(1) uniform extension of all filaments at right angles to the axis of
amount
1 - [sigma]
----------- (1/8)[omega]^2[rho]a^2(3 + [sigma]),
E
(2) radial contraction of amount
1 - [sigma]^2
------------- (3/8)[omega]^2[rho]r^2,
E
(3) contraction along the circular filaments of amount
1 - [sigma]^2
------------- (1/8)[omega]^2[rho]r^2,
E
(4) extension of all filaments at right angles to the axis of amount
(1/E)(1/6)[omega]^2[rho][l^2 - (3_x)^2][sigma](1+[sigma]),
(5) contraction of the filaments normal to the plane of the disk of
amount
2[sigma]
-------- (1/8)[omega]^2[rho]a^2(3 + [sigma])
E
[sigma]
- ------- 1/2 [omega]^2[rho]r^2(1 + [sigma])
E
2[sigma] (1 + [sigma])
+ -------- (1/6)[omega]^2[rho](l^2 - 3z^2)[sigma] -------------.
E (1 - [sigma])
The greatest extension is the circumferential extension near the
centre, and its amount is
(3 + [sigma])(1 - [sigma]) [sigma](1 + [sigma])
-------------------------- [omega]^2[rho]a^2 + -------------------- [omega]^2[rho]l^2.
8E 6E
[Illustration: FIG. 32.]
The longitudinal contraction is required to make the plane faces of
the disk free from pressure, and the terms in l and z enable us to
avoid tangential traction on any cylindrical surface. The system of
stresses and strains thus expressed satisfies all the conditions,
except that there is a small radial tension on the bounding surface of
amount per unit area (1/6)[omega]^2[rho](l^2 - 3z^2)[sigma](1 +
[sigma])/(1 - [sigma]). The resultant of these tensions on any part of
the edge of the disk vanishes, and the stress in question is very
small in comparison with the other stresses involved when the disk is
thin; we may conclude that, for a thin disk, the expressions given
represent the actual condition at all points which are not very close
to the edge (cf. S 55). The effect to the longitudinal contraction is
that the plane faces become slightly concave (fig. 32).
81. The corresponding solution for a disk with a circular axle-hole
(radius b) will be obtained from that given in the last section by
superposing the following system of additional stresses:
(1) radial tension of amount (1/8)[omega]^2[rho]b^2(1 - a^2/r^2)(3 +
[sigma]),
(2) tension along the circular filaments of amount
(1/8)[omega]^2[rho]b^2(1 + a^2/r^2)(3 + [sigma]).