Stiffness tensor in UMAT

[Mange_H]

Dear all,

I'm trying to implement a material model in ABAQUS by using UMAT and I have some trouble with the material stiffeness tensor. I'm not quite shure of what to implement. Let say I have a hyper-elastic material of Fingers tensor i.e. \tau_{ij}=\tau_{ij}(b_{kl}). In the manual the stiffness tensor is defined as

d(\tau)=JC d\epsilon

where d\epsilon is the symmetric part if the spatial velocity gradient.

However, I'm not comfortable with taking the derivative of the Kirchhoff stress with respect of the symmetric part of the spatial velocity gradient ':oops:'. Can someone give me a hint on how to this ':?:'


Mange H

[Jorgen]

Hej Mange,

You are asking a good question, one that in-fact often causes a headache for UMAT coders. Since I don't know the nature of the material model that you are implementing it is difficult to give specific advice, so instead here are some general comments:

:arrow: It is often conceptually easier to use the alternative definition:
[TeX:655bbfef3c]C = \frac{1}{J} \frac{\partial \Delta (J\sigma)}{\partial \Delta \epsilon}[/TeX:655bbfef3c]
when calculating the Jacobian matrix.

:arrow: As I am sure you know, for many advanced materials it can be both difficult and computationally expensive to calculate the exact Jacobian matrix. Instead what is often done, especially for testing and initial development, is that the Jacobian matrix is numerically approximated using numerical differentiation with finite differences.

:arrow: It is important to remember that the accuracy of the Jacobian matrix does not influence the accuracy of the FE solution, but only the numerical stability and the run-time of the simulations. In fact, you can get excellent robustness without losing significant speed with a sophisticated finite difference approach.

Best of luck,
Jorgen