| 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 ω²ρ (l² − 3z²) σ (1 + σ) / (1 − σ). 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. § 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 ω²ρb² (1 − a²/r²) (3 + σ),
(2) tension along the circular filaments of amount
1⁄8 ω²ρb² (1 + a²/r²) (3 + σ).
The corresponding additional strains are
(1) radial contraction of amount
| 3 + σ | { (1 + σ) | a² | − (1 − σ) } ω²ρb², |
| 8E | r² |
(2) extension along the circular filaments of amount