Hysteresis and relaxation model

[miguel_roque]

Hello Dr. Bergstrom,

I have been reading the model that you have developed, because i want to implement it. The problem is that i don't understand how to find the deformation gradient (elastic and viscous) of chain B.

Best regards,

Miguel Roque

[Jorgen]

Hello Miguel,

The way to find the deformation gradients of network B is to use the viscoelastic rate equation ("flow rule") in order to determine how much of the applied deformation is viscoelastic.

Best of luck,
Jorgen

[miguel_roque]

Hello Dr. Bergstrom,

First off all, thank's for your reply.

I have i question, you said that the way to find the deformation gradients of network B is using the "flow rule", but i don't know this rule and how to apply it. Could you please explain it, giving the steps that i have to make?

Thank's in advance,
Miguel Roque

[ashu28]

As an initial step, try explicit scheme. Start assuming everything to be elastic. Calculate gammapdot and Dp (you may find relevant equations in the model you are interested in). Update Fp and calculate stress. This is the general way to go about it, may vary a bit depending on the particular model.


Ashu

[miguel_roque]

Hello Ashu,

I still don't understand how to determine the deformatin gradients for network B. You said that i should start assuming everything to be elastic, but how do i reach to the viscous and elastic deformations gradients for network B? And what do you mean with Dp and Fp, i´m not quiet following. I'm very new on this matter :roll:

Miguel Roque

[ashu28]

Which paper are you exactly refering to ??
Also, I would recommend looking into the refereces is going to help you a lot in understanding the development of a series of models for elastomers and thermoplasts as well.


Ashu

[ashu28]

Which paper are you exactly refering to ??
Also, I would recommend looking into the refereces is going to help you a lot in understanding the development of a series of models for elastomers and thermoplasts as well.
In case you have some specific problems, feel free to ask.

Ashu

[miguel_roque]

The paper i'm refering to, is the thesis of Jorgen Bergstrom, "Large Strain Time-Dependent Behavior of Elastomeric Materials". My problem is i don't know how to find the deformation gradients (elastic and viscous) of network B.
Dr. Bergstrom already said that i must use the "flow rule" but i don't know how to do it. I will try to find some information in the references, but if you could advance something more i would be thankful.

Miguel Roque

[Jorgen]

Miguel,

I agree with ashu28 about first using an explicit solution approach. I have to admit that I was purposely somewhat imprecise in my statement in order to indicate that finding the deformation gradient of the viscous part of network B is somewhat involved. The approach is to figure out how to update Fbv for a given time increment dt with a given F. All you need to perform the calculations are written in my papers, although the equations are written to indicate the theory and not to demonstrate various integration schemes. Perhaps there is a need to write a paper outlining the implementation in more detail.

Feel free to ask specific questions about the implementation of the BB model.

- Jorgen

[miguel_roque]

Dr. Bergstrom,

I have some questions about finding the estimated equilibrium curves/ equilibrium stress. I think this will help to find the deformation gradient of network B.

1- How can i estimate the equilibrium locus, in order to determine the material contants for the eight-chain model?
2- Are the equilibrium curves calculated from the eight-chain model?
3- Time-dependent stress = Total Stress - Equilibrium Stress ?
4- Is network B given by the Time-dependent stress/strain?

Thanks in advance,
Miguel Roque.

[miguel_roque]

Hello,

In my last post i put some questions, but a still don't know how to estimate the equilibrium locus, in order to determine the material constants for the 8-chain model. Is this estimate visual?

I have another question concerning the BB-model. After the determination of all the material constants, for network A and B, how can i use the Cauchy stress (network A and B) with the effective deformation rate of chain B, in order to implement the BB-model?
I don't know how to group that information in order to have the BB-model, could someone help me?

Thanks in advance,
Miguel Roque

[Jorgen]

Miguel,

Here are some more comments:

1- The equilibrium locus can be determined from stress relaxation tests (see for example this data (http://www.polymerfem.com/modules.php?name=Materials_Models&material=1)).
2- The eight-chain model is used to predict the equilibrium curve.
3- Yes, that's the definition of the time-dependent stress
4- Yes

The implementation of the BB-model goes something like this:

- Known values at time t: F, Fbv
- Known values at time t+dt: F

The goal is to calculate the stress and Fbv at t+dt. This can be done as follows:

(1) calculate stress TA at t+dt
(2) calculate the time derivative of Fbv at time t using the "flow rule"
(3) calculate Fbv at time t+dt using some ODE solver
(4) now that Fbv is known, calculate TB at t+dt
(5) calculate T = TA + TB at t+dt
(6) goto (1)

- Jorgen

[miguel_roque]

Hello Dr. Bergström,

In your paper "Large Strain Time-Dependent Behavior of Elastomeric Materials", you modeled network A and B with the generalized Arruda-Boyce model. My questions are:

1. Can the elastic part of network B be modeled with any hyperelastic model, for example Mooney-Rivlin or Ogden, from the stress in network B as a function of applied strain?

