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

From: Konstantinos Poulios
Subject: Re: [Getfem-users] Antwort: Re: 2D nonlinear plane stress
Date: Thu, 4 Jul 2019 21:39:38 +0200

I am attaching my notes on hyperelasticity that include many references by author name and year. I hope it helps.

On Thu, Jul 4, 2019 at 11:24 AM Moritz Jena <address@hidden> wrote:
Hello Kostas,

your approach works perfectly for me.

I checked me code and it seems like there was a little typing error. So what I did at first works too. But your approach seems to be faster. (Some warnings occure in my code.)

I have one last question at the moment. This question concerns the line

md.add_nonlinear_term(mim, "Th*0.5*kappa*sqr(log(Det(F)))+"

Where did you get that _expression_ from? Is there any literatur, where I can find such expressions? Or can you give me an approach to deriving that _expression_?

Best regards,

Moritz Jena

Von:        Konstantinos Poulios <address@hidden>
An:        Moritz Jena <address@hidden>
Kopie:        getfem-users <address@hidden>
Datum:        03.07.2019 23:20
Betreff:        Re: Re: [Getfem-users] Antwort: Re: 2D nonlinear plane stress

Dear Jena,

Without having run your code I would say that your definition of the 3D deformation gradient looks correct. Another way of converting from 2x2 to 3x3 is


but it should be equivalent to your approach.

Could you attach a screenshot of the "strange" solution that you get?


On Wed, Jul 3, 2019 at 9:49 AM Moritz Jena <address@hidden> wrote:
Hello Kostas and all GetFEM-Users,

I finally managed to try your approach for my plane stress problem.

But unfortunately I got a problem with it.
When defining the 3D deformation gradient, I have to add a [2x2] - matrix to a [3x3]- matrix. To work around this problem I tried to rewrite the displacement gradient to a [3x3] - matrix.

So instead of

md.add_macro("F", "Id(3)+Grad_u+epsZ*[0,0,0;0,0,0;0,0,1]")

it tried

md.add_macro("F", "Id(3)+[Grad_u(1,1),Grad_u(1,2),0;Grad_u(2,1),Grad_u(2,2),0;0,0,0]+epsz*[0,0,0;0,0,0;0,0,1]")

To test the approach, I created a little test-script with a simple beam and a force. The calculation works with this work around, but the deformation seems something strange. I think I did a mistake by accessing the matrix-elements of the displacement gradient, but I'm not sure about it.

Can you, or anyone else, help me with my problem?


import getfem as gf

import numpy as np

# Parameter

l = 100

h = 10

b = 1.5

size = 1

E = 203000

nu = 0.3

Lambda = (E*nu)/(1-pow(nu,2))

mu = E/(1+nu)

lawname = 'SaintVenant Kirchhoff'

# Create mesh

m = gf.Mesh('cartesian', np.arange(0, l+size , size), np.arange(0, h+size, size))


mfu = gf.MeshFem(m,2)

mfd = gf.MeshFem(m,1)

mf1 = gf.MeshFem(m,1)

#assign FEM





mim = gf.MeshIm(m, gf.Integ('IM_GAUSS_PARALLELEPIPED(2,6)'))

# Some Infos

print("nbcvs = %d | nbpts = %d | qdim = %d | nbdofs = %d" % (m.nbcvs(), m.nbpts(), mfu.qdim(), mfu.nbdof()))


P = m.pts();

pidleft = np.compress((abs(P[0,:])<1e-6),range(0,m.nbpts()))

pidrigth = np.compress((abs(P[0,:])>l-1e-6),range(0, m.nbpts()))

left = m.outer_faces_with_direction([-1, 0], 0.5)

rigth = m.outer_faces_with_direction([1,0], 0.5)

fleft = m.faces_from_pid(pidleft)

frigth = m.faces_from_pid(pidrigth)

m.set_region(1, fleft)

m.set_region(2, frigth)


md = gf.Model('real')

md.add_fem_variable('u', mfu)

md.add_initialized_data('lambda', Lambda)

md.add_initialized_data('mu', mu)

md.add_fem_variable("epsz", mf1)

md.add_initialized_data('kappa', 68000)

md.add_initialized_data('Th', 0.2)


md.add_macro("F", "Id(3)+[Grad_u(1,1),Grad_u(1,2),0;Grad_u(2,1),Grad_u(2,2),0;0,0,0]+epsz*[0,0,0;0,0,0;0,0,1]")

md.add_nonlinear_term(mim, "Th*0.5*kappa*sqr(log(Det(F)))+Th*0.5*mu*(pow(Det(F),2/3)*Norm_sqr(F)-3)")

# fixed support left side

md.add_initialized_data('DirichletData1', [0,0])

md.add_Dirichlet_condition_with_multipliers(mim, 'u', mfu, 1, 'DirichletData1')

# force on rigth side

md.add_initialized_data('VolumicData', [0,0])

md.add_source_term_brick(mim, 'u', 'VolumicData')

md.add_Dirichlet_condition_with_multipliers(mim, 'u', mfu, 2, 'VolumicData')

# force array

F = np.array([[0,-4], [0,-10], [0,-15], [0,-25], [0,-35], [0,-50]])

nbstep = F.shape[0]

#iterative calc

for step in range(0, nbstep):


    md.set_variable('VolumicData', [F[step, 0],F[step,1]])

    md.solve('noisy', 'max_iter', 50)

    U = md.variable('u')

    s1 = gf.Slice(('boundary',), mfu, 4)

    s1.export_to_vtk('test_%d.vtk' % step, 'ascii', mfu, U , 'Displacement')


Konstantinos Poulios <address@hidden>
Yves Renard <address@hidden>
Moritz Jena <address@hidden>, getfem-users <address@hidden>
05.02.2019 10:05
Re: [Getfem-users] Antwort: Re: 2D nonlinear plane stress

Dear Moritz Jena,

To define a hyperelastic material law for a plane stress problem, the easiest way is to add one extra variable representing the out of plane strain. If mf1 is a scalar fem and mf2 is a vector fem with 2 components, then you can simply do:

md = gf.Model("real")
md.add_fem_variable("u", mf2)           # displacements variable
md.add_fem_variable("epsZ", mf1)        # out of plane strain variable
md.add_initialized_data(’kappa’, kappa) # initial bulk modulus
md.add_initialized_data(’mu’, mu)       # initial shear modulus
md.add_initialized_data(’Th’, Th)       # plate thickness
md.add_macro("F", "Id(3)+Grad_u+epsZ*[0,0,0;0,0,0;0,0,1]") # 3D deformation gradient
md.add_nonlinear_term(mim, "Th*0.5*kappa*sqr(log(Det(F)))+"

Best regards

On Wed, Jan 16, 2019 at 9:40 PM Yves Renard <
address@hidden> wrote:

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 Lamé 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

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,


----- Original Message -----
From: "Moritz Jena" <
To: "yves renard" <
Cc: "getfem-users" <
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

Best regards,


Von:    Yves Renard <
An:     Moritz Jena <
Kopie:  getfem-users <
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,


----- Original Message -----
From: "Moritz Jena" <
To: "getfem-users" <
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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