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Constitutive Modelling of Elastomers

Mohsin Hamzah's picture

Rubber or rubber-like materials, or generally elastomers, sustain large elastic deformations. The problems of such cases are non-linear, the non-linearity came from two sources, the first one due to materials, and the second is geomertrical non-linearity. Elastomers are, also, viscoelastic, i.e. time and temperature dependent.

Modelling elastomers, are difficult task, cause the problem is nonlinear, and time and temperature dependent. So, our object is to build models that satisfy both large deformations behaivour and time and temerature one. 

Any informations conceren elastomers modelling are of our interest.....


Roozbeh Sanaei's picture

Tahoe is a research-oriented, open source platform for the development of numerical methods and material models. it can be use for modelling such materials and it's free.

Roozbeh Sanaei's picture

 Can you give some references about application of this materials modelling?



You are right, mechanical behaviors of rubbers and rubber-like materials are complicated. Therefore, before you start to consider to use existing models or to develop your own model, you need to know what you want to do. Typically, you need to consider the following:

1. Most rubbers or rubber-like materials can sustain very large deformation. As you said, you need to consider large deformation (or finite deformation model);

2. Although rubbers show viscoelastic behavior (time dependent behavior), pure rubber or unfilled rubber usually shows less time dependency than filled rubber. In such a case, you can use hyperelasticity models. There are many existing models, such as Mooney-Rivilin model, Arruda-Boyce model. Here is a very good review of hyperelasticity: Boyce, M.C., Arruda, E.M., "Constitutive Models of Rubber Elasticity: A Review", Rubber Chemistry and Technology, 73, 504-523, 2000.

3. Rubber composites usually shows strong time-dependent behaviors. In mechanical modeling, such a time dependent behavior are usually decomposed into a hyperelastic behaivor (time-independent) and a time dependent behavior. Many models have been developed in the past, including models by Boyce and co-workers, by Govindjee and Simo and their co-workers, by Anand and co-workers, by Lion, by Miehe anc co-workers, by Marckmann and co-workers, and many others.

4. Another interesting behavior of rubbers is the so-called Mullins' effects. It is also known as "softening effects" in rubber composites. It is characterized by the observation that a rubber composite becomes softer after being stretched. There are also some models developed in the past to consider this type of behavior.

Regarding software, ABAQUS is a commercially available software that has many built-in models for hyperelasticity and some models for time-dependent behavior. A nice thing about ABAQUS is that you can use their material model to consider complicated problems, such as contact problems for rubbers.  Of course, you need to pay some license fee to use it. Tahoe is also a good software and it is free.


Zhigang Suo's picture

Dear Jerry:

Thank you so much for this succinct overview of issues concerning the mechanics of rubbers.  Could you please explain why time-dependence is more pronounced in a filled rubber and composites?  I'd like to have a mechanistic picture.

Henry Tan's picture

Dear Jerry and Zhigang,

The nanocomposite material we are studying (polymer filled with nanoscale hard particles) shows an opposite trend: less time dependency for composites.

MichelleLOyen's picture

Well the key question here is "filled with what?" Since Jerry Qi has published a number of papers on thermo-plastic elastomers, the comment about increased time-dependence in "filled rubbers" might be a bit generalized for all reinforced composites but absolutely characteristic for those material systems. 

As I noted previously, the Lakes text "Viscoelastic Solids" has a very nice discussion of this issue of viscoelastic composite responses which is probably worth examining for the 3.5 pages of literature references alone. Note though that again the discussion is using elastic-viscoelastic correspondence and is thus limited to LINEAR viscoelasticity!


There are other important variables that determine the response of viscoelastic composites:

  1. The volume fraction/geometry of the filler material and the matrix.
  2. The frequency of the applied load.
  3. The amount of deformation, etc.

Consider the explosive PBX 9501. This composite consists of 5% (wt) of a elastomeric rubber binder and 95% (wt.) of HMX crystals. Both constituents are polymers but with different glass transition temperatures. At room temperature, the binder is viscoelastic and strongly rate dependent while the crystals ar not. What do you expect the composite response to be? Strongly rate-dependent or not? Turns out that even though the amount of binder is small the composite is strongly rate-dependent at high frequencies. However, one could design materials where this is not necessarily the case.


The frequency of the applied load is important also because there could be some resonant frequencies where the behavior of the composite is quite different from normal, even for linearly viscoelastic materials.


If the deformation is large, the filler phase will probably debond from the matrix and the response of the material will then be viscoelastic-plastic. Also, the linear superposition that is the basis of standard viscoelasticity may no longer be appropriate.


