Benchmark Tests

This section shows the benchmark tests on a single element model with a simple prescribed traction loading. The model is a cube of size 0.5 mm and is meshed with 1 linear brick element with reduced integration (C3D8R). A velocity of 1 mm/s is applied in direction 1 (along the *x*-axis). The element was loaded for 5 cycles at constant max strain amplitude using 10 steps. The velocity was positive (*v* = 1 mm/s) in the odd steps to stretch the element and negative (*v* = −1 mm/s) in the even steps to unload the element. The job was run using double precision. Material parameters are shown in Table 2. Figures 11 and 12 show the typical output for two different cyclic loading path.

**Figure 11.** Test 1: VUMAT output: (**a**) shows the nominal stress versus time; (**b**) shows the nominal strain–stress path with the set of parameters in Table 2.

**Figure 12.** Test 2: VUMAT output: (**a**) shows the nominal stress; (**b**) shows the nominal strain–stress path with the set of parameters in Table 2.

### **5. Discussion and Conclusions**

The new model is based on physical rubber elasticity considerations. The total stress is the weighted sum of two contributions: an amplified, simplified extended tube model, and a hysteretic part. The hysteretic part was implemented using a single Maxwell/Prony element. For compounds with different CB volume fraction the model reproduces the non-linear behaviour under examination: Mullins effect, cyclic stress relaxation, permanent set, and hysteresis. The model requires 10 material parameters. The best-fit sets of parameters show a linear or a polynomial trend with the CB volume fraction. The sensitivity study shows the effect of altering each of the parameters. The FE implementation of the new model reproduces the expected behaviour. Finite Element Analysis was conducted on single-element models and on a realistic component. This preliminary investigation demonstrates that this model is stable when incorporated into a finite element model. The VUMAT works properly with different structures, and it is able to predict the desired non–linear viscoelastic behaviour. The new model was already implemented in the Jaguar Land Rover (JLR) suspension tool. The first job carried out a single cycle at 20 kN on a suspension loaded radially. Preliminary results are reported in Figure 13. Further studies could focus on the modeling of a broader range of deformation rates potentially requiring the use of more Prony elements. Moreover, the idea of a static hysteresis, as implemented using an intrinsic time in [28], deserves further investigation.

**Figure 13.** JLR toolbox: preliminary results.

**Author Contributions:** F.C. developed the curve fitting algorithm, coding and implementation and performed simulation. J.P. developed the non linear elastic model. F.C. and J.P. prepared the first draft of the manuscript. R.W. sponsored the project. J.B. and M.K. supervised the project throughout and contributed to detailed editing of the manuscript. All authors have read and approved the final version of the manuscript.

**Funding:** The project was funded by Jaguar Land Rover, Banbury Road, Gaydon CV35 0RR, UK.

**Acknowledgments:** The author would like to acknowledge the financial support provided by Jaguar Land Rover.

**Conflicts of Interest:** The authors declare no conflict of interest.
