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
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
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