Nonlinear viscoelasticity is a fascinating subject. A couple of references that I use are:

MichelleLOyen's picture

The example given here just goes to show that you have to solve each problem for a viscoelastic composite individually! There are a large number of variables in these problems. To summarize, for any composite system, including one with time-dependent mechanical behavior in the experimental reference frame, the key variables that determine response are:

(1) phase fractions and geometries;

(2) composition of phases and the resulting bonding between the phases

(3) full constitutive law for the mechanical response of each phase, be they elastic, elastic-plastic, viscoelastic, or whatever else. (NOTE: this must be the constitutive law as characterized over the appropriate rate or time regime covered by the applied loading conditions!!! Problem number one with viscoelastic materials is that extrapolation from one time domain to another is frequently impossible.)

(4) applied loading conditions, in terms of magnitude, frequency/time-scale, and state of stress

(5) temperature

Another recommendation for a book covering the mechanics basics nicely is from a slightly different background:

Rheology, by C. Macosko.

Henry Tan's picture

How to characterize, and measure, the bonding between different viscoelastic phases?

Henry Tan's picture


For linearly viscoelastic materials, can the resonant frequencies, that cause the behavior of the composite different from normal, be derived analytically?

huang peng's picture

     I am an engineer. usually,we use Marc to modeling elastomers.As you know,modeling elastomers behaviour is a challenged task.There are so much constitutive model in MARC,such as Mooney,ogden,Gent model,and we can obtain good results  using MARC usually.

     Of course ,you can add your model to marc material model by using subroutine.

Huancheng Tan's picture

strength simulate

Huancheng Tan's picture

Dear all,

Now I use Mooney-Rivlin of marc to simulate a rubber component. I have C10,
C01 and poisson's ratio, at least,marc requires C10,C01 and Bulk Modulus (K).
From marc.out file, I kown "No K is defined. Marc calculates it as 5000 times
inital shear modulus". However, I get "The default value for the Mooney-Rivlin
model represent nearly in compressible condition, which is k=10^4(C10+C01)" from Quick reference Guild.pdf of Msc.I
am confused how to explain the two ways, and how to calculate K from  C10, C01
and poisson's ratio three coefficients? If availble, could you send some
relative materials to my email, especially, the
materials include examples about rubber.

Thank you in advance!

Huancheng Tan

Zhigang Suo's picture

Rubber-like materials are often modeled as incompressible materials.  See a previous thread of discussion.  In finite element method, however, incompressible materials cause numerical issues.  As a result, rubber-like materials are often given some arbitrary small compressibility. 

If you wish to calculate Poisson's ratio from shear modulus and bulk modulus, you can do so by using the standard formula.  However, because the bulk modulus assigned is often just for numerical simulation, and has nothiing to do with the material itself, Poisson's ratio calculated from the fake bulk modulus will have nothing to do with the material.  

Huancheng Tan's picture

Dear Prof. Suo,

Thanks for you quick reply. I have read the "previous thread of discussion" before. In it, I saw "K/G=2(1+v)/(3(1-2v)). So v->0.5 implies large K/G but not large K.".However, I cannot get the K value, just K/G value. Marc requires the K value. From technical support of MSc, I get the formula K=G/(1-2v),G=(C10+C01), the default of V in MSC is 0.49995, so get the K=10^4(C10+C01)=10^4G.In marc.out file that"No K is defined. Marc calculates it as 5000 times inital shear modulus", So we can get K=5000G, not 10^4G. Who is wrong and what is the right formula? From the "Quick Reference Guild.pdf" of MSC, I get "Recommended values for poisson's tator are between 0.490 and 0.495 or higer. Low values may lead to in stabilities", but I use K=G/(1-2v) input to MARC,v=0.485 can get a available stiffness result, which reconciles to experiment result.Is the v=0.485 reasonable? I have calculate many times, if you change the V very little, and input the K value to MARC from the formula K=G/(1-2v), the stiffness tremendously change. The tend is that vary v to smaller, get a small K value.

Thanks in advance!

Huancheng Tan

Ettore Barbieri's picture

Dear Huancheng,

I'm implementing Mooney-Rivlin in my in-house code. The bulk modulus K in MARC (or any other code, really) is more of a "penalty factor" in the Mooney-Rivlin model, in the form of k*(J-1) or k*ln(J) (J is the determinant of the deformation gradient), to enforce incompressibility (which is J=1). For this, see the classic textbook of Belytschko, Moran and Liu or the book by Bonet and Wood.

Of course, there are other ways of enforcing incompressibility (Lagrange multipliers), but penalty methods seem to be the most popular, since it does not involve extra-unknowns. However, this comes at a cost. 

