V
ON THE CONTACT OF ELASTIC SOLIDS
(Journal für die reine und angewandte Mathematik, 92, pp. 156-171, 1881.)
IN the theory of elasticity the causes of the deformations are
assumed to be partly forces acting throughout the volume of
the body, partly pressures applied to its surface. For both
classes of forces it may happen that they become infinitely
great in one or more infinitely small portions of the body, but
so that the integrals of the forces taken throughout these
elements remain finite. If about the singular point we describe
a closed surface of small dimensions compared to the whole
body, but very large in comparison with the element in which
the forces act, the deformations outside and inside this surface
may be treated independently of each other. Outside, the
deformations depend upon the shape of the whole body, the
finite integrals of the force-components at the singular point,
and the distribution of the remaining forces; inside, they
depend only upon the distribution of the forces acting inside
the element. The pressures and deformations inside the sur-
face are infinitely great in comparison with those outside.
In what follows we shall treat of a case which is one of
the class referred to above, and which is of practical interest,¹
namely, the case of two elastic isotropic bodies which touch
each other over a very small part of their surface and exert
upon each other a finite pressure, distributed over the common
area of contact. The surfaces in contact are imagined as
perfectly smooth, i.e. we assume that only a normal pressure
1 Cf. Winkler, Die Lehre von der Elasticität und Festigkeit, vol. i. p. 43 (Prag.
1867); and Grashof, Theorie der Elasticität und Festigkeit, pp. 49-54 (Berlin,
1878).