载荷-位移关系的影响。将结合弹性基板上的板的特征和赫兹接触理论的封闭形式近似与在均匀基板上均匀刚性层的情况下的有限元结果进行比较。在分级具有逆幂律形式的情况下，给出了一些一般结果，并以数值方式研究了截断到有限表面值的影响。最后，考虑了更实用的误差函数分级。在所有情况下，载荷-位移关系比均质情况更接近线性。我们得出的结论是，实验数据可用于以预定的分级形式确定参数，