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Signed von Mises stress when the sign is "zero"

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Joel_Lapointe

Joel_Lapointe

Mechanical
Dec 13, 2023
16
I have a stress state that gives a sign of zero for the signed von Mises stress.
According to most of the literature, the sign is coming from the first invariant I1 of the stress tensor (Hydrostatic stress):
For the general stress state the invariants are :
Invariants_v1_etqamt.jpg


For the principal stresses the invariants are :
Invariants_v2_sif4ny.jpg


The first invariant allows for the sign to be zero. People do add in very small number (e.g. 0.000001) to remove the zero case. I don't like this gerrymandering.
Is it acceptable to use the other 2 invariants instead? Their signification might be totally different hence unusable?

Thank you...
 
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How are you using the sign? What I mean is, how is it relevant to you? All that the sign does is tell you if the hydrostatic stress is positive or negative. Using the other invariants would not be correct.
If you're creating some program or Excel spreadsheet, I'd just default to positive if the hydrostatic stress is zero (which should be pretty rare). Then you're just defaulting to the standard von Mises stress.
 
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the principal stresses are +ve or -ve as the loading dictates. But I recall (maybe incorrectly) that the failure curve for von Mises is curved in two quadrants (where all stresses have the same sign) and linear in the other 4 quadrants (with mixed signs).

but to worry about the sign of zero seems to be more a math (or philosophy) question rather than an engineering question ...

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.
 
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You don't need the first invariant of the stress tensor to compute the equivalent von Mises stress. Normal stresses (s11, s22, s33) and shear stresses (s21=s12, s23=s32, s31=s13) should suffice, and subtractions are squared, so you don't end up with negative values.
 
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