Plastic stretch tensor

**matogether** · 2008-12-30, 15:17

Dear all
I want to use ABAQUS VUMAT to calculate a consitutive models.
The back stress will obtained from the eight-chain model B, in which the plastic strech tensor Vij will be used. But the strech tensor gave in ABAQUS is right strech tensor Uij.
I have saw some references about the Vij, they tell the deformation can be decomposed into stretch and rotation as Fij=Rik*Ukj=Vik*Rkj, but I do not know how to calculate the Vij, and is the right roation equal to the left rotation?
Thank you very much.
Matthew

**Jorgen** · 2009-01-08, 03:41

The Abaqus VUMAT Manual states:

Hyperelastic constitutive models in VUMAT should be defined in a corotational coordinate system in which the basis system rotates with the material. This is most effectively accomplished by formulating the hyperelastic constitutive model in terms of the stretch tensor, U, instead of in terms of the deformation gradient, F. Using the deformation gradient can present some difficulties because the deformation gradient includes the rotation tensor and the resulting stresses would need to be rotated back to the corotational basis.

Hence, you can use U as F, and therefore also V.

-Jorgen

**matogether** · 2009-01-08, 08:26

Dear Jorgen and all
Thank you so much for your reply.
Now I am comfusing how to obtain the strech tensor which will be used to calculate the back stress.
Based on your reply, can I think the U equal to V? As I know, the ABAQUS can provide the right stretch tensor U. If they are equal in VUMAT, maybe I can use the principal values of U to calculate the back stress B, the problem will become easier.
Really appreciate your helps.
Regards,
Matthew

**Jorgen** · 2009-01-25, 04:27

Yes, you can do that. The VUMAT needs to calculate the stress in an intermediate configuration. If you use U as F, then things are OK by default. If you use F, then you need to rotate your calculated stress back to the right configuration using R.

In other words, Abaqus/Explicit takes care of the rotations for you if you use U.

-Jorgen

**mohand** · 2010-07-03, 06:29

Originally Posted by Jorgen

Yes, you can do that. The VUMAT needs to calculate the stress in an intermediate configuration. If you use U as F, then things are OK by default. If you use F, then you need to rotate your calculated stress back to the right configuration using R.

In other words, Abaqus/Explicit takes care of the rotations for you if you use U.

-Jorgen

Dear users,

I have two little questions to fix my ideas. The questions concern the use of the deformation gradient F when implementing a Vumat for hyperelastic material (Arruda Boyce).

If one need to use this tensor (F) directly when implementing the model (without using stretch tensor U), one needs to rotate the stress back to corotational frame (Abaqus documentation).

I think the first step is to calculate the rotation tensor R (not provided in the Vumat) using the relation: F=R*U (R=f*U-1), then apply the relation: TR*Stress*R (TR is the transposed matrix of R). Am I right?

Is there any way to do rotate the stress back using the relative spin increment Dt*(W-Omega) already provided in the Vumat?

Thank you very much.