Material Jacobian

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:

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