Rubber materials, just like all other polymers, suffer from some amount of permanent set after being unloaded from a loaded configuration. The amount of residual strain depends on the chemistry of the polymer, the load environment, and the time under load. In many applications, including seals and o-rings, this behavior is undesirable and one of the goals for the design engineer is often to either minimize the permanent set or to design the product such that the residual displacements and forces are acceptable for the product performance. In this article I will discuss what experimental tests are needed and what material models are capable of accurately predicting the permanent set of rubbers.
What experiments to use?
What material models to use?
For this case study I will use the following experimental data for a nitrile rubber:
- Uniaxial compression at 3 strain rates.
- Multiple stress relaxation segments. The strain was held constant for up to 5 minutes at each relaxation level.
Is this experimental data sufficient to accurately predict the compression set?
Experimental Compression Set Testing - ASTM D395
Traditional cyclic tests do not provide sufficient information for accurate predictions of the permanent set of rubbers. An easy and accurate way to get the additional needed experimental data is to perform the tests specified in ASTM D395 (“Standard Test Method for Rubber Property – Compression Set”). This useful low-tech method compresses rubber specimens to a pre-defined strain level, and then holds that strain for an extended period of time. After the specimens have been held at the target strain and temperature, they are then unloaded and the residual strain measured.
In this case study I performed the following compression set experiments:
- I applied a 15% engineering strain for 6 months at room temperature. The residual strain after unloading was 5.0%.
- I applied a 30% engineering strain for 6 months at room temperature. The residual strain after unloading was 8.0%.
The permanent set is defined as: (residual strain) / (compressive strain) * 100%
Material Model Calibration Approach
MCalibration directly supports the ASTM D395 Permanent Set test. All you need to do is to define a “Permanent Set” load case, and enter the specific details of your test. Note that you can use multiple Permanent Set load cases at the same time in MCalibration. The figure to the right shows how to enter the data for the 15% compression test that I performed.
Here is a screenshot shows the different load cases that I used for the material model calibration. Note that I used all experimental data at once for the material model calibration.
Calibration Results: Yeoh Hyperelasticity
Since hyperelasticity is a non-linear elastic material model, it cannot predict either permanent set or the viscoelastic behaviors that are exhibited by the rubber.
Calibration Results: Yeoh Hyperelasticity with Mullins Damage
Adding Mullins damage to a hyperelastic material model still does not allow you to predict either the permanent set of the viscoelastic behaviors that are exhibited by the rubber.
Calibration Results: Yeoh Hyperelasticity with Linear Viscoelasticity
Linear viscoelasticity typically recovers too fast to be useful for predicting permanent set of rubbers. The figure shows that the predicted errors are no better than for a basic hyperelastic model.
We can do better!
Calibration Results: Bergstrom-Boyce Model
The Bergstrom-Boyce (BB) model is one of my favorite models for rubbers, which should come as no surprise since I developed it 🙂
In this case, however, it cannot predict both the permanent set and the viscoelastic response of the material. The figure shows that my calibration very accurately predicts the permanent set, but clearly misses the viscoelasticity.
Calibration Results: Bergstrom-Boyce Model with Mullins Damage
Switching on Mullins damage with the Bergstrom-Boyce model works really well in this case. The average error in the model predictions is 4.8%. That is quite impressive!
This is the model to use.
Calibration Results: Abaqus Parallel Rheological Framework (PRF2YPM)
This figure shows the predictions from the Abaqus Parallel Rheological Framework (PRF) model with 2 networks, Yeoh hyperelasticity, power-law flow, and Mullins damage.
The predictions looks almost as good as for the PolyUMod Bergstrom-Boyce-Mullins model shown above.
Calibration Results: Ansys Bergstrom-Boyce with Mullins Damage
Also the Ansys Bergstrom-Boyce model with Mullins damage works very well in this case, of course. The average error in the model predictions is about 5.3%.
Calibration Results: Abaqus Yeoh Hyperelasticity with Mullins Damage and Isotropic hardening Plasticity
This model combination of hyperelasticity, Mullins damage, and metal plasticity, is sometimes called the FeFp model. When using this model combination the only reason to add the plasticity is to enable predictions of permanent set. As shown in the figure, this model is slightly less accurate than the Bergstrom-Boyce-Mullins models.
Accurately predicting the permanent set of rubbers requires:
- Stress-strain data at different strain rates (preferable cyclic data with stress relaxation segments)
- Permanent set data, for example from ASTM D395
- A viscoplastic material model, for example, the Bergstrom-Boyce model with Mullins damage
Note that the MCalibration software can quickly calibrate the material model using all of the experimental data.