Question

A radial crack of 3mm in length has been detected in a large hollow disc of 0.36m outside radius and 0.16m of inside radius. Calculate the rotational speed (in rad/s) at which the disc will break. Data: E=200GPa, density 7800Kg/cm³, Kic=55MPa.m^0.5., Poisson's ratio=0.33 Consider that the geometric factor Y is given by [$1.12 - 0.23 \left(\frac{a}{b}\right) + 10.55 \left(\frac{a}{b}\right)^2 - 21.72 \left(\frac{a}{b}\right)^3 + 30.39 \left(\frac{a}{b}\right)^4$] Where \"a\" is the crack length and \"b\" is the difference of the external and internal radius And that the hoop stress is given by: $\sigma_\theta = \frac{3+\nu}{8} \rho \omega^2 \left[ \frac{r_2^2 + r_1^2}{r^2} - \frac{r_1^2 r_2^2}{r^4} - \left(\frac{1+3\nu}{3+\nu}\right) \frac{r_2^2}{r^2} \right]$ Where r1= internal radius r2 = external ratios v = Poisson's ratio r is the generic radius and p is the density Hint: Calculate the hoop stress at the crack tip

          A radial crack of 3mm in length has been detected in a large hollow disc of 0.36m outside
radius and 0.16m of inside radius. Calculate the rotational speed (in rad/s) at which the disc will
break.
Data: E=200GPa, density 7800Kg/cm³, Kic=55MPa.m^0.5., Poisson's ratio=0.33
Consider that the geometric factor Y is given by
[$1.12 - 0.23 \left(\frac{a}{b}\right) + 10.55 \left(\frac{a}{b}\right)^2 - 21.72 \left(\frac{a}{b}\right)^3 + 30.39 \left(\frac{a}{b}\right)^4$]
Where \"a\" is the crack length and \"b\" is the difference of the external and internal radius
And that the hoop stress is given by:
$\sigma_\theta = \frac{3+\nu}{8} \rho \omega^2 \left[ \frac{r_2^2 + r_1^2}{r^2} - \frac{r_1^2 r_2^2}{r^4} - \left(\frac{1+3\nu}{3+\nu}\right) \frac{r_2^2}{r^2} \right]$
Where
r1= internal radius
r2 = external ratios
v = Poisson's ratio
r is the generic radius and p is the density
Hint: Calculate the hoop stress at the crack tip

A radial crack of 3mm in length has been detected in a large hollow disc of 0.36m outside
radius and 0.16m of inside radius. Calculate the rotational speed (in rad/s) at which the disc will
break.
Data: E=200GPa, density 7800Kg/cm³, Kic=55MPa.m^0.5., Poisson's ratio=0.33
Consider that the geometric factor Y is given by
[1.12 - 0.23 ((a)/(b)) + 10.55 ((a)/(b))^2 - 21.72 ((a)/(b))^3 + 30.39 ((a)/(b))^4]
Where äïs the crack length and b̈ïs the difference of the external and internal radius
And that the hoop stress is given by:
σθ = (3+ν)/(8)ρω^2 [ (r2^2 + r1^2)/(r^2) - (r1^2 r2^2)/(r^4) - ((1+3ν)/(3+ν)) (r2^2)/(r^2)]
Where
r1= internal radius
r2 = external ratios
v = Poisson's ratio
r is the generic radius and p is the density
Hint: Calculate the hoop stress at the crack tip

Added by Esther L.

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