148
CONTACT OF ELASTIC SOLIDS
Then we may write the equations of the two surfaces
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≈₁ = A₁x² + Cxy +В₁y², z₂=A¸x²+Cxy + B₂y³,
and we have for the distance between corresponding points
of the two surfaces 21-2, = Ax²+By², where A = A₁ — A¸,
B=B₁ - B₂, and A, B, C are all infinitesimal.¹ From the
meaning of the quantity z₁—z, it follows that A and B have
the like sign, which we shall take positive. This is equivalent
to choosing the positive z-axis to fall inside the body to
which the index 1 refers.
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Further, we imagine in each of the two bodies a rect-
angular rectilinear system of axes, rigidly connected at
infinity with the corresponding body, which system of axes
coincides with the previously chosen system of xyz during the
mathematical contact of the two surfaces. When a pressure
acts on the bodies these systems of coordinates will be shifted
parallel to the axis of z relatively to one another; and their
relative motion will be the same in amount as the distance by
which those parts of the bodies approach each other which
are at an infinite distance from the point of contact.
plane z = 0 in each of these systems is infinitely near to the
part of the surface of the corresponding body which is at a
finite distance, and therefore may itself be considered as the
surface, and the direction of the z-axis as the direction of the
normal to this surface.
The
Let §, n, be the component displacements parallel to the
axes of x, y, z; let Y, denote the component parallel to Oy of
the pressure on a plane element whose normal is parallel to
Ox, exerted by the portion of the body for which x has
smaller values on the portion for which x has larger values,
and let a similar notation be used for the remaining com-
1 Let P1, P12 be the reciprocals of the principal radii of curvature of the sur-
face of the first body, reckoned positive when the corresponding centres of
curvature lie inside this body; similarly let P21, P2 be the principal curvatures
of the surface of the second body; lastly, let w be the angle which the planes
of the curvatures P₁1 and p21 make with each other. Then
2(A+B) = P11+P12+ P21 + P22,
2(AB)=√(P11-P12)²+2(P11 ~ P12)(P21 − P22) cos 2w + (p21 – P22)².
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If we introduce an auxiliary angle by the equation cos 7 = (A − B)/(A + B), then
2A=(P11+P12+ P21 + P22) Cos²; 2B (P11+P12+ P21+ P22) sin²
T
2'