From bouncy balls to automotive tires, hose pipes to suspension bushings, the bewildering behavior of rubber has been part of lives since Aztec times. Thus, it is important that accurate simulation methods can be created to facilitate engineering design decisions to be made for such components. However, this is not a trivial exercise and much time is spent on characterizing the response.
The complex interplay of the long-chain spaghetti like molecules and the mild cross-linking lead to a wide range of mechanical responses that are a function of temperature, time and history. Some typical responses are listed below:
- Large strain elasticity that can be captured with an energy based hyperelastic response best fit to multiple deformation modes.
- Rate/time dependency captured with viscoelasticity. This may be linear or nonlinear viscoelasticity.
- Damage and/or permanent set on the application and removal of large strains
Abaqus can capture many of these responses, but there is one rubber response that has presented a challenge – the Payne effect (or perhaps more correctly the Fletcher-Gent effect). Classic linear viscoelasticity allows for a storage and loss modulus that are functions of frequency, a common requirement. But, the experimental tests of Payne on carbon black (Carbon black is commonly used in rubbers to increase volume, strength and vulcanization) filled rubbers demonstrated that the viscoelastic response, that controls the storage and loss modulus, is also a function of strain amplitude. Payne's results are shown below
| A demonstration of the Payne effect from Gent and Scott, Engineering with Rubber - How to Design Rubber Components |
The images below shows the storage and loss modulus for two rubber compounds, a natural rubber with 75 phr (parts per hundred) carbon black filler and a silica filled silicone rubber. The experimental data in the plots comes from the work of Chazeau et al, 2000: Modulus Recovery Kinetics and Other Insights into the Payne Effect for Filled Elastomers (Polymer Composites, vol21, No 2).
| Natural Rubber | |
| Silicon Rubber | |
The Payne effect is not easy to simulate within Abaqus:
- linear viscoelasticity does not capture it – the storage and loss modulus will remain constant as a function of strain amplitude
- non-linear viscoelasticity does not capture it, though the Bergstrom-Boyce creep model within the PRF goes some way.
Instead, Bergstrom proposed an extension to the Bergstrom Boyce model, termed the Dynamic Bergstrom Boyce (DBB) model. His paper at the 2008 Abaqus User Conference demonstrated its ability to capture the Payne effect, calibrating well to experimental data.
The DBB model makes the stiffness in the hyperelastic component of the equilibrium network a function of the strain amplitude. Bergstrom’s original formulation made the stiffness a function of
| Effective mises strain | |
| Current stiffness in the equilibrium network | |
| Stiffness material constant | |
| Stiffness material constant | |
| Strain material constant |
Leaving the current stiffness taking the following form
The implementation of this that we share here uses a modified approach – the strain terms are replaced with the current strain energy or more correctly the square root of the current strain energy, which is analogous to the strain in simple linear elasticity, and an appropriate material constant.
This was implemented within a user subroutine, and then a traditional Isight workflow was used to determine suitable material constants. The images below show the match between the energy based DBB model and the experimental data for natural and silicon rubbers
| Silicon Rubber | |
A comparison of the storage and loss modulus predicted by this new material model and the experimental data for silicon rubber
| Natural Rubber | |
A comparison of the storage and loss modulus predicted by this new material model and the experimental data for natural rubber
This response will be important for any of your customers that are working with the cyclic loading of rubber, typically tires, rubber bushes etc.
My thanks to the technical leadership from the SIMULIA RnD team that made this possible. This work was implemented by the SIMULIA HUB - the services delivery organization of the SIMULIA Brand. We are usually working on exciting projects that drive our customers forward and are therefore covered by the appropriate confidentiality agreements. However, this was some internal work, using publicly available data, as a consequence of a number of enquiries - therefore we thought it was worth sharing.
Please get in touch if you are interested in how this or any other SIMULIA services implementation can benefit you business, stuart.nixon@3DS.com
