## Introduction Parallel Rheological Framework

In this tutorial I will show how to first select a proper Abaqus Parallel Rheological Framework (PRF) model structure, and then demonstrate how to quickly and easily calibrate both a basic PRF model and a temperature-dependent PRF model.

As will be shown below, there are certain PRF model elements and structures that should not be used!

When you use a Parallel Rheological Framework (PRF) model in an Abaqus simulation you need to use one of the following:

- Abaqus Explicit
- Abaqus Standard with *VISCO steps
- Abaqus Standard with *Dynamic, Application=Quasi-Static steps

Note that you cannot use a PRF model in a static analysis in Abaqus.

## Parallel Rheological Framework Model Introduction

Let’s start with some of the basics:

- The PRF model is a multi-network model where each network consists of one hyperelastic element and one flow element.
- All networks have to have the same hyperelastic element type and the same flow element type.
- All standard hyperelastic types in Abaqus are supported (I particularly like the Yeoh model).
- The PRF model is a simplified version of the PolyUMod Parallel Network (PN) model, and has similar features to the PolyUMod TN and TNV models.
- The PRF model networks can also include Mullins damage and pure plasticity. It is almost never necessary to use the plasticity option in the PRF model.
- Use 2 networks when modeling rubber-like materials.
- Use 3 networks when modeling thermoplastics.
- It is almost never needed to use more than 3 networks.

## Flow Element Components

**LAW=POWER LAW**

The best flow element type for thermoplastics and thermosets is the **Power-Law Strain Hardening** model:

\(\displaystyle\dot{\varepsilon}^{cr} = \dot{\varepsilon}_0 \left( \left( \frac{\tilde{q}}{q_0+a\langle p\rangle} \right)^n \left[ (m+1) \varepsilon^{cr} \right]^m \right)^{1/(1+m)}. \)

**LAW=STRAIN**

Another available option is the **Strain Hardening** model:

\(\dot{\varepsilon}^{cr} = \left( A \tilde{q}^n \left[ (m+1) \varepsilon^{cr} \right]^m \right)^{1/(1+m)}. \)

This flow model should not be used due to how the pre-factor A is defined.

**LAW=HYPERB**

A third flow model is the **Hyperbolic-sine** model:

\( \dot{\varepsilon}^{cr} =A\left( \text{sinh}(B\tilde{q})\right)^n. \)

This model is also less useful since it does not contain any strain-dependence of the flow rate.

**LAW=BB POWER LAW**

The final flow model that is the **Bergstrom-Boyce (BB)** model. This model, of course, is really cool since it is based on work that I did during my PhD research at MIT.

\( \displaystyle\dot{\varepsilon}^{cr} = \dot{\varepsilon}_0 \left( \lambda^{cr} – 1 + E \right)^C \left[ \frac{\tilde{q}}{q_0} \right]^m. \)

This model is a good choice for rubber-like materials.

**LAW=BERGSTROM-BOYCE**

There is also an older version of the Bergstrom-Boyce flow equation available in Abaqus. That version was based on the following flow equation:

\(\dot{\varepsilon}^{cr} = A (\lambda^{cr} – 1 + E)^C \tilde{q}^m.\)

This is a poor formulation and should not be used.

In summary, use the Power-Law strain hardening model, or the BB Power Law model.

## Material Model Calibration

The parallel rheological framework model can be quickly and accurately calibrated using MCalibration. As shown in our other tutorials, first read in the experimental data, then select the desired PRF model:

The run the calibration as normally done using MCalibration. Note that MCalibration has its own internal implementation of the listed PRF models. That makes the calibration very quick. Once the calibration has completed, you can export the calibrated material model to an Abaqus inp-file.

` ````
```*Material, name=MCal_Mat
** Calibrated with MCalibration
** Units: [length]=millimeter, [force]=Newton, [time]=seconds, [temperature]=Kelvin
*Density
1e-09
*Hyperelastic, Yeoh, Moduli=instantaneous
1288.01, 0, 0,0.00016773, 0, 0
**
*Viscoelastic, Nonlinear, NetworkId=1, SRatio=0.789162, Law=Power Law
22.8048, 7.11344, -0.248253, 0.047694, 1
**
*Viscoelastic, Nonlinear, NetworkId=2, SRatio=0.20867, Law=Power Law
79.8319, 14.9972, -0.499959, 5.41635, 1

## Calibrate a Temperature-Dependent PRF Model

To calibrate a temperature-dependent parallel rheological framework (PRF) model simply calibrate the same PRF model structure to each individual temperature. Then export the calibrated material models into individual inp-files (as shown above). You can then combine the individual calibrations into a temperature-dependent PRF model by opening a new MCalibration window, selecting the material model dialog box, and selecting “Temperature-Dependent PRF”.

The software will then ask you for the already exported inp-files. MCalibration will then combine the models into one temperature-dependent model:

The final step is to specify the testing temperatures for the individual calibrations:

That is it! You can now export the combined model and use it in Abaqus. Life is good!

## 5 thoughts on “Abaqus Parallel Rheological Framework (PRF)”

AlexandreHi Jorgen,

Thanks a lot for this post. It will be very useful in my next work.

I always use viscous steps with PRF model. But this time (curiosity) I have run a simulation with PRF model in Abaqus using static analysis.

I’m able to run this analysis, but as expected, the viscous response is neglected.

I have run three analysis and I have used a node to monitor the strain (A – static, B – viscous 100s, C – viscous 1000s). For the same load the obtained strains are: e_C > e_B > e_A. Can we conclude that the static analysis is a kind of “fast viscous analysis” or this conclusion does not make sense?

Regards,

Alex

JorgenHello Alex, The MCalibration exported PRF models have the definition:

*Hyperelastic, Yeoh, Moduli=instantaneous

So, YES, your conclusion is correct. That is a good way to think about it!

/Jorgen

imendiaHello Jorgen,

I am trying to calibrate a PRF model for a rubberlike material with the recommended 2nd order Yeoh-BB model. Now I am carrying out the experimental characterization of the material with compression, traction and simple shear tests. I would like to know what are the most suitable tests for a good calibration of a PRF model. I am interested in modelling the hyperelasticity and amplitude and frequency dependence of the material.

Thank you very much,

Irati

JorgenHello Irati,

Great question. I would recommend something like what is discussed in this article, with the addition of DMA frequency tests.

https://polymerfem.com/smart-mechanical-testing-of-polymers/

Thanks,

Jorgen

bh1g12Thanks for the helpful video Jorgen, I have a question about the language you use to describe the viscous element in the PRF.

In the video you appear to use the term “viscoelastic” and “viscoplastic” interchangeably when referring to selection of the viscous element. In the case of the PRF model, strictly speaking isn’t it viscoelasticity rather than viscoplasticity that is captured in networks 1 to N? (with elasto-plasticity captured in network 0).

Thanks!