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geometrical non-linear problems

I have a mechanical problem it is a part of my studies, this problem relates to mechanics of material (geometrical non-linear problems by using Newton Rahpson method (If I was right) I am struggling to solve it. The thing is because of my poor knowledge I could not find the right reference. All I want is a volunteer someone I will send him the problem then he advice me where should I go to find similar examples or at least the right sources

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yawlou's picture

You may like to look at information in the following two blogs.   In the blogs, some suggested references are provided which may help you learn to do Newton-Raphson iterations and a geometrically nonlinear analysis.

http://imechanica.org/node/4995

http://imechanica.org/node/5127

Alejandro Ortiz-Bernardin's picture

I am going to recommend you the same steps I did to understand the subject. I spent like 1 week reading some books. Here are the steps:

1) Start reading the book by Peter Wriggers, Nonlinear Finite Element Methods , in the following order: Appenix A, Chapters 1, 3.1, 3.2, 3.4, 3.5

2) Then follow with the book by Bonet and Wood, Nonlinear Continuum Mechanics for Finite Element Analysis , Chapters 1.4,  2.3,4.12, 4.13, 4.14, 5, 6.1, 6.2, 6.3, 6.4, 8.

3) Now, try to recognize what you have learned from 1) and 2) in the book by Kojic and Bathe, Inelastic Analysis of Solids and Structures

There is an intermediate step that I followed between 2) and 3). I read the book by Bathe, Finite Element Procedures , Chapters 6.1, 6.2. But then I figured out that the book in 3) is basically the same, but simpler to understand. So, I think you will not need to do the same.

Once you have read the above suggested literarature, you will have some basis of the linearization of the Principle of virtual work using Newton-Raphson method. Then, you should pick any of the references in 1), 2) or 3) to read about the discretization procedure.

I'd also like to point you to my notes at

 http://en.wikiversity.org/wiki/Nonlinear_finite_elements

-- Biswajit 

Alejandro Ortiz-Bernardin's picture

Dear Biswajit,

Thanks for pointing out your notes. Actually, there is something that maybe you can clear up from my mind. I have seen in different books of the subject the use of distinct names for the formulations in the current configuration. Namely, in Bathe's book the term updated-Lagrangian is employed, while in Bonet's book Eulerian formulation is employed (Wrigger's book refers to the same simply as formulation in the current configuration). I tend to associate the Eulerian formulation to the field of fluid mechanics, being an Eulerian mesh a fix mesh in the space where the evolution of quantities is studied. The latter is what I understand from what you have depicted in your notes. On the other hand, it is true that in the updated-Lagrangian formulation the quantities are referred to the last known configuration, thus they become spatial quantities. Is there any subtle difference between updated-Lagrangian and Eulerian formulation? Or is it due to the spatial character of the updated-Lagrangain that some authors refer to it as Eulerian formulation?

Regards,

Alejandro.

I think Bonet's book uses the total Lagrangian approach. But total and updated Lagrangian aproaches are essentially the same thing - you're still sitting at a material point in the body.  Eulerian descriptions require you to move out of the body and watch it from a distance.

You could think of the updated Lagrangian momentum equations as Eulerian equations for a control volume.  But in that case the Eulerian conservation of mass has to be added to the mix and Reynold's transport theorem has to be used appropriately to get the correct description.

So updated Lagrangian .ne. Eulerian . 

-- Biswajit 

 

Alejandro Ortiz-Bernardin's picture

Dear Biswajit,

That is exactly what I understand by an Eulerian formulation. Indeed, for me it is more applicable to fluid mechanics, where the FE mesh is fixed in the space and the fluid passes where the mesh is located. In the Updated Lagrangian the mesh is not fixed but rather the mesh is continuously updated. With respect to Bonet's book, both Total Lagrangian and Eulerian formulation are described. If you see the chapters I pointed out in the previous post, you will see that they use "Eulerian formulation" when they are talking about the formulation in the current configuration and that when they linearize the virtual work they obtain exactly the same linearized virtual work for the Updated Lagrangian formulation, as one can easily compare with Bathe's book. So far, we agree that Updated lagrangian formulation is not the same as Eulerian formulation, but I don't think that Professors Bonet  and Wood have made such a mistake in naming the formulation. There might be another reason, I think it is just because in both formulations, Updated Lagrangian and Eulerian, the quantities are written in spatial form, and thus my original question was if there exist another reason for the same.

Regards,

Alejandro.

Hisham's picture

Dear all,

thanks a lot for a useful discussion. I have a question: In fluid-solid interaction problems and aside from using a coupled Eulerian-Lagrangian mesh, is it possible to find litrature that treats history dependant material, e.g. geomaterials, with an Eulerian mesh??? So that the problem can be totaly solved with a unified mesh! 

Thanks in advance,

Hisham

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