The stress concentration factor was obtained using a finite element model built in COMSOL (file is
attached). The model was solved for values of the major axis, a, of 0.01, 0.02, 0.05, and 0.1. The stress
concentration factor was calculated using Equation 1. The stress concentration factors for each a/b
ratio are shown in Table 1.1.
σ
𝑆 = σ max
(1)
nom
The stress concentration factor was then obtained for the same a/b ratios using a Formula given in
Roark’s Formulas for Stress and Strain (Maple file is attached). It is evident that the percent error in the
finite element solution increases as the ellipse aspect ratio increases.
a/b
1
2
5
10
S_exact
2.864
4.514
8.536
13.178
S_numeric
2.67
4.48
10.2
22.4
% Err
6.773743
0.753212
19.49391
69.98027
Using the data from the first sheet, 𝐶 = 𝜎𝐹 ∙ √𝑎 = 480𝑀𝑃𝑎 ∙ √0.02𝑚 = 67.88
Therefore, 𝜎𝐹 =
67.88
√0.05𝑚
= 303.58𝑀𝑃𝑎
The Paris equation can be integrated via separation of variables, which results in:
14.15 = 𝐶(∆𝜎)3 (𝜋)3/2 𝑁𝑓
If the initial crack size is reduced to 0.001m, the Paris equation yields:
40.88 = 𝐶(∆𝜎)3 (𝜋)3/2 𝑁𝑓 ′
Since the unknown variables are all constants, other than 𝑁𝑓 , the increase in the number of cycles to
failure is
40.88
= 2.89
14.15