Modified Hertz theory for spherical indentation

The elastic field in a half-space indented by a rigid sphere is accurately described by the Hertz equations if the contact radius is small compared with that of the sphere. For wider contact the equations are not expected to be reliable, but are nevertheless often used for convenience. This paper describes some of the errors which may develop as contact widens, and presents a second elastic field which provides a first-order correction. From this it is possible to estimate the probable error introduced in a particular variable by applying the Hertz equations beyond the recommended range. Useful relations between some familiar elastic indentation fields are also reported.