Elastomer modeling comparison

[Jorgen]

This paper outlines the current state of the art in finite element modeling of elastomers, and exemplifies the predictive capabilities of modern constitutive theories for filled elastomers.

[burgeand]

Dear Jorgen,

I have read your summary of constitutive models of Elastomers.

In the section where you discuss the BB model you state that the other models "are all inherently incapable of predicting rate-dependence, hysteresis, and the response during cyclic loading". I am especially refering to the Qi-Boyce model. Qi-Boyce have developed a constitutive model for PU (2005) that builds upon the model you looked at in your comparision. Is it correct that this newer model overcomes some of the short comings you mentioned in your document?

Kind regards,
Andreas

[Jorgen]

I am familiar with the Qi-Boyce model for PU, and yes, that model is certainly capable of capturing both rate-dependence and hysteresis. Good point.

-Jorgen

[burgeand]

Hi Jorgen,

Another question concerning the BB (and the Qi-Boyce) model:

In my project I'm also interested in the long-term creep/relaxation behaviour of PU. Both the BB and the QB model include compression tests with short term strain holding periods (relaxation) to indentify the equilibrium stress-strain response. Since the material is not completely relaxed after these short periods I was wondering if I could combine these constitutive models with some sort of "pure" viscoelastic model (coefficients derived from either long-term creep/relaxation experiments or from DMA). Or do you think it is possible to lengthen the short holding periods?

Thanks very much for your help.
Andreas

[Jorgen]

My recommendation is to lengthen the holding periods. You can make them as long as you want. The BB and the QB models should work well also for long-term creep/relaxation behavior.

- Jorgen

[burgeand]

Thanks for the help.

Andreas

[burgeand]

Hi Jorgen,

I have another question concerning the BB model.

Did you use constant true strain rates in the mechanical experiments? The reason I'm asking is because the Qi-Boyce model uses true stress/strain together with constant true strain rates. In the BB model you also use true stress/stress so I figured you probably also used true strain rates.

I've been trying to figure out if constant engineering strain rates could instead be used. I haven't come to a conclusion yet if it matters.

Thanks
Andreas

[Jorgen]

Yes, I also used constant true strain rate. There is no reason, however, why you could not use constant engineering strain rate. I recommend that you use whatever is easier for you to use.

-Jorgen

[burgeand]

Hi Jorgen,


I have three questions regarding the BB model:

1)
----------------------------------
I have a question regarding the effective stress formulation for network B:

I have found about three different versions of it:

1.[Bergstroem-Boyce 2001]
tau_b=sqrt(tr(T_b'*T_b'))
N_b=T_b'/tau_b

2. [Bergstroem: Hysteresis and Stress Relaxation in Elastomers: from
polymerfem.com]
tau_b=sqrt(tr(T_b'*(T_b')^T))
N_b=T_b'/tau_b

3. [Bergstroem-Boyce 1998]
tau_b=sqrt(0.5*tr(T_b'*T_b'))
N_b=T_b'/(sqrt(2)*tau_b)

To my understanding, the Frobenious Norm is defined as follows:
||A||_F=sqrt(tr(A*A^T)).

1 and 3 would result in the same driving stress N_b but 2 would give a different N_b. However, the effective strain rate would be different for all three.... Or am I missing something and tau_b in 1-3 are equal?

2)
------------------------------
Another question concerns the formulation of the above T_b'

In your papers (ones mentioned above) you are using the following formula for the driving stress state on the relaxed configuration convected to the current configuration:

T_b'=T_b-1/3tr(T_b)I=dev(T_b)

In the Qi-Boyce model for PU (2005) they use the following formula for the stress acting on the viscoelastic-plastic component convected to its relaxed configuration:

T_v^-'=dev(T_v^-)
where: T_v^-=R^ve(Transp)*T_v*R^ve (page 829)

They use T_v^-' where you use T_b'

Where exactly is the difference coming from?

3)
--------------------------
In the formulation of the effective deformation rate:

Is the following equation correct for the term lambda_b^p which is used in the effective deformation rate (Paper 2001, Eq.3)?:

lambda_b^p=sqrt((1/3)*tr(B_b^p))
where: B_b^p=F_b^p*F_b^p^(transp)

Thanks a lot
Andreas

[Jorgen]

[1] Isn't T_b' symmetric, and hence the expressions are the same

[2] I don't have the Qi paper in with me so I will get back to you on that one...

[3] seems OK to me. Do you not agree?

-Jorgen

[burgeand]

thanks for your reply.

[1]
------------------------
that is a good point :) I overlooked that. Nevertheless, the effective strain rate would be different for 3:

All three use the following formula for the strain rate:

gamma_dot=gamma0_dot*((lambda_b^p)-1)^c*(tau_b/tau_base)^m

but tau_b for 3 (BB 1998) is different then tau_b for 1 and 2 (BB 01, Bergstroem). So in my view that would result in two different gamma_dot (one for 1,2 and one for 3).

[2]
---------------------
that would be great if you could get back to me on that one. Thanks!

[3]
--------------------
I do. I was just trying to clarify that I do not have to multiply B_b^p with (J_b^p)^(-2/3), like it is done for the chain stretch in T_A [lambda^(*): from B_(*)=J^(-2/3)*B] and in T_B [lambda_b^(e*): from B_b^(e*) =(J_b^e)^(-2/3)* B_b^(e)] but I just realized that J_b^p=1....or is it is this only in the very beginning when lambda+0

Thanks

[MarioJr]

I'm a college student and recently we did biaxial tension (and uniaxial tension) experiments on Natural Rubber. It was using a simple tester we just designed. It generated stresses using weights (by potential energy). Basically, we loaded weights of specific increments on a platform and measure the changes in lengths of the specimen and the fall of the platform.

We were supposed to plot Engineering Stress against Engineering Strain, modelling Stress through the YEOH model. We just needed an expression for the Strain Energy (for the Cauchy Stress). We assumed that this Strain Energy was equal to the Potential Energy but the results weren't that good (compared to PolymerFEM experimental data).

I hope you could enlighten me on this matter. Perhaps a guide to a better expression for Strain Energy or a comment on the assumptions or the procedure. Thanks.

Mario

[Jorgen]

The Yeoh model typically works relatively well. How bad was your prediction?

I think your approach sounds fine. Are you sure you don't have other experimental errors, for example from friction.

-Jorgen

[MarioJr]

Thanks for your response Dr. Jorgen.

The experiment was within a limited range of Engineering Strain [it was intended just for instructional purpose]. The errors were some 5% to 20%, but then increases to as much as 150% with increased Engineering Strain (between 0.15 and 0.30). I am inclined to think that the errors are more likely due to friction or maybe from the length measurements. Our instructor, however, thought that the expression of Strain Energy [assumed to be equal to the Total Energy, mass x g x height displaced by platform] could be a mistake in the first place.