Simulating neo-hookean material

General Discussion
1

Dear all,
I am using the example given in the FF documentation about “Compressible Neo-hookean materials” in the “Mathematical Models” menu.
I am wondering why the program makes use of only one Lame coefficient, mu (declared line 238).
The other one, D, is declared (line 239) but never used.
Thanks a lot for any clue!

2

Just to play
I rebuild the test nl-elas-neo-Hooken
nl-elas-neo-Hookean-v2.edp (2.9 KB)

with new set of macro

ElasticLaw2d.idp (2.9 KB)

With a clean set of Lame coefficient

3

Thank you very much.
I will try that.

4

Hi Frédéric,

I also used this neo-Hookean mechanical model. I checked some literature and found that the neo-Hookean model with only one parameter means it is an incompressible model. So I want to make sure whether the example code given in the documentation is compressible or incompressible?

Thank you!

5

I sorry I miss this message,
I do not now , I am not specialists in solide mechanics

6

Thank you for your reply. I will try both and see if the differences are large.

7

Hi Frederic,

I am reading your script with the new macro in the .idp file. I can derive the first-order derivative of the right Cauchy-green tensor dC2. But when it comes to ddC2 the second-order derivative, the expression is not very clear to me, as my hand-derivation leads to a different result. Could you maybe point me to any derivation process of the expression?

Thank you in advance!

Best wishes
Yue

8

The derivation looks correct to me. We have:
C(u)=F^T(u)F(u)=(I+\nabla{}u)^T(I+\nabla{}u)=I+\nabla{}u^T+\nabla{}u+\nabla{}u^T\cdot\nabla{}u
Then the first Frechet derivative is:
dC(u,U)=(dF(u))^TF(U)+F^T(U)(dF(u))=\nabla{}u^T(I+\nabla{}U)+(I+\nabla{}U)^T\nabla{}u
...=\nabla{}u^T+\nabla{}u+\nabla{}u^T\cdot\nabla{}U+\nabla{}U^T\cdot\nabla{}u
Then the second is:
ddC(u,U)=(dF(u))^T(dF(U))+(dF^T(U))(dF(u))=(\nabla{}u)^T\nabla{}U+\nabla{}U^T\nabla{}u
...=\nabla{}u^T\cdot\nabla{}U+\nabla{}U^T\cdot\nabla{}u