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For example here, it is mentioned:

The von Mises criterion is NOT a valid yield (failure) criterion for materials with different yield stress values in tension and compression.

I can't really understand why this is true. I can't really think of a reason why it should not be true either, but I would like some rigorous reasoning behind this statement.

I also have some lecture notes (in my native language, not English) that state that to be able apply von Mises criterion, the compressive limit has to be slightly bigger than the tensile one. So which of these statements are correct and why?

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This is more of an opinion on the question rather than a definitive answer.

TL;DR: I believe Von Mises is not really suitable when there are different yield stresses, however it's a good enough compromise

I believe that there is a misunderstanding that there is a "Von Mises stress". What actually the Von Mises criterion estimates is the strain energy of a infinitesimal cube. It just happens to be in the same units as "stress". Since it can be conveniently compared to the yield stress, has led to the widespread belief that this quantity is a stress.

Now, the problem you mention, regarding different compressive and tensile yield, I believe it has to do with the formulation. Basically, the strain energy is calculated based on the principal stresses (or equivalently normal and shear stress at a given stress state). So the yield stress only comes when comparing. So the question is, what should I compare to the tensile or the compressive tensile yield stress?

In that aspect, I believe the article states that the von Mises criterion is not valid when compressive and tensile yield stresses differ

Additionally, the article you refer to, presents other failure criteria (not widely known or used) and IMHO is trying to showcase their competitive advantage (which I don't disagree that they have).

The downside is that they introduce additional constants which - in my experience- are not widely available. (To be honest most of the times - when I was younger- I felt lucky whenever I got to find a Young's modulus and a UTS value for the specific material I was looking for). Even tensile yield stress is not widely available (let alone compressive yield stress). So, developing extra accurate criteria that need additional tests can be quite excessive and you can get along with a more conservative criterion in most cases. That probably is the reason that you've been taught that when compressive yield stress are little higher than tensile yield stress its ok to use the von Mises criterion (with the tensile yield stress)

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