*4.1. Sensitivity Analysis*

This section shows how each parameter effects the output stress. The initial parameters shown in Table 2 are used, it is an arbitrary set. The parameter *<sup>X</sup>*max,0 = *<sup>X</sup>*max(*<sup>t</sup>* = 0) is the boundary condition of the differential Equation (8).

**Table 2.** Material parameters used in the sensitivity analysis and in the VUMAT.


Figure 4 shows the effect of three of those parameters *G*c, *G*e, *<sup>X</sup>*max,0 on the predicted outputs. Figure 5 shows the effect of parameters *χ*, *C* and *c*1. Figure 6 shows the effect of the parameters *c*2, *n* and *τc*. Figure 7 shows the effect of the parameter *φ*. *G*c and *G*e, which respectively represent the crosslink modulus and entanglement modulus in the extended tube model, influence the stiffness of the material. The parameter *<sup>X</sup>*max,0 determines the value of *X*max before any deformation in the system, and it influences cyclic stress relaxation phenomena. The parameters *χ* and *C* characterise the power law distribution of the amplification factor. They influence the response of the material in the loading paths, modifying the the level of stress and the dissipated energy and defining the level of continuum damage due to cyclic stress softening. *n*, *<sup>c</sup>*1,and *c*2 define the relaxation of the maximum amplitude factor to predict the continuum damage, so they operate on the response of the material at a strain higher than 1 and envelop the cyclic stress relaxation process. The constant *c*2 scales the zero-load relaxation of the material (which is usually very small), and thus determines, together with *c*1 the stress threshold to overcome before softening starts. The parameter *φ* balances elastic stresses (including softening) and viscous effects. The timescale *τc* defines the amount of dissipated energy.

### **Figure 4.** *Cont.*

**Figure 4.** Influence of parameters. (**<sup>a</sup>**,**b**) show the effect of the parameter *G*c; (**<sup>c</sup>**,**d**) show the effect of the parameter *G*e; (**<sup>e</sup>**,**f**) show the effect of the parameter *<sup>X</sup>*max,0.

**Figure 5.** Influence of parameters. (**<sup>a</sup>**,**b**) show the effect of the parameter *χ*; (**<sup>c</sup>**,**d**) show the effect of the parameter *C*; and, (**<sup>e</sup>**,**f**) show the effect of the parameter *c*1.

**Figure 6.** Influence of parameters. (**<sup>a</sup>**,**b**) show the effect of the parameter *c*2; (**<sup>c</sup>**,**d**) show the effect of the parameter *n*; and, (**<sup>e</sup>**,**f**) show the effect of the parameter *τc*.

**Figure 7.** Influence of parameters. (**<sup>a</sup>**,**b**) show the effect of the parameter *φ*.

### *4.2. Model Fit to Experimental Data*

Model parameters were determined by minimising the mean square error between the model predictions and experimental data. Figures 8a–c show comparisons of model predictions for a selection of materials and loading configurations. For compounds with different amounts of carbon black, the model accurately reproduces the initial loading curve, relaxation cycles, the Mullins effect and the pre-strain effect.

**Figure 8.** *Cont.*

**Figure 8.** Comparison of the new model with experimental data loaded with cyclic uniaxial test with uniform maximum strain amplitude [14]. Subfigures (**<sup>a</sup>**–**<sup>c</sup>**) show respectively the behaviour for NR2, NR40 and NR60.

### *4.3. Effect of Carbon Black Content on the Parameters Used in the New Model*

The uniaxial tests, where the compound were stretched cyclically with the step up of maximum strain amplitude reached, represent a perfect test to highlight all the non linear viscoelastic behaviour under examination (Mullins effect, cyclic stress softening and permanent set) (Figure 9). The seven compounds were fitted starting from the material with the smaller amount of filler and using the new set of parameters as the guess set for the following compound. The best-fit parameters for different compounds loaded with this strain history show a robust trend with the true carbon black filler volume.

**Figure 9.** Comparison of the new model with experimental data for NR30 loaded with cyclic uniaxial stress with different maximum strain amplitude in step up. Subfigure (**a**) shows the Nominal Stress versus time. Subfigure (**b**) shows the Nominal Stress versus Nominal Strain.

The trend of the parameters with the true Carbon Black (CB) filler volume is presented in Figure 10.


Moreover, it can be seen that the parameters *G*e, *C*, *c*1 , *φ* and *τ*c show some kind of discontinuity in the range of 9–13 vol. % carbon black loading. This roughly corresponds to the percolation threshold of the filler network inside the polymer matrix [37].

**Figure 10.** *Cont.*

**Figure 10.** Trend of he parameters with the true carbon black filler volume *Vf* . Subfigures (**<sup>a</sup>**–**j**) show respectively the trend of *G*c, *G*e, *<sup>X</sup>*max,0, *χ*, *C*, *c*1, *c*2, *φ*, *τ*c and *n*.

### *4.4. Finite Element Analysis*
