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

The answer is "
`\bold{87.3906 \ MPa √(m)}`".

Step-by-step explanation:

Given value:

`\sigma = 1400 \ MPa \ \ \ \ \ \ \ \ where \ \sigma = failure \ strength\\\\a = 1 \ mm = 1 * 10^(-3) \ m \ \ \ \ \ \ \ \ \ \ where\ a = crack\ length\\\\b= 12.5 \ cm = 125 \ mm = 0.125 \ m\\\\`

`\to \alpha = (a)/(b) =(1)/(125) = 8 * 10^(-3)\\\\`

`k_(b) = (1.12 + \alpha (2.62 \alpha -1.59))/(1-0.7 \alpha)\\`

`= (1.12 + (8* 10^(-3)(2.62(8* 10^(-3)) -1.59)))/(1-(0.7 * 8* 10^(-3)))\\\\= (1.12 + (8* 10^(-3)(0.02096 -1.59)))/(1-(0.7 * 8* 10^(-3)))\\\\= (1.12 + (8* 10^(-3)(-1.56904)))/(1-(0.0056))\\\\= (1.12 + (-0.01255232))/(0.9944)\\\\= (-1.10744768)/(0.9944)\\\\= -1.11368431\\\\`

`k_(ic) = \sigma √(\pi a) \ y_b`

`=1400 * \sqrt{\pi * 1 * 10^(-3) } * -1.11368431\\\\=1400 * 0.00177200451 * -1.11368431\\\\=87.3906 \ MPa √(m)`