Hello everyone,
I found that there is no comprehensive topic about how to calculate material jacobian matrix for UMAT while it's one of the most complicated and important parts of writing a subroutine and a lot of the researchers deal with it.
I think it would be great if experts share their sources and experiences here for all. I personally, try to provide the basics and essential materials as well.
Here is an essential available PDF on Imechanica:
http://imechanica.org/files/Writing%20User%20Subroutines%20with%20ABAQU…
Material Jacobian Matrix for hyperelastic materials
Although the methods for calculating jacobian matrix for well-known hyperelastic materials such as Neo-Hookean model is available online, there is no appropriate and comprehensive source for the more complicated formulations.
One of the most famous formulations recently is following:
Strain Stored energy (W)=W(λi)
while λ is the eigenvalue of deformation gradient or principle stretch and n is the eigenvector of deformation gradient or principle direction
then one can find the stress using following equation:
σ=(λi ∂W/∂λi ni ×ni) /J
while × is dyadic product and J is λ1 *λ2 *λ3
Now I have the following questions:
1- what is the explicit equation in terms λ to find jacobian matrix?
2- do we need to subtract the hydrostatic pressure from the σ and then use new stress tensor and calculate material jacobian regarding new stress tensor?