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# STRESS TRIAXIALITY

Dear ALL,

Which reading material/source that describes the good understanding of stress triaxiality?

What is the role of stress triaxiality for FE analyst?

I have read a lot of papers/articles but still have confusing me on the fundamental of stress triaxiality.

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## Comments

## On stress triaxiality

Greetings!

I have come across stress triaxiality when working on elsatic-plastic fracture mechanics.

Stress triaxiality is a concept that you pick up with regard to "state of stress": it usually impacts the way you look at strains, and hence plastic deformation.

First I am sure you are familiar with Mohr's circle, and the 2D state of stress: sigma-x, and sigma-y, and the in -plane shear stress tau-xy.

From that you should appreciate that stress is a tensor: something of higher order and more complicated than a vector.

Now, there are cases when assumption of 2D stress state is valid: hence you only require the three terms: sigma-x, sigma-y, and tau-xy (or alternatively the principal stresses, sigma-1, and sigma-2, and the orientation angle (or half-angle!) theta..

Examples are many, such as stresses in thin shells, etc.

However, many times this is not possible due to existense of 3D state of stress, and hence you need six terms: sigma-x, sigma-y, sigma-z, tau-xy, tau-xz, tau-yz.

This required a 3D model needs to solve.

At times, a 2D state of stress approximation is a good starting point, and less complicated; to evaluate how a stress state is close to a 2D assumption/approximation: in other cases a 2D state of "strain" is much easier to work with than a 3D state of stress, in this case you would only need eta-x, eta-y, and gamma-xy: this case is applicable in thick structures, and where you have high stress "triaxiality".

To evaluate whether its better to work with 2D state of stress (plane-stress) which is one extreme, or to work with 2D state of strain (plane-strain) another extreme, you evaluate stress triaxiality....

In fracture mechanics, very low stress triaxiality implies a 2D plane stress assumption is valid, while a high stress triaxiality implies a 2D plane "strain" assumption is valid.

In fracture mechanics, one often describes trixiality as "constraint" as high stress triaxiality limits plastic deformation in one axis, leading to a 2D strain (plasticity) problem; things get interesting when you good lower constraint, and this leads to a lot of interesting research topics.

## erratum

correction:

"..you would only need eta-x, eta-y, and gamma-xy.."

"eta" shall be "epsilon"!

## Stress Triaxiality

Thanks Aladdin for your inputs!

Hope this helps me to grow my understanding in stress triaxiality.

Could you share any reading materials that could help me as well, as you understand this good knowledge.

## References

I am sure that you can find numerous literature on the topic available.

During my thesis work, I have selected these two, as I have found them relevant:

- Thaulow C, Østby E, Nyhus B, Olden V, Zhang Z. The Philosophy Of Constraint Correction.

2nd International Symposium on High Strength Steel, April 2002, Stiklestad, Norway.- Schwalbe K-H, Newman Jr. J, Shannon Jr. J.Fracture Mechanics Testing on Specimens with Low Constrai Standardisation activities within ISO and ASTM,

Engineering Fracture Mechanics 72, pp. 557-576, 2005.Note that these were selected with experimentation in mind, as my M.Sc. thesis was mostly experimental.

I have come across many other good papers, but regretably do not have the references direct at hand.

## ON STRESS TRIAXIALITY

STRESS TRIAXIALITY is the ratio of the hydrostatic part of the stress to the equivalent stress.

In 1-D tensile test, STRESS TRIAXIALITY is 1/3 upto necking, as necking starts the state of stress is not 1-D, so TRIAXIALITY is no longer 1/3.

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