## Q. 7.11

Design of a Fatigue Resistant Plate
A high-strength steel plate (Figure 7-20), which has a plane strain fracture toughness of 80 MPa$\sqrt{m}$ is alternately loaded in tension to 500 MPa and in compression to 60 MPa. The plate is to survive for ten years with the stress being applied at a frequency of once every five minutes. Design a manufacturing and testing procedure that ensures that the component will serve as intended. Assume a geometry factor f = 1.0 for all flaws.

## Verified Solution

To design our manufacturing and testing capability, we must determine the maximum size of any flaws that might lead to failure within the ten-year period. The critical crack size using the fracture toughness and the maximum stress is

$K_{Ic}= f \sigma\sqrt{\pi a_{c}}$
80 MPa$\sqrt{m}$ = (1.0)(500 MPa)$\sqrt{\pi a_{c}}$
$a_{c}$ = 0.0081 m = 8.1 mm

The maximum stress is 500 MPa; however, the minimum stress is zero, not 60 MPa in compression, because cracks do not propagate in compression. Thus, $\Delta \sigma$ is

$\Delta \sigma =\sigma _{max} -\sigma _{min} = 500-0=500$ MPa

We need to determine the minimum number of cycles that the plate must withstand:
N = [1 cycle/(5 min)](60 min/h)(24 h/day)(365 days/yr)(10 yr)
N = 1,051,200 cycles
If we assume that f = 1.0 for all crack lengths and note that C = 1.62 × $10^{-12}$ and n = 3.2 from Figure 7-20 in Equation 7-20, then

$N= \frac{2[(a_{c})^{(2 \ – \ n)/2} \ – \ (a_{i})^{(2 \ – \ n)/2}]}{(2 \ – \ n)C f^n \Delta \sigma ^n \pi ^{n/2}}$      (7-20)

1,051,200 = $\frac{2[(0.0081)^{(2 \ – \ 3.2)/2} \ – \ (a_{i})^{(2 \ – \ 3.2)/2}]}{(2 \ – \ 3.2)(1.62 \ × \ 10^{-12}) (1)^{3.2} (500)^{3.2} \pi ^{3.2/2}}$

1,051,200 = $\frac{2[18 \ – \ a_{i})^{-0.6}]}{(-1.2)(1.62 \ × \ 10^{-12})(1) (4.332 \ × \ 10^{8})(6.244)}$

$a_{i}$ = 1.82 × $10^{-6}$ m = 0.00182 mm for surface flaws
2$a_{i}$ = 0.00364 mm for internal flaws

The manufacturing process must produce surface flaws smaller than 0.00182 mm in length. In addition, nondestructive tests must be available to ensure that cracks approaching this length are not present.