In the penalty methods, k is an "arbitrary" large number. However, too "large" can cause ill-conditioning of the tangent stiffness matrix, as prof Suo pointed out. Therefore, k = 1e3(C10+C01) it's a reasonable "rule of thumb", to have a high penalty, but not much larger than the other material constants. The bulk modulus calculated in this way is only a penalty factor, without being necessarily a material constant: it will just to the "job" of imposing the constraint J=1. You should get, as a consequence, a Poisson ratio "close" to 0.5. 

So, if you instead of k=1e3(C10+C01) you choose a "Poisson ratio-based" rule, you risk having a very very high penalty factor (at the limit, infinite, as it approaches 0.5). Therefore, you don't need to specify a Poisson ratio, as long as what you get eventually is J = 1, or approximately J=1, which automatically implies a Poisson ratio = 0.5. 

Huancheng Tan's picture

Dear Ettore,

Tanks for your full reply. The question is that the stiffness that is calculated with k=1e3(C10+C01) by marc is not consistent with experimental stiffness. The calculated stiffness is  three times larger than experimental stiffness.Maybe my experiment is not accurate.


Huancheng Tan



If I am wrong, discard my words.   

Since you have an experimental data of ruber, why don't you fit the incompressible Moone-Rivin model to your experimental data and find out all the parameter of model? I assumed you have a data of 3-d volume-pressure data to find out bulk modulus.  

I think K(J-1), the penalty factor is the same as the lagrange multiplier. 

 If your ruber modulus K is not high, why do you always focus on incompressible hyperelastic model? I may think compressible hyperelatic model could also be a good choice. The strain energy function then has one or several parameters to capture compressibility of materials. In this case, in my opinion, the bulk modulus is keeping changing during loading. This hyperelastic bulk modulus will go back to the bulk modulus of linear elasticity as the whole things you derived. Its special case is J = 1 for incompressible hyperelastic materials, or \epsilon_ii = 0 for incompressible linear elasticity (possion ratio = 0.5).

Hope it has some help to you!


Lixiang Yang

Mohsin Hamzah's picture

Hello Huang

I know that Marc is an excellent  FEM software, but, first, I have no copy of this software, second how can I learn to use it? If you can help with this I will be gratful



verron's picture

Classically, the time-dependent stress strain response of elastomers is considered as viscous (or viscoelastic), if the Mullins effect is not considered. So it leads to a large number of constitutive equations in integral form (in the 70s or 80s, inspired by the KBKZ approach) or with internal variables (papers of Lion, Bergstrom and Boyce ...).

Nevertheless, recent experimental works performed by polymer physicists have demonstrated that the well-known stress-strain hysteresis and the relaxation phenomena observed in rubberlike materials are closely related to strain-induced crystallization (see the works of Toki et al., and Rault et al. published between 2000 and 2007, it was also partly observed in the 60s).

In these works, the influence of fillers on the viscoelastic response is explained by considering that fillers are nucleation sites for crystallites (because strain is larger than the macroscopic strain level in the neighbourhood of fillers). Thus, the kinetics of crystallization is amplified by fillers.

Finally, we can say that there are only few models dedicated to strain induced crystallization in rubber. 


Henry Tan's picture

Jinglei Yang’s ( PhD work (2006) on TiO2/polyamide 66 composite materials has demonstrated that fillers are nucleation sites for crystallites.

Bridging segments forms between the filler and the polymer matrix. Particles and the bridging segments could form a huge and dense network.

This network, together with the crystallized polymer chains, could enhance the capability to bear load, markedly at elevated temperatures.


Dear all,

 I am a student working in crack development in filled elastomer

Concerning Mullins effect : Most papers I read mention the problems of cohesion between fillers and the matrix, but nobody was able to give a direct experimental proof of what is happening.

The problem, i think, is that we can't measure the cohesion energy between fillers and matrix (but we can calculate it with the bond energies).

I was wondering if you knew papers concerning Mullins effect (it is possible that i have forgotten the most interesting papers). I know papers trying to modelise this effect, but which one is the most caracteristic ?  

There is another phenomenon that i read in one or two Phd memories, it is the continuous softening, that is to say : when you apply a cycling sollicitation (for example 10 cycles) on a sample(let's say a uniaxial sollicitation) at the same strain amplitude, we have of course, between the first and the second cycle, the Mullins effect, and then when you see the 8 other cycles, the stress amplitude always decrease. And this would be due to an accumulation of damage. But i read also that this phenomenon was debate and some people say that it is not a softening. Have you already heard about this phenomenon and do you think it is really due to a damage ?

 Thank you for letting a student like me reading your discussions that are very rich.


Dear All,

 I am a Masters Student in Material Science, I am supposed to do the Finite

Element modeling of PTFE material with few content of short carbon fibers. Could any of you please suggest the best,

 1.Modeling method, its

2.Material parameters and 

3.References of that best modeling method.


Would be really thankful and would be helpful for my studies as well. 



Raghu Raman Rajagopal



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