Question

A rectangular bar has a edge crack at the bottom and is subjected to a pure bending moment. The crack length is a = 1 mm. The height of the bar is b = 12.5 cm. Knowing that the failure strength of the material is Sigma = 1,400 MPa, what is the fracture toughness of the material, K_ic.

Answer 1

Answer:

The answer is "$\bold{87.3906 \ MPa \sqrt{m}}$".

Explanation:

Given value:

$\sigma = 1400 \ MPa \ \ \ \ \ \ \ \ where \ \sigma = failure \ strength\\\\a = 1 \ mm = 1 \times 10^{-3} \ m \ \ \ \ \ \ \ \ \ \ where\ a = crack\ length\\\\b= 12.5 \ cm = 125 \ mm = 0.125 \ m$

$\to \alpha = \frac{a}{b} =\frac{1}{125} = 8 \times 10^{-3}$

$k_{b} = \frac{1.12 + \alpha (2.62 \alpha -1.59)}{1-0.7 \alpha}$

    $= \frac{1.12 + (8\times 10^{-3}(2.62(8\times 10^{-3}) -1.59))}{1-(0.7 \times 8\times 10^{-3})}\\\\= \frac{1.12 + (8\times 10^{-3}(0.02096 -1.59))}{1-(0.7 \times 8\times 10^{-3})}\\\\= \frac{1.12 + (8\times 10^{-3}(-1.56904))}{1-(0.0056)}\\\\= \frac{1.12 + (-0.01255232)}{0.9944}\\\\= \frac{-1.10744768}{0.9944}\\\\= -1.11368431$

$k_{ic} = \sigma \sqrt{\pi a} \ y_b$

     $=1400 \times \sqrt{\pi \times 1 \times 10^{-3} } \times -1.11368431\\\\=1400 \times 0.00177200451 \times -1.11368431\\\\=87.3906 \ MPa \sqrt{m}$