2. If yes, how can i find the material constants for that hyperelastic model? How can i see the elastic part of network B from the stress as a function of applied strain?

3. The stress-strain data can be engineering or must be true?

Thanks in advance!

[Jorgen]

Here are my answers:

(1) Yes, you can use any hyperelastic model. Both M-R and Ogden are perfectly valid to use.

(2) You can find the material parameters by considering the equilibrium stress and the time-dependent stress. You can also find the parameters by "fitting" the model to the data.

(3) You can use either engineering or true data (as long as you are consistent).

- Jorgen

[miguel_roque]

Dr. Bergström,

Thank you for your answers, they were very helpful :)

I just want to clarify something that i didn't understand very well, that is: when we subtract the equilibrium response from the experimental data we find the stress-strain response of network B, or in other words, the time-dependent response, right? So the stress-strain of network B as an elastic and a time-dependent components.

My doubts are:

1. how can i see in this response (stress-strain of network B) the corresponding elastic part?

2. is the elastic part of network B linear?

Thank you very much in advance!

[Jorgen]

You can get information about the elastic response of network B by looking at the slope of the stress-strain of network B at zero strain and at strain reversals.

The elastic part of network B is typically not stretched very much and its response is almost linear. I often use the neo-hookean or the eight-chain model to represent network B.

- Jorgen

[miguel_roque]

Dr. Bergström,

You have mentioned in one earlier post, that i could use engineering or true data, but after all the plots and calculations done with engineering data, the material paraemters for the "flow-rule" didn't get right. For example the parameter C2, wich is suposed to be closer to -1, to me is 1.5 :?

I used the data from Viton, available in the polymerfem site, but i don't know if the material parameters should be taken from true data instead of engineering. In your paper, you work only with true data but i need to work with nominal because of abaqus.

1. Could you please explain how the material parameters (for the eight-chain and BB-model) are taken from engineering data or if they are equal to those taken from true data?

2. How can i calculate the effective creep strain rate in network B?

In the page where the material data of Viton is downloaded, the first picture of the BB-Model as the values to the material parameters, but only the values. Is it possible to connect those values with the material parameters? It will help.

Thank you very much!

[Jorgen]

You can use either engineering or true stress and strain when finding the material parameters. However, you need to make sure that you are consistent. If you use engineering stress and strain, then you need to also make sure your stress calculations use engineering stress and strain.

(1) One of the reasons I often use true stress and strain is that ABAQUS internally use these quantities (and not the engineering values). If you want to use engineering stress and strain, then you need to convert all experimental and model predicted stresses and strain to engineering.

(2) If you have the parameters for network B, then you can determine its creep rate. So I guess your questions is how to find those parameters. One approach that I often take is to find the parameters by fitting the model to a combination of stress relaxation, or creep, or monotonic uniaxial tests at different rates.

(3) The given parameters for the BB model that are given on this web site are as follows:

1. muA
2. lamLockA
3. kappaA
4. Tbase (for temperature dependence)
5. sB
6. C
7. m
8. n (for temperature dep.)
9. tauBase
10. Delta

- Jorgen

[miguel_roque]

Hello Dr. Bergstrom,

1. In that case the material parameters are independent of the data (engineering/true) used, is it?

2. What is the meaning of sB, tauBase and Delta in your last post?

3. Thus the Arruda-Boyce model implemented in ABAQUS give the same results of the Eight-Chain model that you used in your thesis?



The implementation of the BB-model goes something like this:

- Known values at time t: F, Fbv
- Known values at time t+dt: F

The goal is to calculate the stress and Fbv at t+dt. This can be done as follows:

(1) calculate stress TA at t+dt
(2) calculate the time derivative of Fbv at time t using the "flow rule"
(3) calculate Fbv at time t+dt using some ODE solver
(4) now that Fbv is known, calculate TB at t+dt
(5) calculate T = TA + TB at t+dt
(6) goto (1)

- Jorgen

4. How do i know Fbv at time t ?

5. At (3), is the ODE the "flow rule"? If yes how can i calculate Fbv at time t+dt if i don't know TB(t+dt) yet?

Thank you very much in advance.

Miguel

[Jorgen]

1. I don't quite understand your question. The material parameters are determined by the experimental data. You can use either engineering or true stress & strain, as long as you use the data appropriately.

2A. sB is defined by muB / muA
2B. tauBase and Delta are describe in my papers (http://www.polymerfem.com/modules.php?name=Downloads&d_op=viewdownload&cid=2) about the BB-model.

3. Yes the ABAQUS Arruda-Boyce model is the same as the Eight-chain model that I often use.

4. You know Fbv at time t from the state variables that was calculated in the previous time increment.

5. Yes the ODE is given by the flow rule. Note, however, that you have a system of ordinary differential equations. One equation for each component of the tensor Fbv. You can perform the calculation in different ways depending on what ODE solution approach that you want to use.

- Jorgen