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✔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

  • Phi(r,theta) = f_1(r) + f_2(r) cos 2 theta
  • f_1(r) = C_1 r^2 ln r + C_2 r^2 +C_3 ln r +C_4
  • 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

  1. The stresses are finite as r→∞ ⇒ c_1 = c_6 = 0.
  2. C4 is arbitrary.
  3. At r=a: σr=0 and τrθ=0.
  4. As r→∞, σx = 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⇒

2 C_2 + {1/r^2} C_3= {sigma/2}- {sigma a^2}/{2 r^2}⇒

  • 2 C_2 = sigma/2 ⇒ C_2 = sigma/4
  • C_3 = -{sigma a^2}/2

2 C_5 + {6/r^4} C_7 + 4/r^2 C_8 = sigma/2 + {3 sigma a^4}/{2r^4} - {4 sigma a^2}/{2r^2}⇒

  • -2 C_5 = sigma/2 ⇒ C_5 = -{sigma/4}
  • -6 C_7 = {3 sigma a^4}/2 ⇒ C_7 = -{{sigma a^4}/4}
  • -4 C_8 = -2 sigma a^2 ⇒ C_8 = {sigma a^2}/2

Phi = f_1(r) +f_2(r)cos 2theta = C_1 r^2 ln r + C_2 r^2 +C_3 ln r +C_4 +(C_5 r^2 +C_6 r^4 + {1/{r^2}}C_7 +C_8)cos 2theta=

Phi = sigma/4 r^2 -{sigma a^2}/2 ln r +C_4 + ( - {sigma/4} r^2- { {sigma a^4}/4} {1/{r^2}} +{sigma a^2}/2) cos 2 theta

Check It

sigma_theta = {partial^2 Phi}/{partial r^2} =-{(r^4+3 a^4) sigma cos 2 theta+(-r^4-a^2 r^2) sigma}/{2 r^4} OK.


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