Dear Adityia,

yes, it can be higher than one. I´m referring as stress triaxiality understood as the ratio between the hydrostatic pressure and the Von Misses stress. Imagine a situation on which the different principal stresses are very similar, or identical (e.g., material deep inside a fluid): in such case, the stress triaxiality tends to infinite.

Regards.

Good afternoon, dear Aditya!

I'm very glad to the opportunity to attempt again to help you.

In my opinion, your last question isn't entirely correct.

However, perhaps the problem consists in the usage of different terminology.

Because of the lack of concretization of the problem essence, it can be understood in different ways.

For example, it may be perceived as a question under what conditions the relative value of the stress tensor sij (by the way, relative to what?) turns out to be more of... what - unit or one?

Another possible interpretation is under which conditions the same average normal stress is more of, again, what - unit or one?

Without touching the question relatively of which the stress is considered, in the first case, for the answer you can use, for example, the Mohr diagram (the circle), in the second - to analyze when the average stress

               sav = (sxx + syy + szz) / 3 > 1 (ore one unit, for example, sxx).

You can also use the stress ellipsoid.

If I have understood the essence of your question correctly, it may be because of the variety of reasons: from a loading to relaxation of internal stress, or, speaking more simply, at any statically or dynamically nonequilibrium state.

Thank you Prof Eugene

Thank You Dear Sergio Cicero for your nice answer.

Aditya,

Just one point to add to the above answers: high values of stress triaxiality at some points on the specimen essentially mean that plastic deformation is hindered at those points. If the stresses are compressive (as in Dr. Cicero's example of a ball being submerged into a fluid), this will have theoretically not cause failure. But in case of tension- which you may be interested in- high stress triaxiality will result in microvoid formation and coalescence, thus resulting in failure.

In a tension test, till necking the stress triaxiality is theoretically constant, but once necking starts, depending on the geometry of neck (a and R in Bridgmann's formula), it may take various values >1 at different points. As I understand, the failure will happen wherever this value is highest.

Do have a look at this article for a better explanation :

https://www.researchgate.net/publication/241144369_A_CRITICAL_REVIEW_OF_TRIAXIALITY_BASED_FAILURE_CRITERIA

Article A CRITICAL REVIEW OF TRIAXIALITY BASED FAILURE CRITERIA

Thank You @ Aashranth Sir.

I am not very sure whether stress triaxiality can be greater than 1. For Segieo 's example, the stress triaxiality is negative infinite. For pure uniaxial tensile loading (plane stress), the triaxiality is 1/3. not very sure if the traaxiality factor can be greater than 1/3.

Cheers

Sorry to disagree. There are two ways how stress triaxiality can build up at the material point: under quasi-static loading it is the result of the stress gradient associated to geometry changes and stress concentration. The upper bound value for this is 3. The second way is because of stress waves interactions in dynamic impact. Under planar impact (uniaxial strain condition) stress triaxiality increases linearly with the uniaxial strain. Depending on the ratio of the bulk modulus and the flow stress. To give you an idea in pure copper it reaches 22 and more. Under ideally hydrostatic condition it becomes infinite ;)

Thank you Prof. Nicola Bonora

Nicola Bonora :

Thanks for an interesting response, especially for adding the dynamic impact part.

Nevertheless, it isn't clear to me how the stress triaxiality (defined as ratio of hydrostatic stress to equivalent stress) is rate-dependent; it seems like a purely geometrical construct. Unless of course, you use some constitutive/failure equation to get the equivalent stress, and this equation contains some rate-dependent terms (JC model for instance). Please elaborate on this, would be nice if you can share some literature on the dynamic impact aspect.

As a secondary point, I think the stress triaxiality under hydrostatic compression will be infinite irrespective of the strain rate (quasi-static/dynamic), simply because the equivalent stress (i.e. the denominator) goes to zero (using von Mises formulation for example).

That can be a mathematical aspect all together, eventually, in real system even in stress triaxiality under hydro-static compression the value can not go infinity. I think, we need some discussions on this issue.

Aashranth B. Nicola Bonora

Certainly; in real world systems the constraint comes from how compressible the system is. This is where bulk modulus as mentioned by Nicola Bonora comes into play. For simplification, we often consider solids to be incompressible, so even if you apply perfect hydrostatic stress, they shouldn't shrink in volume. On the other hand, if you consider compressible systems, like an air-filled balloon submerged inside a water tank, you will generate a net von Mises equivalent stress of 0 (because all 3 principal stresses are equal in magnitude). In such a case, you will not have material flow (shear) or shape change, just a reduction in volume (as much as the bulk modulus of air and hydrostatic pressure due to water permit). So in this case, stress-triaxiality loses meaning, and mathematically becomes infinite. (Remember, stress-triaxiality causes a material to fracture rather than shear; if there is no question of shearing, then the high triaxiality has little meaning).

A non-trivial example would be densifying a powder through isostatic pressing, but this is again a bit of an approximation, because the system is not a continuum (i.e. it contains powder and entrapped air).

Thank you Aashranth B. Sir