Polycrystal Shape Memory Alloys

[NazBaba]

Hi all,

I am modelling polycrystal shape memory alloys upon uniaxial loading using Abaqus UMAT. I modelled single crystal which are unrotated or rotated successfully. But when I model polycrystal shape memory alloys I get always this error in Abaqus message file: "***NOTE: THE SOLUTION APPEARS TO BE DIVERGING. CONVERGENCE IS JUDGED UNLIKELY. ***ERROR: TOO MANY ATTEMPTS MADE FOR THIS INCREMENT" . I know that this problem occurs because of the material Jacobian. I calculated my Jacobian numerically using finite difference.

I used this algorithm which Matsgd wrote for me in another thread:

1. Calculate the stress at time t+dt based on the strain at t+dt (and the state variables at time t).
2. Update STRESS and STATEV in the UMAT. Do not modify these when calculating DDSDDE.

3.

Define SIX strain increment vectors:
de1 = alpha*[1 0 0 0 0 0]
de2 = alpha*[0 1 0 0 0 0]
de3 = alpha*[0 0 1 0 0 0]
de4 = 2*alpha*[0 0 0 1 0 0]
de5 = 2*alpha*[0 0 0 0 1 0]
de6 = 2*alpha*[0 0 0 0 0 1]

If your constitutive model uses engineering shear strains, ignore the factor "2" above.

For each of the strain increment vectors above, calculate a stress vector, using the known strain epsilon at time t+dt, and the strain increment vectors above. In other words, calculate stresses just the same way you would do normally, only this time based on the SIX strain vectors:

(epsilon + de1) -> sigma1
(epsilon + de2) -> sigma2
(epsilon + de3) -> sigma3
(epsilon + de4) -> sigma4
(epsilon + de5) -> sigma5
(epsilon + de6) -> sigma6

The first column of C is then given by
(sigma1-sigma)/alpha.
The second is given by
(sigma2-sigma)/alpha





This algorithm is useful for single crystal problems. However, when I use polycrystal model, my solution does not converge. What can I do? Can it be an another problem different from Jacobian calculation? or Can I try different things in calculation of Jacobian?

Thank you.
Nazim Babacan

[Frank]

Hello

once more a few hints:

1) Regarding the factor of 2, check out the equation in the User's Manual, chapter 1.2.2 Conventions:
Shear strains
Abaqus always reports shear strain as engineering shear strain,

2) In a polycrystal you have to define the local orientation of each grain with respect to the global coordinate system. However, neglecting this is unlikely to be the cause for your current problem stated in the posting.

3) Can you be sure that these messages can be caused exclusively by an incorrect Jacobian ?

4) you may try the ABAQUS mailing list
http://tech.groups.yahoo.com/group/Abaqus/

[NazBaba]

I want to add about orientation in my code:
I don't use *orientation option in Abaqus. I am using rotation matrix in my code and I rotate only my transformation strain local coordinate to global coordinate because I know transformation strain values in local coordinate. According to our model, only difference between local and global coordinate is transformation strain so I think rotating transformation strain is adequate. Can you give me any opinion about these situations?

[Frank]

Try to find publications discussing the implementation of other 3d SMA models.
Or post on iMechanica.org

[matsgd]

Here's a suggestion:

Print, to a text file, the stress at some time t+dt, and the six stress vectors corresponding to the perturbed states. Then post them here. Let's take a look at it all.
Also, print the components of the calculated Jacobian (all thirty-six components, with indices, please).

A note: Single vs. polycrystal - doesn't matter. The perturbation scheme does not care. It only knows of finite differences...

Also, as Frank points out, check the definition of the shears. Whatever you do has to be consistent with what ABAQUS does. Perhaps my previous outline of the algorithm was unclear / wrong about this. If this is wrong, you would have problems also with the single crystal case, if you subjected an element to shear.

M