== ✔ME 5463-001 Fracture Mechanics, Homework #3 ==
 --- //[[honeypot@handbasket.org|David Wagner]] 2009/09/07 15:28//

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

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

Determine the constants.

  * <m>sigma_r = {1/r^2}{{partial^2 Phi}/{partial theta^2}}+{1/r}{{partial Phi}/{partial r}}</m>=<m>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</m>
  * <m>sigma_theta = {partial^2 Phi}/{partial r^2}</m>=<m>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</m>
  * <m>tau_{r theta} = {1/r^2}{{partial Phi}/{partial theta}}+{1/r}{{partial^2 Phi}/{partial r partial theta}}</m>

Boundary Conditions

  - The stresses are finite as r→∞ ⇒ <m>c_1 = c_6 = 0</m>.
  - C<sub>4</sub> is arbitrary.
  - At r=a: σ<sub>r</sub>=0 and τ<sub>rθ</sub>=0.
  - As r→∞, σ<sub>x</sub> = σ.
     * <m>sigma_r = {1/2}(1+cos 2 theta)</m>
     * <m>sigma_theta = {1/2}(1-cos 2 theta)</m>
     * <m>tau_{r theta} = -{1/2}sigma sin 2 theta</m>

Solutions

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

====== Find the Airy Function ======

Calculate the Constants

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

<m>2 C_2 + {1/r^2} C_3</m>= <m>{sigma/2}-  {sigma a^2}/{2 r^2}</m>=>
  * <m>2 C_2 = sigma/2</m> => <m>C_2 = sigma/4</m>
  * <m>C_3 = -{sigma a^2}/2</m>

<m>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}</m>=>
  * <m>-2 C_5 = sigma/2</m> => <m>C_5 = -{sigma/4}</m>
  * <m>-6 C_7 = {3 sigma a^4}/2</m> => <m>C_7 = -{{sigma a^4}/4}</m>
  * <m>-4 C_8 = -2 sigma a^2</m> => <m>C_8 = {sigma a^2}/2</m>

<m>Phi = f_1(r) +f_2(r)cos 2theta</m> = <m>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</m>=

|<m>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</m>|

====== Check It ======

<m>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}</m> OK.