Question

Derivative of deformation gradient with respect to Green-Lagrangian strain?

Answer 1

For hyperelastic material, the elastic energy Ψ is related to the deformation gradient F and other internal variables (e.g. temperature θ)

In many literatures (including Malvern's and Belytchko's), however, their derivatives (especially Hessian) are usually derived in terms of left Cauchy-Green tensor C=FtF.

i.e. 2nd PK stress Sij=∂Ψ∂Eij, and CSEijkl=∂2Ψ∂Eij∂Ekl

I can convince myself that such derivation may help simplify the steps as the materials are usually represented by tensor C, but what I'm having truble now is a possibility of other ways, such as:

S=F−1∂Ψ∂F

D=∂S∂F∂F∂E

To me it looks ∂F∂E should be straitforward (as both ∂S∂F and ∂S∂E is attainable).

But it makes me perplexed is that its inverse ∂E∂F is not invertible as:

∂Eij∂Fkl=∂(FpiFpj)∂Fkl=(Fkiδlj+Fkjδli),

which is a kind of Sylvestre equations. I think there is an alternative way to bridge these two equations using tensor manipulation, but I'm at a loss.

Any comments about what I am missing would be greatly appreciated. Thanks in advance!

EDIT

In short, my question is whether it is possible to compute ∂F∂E

it might help getting CSEijkl from ∂Sij∂Fij, which is sometimes conveinient when compared to ∂Sij∂Eij.

Answer 2

heya ❕ here's your answer
For hyperelastic material, the elastic energy ΨΨ is related to the deformation gradient FF and other internal variables (e.g. temperature θθ)

In many literatures (including Malvern's and Belytchko's), however, their derivatives (especially Hessian) are usually derived in terms of left Cauchy-Green tensor C=FtFC=FtF.

hope it helps you