A large high carbon steel plate with a thumbnail crack for which [latex]K_{max} = 1.2σ \sqrt{πa}[/latex] has a fracture toughness of [latex]72 MN/m^{3/2}[/latex] and [latex]σ_{γ} = 1450 MN/m²[/latex]. If [latex]σ = \frac {2}{3} σ_{γ}[/latex], determine the critical initial crack size, assuming linear elastic material.

At fracture, with [latex]\sigma = \frac{2}{3} \sigma _{\gamma } = \frac{2 x 1450}{3} \frac{MN}{m^{2}}[/latex] and [latex]K_{IC} = 72 \frac{MN}{m \frac{3}{2} }[/latex] Then, [latex]72 \frac{MN}{m \frac{3}{2} } = 1.2x \left(\frac{(2 x 1450)}{3} \sqrt{\pi a_{crit}} \right)[/latex] Therefore, [latex]a_{crit} = \frac{1}{\pi } \left(\frac{72 x 3}{1.2 x 2 x 1450} \right)^{2} m[/latex] [latex]= 1.226 \times 10^{- 3} m[/latex] = 1.226 mm If the material was mild steel, with [latex]\sigma _{\gamma } = 210 \frac{MN}{m^{2}}[/latex] and [latex]K_{IC} = 200 \frac{MN}{m^{\frac{3}{2} }}[/latex] Then [latex]a_{crit } = 0.451 m = 451 mm[/latex] i.e. it is much more likely to be detected during inspection!

Question 3.15: A large high carbon steel plate with a thumbnail crack for w...

An Introduction to Mechanical Engineering

A large high carbon steel plate with a thumbnail crack for which $K_{max} = 1.2σ \sqrt{πa}$ has a fracture toughness of $72 MN/m^{3/2}$ and $σ_{γ} = 1450 MN/m²$ .

If $σ = \frac {2}{3} σ_{γ}$ , determine the critical initial crack size, assuming linear elastic material.

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At fracture, with

\sigma = \frac{2}{3} \sigma _{\gamma } = \frac{2 x 1450}{3} \frac{MN}{m^{2}}

and

K_{IC} = 72 \frac{MN}{m \frac{3}{2} }

Then,

72 \frac{MN}{m \frac{3}{2} } = 1.2x \left(\frac{(2 x 1450)}{3} \sqrt{\pi a_{crit}} \right)

Therefore,

a_{crit} = \frac{1}{\pi } \left(\frac{72 x 3}{1.2 x 2 x 1450} \right)^{2} m

= 1.226 \times 10^{- 3}  m

= 1.226 mm

If the material was mild steel, with

$\sigma _{\gamma } = 210 \frac{MN}{m^{2}}$ and $K_{IC} = 200 \frac{MN}{m^{\frac{3}{2} }}$

Then

a_{crit }  = 0.451  m = 451  mm

i.e. it is much more likely to be detected during inspection!

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Question 3.15: A large high carbon steel plate with a thumbnail crack for w...

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