Work conjugacy error in commercial finite-element codes: its magnitude and how to compensate for it

Most commercial finite element (FE) programs use the Hencky (or logarithmic) finite strain tensor for which the associated (or work-conjugate) objective stress rate is the Jaumann rate of Kirchhoff stress. However the stress rate of FE program misses the volumetric term and the error due to this term violates energy balance.

The error in using the "incorrect" Jaumann rate of Cauchy stress was pointed out in the literature long ago, are but has either been ignored or thought to cause only negligible errors. Indeed, the error is generally less than 0.1 per cent for metals and other materials that are inelastically incompressible. It now appears, though, that this is not true for highly compressible inelastic materials.

http://rspa.royalsocietypublishing.org/content/early/2012/05/22/rspa.2012.0167.abstract

How to create automatically the user subroutine UMAT

Another way to create automatically the user subroutine UMAT is by automatic UMAT code generation procedure for pseudoelastic materials using the symbolic mathematics package Mathematica, described in the article "Automatic Generation of User Material Subroutines for Biomechanical Growth Analysis", here, and also here.

We found the PhD thesis where many interesting details of the Mathematica implementation for FORTRAN UMAT subroutine generation and examples of the subroutines itself are available.

In this thesis the author claim that they have published the Mathematica notebooks in the Journal of Biomechanical Engineering ASME as supplemental material, where it is free to access and download. Unfortunately, we failed to find these notebooks on WEB. If you know how to get it - please comment it.