✔ME 5463-001 Fracture Mechanics, Homework #3

— David Wagner 2009/09/07 15:28

Determine the Airy function Φ(r,θ) for the case of the plate with a central hole subjected to a simple tensile load.

Given

$f_2(r) = C_5 r^2 +C_6 r^4 + {1/{r^2}}C_7 +C_8$

Determine the constants.

$sigma_r = {1/r^2}{{partial^2 Phi}/{partial theta^2}}+{1/r}{{partial Phi}/{partial r}}$ = $C_1(1+2 ln r) + 2 C_2 + {1/r^2} C_3 - (2 C_5 + {6/r^4} C_7 + 4/r^2 C_8) cos 2 theta$
$sigma_theta = {partial^2 Phi}/{partial r^2}$ = $C_1(3+2 ln r) + 2 C_2 + {1/r^2} C_3 + (2 C_5 + 12 C_6 r^2 + 6/r^4 C_7) cos 2 theta$
$tau_{r theta} = {1/r^2}{{partial Phi}/{partial theta}}+{1/r}{{partial^2 Phi}/{partial r partial theta}}$

Boundary Conditions

The stresses are finite as r→∞ ⇒ .
C₄ is arbitrary.
At r=a: σ_r=0 and τ_rθ=0.
As r→∞, σ_x = σ.
- $sigma_r = {1/2}(1+cos 2 theta)$
- $sigma_theta = {1/2}(1-cos 2 theta)$
- $tau_{r theta} = -{1/2}sigma sin 2 theta$

Solutions

$sigma_r = {sigma/2}(1-{a^2}/{r^2}) + sigma/2 (1 + {3 a^4}/r^4 - 4a^2/r^2)cos 2 theta$
$sigma_theta = {sigma/2}(1+{a^2}/{r^2}) - sigma/2 (1 + {3 a^4}/r^4 )cos 2 theta$
$tau_{r theta} = -sigma/2 (1 - {3 a^4}/r^4 + 2a^2/r^2)sin 2 theta$

Find the Airy Function

Calculate the Constants

sigma_r = $2 C_2 + {1/r^2} C_3 - (2 C_5 + {6/r^4} C_7 + 4/r^2 C_8) cos 2 theta$ = ${sigma/2}(1-{a^2}/{r^2}) + sigma/2 (1 + a^4/r^4 - 4a^2/r^2)cos 2 theta$ ⇒