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A Finite Element Method for Transient Analysis of Concurrent Large Deformation and Mass Transport in Gels

Hanqing Jiang's picture

A gel is an aggregate of polymers and solvent molecules.  The polymers crosslink into a three-dimensional network by strong chemical bonds, and enable the gel to retain its shape after a large deformation.  The solvent molecules, however, interact among themselves and with the network by weak physical bonds, and enable the gel to be a conduit of mass transport.  The time-dependent concurrent process of large deformation and mass transport is studied by developing a finite element method. 

We combine the kinematics of large deformation, the conservation of the solvent molecules, the conditions of local equilibrium, and the kinetics of migration to evolve simultaneously two fields: the displacement of the network and the chemical potential of the solvent.  The finite element method is demonstrated by analyzing several phenomena, such as swelling, draining and buckling.  This work builds a platform to study diverse phenomena in gels with spatial and temporal complexity.

Revision: ABAQUS UEL Fortran source code is attached with limited comments. Please contact me for questions/comments/discussions.

PDF icon Jiang Hydrogel.pdf1.03 MB
Plain text icon GelUEL.for_.txt45.16 KB


Rui Huang's picture

Hi Hanqing

Thanks for posting your very interesting work! Could you give some details about the User-Element subroutine you used in ABAQUS? As I remember, in an earlier work by Wei Hong et al., they used UHYPE (user-defined hyperelastic material) subroutine to analyze the equilibrium deformation of hydrogels. Something different has to be done for the transient analysis as shown by the numerical examples in this manuscript.



Hanqing Jiang's picture

Dear Rui:

Regarding ABAQUS UEL, we follow the standard procedure to formulate element stiffness matrix, starting from the shape function, the integration scheme, material matrix, to stresses. In fact, it does not have big difference between steady state or transient problem, if you use UEL. The main difference is that chemical potential is a field varibale in transient analysis, while it is a constant (or uniform parameter) for steady state analysis. For the examples in the manuscript, yes, we do need to use some "tricks", such as the modificatioin of the stiffness matrix. 



Hanqing Jiang's picture

Dear Rui:


I just posted the Fortran source code.

Hi Hanqing,

This is Chunguang Xia from UIUC. How are you? Thanks for sharing your nice work. In your paper, you applied Prof. Sou's theory to dry sample, but the incompressbity condition in his paper, 1+aC=det(F) is not for dry sample. Because there is deformation in dry area, but the concentration C=0. The phenomenons in your simulation examples can also be predicted by most of others. In order to test Prof.Suo's theory, I think another very important aspect is the time scale, can his theory predict the swelling time scale more accurate than others(Li-Tanaka,Wang-Li-Hu,Tatsuya-Doi,etc). 



Hanqing Jiang's picture

Dear Chunguang:

Thanks. Firstly of all, unlike biphasic theory or coupled solid/fluid mixture theory, Prof. Suo's theory does not have "dray area". The deforamtion gradient F is for the mixture (i.e. gel).  If a dry gel, which has no solvent molecules or C =0, is taken as the reference configuration, det(F) will be zero. Also, I thank you for pointing our the time scale issue. The framework proposed by Prof. Suo does not depend on material model, namely free energy function and mobility tensor. To more accurately predict the time scale, a more accurate material model is needed.

PS. By the way, I will be at UIUC next Tuesday. Maybe we can meet



Hi Hanqing, i am not good at thermodynamics and  thank you and wei for  helping me out the confusion. I am free all the next Tuesday. Let me know your plan.




Wei Hong's picture

Hi Chunguang,
There are two things I might need to comment a little bit more:

  1. Our theory does not need  1+aC=det(F) at the first place.  The condition is just a simplification based on the incompressibility of the material.  For incompressilbe material, even when it is dry, C=0, det(F)=0 should be satisfied.  So there is no problem at all applying the theory to dry network.
  2. At the current stage of research on hydrogels, I don't think there is any theory that could "predict" the behavior of a gel.  It is more or less a curve fitting or extracting of material constants from observations.  The theory presented in our recent paper is rather a theoretical framework than a material model.  It would not predict anything without experimental input.  In terms of material model, we merely adopted Flory-Huggins theory for the static behavior, and linear kinetic law for time-dependent behavior, as an example in that paper.  If fitting a single curve is regarded as prediction, then all existing theories could "predict" equally well.

Hope these comments will resolve your concern. Wei

Hi Wei, I quite agree with you on the current theories on hydrogel. And I am desperately looking a field theory that can explain our experiments. And that is why I am excited about yours and hanqing's work. I am very happy to see if your theory can explain and predict my experiments.



Wei Hong's picture

Hi Chunguang,

I will be in Champaign next week, too.  Let three of us find a time to meet.

Tuesday during lunch time or in the evening will be good for me.


hi wei, sorry that i did not follow up with this post. too late to see the post.

Hua Li's picture

Very nice to see you again. Thanks for Zhigang's email to let me know the latest output. It is really exciting work, and also challenges me to gain an insight
into. I am still on the surface and will learn from you.

you said that "the solvent molecules, however, interact among themselves
and with the network by weak physical bonds, and enable the gel to be a conduit
of mass transport" (see in Abstract), and also that "Elements of the
gel in different locations may not be in equilibrium with each other, and this
disequilibrium motivates the solvent to migrate" (see Page 6)”.  As
such, I didn't get your point – what is a driving source for mass
transport? Is it possible to give more detailed info.

minor suggestions:  (1) you use "Nk(X)" representing the
unit vector normal to the element of area, and "Na(X)" the shape
functions, which may confuse some readers due to the quite similar symbols. (2)
It could be much better to add the reference citations for the parameter values
(in Page 10) you use as input data, and then we can follow up. 

Thank you for your effort,


Hanqing Jiang's picture

For your question 1, the driving force for mass transport is the gradient of chemical potential. Meanwhile, the chemical potential also depends on mechanical deformation. Therefore, for concurrent deformation and mass transport, as stated in the manuscript, "
", explicitly, disequilibrium of mechanical field and chemical potential, drives the mass transport.

We will make it clear for your question 2 when we revise the manuscript. Thanks.

I want to model a RCC beam using ABAQUS software. plesae help me....



subhajit mondal

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