--- //[[honeypot@handbasket.org|David Wagner]] 2009/10/02 21:08//
  * {{:me5463:hw_9.xls|Spreadsheet}}
====== √ME 5463-001 Fracture Mechanics, Homework #9 ======
<box 100%>
**3.15**
Using the Newman-Raju empirical equations for a semi-elliptical surface flaw
[Eq.(3.91)], tabulate and plot the distribution of //Y(a/W)// around the flaw border
(//i.e.// as a function of the flaw parameter, φ) for both tension and pure bending
for the following flaw-geometry parameters:

**(a)**
//a/c// = 1.0 and //a/t// = 0.3
</box>

Eq.(3.91)
<m>K = (sigma_t + H sigma_b)sqrt{{pi a} /Q} F({a/t},{a/c},{c/W},phi)</m>

<m>K = sigma sqrt{pi a} Y(a/W)</m>


<m>sigma_b = {6M}/{Wt^3}</m>

//a/c//=1 => <m>Q = Phi^2 = 1+ 1.464(a/c)^{1.65}</m> = 2.464

==== Find F ====
//a/c// = 1.0 and //a/t// = 0.3

  * <m>M_1 = 1.13 - 0.09(a/c)</m> = 1.04
  * <m>M_2 = -0.54 +{{0.89}/{0.2+{a/c}}} = {{0.89}/{1.2}} -0.54</m> = 0.20167
  * <m>M_3 = 0.5 - {{1.0}/{0.65 +{a/c}}} + 14(1.0-{a/c})^24</m> = <m>0.5 - {{1.0}/{0.65 +1}} + 14</m> = <m>0.5 - {{1.0}/{1.65}} + 14</m> = 13.8939
  * <m>f_phi = [(a/c)^2 cos^2phi sin^2phi]^{1/4}</m> = <m>root{4}{cos^2phi sin^2phi}</m>
  * <m>g = 1+[0.1+ 0.35(a/t)^2](1-sin phi)^2</m> = <m>1+[0.1+ 0.35(0.3)^2](1-sin phi)^2</m> = <m>1 + 0.1315(1-sin phi)^2</m>
  * <m>f_W = sqrt{ sec({{pi c}/{W}} sqrt{a/t}) }</m> = <m>sqrt{ sec({{0.5477 pi c}/{W}} ) }</m>

<m>F = [M_1 + M_2(a/t)^2 +M_3(a/t)^4] f_phi g f_W</m>
= 
<m>[1.04 + 0.20167(0.3)^2 + 13.8939(0.3)^4]
 root{4}{cos^2phi sin^2phi}
 (1 + 0.1315(1-sin phi)^2)
 sec^{1/2}({{0.5477 pi c}/{W}} )</m>
=
<m>[1.04 + 0.01815 + 0.11254]
 root{4}{cos^2phi sin^2phi}
 (1 + 0.1315(1-sin phi)^2)
 sec^{1/2}({{0.5477 pi c}/{W}} )</m>

= F =
<m>1.1707
 root{4}{cos^2phi sin^2phi}
 (1 + 0.1315(1-sin phi)^2)
 sec^{1/2}({{0.5477 pi a}/{W}} )</m>

==== Find H ====
//a/c// = 1.0 and //a/t// = 0.3

  * <m>P = 0.2+ {a/c}+ 0.6{a/t}</m> = <m>0.2+ {1.0}+ 0.18</m> = 1.38
  * <m>H_1 = 1 - 0.34{a/t} -0.11{a/c}{a/t}</m> = <m>1 - 0.102 -0.033</m> = 0.865
  * <m>G_1 = -1.22- 0.12{a/c}</m> = -1.34
  * <m>G_2 = 0.55 - 1.05(a/c)^{3/4} + 0.47(a/c)^{3/2}</m> = <m>0.55 - 1.05 + 0.47</m> = -0.03 
  * <m>H_2 = 1 + G_1{a/t} + G_2(a/t)^2</m> = <m>H_2 = 1 -0.402 -0.0027</m> = 0.5953

<m>H = H_1 +(H_2 - H_1)sin^P phi</m>
=
<m>0.865 +( 0.5953 - 0.865)sin^{1.38} phi</m> = <m>0.865 -0.2697sin^{1.38} phi</m>

==== Find Yt(a/w) for Pure Tension ====

<m>K = sigma_t sqrt{{pi a} /Q} F({a/t},{a/c},{c/W},phi)</m>
=
<m>sigma_t sqrt{pi a} Y_t(a/W)</m>
=>
<m>Y_t(a/W) = 1/sqrt{Q} F</m>
=
<m>1/sqrt{2.464} 1.1707
 root{4}{cos^2phi sin^2phi}
 (1 + 0.1315(1-sin phi)^2)
 sec^{1/2}({{0.5477 pi a}/{W}} )</m>

=
<m>Y_t(phi, a/W) = 0.7458
 root{4}{cos^2phi sin^2phi}
 (1 + 0.1315(1-sin phi)^2)
 sec^{1/2}({{0.5477 pi a}/{W}} )</m>


==== Find Yb(a/w) for Pure Bending ====

<m>K = H sigma_b sqrt{{pi a} /Q} F({a/t},{a/c},{c/W},phi)</m>
=
<m>sigma_b sqrt{pi a} Y_b(a/W)</m>
=>
<m>Y_b(a/W) = H/sqrt{Q} F</m>
=
<m>{0.865 -0.2697sin^{1.38} phi}/sqrt{2.464} 1.1707
 root{4}{cos^2phi sin^2phi}
 (1 + 0.1315(1-sin phi)^2)
 sec^{1/2}({{0.5477 pi a}/{W}} )</m>

=
<m>Y_b(phi, a/W) = {(0.6451 -0.2011) sin^{1.38} phi}
 root{4}{cos^2phi sin^2phi}
 (1 + 0.1315(1-sin phi)^2)
 sec^{1/2}({{0.5477 pi a}/{W}} )</m>