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Perturbation technique based on forward difference approximation


 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!



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