# OGDEN, N=2 and N=3 Material Model

Hi,

I have tried one verification file of Ogden material model N=2 and N=3 under Uniaxial, Biaxial and Planar Tension.
Pending further configurations, It performs well but I have experience the intrinsic ccx limitation about pure incompressibility.

I’m not sure exactly up to which point Ogden would be usable but seems to be the point where the 3rd principal Strain becomes -0.5.
At that point, the Lagrange strain freezes, and the strain energy and Jacobian are not properly computed. Jacobian grows no matter how small I set up the Di coefficients, specially in Biaxial tension.

¿Should I consider that -0.5 Strain the limit point of validity for the Ogden Material model in ccx or is there some other considerations to should take into account?

Can you share the constants of those models and the strain range of interest (e.g. -0.5 to 0.5) ? I would use Abaqus to generate the curves and use an automatic stability check.

Thanks Calc-em. You always so fast.
Where could I find to learn how to do that?

*HYPERELASTIC,OGDEN, N=3

** µ1 , α1 , µ2 , α2 , µ3 , α3 , D1, D2
** D3 , Temperature1.

** µ1= 160,000,000 [Pa]
** α1= 2.00 [ ]
** µ2= 40,000,000 [Pa]
** α2=-2.00 [ ]
** µ3= 0 [Pa]
** α3= 1.00 [ ]
** D1=1.00E-09 [1/Pa]
** D2=1.00E-09 [1/Pa]
** D3=1.00E-09 [1/Pa]

You will find µ3= 0. That’s because I’m tring to see if N=3 can really cover N=1 and N=2 simplifing my input files.
I’m pushing up to Principal Lagrangian Strain=17.5 or it’s equivalent Principal Stretch= 6

Basically, you just have to define a material with hyperelasticity, then right-click its name and choose Evaluate. Select load cases like uniaxial, biaxial, planar and so on and specify strain ranges, then click Ok and wait for the results.

Here are the curves for uniaxial, biaxial and planar test with strain ranges -0.9 to 5. The material is shown to be stable for the whole range.

Thank you very much Calc_em. ¿Could you show how do they look like from 0 up to 10 ?
Values in compression are so huge that response under tension is Indistinguishable.
I would generate ccx curves to compare with Abaqus.

Sure, here’s what I got for that range:

Beautiful. Thank you very much.

I thought you showed me all the possible existing Strains but there seems to be a new one.

Nominal Strain……

According to Science Direct means: The nominal strain is defined by the length change divided by the original length.

That is the Engineering Strain. I guess it will be the same for Nominal Stress = Engineering Stress.

It would have a lot of sense to me that Abaqus has stablish a link between experimental data (Engineering Stress Strain/Curves and the corresponding Constants to be used for FEA.)

I will try to learn how to get those shapes or perform the regression…

Yes, nominal strain is just a different name for engineering strain. On the other hand, logarithmic strain may also be called true, Hencky or natural strain.

Hi,

I have finished one verification file of ARRUDA-BOYCE Material model. Unfortunately , this is the only hyperelastic material model that I have been able to find in the ABAQUS documentation for which the constants are provided.

This are the two reference pages I have been following.

http://130.149.89.49:2080/v6.11/books/bmk/default.htm?startat=ch03s01ach165.html

http://130.149.89.49:2080/v6.11/books/usb/default.htm?startat=pt05ch21s05abm07.html#usb-mat-chyperelastic

This are the constants I have used:

** µ1= 328000 [Pa]

** α1= 5.24 [.]

** D1=1.00E-09 [1/Pa]

328000 , 5.24 , 0.0000000001,275.0

328000 , 5.24 , 0.0000000001,375.0

I have set up all three recommended test, Uniaxial, Planar and Biaxial and compare results with Treloar Experimental data and Abaqus curves produced by those constants (Curves Nominal Stress, Nominal Strain).

I have compute Jacobian during the whole stretching process, and it has maintained below a maximum value of 1.001 respecting the incompressibility requirement requested in the problem statement.

Even known the ccx elements limitation regarding incompressibility (C3D8R) , result compare well with ABAQUS.

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According to the original reference: R.W.Ogden, Proceedings of the Royal Society of London SeriesA-Mathematical and Physical Sciences 326 (1972) 565–584, it has to be

Did you get good results with the above values?

on top of that, in the manual v2.20 Ogden model is written:

while the original formula shows:

I think is a misprint but could be a different formulation… (wikipedia shows same formulation as the original paper from Ogden)

right, it seems there are more than one Ogden formulation and the one used by ccx and abaqus is not the original Ogden’s formulation …Hyperelastic behavior of rubberlike materials

Hi, Juan.

I assume you are referring specifically to the imposition that the product of coefficients be positive. I am familiar with that condition and I have to say that I have already come across several papers that do not meet it.
I specifically referenced one of them a few days ago. "A study of balloon type, system constraint and artery constitutive model used in ﬁnite element simulation of stent deployment " by Mr. Alessandro Schiavone and Mr.Liguo Zhao.

(I consulted the author some weeks ago but I still have no answer).
I think that the background of that condition is nothing more than the need for the strain energy density to be always positive. That condition in the Abaqus formulation (ccx) carries over only to the Mu’s as the alphas are squared.
I believe that these functions (with their coefficients) are still local adjustments of the material response. It is possible that some coefficient are negative, but you are still working in a stretching zone where the general principle of positive energy is respected. I think that generalizing that condition can be quite restrictive.

I also have to say that the model of the paper diverges when I add the layer (Artery).
I am still looking for some reference to validate/verify the model while I go a little deeper into it, but I can’t find any analytical solution of it.

-Manual formula agrees with the Abaqus theory manual v6.6 formula (4.6.1-14).

https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/stm/default.htm?startat=ch04s06ath123.html

I have realized that Mr.Ogden original text says “sufficient” but doesn’t say “necessary”.

search for Drücker Condition Stability Check …seems to be difficult to assess validity of data…

I have a file with three equal small cubes and another with three equal small shells , each one under the main loading conditions, Axial, biaxial and Planar. You can figure it on the first picture of the post. I can put my material model and extract automatically the Nominal Stress Strain curves before going to the bigger model. It helps to check if coefficients have sense and which stress state could give me problems. One can anticipate the stability region too.
It has help me to progress relatively quick on the peculiarities of the different hyperplastic models.

Yes, that’s a good approach. I have those checks in the original publication but for the original Ogden’s model, so I have to figure out if the conversion

is correct.

Are you sure you can translate those coeficients from one formulation to other?. Those alfa_i are at the exponents too.
Probably comparing the Nominal Stress-Strain Curves to see if you end up with the same behaviour.¿?¿

Not sure, but looks like it’s what radioss documentation is suggesting…

Hi Juan,

I have found some verification files that provide the Material coeficients according to the Abaqus formulation (ccx). OGDEN N=2 is available.

Ogden coefficients (N=2):

I will asume this are [MPa] and [1/MPa] for D1.

Unfortunately they don’t satisfy the necessary condition (26) for Ogden N2.
Point X. If someone animates to double check we can later compare. I will be doing Neo Hookean and Mooney-Rivlin too as I’m still involved to clarify about this two models through other channel.

https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/ver/default.htm?startat=ch02s02abv132.html