Elasticity tensor calculation
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?
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??
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!