## Q. 7.3

Design of a Ceramic Support
Determine the minimum allowable thickness for a 3-in.-wide plate made from sialon that has a fracture toughness of 9000 psi $\sqrt{in.}$. The plate must withstand a tensile load of 40,000 lb. The part will be nondestructively tested to ensure that no flaws are present that might cause failure. The minimum allowable thickness of the part will depend on the minimum flaw size that can be determined by the available testing technique. Assume that three nondestructive testing techniques are available. X-ray radiography can detect flaws larger than 0.02 in.; gamma-ray radiography can detect flaws larger than 0.008 in.; and ultrasonic inspection can detect flaws larger than 0.005 in. Assume that the geometry factor f = 1.0 for all flaws.

## Verified Solution

For the given flaw sizes, we must calculate the minimum thickness of the plate that will ensure that these flaw sizes will not propagate. From Equation 7-5, with f = 1.0,

$\sigma=\frac{K}{f\sqrt{\pi a}}$   (7-5)
$\sigma_{max}=\frac{K_{Ic}}{\sqrt{\pi a}}=\frac{F}{A_{min}}$
$A_{min}=\frac{F \sqrt{\pi a}}{K_{Ic}}=\frac{(40,000 \ lb)(\sqrt{\pi })(\sqrt{ a})}{9000 \ psi \ \sqrt{in.}}$
$A_{min}=7.88 \sqrt{ a} \ in.^2$ and thickness $= [7.88 \ in.^2/(3 \ in.)]\sqrt{ a} =2.63 \ \sqrt{ a}$

Our ability to detect flaws, coupled with our ability to produce a ceramic with flaws smaller than our detection limit, significantly affects the maximum stress than can be tolerated and, hence, the size of the part. In this example, the part can be smaller if ultrasonic inspection is available.
The fracture toughness is also important. Had we used $Si_3N_4$, with a fracture toughness of 3000 psi $\sqrt{in.}$ instead of the sialon, we could repeat the calculations and show that, for ultrasonic testing, the minimum thickness is 0.56 in. and the maximum stress is only 24 ksi.

 Nondestructive Testing Method Smallest Detectable Crack (in.) Minimum Area (in.²) Minimum Thickness (in.) Maximum Stress (ksi) X-ray radiography 0.020 1.11 0.37 36 γ-ray radiography 0.008 0.70 0.23 57 Ultrasonic inspection 0.005 0.56 0.19 72