Elasticity tensor calculation

[Paul_S]

Hello everybody,

I am trying to write UMAT subroutine for Abaqus and have a great problem now to understand the principle of Elasticity tensor definition (ET). Let as consider material description of ET that is defined as

ET(i,j,k,l) = 4 * d[ d[psi]/d[Cij] ] / d[Ckl]

where: d[A]/d[b] mean derivation of A with respect of b
psi means energy density function
Cij mean (i,j) components of Right C-G tensor

What I really don't understand is how to do derivation (e.g. analyticaly) to respect the SYMMETRY OF C! Let me consider the following HYPOTHETICAL example:

psi = psi (C11,C12,C21,C33) = 2*C12 + 3*C12*C21 + 5*C22
(note that psi is symmetric with respect to C12 and C21)

HOW TO COMPUTE THE ET components? I did e.g. the following:

ET(1,2,1,2) = 4* d [ d[psi]/d[C12] ] / d[C12] = 4* d [3*C21] / d[C12] = 0
ET(1,2,2,1) = 4* d [ d[psi]/d[C12] ] / d[C21] = 4* d [3*C21] / d[C21] = 4*3 = 12

HOWEVER this result does not corresponds to my expectation of symmetry of ET tensor :-(
I think that the following relation should be held: ET(1,2,1,2) = ET(1,2,2,1)

May be I could consider symmetry by setting C21=C12 and re-express the form of psi to the form of psi=psi(C11,C12,C22) and calculate only componets of ET regarding to C12. However I am not sure if this approach is correct (gives me results that does not correspond to my refference results - factor of 2 or 4 higher).

Could anybody, please, give me an advice how to correctly do the general analytical calculation of ET based on components of C?

Paul

[Jorgen]

Well, isn't your hypothetical example fishy in the sense that the strain energy function is not symmetrical in C12 and C21? Do you claim that your function psi is physical??

- Jorgen

[Paul_S]

Thanks a lot for your response, Jorgen,

I have to apologize!!! - I made a typing mistake .... the correct form of (hypothetical) strain energy function should be:

psi = psi (C11,C12,C21,C33) = 2*C11 + 3*C12*C21 + 5*C22
... that is (now really) symmetrical in C12 and C21.

I am sorry to make you confused and hope that this is the only mistake in my above contribution.

However the problem is still the same:
ET(1,2,1,2) = 0
ET(1,2,2,1) = 12
that is not symmetrical!

Paul