V
149
CONTACT OF ELASTIC SOLIDS
1
ponents of pressure; lastly let K₁0, and K,0,¹ be the respective
coefficients of elasticity of the bodies. Generally, where the
quantities refer to either body, we shall omit the indices. We
then have the following conditions for equilibrium :-
1. Inside each body we must have
=²n+(1+20)
до
до
0 = ▼²+(1+20) ~~
'dx'
0 = ▼³n + (1 +20) бy'
до
de an as
0 = ▼²5+ (1 + 20) ~
'dz'
σ= + + ;
эх ду Əz
and in 1 we have to put ₁ for 0, in 2 0, for 0.
2. At the boundaries the following conditions must hold :—
(a) At infinity §, n, vanish, for our systems of co-
ordinates are rigidly connected with the bodies there.
(b) For 20, i.e. at the surface of the bodies, the tan-
gential stresses which are perpendicular to the z-axis must
vanish, or
ô×
= =
Y, - -K(+35)-0, X.--K(
=
=
(+)
= 0.
(c) For z = 0, outside a certain portion of this plane, viz.
outside the surface of pressure, the normal stress also must
vanish, or
Inside that part
=
θα = =
Z. - 2K (25+0) - 0.
Z₁₁ = Z₂2
We do not know the distribution of pressure over that part,
but instead we have a condition for the displacement over it.
(d) For if a denote the relative displacement of the two
systems of coordinates to which we refer the displacements,
the distance between corresponding points of the two surfaces
after deformation is Ax²+ By²+5₁₂-a, and since this
distance vanishes inside the surface of pressure we have
5₁₂ = a - Ax² - By² = a - %, +%2.
51-52
(e) To the conditions enumerated we must add the con-
1 [Kirchhoff's notation, Mechanik, p. 121.-TR.]