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# J2 Plasticity

Good day, everyone.

I would like to ask this question although others may find this question a bit silly. The situation is this: I have a steel element which I modeled using a degenerated shell. The element is initially subjected to longitudinal stresses due to the shear friction transfer. At some point, the element yielded and the majority of the load is due to longitudinal direction. Then, the shell element was subjected to lateral expansion making the stress in the element biaxial. As the lateral expansion increases, the longitudinal stress decrease while the lateral stress increase.

The question is, the problem is plane stress as described by the loading above, is it really possible for a yielded material to change its stress components from say q22 dominant into a q11 dominant?

Thank you for your response.

Sincerely,

Rodolfo.

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## Comments

## J2 Plasticity

It is possible for the material being yielded along 22 direction to have excessive deformation along 11 direction due to lateral expansion and then the rotation of some elements cause the stress along 11 to be projected along 22 and increase the strength in that direction. In some sense geometric nonlinearity is involved here. Excessive rigid body rotations of some elements produce contribution along yielded direction. This perhapse leads to the stress along 11 to increase to compensate for 22 direction.

Just a rough idea.

Mohsen

## Thank you for your reply and

Thank you for your reply and the idea. So you were saying that this kind of behavior is something that could happen? My developed shell program is equipped with geometric nonlinearity, however, based on deformed behavior, rotations are not enough to make the switch in stress from 22 to 11. Can you elaborate your idea?

## Actually I do not have any

Actually I do not have any idea about the deformation of your model. However, if it is a 3D model the redirtribution of stresses in unyielded surrounding part can also help to control excessive deformation. As a suggestion, one can monitor the variation of S11 and S22 as a function of deformation. It can illuminate at what step we have switch from 22 to 11. Also, the component of rotation can be expressed as a function of deformation. Based on such variations, one has a better idea into the physics of problem.

Mohsen

## Thank you for the idea. The

Thank you for the idea. The suggestions are very much appreciated.

## Re: J2 plasticity

In my opinion, the best way to understand the cause of what you see is to work out the details using the equations. I haven't done that but it should be easily doable. To see what the equations are, look at my articles at the following links (the articles deal with Drucker-Prager but conversion to J2 is straightforward):

http://www.parresianz.com/mechanics/plasticity/algorithm/plane-stress-drucker-prager/

http://www.parresianz.com/mechanics/plasticity/algorithm/plane-stress-drucker-prager-return/

http://www.parresianz.com/mechanics/plasticity/algorithm/plane-stress-drucker-prager-return-part-3/

http://www.parresianz.com/mechanics/plasticity/algorithm/plane-stress-drucker-prager-return-part-4/

http://www.parresianz.com/mechanics/plasticity/algorithm/plane-stress-drucker-prager-return-part-5/

-- Biswajit

## Thank you

Thank you for your reply, Prof Banerjee. As a clarification, so you were saying that this kind of behavior/phenomenon is normal? That is, if a material already yielded with most of the stress coming from 22-direction, it can be "unyielded" and yield again with dominant stress contribution in 11-direction?

## Re: Stresses in yielded shell

There is no physical reason why that a shift of the major principal stress from the 22 to the 11 direction should be impossible. However, whether that actually can happen depends on the exact material properties (e.g., hardening modulus if any, isotropic/kinematic hardening), load duration and magnitudes, etc. for the particular problem at hand.

The only way to be sure is to work out the equations for your particular problem, explore the stress-paths in a few spaces (e.g., J2-I1) and then identify if a switch of principal direction is possible.

-- Biswajit

## Thank you very much,

Thank you very much, Professor.