| 1 | p1 − p0 | r0³r1³ | , | ||
| 4μ | r0³ − r1³ | r³ |
where μ is the modulus of rigidity of the material, = ½E / (1 + σ). The volume included between the two surfaces of the body is increased by the fraction (p1r1³ − p0r0³) / k(r0³ − r1³) of itself, and the volume within the inner surface is increased by the fraction
| 3 (p1 − p0) | r0³ | + | p1r1³ − p0r0³ | |
| 4μ | r0³ − r1³ | k (r0³ − r1³) |
of itself. For a shell subject only to internal pressure p the greatest extension is the extension at right angles to the radius at the inner surface, and its amount is
| pr1³ | ( | 1 | + | 1 | r0³ | ); | |
| r0³ − r1³ | 3k | 4μ | r1³ |
the greatest tension is the transverse tension at the inner surface, and its amount is p (½ r0³ + r1³) / (r0³ − r1³).
77. In the problem of a cylindrical shell under pressure a complication may arise from the effects of the ends; but when the ends are free from stress the solution is very simple. With notation similar to that in § 76 it can be shown that the stress at a distance r from the axis consists of
(1) uniform tension in all directions at right angles to the axis of amount
| p1r1² − p0r0² | , |
| r0² − r1² |
(2) radial pressure of amount