Hi Arnold,
When I compare the documentation to step-44 that you quote to the code itself,
it would seem like there’s a typo there. The second term “-2 p \mathcal{I}” of
the part that underlined should actually read “-2 p \mathcal{S}” since this
really is the symmetric fourth order identity tensor that holds the symmetry
properties that you also quoted.
To get from the Holzapfel equation that you quoted to the one used in the
tutorial, you still need to “push forward” these quantities. The tensor 6.166
is the volumetric tangent described in the reference configuration, while the
tutorial / Miehe describe a tangent in the spatial configuration. The
appropriate transformation is the Piola transformation
<https://dealii.org/developer/doxygen/deal.II/namespacePhysics_1_1Transformations_1_1Piola.html#a83adfd1f3e4a83fd040da52b803a0723>.
As for your derivation, I’m not quite sure that I follow what you’re trying to
do here. The identity that is applied at the equation below Miehe (57) — let’s
take it at face value that its true — is used to simplify the calculation in
(58) where the formal definition of the tangent involves the contraction from
both sides of the derivative of the energy function by b, the left Cauchy-Green
tensor. Since the identity involves produces to b^{-1} on either side of the
symmetric fourth order unit tensor, they cancel out in (58). To verify your
result, you’ll need to compute the LHS of the equation, i.e. \partial_{b}
b^{-1}. But, in fact, this entire identity comes from the statement
0 = d/db (I) where I is the 2nd order identity tensor
= d/db (b^{-1} . b) where the “.” denotes a single contraction
= [db^{-1}/db] . b + b^{-1} . [db/db]
= [db^{-1}/db] . b + b^{-1} . S where S is the forth order unit identity
tensor (this is a standard derivation, specialised for the case of b being
symmetric)
—> contracting by b^{-1} on the right and rearranging —>
db^{-1}/db = - b^{-1} . S . b^{-1}
I must say that, for me at least, trying to get straight to the result as Miehe
does is not easy to understand unless you know some rather unusual (maybe?)
identities. I prefer the to start in the reference configuration, as Holzapfel
does, and then push forward all results to the spatial configuration. That way,
the individual step are, in my opinion, quite clear albeit lengthy. There are
some notes in Appendix A.3 of my dissertation
<https://open.uct.ac.za/handle/11427/9519> that sketch it out.
I hope that helps you a little.
Best,
Jean-Paul
> On 6. Oct 2021, at 23:37, 'Arnold' via deal.II User Group
> <dealii@googlegroups.com> wrote:
>
> Hello,
> I don't get the formula for Jc_vol in step-44 (marked purple), particulary
> the fourth order unit tensor in there. I looked up the source that is quoted
> in step-44 (Miehe 1994), and the formula for db^-1/db uses a unit tensor as
> well (see picture for details).
>
> I'm not sure however if the formula Miehe uses is the correct one for
> _symmetric_ tensors. It should theoretically be equal to the formula below
> from "Nonlinear Solid Mechanics", but when I try to evaluate the Miehe
> formula in index notation I get a weird result that is much different from
> the Holzapfel formula.
>
> Can someone confirm with certainty that these two formulas (marked red) are
> actually the same thing? Or even better, locate the error in my derivation
> attempt?
>
> Best Regards,
>
> Arnold
>
>
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