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## Re: [Getfem-users] Antwort: Re: 2D nonlinear plane stress

 From: Yves Renard Subject: Re: [Getfem-users] Antwort: Re: 2D nonlinear plane stress Date: Wed, 16 Jan 2019 21:39:46 +0100 (CET)

```
Dear Jena,

For an hyperelastic law, the weak form of the static elastic problem can be
written in the weak form language

"Def F := Id(meshdim)+Grad_u; (F * S) : Grad_test_u"

for u the displacement, F the deformation gradient and S has to contains the
expression of the second Piola-Kirchhoff stress tensor (you can of course
express it in term of PK1 also). For instance for a St Venant Kirchhoff law,
you can write

"Def F := Id(meshdim)+Grad_u; Def E := 0.5*(F'*F-Id(meshdim));  (F *
(lambda*Trace(E)+2*mu*E)) : Grad_test_u"

where E will be the Green Lagrange deformation tensor and lambda, mu the LamÃ©
coefficients.

This gives you some examples of construction of hyperelastic laws. The weak
form language gives you access to some standard operators (Trace, Det ...) see
http://getfem.org/userdoc/model_nonlinear_elasticity.html#high-level-generic-assembly-versions

So, if you have the expression of your law in plane stress, it should not be
very difficult to implement it. But of course you need the expression of the
law in plane stress. On the construction itself of plane stress hyperelastic
law, now, I don't know a good reference, unfortunately.

Best regards,

Yves

----- Original Message -----
From: "Moritz Jena" <address@hidden>
To: "yves renard" <address@hidden>
Cc: "getfem-users" <address@hidden>
Sent: Wednesday, January 16, 2019 2:17:01 PM
Subject: Antwort: Re: [Getfem-users] 2D nonlinear plane stress

Hello Yves,

thank you for your answer.

I'm afraid I'm not into the topic weak form language and I'm not sure
where to start with this problem.

I looked at the chapter in the documentation, however I don't know how to
describe a plane stress material model with it.

I also studied the examples, that come with the MATLAB-Interface. There
are a few examples, how to declare a material model with this weak form
expressions. But I still don't know, how to build this expressions.

Can you give me a approach for this problem or where I can find
expressions for such a problem?  Is there any literature that you can
recommend?

Best regards,

Moritz

Von:    Yves Renard <address@hidden>
An:     Moritz Jena <address@hidden>
Kopie:  getfem-users <address@hidden>
Datum:  09.01.2019 17:17
Betreff:        Re: [Getfem-users] 2D nonlinear plane stress

Dear Moritz Jena,

No, unfortunately, plane stress versions of Hyperelastic laws has not been
implemented yet.

I would not be so difficult, but as to be made. If you need one in
particular and have the expression, it is no so difficult to describe it
with the weak form language.

Best regards,

Yves

----- Original Message -----
From: "Moritz Jena" <address@hidden>
To: "getfem-users" <address@hidden>
Sent: Tuesday, January 8, 2019 4:11:48 PM
Subject: [Getfem-users] 2D nonlinear plane stress

Dear GetFEM-Users,

I use the MATLAB-Interface of GetFEM to create a program that
automatically solves different models of the same problem.

The problem is three-dimensional, but can be reduced by plain stress
approximation. (to reduce computing time).

I want to define a nonlinear material with the brick

gf_model_set(model M, 'add nonlinear elasticity brick',
[...])

For this nonlinear command it is specified in the description, that in 2D
always plain strain is used.

So my question is: Is there a way to define a nonlinear material with the
plain stress approximation? Or is it planned to install such an option in
a future release?

I hope you can help me with my problem.

Best regards,

Moritz Jena

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