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# Perturbation technique based on forward difference approximation

Hi,

I am trying to implement (in Abaqus) the elasticity tensor using a numerical approach. I have come across a paper by C.Miehe (Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity, 1996).

There are a few things I don't quite understand.

In some equations he uses Tensors in indices form and Tensors without indices (same equation). For example: Fe(CD)=F+deltaFe(CD) which is the perturbated deformation gradient (CD are indices and e is just a letter showing it is different from F). Would that mean that to every element of F deltaF is added? If so, why not using indices for F as well? The actual calculation of the elasticity tensor is also not quite clear:

C(ABCD)= (1/eps)*(Spert(AB)(Fe(CD))-S(AB)) (Eq. 2.10)

where eps is the perturbation constant, S is the 2PK, and SpertAB is perturbated 2PK as a function of Fe(CD)

ABCD are indices

So say I wanted to calculate C(1234) than I would have have to calculate Sspert12 as a function of Fe(34) subtract S12 which is known. Would that make sense? But how do I calculate a component of Sspert12 as a function of component of Fe ? One idea I had is to calculate the perturbated deformation tensor Fe by adding the deltaF to each component of the original F (from Abaqus) and basically use this new deformation tensor Fe as input into my constitutive equation and calculate a "perturbated stress" for each increment. But then how to I get C from that using 2.10?

I'm quite confused here.

Has anyone ever used this approach?

Any thoughts are highly appreciated!

Thanks,

Andreas

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