*2.2. Stress Softening*

In [28] the maximum amplification factor is defined to be a monotonically decreasing function of the all-time maximum of the modified first invariant ˜ *I*1. This naturally introduces immediate stress softening if the material surpasses the previous maximum strain. As described in the introduction stress softening is not immediate, but proceeds logarithmically or according to a slow powerlaw. Generally, logarithmic relaxation in stress *f*∼− log *t* can be generated by differential equations of the form d*f* /d*t*∼− exp | *f* |. From a physical point of view this can be understood in terms of force-induced hopping over a potential barrier, as formulated by Kramers [35]. It is reasonable to assume that softening is predominantly happening in the most stretched domains. A crude approximation of the stress-scaling in this regime gives

$$f \sim \frac{1}{\langle \lambda \rangle} \mathcal{W}(X\vec{I}\_1, X\vec{I}^\*) \approx \frac{1}{\sqrt{I\_1}} \frac{I\_1 \, X\_{\text{max}}}{1 - \frac{1}{n} I\_1 \, X\_{\text{max}}} \tag{7}$$

where the entanglement part of Equation (1) was neglected, because its influence is small at high strains. The prefactor *λ* = ˜*I*1 is a frame independent measure of the systems stretch. Heuristically, stress is also directly scaled by the amplification factor, such that d*f* /d*t*∼ d*X*max/d*t*. Using again the logarithm-generating differential equation d*f* /d*t*∼− exp | *f* | and introducing scaling constants *c*1 and *c*2 this gives

$$\frac{\mathrm{d}X\_{\mathrm{max}}}{\mathrm{d}t} = -\exp\left(-c\_2 + \frac{1}{c\_1} \frac{\sqrt{I\_1} \, X\_{\mathrm{max}}}{1 - \frac{1}{n} I\_1 \, X\_{\mathrm{max}}}\right) \cdot \frac{1}{\mathrm{s}}\tag{8}$$

The timescale of zero-load softening given as *τ*soft = exp(*<sup>c</sup>*2) s was put in the exponential for numerical reasons. In practice, it should be sufficiently long that no significant softening happens on the timescale of the simulation, if no load is imposed. This means that for ˜ *I*1 → 0 the result of Equation (8) is usually small and the maximum amplification factor *X*max stays almost constant. When the model approaches divergence at ˜ *I*1 *X*max/*n* → 1 the second summand inside the exponential increases and induces a decrease of *X*max. It shall be noted that a more careful derivation was carried out in [32] which relates *c*1 and *c*2 to temperature and allows the modeling of Mullins effect recovery [6] at elevated temperatures.
