Tools › Structures › Stress Concentrations › Elliptic Inclusion › Example
Hardiman's elliptic inclusion tool in the empty-hole limit: Inglis K_t = 1 + 2a/b = 5
| Tool | Elliptic Inclusion |
| Source | Hardiman, N. J. (1954). "Elliptic Elastic Inclusion in an Infinite Elastic Plate." Quarterly Journal of Mechanics and Applied Mathematics, Vol. 7, No. 2, pp. 226–230; Inglis, C. E. (1913). "Stresses in a Plate Due to the Presence of Cracks and Sharp Corners." Transactions of the Institution of Naval Architects, Vol. 55, pp. 219–241. |
| Level | Graduate |
Problem
A 0.1 in plate (E_1 = 10 Msi, ν_1 = 0.33) carries 15,000 lb/in in y and contains an elliptic inclusion with semi-axes 0.25 in along x and 0.125 in along y whose material is removed (empty hole). Find the hoop-stress concentration factor on the hole edge and compare it with Inglis.
Inputs to Enter
- N_xx = 0, N_yy = 15000 lb/in, N_xy = 0, t = 0.1 in
- a = 0.25 in, b = 0.125 in, β = 0; plane stress; E_1 = 10 Msi, ν_1 = 0.33; inclusion type: hole
- r = 0.25 in, θ = 90°
Answers
- K_hoop = 4.9998 against Inglis 1 + 2a/b = 5 (collocation series, adaptive order)
- σ_edge,max = K_hoop S_1 = 750 ksi at the ends of a
- Convergence recorded at the build: error 4 ksi at order 4, 0.06 at 8, 1.2e-3 at 12, 5e-6 at 16
Notes
Change the loads to N_yy alone and the inclusion type to hole. With a = b the tool equals the Circular Inclusion tool to 14 digits; the default elastic case (a/b = 2, E_2/E_1 = 3) gives K_I = 1.163, K_rad = 1.163, K_hoop = 0.461; rigid at β = 30° in plane strain gives K_rad = 1.356.