**2. Theory**

### *2.1. Free Energy Density*

The underlying free energy density is derived on the basis of the extended non-affine tube model of rubber elasticity [29,30] which was shown to be the best compromise between fitting quality and number of parameters [31]. In a simplified form it reads

$$\mathcal{W}(\tilde{I}\_1, \tilde{I}^\*) = \frac{G\_\mathbb{C}}{2} \frac{\tilde{I}\_1}{1 - \frac{1}{\tilde{\Pi}} \tilde{I}\_1} + 2G\_\mathbb{B} \,\, \tilde{I}^\* \tag{1}$$

with

$$I\_1 = I\_1 - 3 = \lambda\_1^2 + \lambda\_2^2 + \lambda\_3^2 - 3 \tag{2}$$

$$I^\* = I^\* - 3 = \lambda\_1^{-1} + \lambda\_2^{-1} + \lambda\_3^{-1} - 3 \tag{3}$$

being modified invariants of the left Cauchy Green tensor, expressed by the principal stretches *λi*. The straightforward simplifications done to arrive at Equation (1) are outlined in [32]. The parameter *G*c is called crosslink modulus and scales the contribution of crosslinks to the mechanical response. Accordingly, *G*e scales proportionally to the number of trapped entanglements of the polymer [33]. The third parameter *n* measures the number of statistical segments between network nodes, which is a measure of elastically effective chain length. For example, natural rubber has a segmen<sup>t</sup> length of about 0.934 nm [34], corresponding to roughly two isoprene units. Fillers are introduced by assuming that they amplify strain heterogeneously within the matrix by local amplification factors *X* [28]. To avoid problems arising from frame references the amplification factors directly act on the invariants. In the case of ˜*I*1 this is reasonable, as it represents the norm of a hypothetical length within the material. The amplified energy density is then calculated as a superposition of differently amplified domains

$$\mathcal{W}\_{\mathcal{X}}(\vec{I}\_{1}, \mathcal{X}\_{\text{max}}, \mathcal{X}\_{\text{min}}) = \int\_{\mathcal{X}\_{\text{min}}}^{\mathcal{X}\_{\text{max}}} \text{d}X \, P(\mathcal{X}) \, \mathcal{W}(\mathcal{X}\vec{I}\_{1}, \mathcal{X}\vec{I}^{\*}) \tag{4}$$

where the distribution of amplification factors is given by

$$P(X) = (X + \mathbb{C})^{-\chi} \cdot \frac{\chi - 1}{(X\_{\text{min}} + \mathbb{C})^{1 - \chi} - (X\_{\text{max}} + \mathbb{C})^{1 - \chi}} \tag{5}$$

The parameter *χ* gives the width of the distribution and *C* defines a plateau value around *X* = 1. For *C* = 0 Equation (5) is equivalent to its equivalent in [28]. It is normalized to the interval [*X*min,*X*max], which is motivated from conservation of the number of rubber-filler structures. Carrying out the integral given by Equation (4) generates hypergeometrical functions. Additive splitting of the integrand, as is explained in [32], gives a good approximation containing only elementary functions:

$$\begin{array}{l} \mathcal{W}\_{\mathcal{X}}(X\_{\text{max}}, X\_{\text{min}}) \approx \frac{1}{2} \frac{1}{(X\_{\text{min}} + \mathcal{C})^{1-\chi} - (X\_{\text{max}} + \mathcal{C})^{1-\chi}}. \\\ \left[\frac{\mathbb{C}\_{\text{c}} I\_{1} + 4\mathcal{C}\_{\text{c}} I^{\*}}{\chi - 2} \left( (\mathcal{C} + X\_{\text{min}})^{1-\chi} (\mathcal{C} + (\chi - 1)X\_{\text{min}}) - (\mathcal{C} + X\_{\text{max}})^{1-\chi} (\mathcal{C} + (\chi - 1)X\_{\text{max}}) \right) \right. \\\ \left. + \mathcal{G}\_{\text{c}} n^{2} (\chi - 1) \operatorname{I}\_{1}^{X^{-1}} \left( \mathcal{C} \operatorname{I}\_{1} + n \right)^{-\chi} \log \left( \frac{I\_{1} X\_{\text{min}} - n}{I\_{1} X\_{\text{max}} - n} \right) \right] \end{array} \tag{6}$$

From Equation (6) the hyperelastic stresses can be derived, if *X*min and *X*max are known. The minimum amplification factor is set to *X*min = 1, which corresponds to the assumption that there are non-amplified domains (e.g., without filler) within the material.
