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ABAQUS UMAT for hyperelastic material

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Hi all,

I have gone through the theory manual, subroutines manual and examples manual of ABAQUS documentation. There is an example of UMAT implementation for Neo-Hookean material at the following link starting on page L6.44.

[url] [/url] User Subroutines with ABAQUS.pdf

it may show up as

[url] [/url]

They give an explicit expression for material Jacobian C_ijkl for Neo-Hookean material on page L6.46. I tried to derive this expression based on the information provided in Abaqus Theory Manual Section 4.6.1. The problem I am facing in this derivation is that that I dont understand where does the term [(2/9)(delta_ij)(delta_kl)(BBAR_mm)] come from in the expression of C_ijkl. This is assuming that the strain energy density function is expressed in terms of the invariants of B (left Cauchy-Green tensor) and not BBAR(left Cauchy-Green tensor of deviatoric deformation). If the strain energy function is expressed in terms of the invariants of BBAR then the above mentioned term factors in but with some additional terms which dont match with the expression for C_ijkl.

Can anyone throw some light on this derivation/issue? It will be really useful for me as well as the community.

Thanks in advance.