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Calculating material jacobian matrix for ABAQUS UMAT subroutine

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:



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 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 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?  

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