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Issue in computing first invariant of cauchy-green tensor

Posted 23.01.2018, 09:43 GMT-5 Parameters, Variables, & Functions, Structural Mechanics Version 5.2a 5 Replies

Eric Harper
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I am trying to understand how COMSOL computes the invariants of the Cauchy-Green tensor, and I am getting errors that I cannot understand.

I worked through the equation view for the hyperelastic equation model, and identify the following workflow:

solid.I1Cel=solid.Cel11+solid.Cel22+solid.Cel33 solid.Cel11=(solid.Fiil11*solid.Cl11+solid.Fiil21*solid.Cl12+solid.Fiil31*solid.Cl13)*solid.Fiil11+(solid.Fiil11*solid.Cl12+solid.Fiil21*solid.Cl22+solid.Fiil31*solid.Cl23)*solid.Fiil21+(solid.Fiil11*solid.Cl13+solid.Fiil21*solid.Cl23+solid.Fiil31*solid.Cl33)*solid.Fiil31 ... solid.Cl11=solid.Fdlx1^2+solid.Fdly1^2+solid.Fdlz1^2 ...

So that I understand the underlying math, I've created a variable set under the definitions of my component, copy-pasting and replacing each variable e.g.

I1Cel=solid.Cel11+solid.Cel22+solid.Cel33

then

Cel11=(solid.Fiil11*solid.Cl11+solid.Fiil21*solid.Cl12+solid.Fiil31*solid.Cl13)*solid.Fiil11+(solid.Fiil11*solid.Cl12+solid.Fiil21*solid.Cl22+solid.Fiil31*solid.Cl23)*solid.Fiil21+(solid.Fiil11*solid.Cl13+solid.Fiil21*solid.Cl23+solid.Fiil31*solid.Cl33)*solid.Fiil31

This works until I get to defining my own values for Cl11, Cl12, etc. When I create my own entries for these, instead of reproducing the same simulations, I get "No convergence", and I can't understand why.

I'm happy to provide any files, etc.

Thanks in advance

5 Replies Last Post 02.02.2018, 02:44 GMT-5
Henrik Sönnerlind COMSOL Employee

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Posted: 7 years ago 23.01.2018, 16:27 GMT-5

Hi Eric,

I guess you are trying to feed the user defined hyperelastic material with your own expressions. If you check out the theory chapter, you will find that such expressions are only allowed to depend on a limited set of built in variables (C-tensor, principal stretches, strain tensor).

Regards,

Henrik

-------------------
Henrik Sönnerlind
COMSOL
Hi Eric, I guess you are trying to feed the user defined hyperelastic material with your own expressions. If you check out the theory chapter, you will find that such expressions are only allowed to depend on a limited set of built in variables (C-tensor, principal stretches, strain tensor). Regards, Henrik
Eric Harper
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Posted: 7 years ago 24.01.2018, 08:45 GMT-5

Thanks for your quick response.

I am indeed trying to define a strain energy density that is a function of both

and

From looking at the theory chapter and from what you are saying, it appears this is not possible. Is that correct? Will this functionality be added in a future release?

Thanks

Thanks for your quick response. I am indeed trying to define a strain energy density that is a function of both B=F F^T and C=F^T F From looking at the theory chapter and from what you are saying, it appears this is not possible. Is that correct? Will this functionality be added in a future release? Thanks
Henrik Sönnerlind COMSOL Employee

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Posted: 7 years ago 25.01.2018, 02:54 GMT-5

Hi Eric,

It is currently not possible to use an expression in terms of the B tensor directly. Maybe you can to re-formulate the the strain energy density function in terms of the C tensor?

Since the complete formulation of the hyperelastic material is based on a 'material frame' formulation with stresses (2nd Piola-Kirchhoff) related to material directions it is more natural to express the strain energy density function in terms of C which is a tensor in the material frame. B, however, has both indices in the spatial orientations, so dW/dB will give lead to a Cauchy type stress.

We will investigate if it is possible to allow other formulations too.

Regards,

Henrik

-------------------
Henrik Sönnerlind
COMSOL
Hi Eric, It is currently not possible to use an expression in terms of the B tensor directly. Maybe you can to re-formulate the the strain energy density function in terms of the C tensor? Since the complete formulation of the hyperelastic material is based on a 'material frame' formulation with stresses (2nd Piola-Kirchhoff) related to material directions it is more natural to express the strain energy density function in terms of C which is a tensor in the material frame. B, however, has both indices in the spatial orientations, so dW/dB will give lead to a Cauchy type stress. We will investigate if it is possible to allow other formulations too. Regards, Henrik
Eric Harper
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Posted: 7 years ago 01.02.2018, 16:34 GMT-5

Henrik,

I've been reconsidering my problem, and am wondering if COMSOL's PDE interface will allow me to compute what I need. I can express the weak form of the hyperelastic equation I need, but am running into issues in actually computing the quantities I need. Does COMSOL support any kind of vector/matrix/tensor math, or will I need to calculate each component of the tensor myself?

Thanks in advance

Henrik, I've been reconsidering my problem, and am wondering if COMSOL's PDE interface will allow me to compute what I need. I can express the weak form of the hyperelastic equation I need, but am running into issues in actually computing the quantities I need. Does COMSOL support any kind of vector/matrix/tensor math, or will I need to calculate each component of the tensor myself? Thanks in advance
Henrik Sönnerlind COMSOL Employee

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Posted: 7 years ago 02.02.2018, 02:44 GMT-5

Hi Eric,

If you implement it using the Physics Builder (which is the tool we use for our internal development), you have access to tensor math and other goodies. For example automatic transformations between different coordinate systems and frames, and consistency checks of various kinds (you cannot by mistake do math on tensors which do not have the same base system).

It may take a couple of days to learn the Physics Builder, but it is well worth the effort if you are going to do serious own implementations. Start with the tutorial in the Physics Builder Manual. That will give you a rather quick idea of what it is all about.

Regards, Henrik

-------------------
Henrik Sönnerlind
COMSOL
Hi Eric, If you implement it using the Physics Builder (which is the tool we use for our internal development), you have access to tensor math and other goodies. For example automatic transformations between different coordinate systems and frames, and consistency checks of various kinds (you cannot by mistake do math on tensors which do not have the same base system). It may take a couple of days to learn the Physics Builder, but it is well worth the effort if you are going to do serious own implementations. Start with the tutorial in the Physics Builder Manual. That will give you a rather quick idea of what it is all about. Regards, Henrik

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