# Angular speed that breaks a spinning body apart with inertial stress

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## Main Question or Discussion Point

How to find the angular speed, on which a spinning hollow cylindrical body breaks due to inertial stress(force)?
I found 2 sources(http://www.roymech.co.uk/Useful_Tables/Cams_Springs/Flywheels.html (last 2 equations) , https://www.engineersedge.com/mechanics_machines/solid_disk_flywheel_design_14642.htm (eq. 4) ) that offer different formulas to calculate tangential and radial stresses. One of them claims that maximum radial stress is at $r=\sqrt{r_1*r_2}$ and other one that it occurs at $r=0$. Which one of these is correct?
Is it that the body breaks if
$\sigma_{tangential}>\sigma_{ultimate\ stress}\ or\ \sigma_{radial}>\sigma_{ultimate\ stress}$
or if
$(\sigma_{tangential}^2+\sigma_{radial}^2)>\sigma_{ultimate\ stress}^2$

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More generally how determine if a body breaks if I know stress tensor appllyied on that body and mechanical propetis of that body?

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I heard that it has something to do with principal stresses. Can someone explain that to me with more details how to determine whether body breaks based on principal stresses applied on the body?

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More generally how determine if a body breaks if I know stress tensor appllyied on that body and mechanical propetis of that body?
What about simple case when it pushed from 2 sides so that the stress tensor is:
$\begin{bmatrix} \sigma_{xx}&0 &0 \\ 0&\sigma_{yy}&0\\ 0&0&0 \end{bmatrix}$
How to know if a body breaks under this stress?

You will need to know something about the material properties (is it brittle or will it deform plastically?) and you will need a failure theory. The inputs to most failure theories are the principal stresses, so you are part way there.

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you will need a failure theory.
Which criterion should In use to determine whether a metal body breaks into pieces or not?

Is one of these good approximation?
body breaks if
$\sigma_{tangential}>\sigma_{ultimate\ stress}\ or\ \sigma_{radial}>\sigma_{ultimate\ stress}$
or if
$(\sigma_{tangential}^2+\sigma_{radial}^2)>\sigma_{ultimate\ stress}^2$

The question is too broad. You have to specify what the metal is and what its state is (annealed, hardened, drawn, etc.). This is not a simple question, and not the sort of thing that can be adequately answered on PF.