Implementing hyper-viscoelastic constitutive model in UMAT

[VIPer]

Greetings,
I've recently developed a nonlinear viscoelastic constitutive equation which is supposed to model the mechanical properties of HDPE in a spectrum of elastic deformations.

The constitutive model is based on the methodology developed by Pioletti et. al:

http://infoscience.epfl.ch/record/89...(Pioletti).pdf

To be specific: the total stress is the sum of the elastic response, viscous stress and so-called "long term response" modelled with a heridetary integral (see Pioletti's paper for the details). Somethin'g like Fung's quasi-linear viscoelasticity but with an extra viscous potential Wv to model strain rate dependency.

Now I'd like to conduct some one finite element tests in Abaqus. I'm not very experienced in this matter but for whole I know I should use UMAT subroutine.

I should determine the tangent moduli tensor for UMAT. And here comes my dilema becouse in this kind of constitutive relation I have to potential funtions: We (elastic) and Wv (viscous) so I can compute two various tangent moduli tensors. I have quite no idea how to deal with it.

In case of normal quasi-linear viscoelasticity I would determine the tangent moduli from the elastic potential We. Then, if I'm correct, I would get the elastic stress which then would be incorporated in numerical integrating of a heridetary integral.

But there is this viscous potential... Maybe somebody here has some clue to this (recommended reading for instance).

Regards,

VIPer

[Jorgen]

I am not familiar with the details of that specific model, but your approach sounds right. You need a UMAT and your need the tangent stiffness (Jacobian). Note that implementing that type of model will not be easy or fast. There is no simple way to reach the solution. My recommendation is that you study continuum mechanics very carefully. Check out Holzapfel's book.

Also, I have done a lot of work on HDPE and I have developed some very powerful material models for that material. For example, the Three-Network Model, and the Parallel Network Model. Both of these models are commercially available from Veryst Engineering.

-Jorgen

[VIPer]

Thanks for your reply.

I'm familiar with some of your papers. You use Saint-Venant-Kirchhoff model together with evolution equations of several state variables. And by this way you achieve good modeling of nonlinear elasticity, plasticity, strain rate dependency, different loading-unloading behavior etc.

Personally I'm a big fan of this Fung / Pioletti framework. I can model nonlinear elasticity, strain rate dependency and stress relaxation which is quite enough for my purposes.

Note that implementing that type of model will not be easy or fast. There is no simple way to reach the solution.

You mean implementing the equation in UMAT, or generally developing a constitutive equation?

I think that UMAT code for my model should be something similar to that example of a viscoelastic model from Abaqus' documentation.

I have elastic stress tensor and viscous stress tensor. The total stress is a sum of these two. I can imagine computin a Jacobian for it. What bothers me is that I need pure elastic stress for the hereditary integral. Would I have to extract it somehow from the total stress calculated with the Jacobian... ? Maybe somebody did something of that sort in the past. Anyway, thanks for your remarks. I'll check Holzapfel's book. I've been considering buying it for some time.

[Frank]

Hello VIPer,


regarding
"I think that UMAT code for my model should be something similar to that example of a viscoelastic model from Abaqus' documentation."

This example in the Abaqus manual for a viscoelastic UMAT starts with 1D linear standard solid model. Then it is generalized to the threedimensional form and introduces three more constants. It is unclear from the manual how these coefficients relate to spring and dashpot coefficients.

The threedimensional solution is detailed in my PhD thesis, chapters 3.4, 6 and Appendix B:

Upsetting and Viscoelasticity of Vitreous SiO2: Experiments, Interpretation and Simulation
http://opus.kobv.de/tuberlin/volltex...179/index.html

I couldn't hold back at the time and expressed my frustration in the paragraph starting with 'The code was straightforward to program once the obstacles...'.

So much for that.

Actually, I am now also in biomechanics and read Pioletti's thesis. Note that the FEM in his thesis is a simpler form of the model that had been established before as being appropriate.

My conjecture is that you have to implement his equation 4.13. It is not related to a spring-dashpot model.

Note that there is also the possibility to program a UHYPER subroutine. I do not know whether or not this is more appropriate, given that you also have to implement the viscous term.

My current project is still somewhat undefined and I do not know if I will also try to implement this model in ABAQUS.

Good luck

Frank

[VIPer]

Hello Frank,

Thanks for your reply. I've already downloaded your thesis and I have read the discussion regarding it. It encouraged me to register to the forum.

When the general constitutive equation proposed by Pioletti is limited to the sum of elastic and viscous stress (without the heridetary integral) it is very similar to the one-dimensional constitutive model derived from Kelvin-Voigt spring and dashpot model.

Thus I deduce that implementing in Abaqus that constitutive model which uses both elastic and viscous potential should be similar to that implementation discussed in UMAT manual. Although the model discussed in the manual is not Kelvin-Voigt model but Zener model if I'm correct.

There are numerous papers regarding derivation of such constitutive equation. But so far I haven't come across anybody who actually did implement it in FEM. As you say, even Pioletti didn't implement the constitutive equation he derived. He limited himself to implementing pure hyperelastic model.

Here is a paper dating 2010, where some guys claim that they are planning to implement Pioletti's type of constitutive equation in the future:

http://publications.lib.chalmers.se/...ext/128557.pdf

They want to use it for the purpose of muscle modelling. I browsed through their paper and noticed that equation (2.30) is some kind of expression for a tagent moduli related to the hereditary integral. They derived (2.30) from Taylor's algorithm for numerical integration of hereditary integral.

My conjecture is that you have to implement his equation 4.13. It is not related to a spring-dashpot model.

Do you refer to (4.13) form Pioletti's thesis? It's one-dimensional simple tension equation derived from pure hyperelastic model.

I think that for this entire Pioletti's nonlinear viscoelastic equation with viscous potential etc., UMAT is the only option. For whole I know UHYPER is just for hyperelasticity and there is no way you can implement the viscous part and the hereditary integral in this kind of subroutine.

Good luck to you too!

VIPer