Tools › Structures › Stress Concentrations › Spherical Cavity › Example
Spherical cavity under remote uniaxial tension (Southwell and Gough 1926; Goodier 1933): K_t = (27 − 15ν)/(2(7 − 5ν))
| Tool | Spherical Cavity |
| Source | Southwell, R. V., and Gough, H. J. (1926). "On the Concentration of Stress in the Neighbourhood of a Small Spherical Flaw; and on the Propagation of Fatigue Fractures in Statistically Isotropic Materials." Philosophical Magazine, Series 7, Vol. 1, No. 1, pp. 71–97; Goodier, J. N. (1933). "Concentration of Stress Around Spherical and Cylindrical Inclusions and Flaws." Transactions of the ASME, Journal of Applied Mechanics, Vol. 55, APM-55-7, pp. 39–44. |
| Level | Graduate |
Problem
An infinite solid with Poisson's ratio 0.3 carries a remote tensile stress of 50 ksi and contains a spherical cavity of 0.05 in radius. Find the peak stress on the equator, the stress at the poles, the axial stress on the equatorial plane at twice the radius, and compare the factor with the round hole in a plate.
Inputs to Enter
- S = 50 ksi (remote tension)
- a = 0.05 in (cavity radius)
- r = 0.05 in (cavity surface, equatorial plane), ν = 0.3
Answers
- Equator: K_t = (27 − 15ν)/(2(7 − 5ν)) = 2.0455, σ = 102.3 ksi
- Poles: −(3 + 15ν)/(2(7 − 5ν)) = −0.6818, σ = −34.09 ksi
- ν = 0 gives 1.9286; the plate hole gives 3
- At r = 2a on the equatorial plane σ_z/S = 1 + (4−5ν)a³/(2(7−5ν)r³) + 9a⁵/(2(7−5ν)r⁵) = 1.0540
Notes
The tool's default inputs are this case; r = a reproduces K_t as a built-in check. Figure 2 plots both factors against ν.