Characteristics of local compliance of an elastic body under a small punch indented into the plane part of its boundary

An asymptotic solution of the contact problem of an elastic body indented (without friction) by a circular punch with a flat base is obtained under the assumption of a small relative size of the contact zone. The resulting formulas involve integral characteristics of the elastic body, which depend on its shape, dimensions, fixing conditions, Poisson's ratio, and location of the punch center. These quantities have the mechanical meaning of the coefficients of local compliance of the elastic body. Relations that, generally, reduce the number of independent coefficients in the asymptotic expansion are obtained on the basis of the reciprocal theorem. Some coefficients of local compliance at the center of an elastic hemisphere are calculated numerically. The asymptotic model of an elastic body loaded by a point force is discussed.