CONTACT OF ELASTIC SOLIDS
147
acts between the parts in contact. The portion of the surface
which during deformation is common to the two bodies we
shall call the surface of pressure, its boundary the curve of
pressure. The questions which from the nature of the case
first demand an answer are these: What surface is it, of
which the surface of pressure forms an infinitesimal part?¹
What is the form and what is the absolute magnitude of the
curve of pressure? How is the normal pressure distributed
over the surface of pressure? It is of importance to determine
the maximum pressure occurring in the bodies when they are
pressed together, since this determines whether the bodies will
be without permanent deformation; lastly, it is of interest to
know how much the bodies approach each other under the
influence of a given total pressure.
We are given the two elastic constants of each of the
bodies which touch, the form and relative position of their
surfaces near the point of contact, and the total pressure. We
shall choose our units so that the surface of pressure may be
finite. Our reasoning will then extend to all finite space;
the full dimensions of the bodies in contact we must imagine
as infinite.
In the first place we shall suppose that the two surfaces
are brought into mathematical contact, so that the common
normal is parallel to the direction of the pressure which one
body is to exert on the other. The common tangent plane
is taken as the plane xy, the normal as axis of z, in a rect-
angular rectilinear system of coordinates. The distance of
any point of either surface from the common tangent plane
will in the neighbourhood of the point of contact, i.e. through-
out all finite space, be represented by a homogeneous quad-
ratic function of x and y. Therefore the distance between
two corresponding points of the two surfaces will also be
represented by such a function. We shall turn the axes of x
and y so that in the last-named function the term involving
xy is absent.
1 In general the radii of curvature of the surface of a body in a state of strain
are only infinitesimally altered; but in our particular case they are altered by
finite amounts, and in this lies the justification of the present question. For
instance, when two equal spheres of the same material touch each other, the
surface of pressure forms part of a plane, i.c. of a surface which is different in
character from both of the surfaces in contact.