## 3 Results  
To explore whether, as a matter of principle, the previous framework was able to reproduce the classic evolution of the nerve stiffening phenomenon, the range of variation of the main theoretical curves was studied. First, the range of variability of Equation (9), accounting for the stress due to the matrix contribution (Figure 2a), was investigated for increasing values of \(K_{1}\). In particular, the values of \(\sigma_{mat}(\lambda,1+\Delta K_{1})\) were computed for \(\Delta K_{1}\in Q\). For a small stretch (i.e., \(\lambda\to 1\)), Equation (9) reduced to \(\sigma_{mat}=6K_{1}(\lambda-1)\), so the initial slope of the curve was \(\partial_{\lambda}\sigma_{mat}(1,K_{1})=6K_{1}\). Then, Equation (10), accounting for the stress derived from the collagen fibrils' contribution, was investigated for \(D=1\) and \(A=1+\Delta A\), \(\Delta A\in Q\). Larger values of \(A\) resulted in larger values of \(\sigma_{fib}(\lambda,A,1)\) (as shown in logarithmic scale in Figure 2b), while for \(\lambda\to 1\), its slope was \(\partial_{\lambda}\sigma_{fib}(\lambda,A,1)=0,\forall A\in\mathbb{R}\). The contribution of fibrils was also investigated for \(A=1\) and for different values of \(D=1+\Delta D\), \(\Delta D\in Q\), through the evolution of the natural logarithm of \(\sigma_{fib}(\lambda,1,D)\) (Figure 2c). More specifically, the larger the value of \(D\) was, the more the value of fibril stress increased, while for \(\lambda\to 1\), its slope was \(\partial_{\lambda}\sigma_{fib}(\lambda,1,D)=0,\forall D\in\mathbb{R}\). Figure 2: (**a**) Steepening of the slope of \(\sigma_{mat}\) for increasing values of \(K_{1}\) (arrow). (**b**) Evolution (logarithmic scale) of \(\sigma_{fib}\) for increasing values of \(A\) (see legend) and D = 1. (**c**) Evolution (logarithmic scale) of \(\sigma_{fib}\) for increasing values of \(D\) (see legend) and \(A=1\). (**d**) Dimensionless extrinsic curvature of \(\sigma_{mat}\), all curves are superimposed. (**e**) Dimensionless curvature of \(\sigma_{fib}\) for D = 1, all curves are superimposed. (**f**) Dimensionless curvature of \(\sigma_{fib}\) for increasing values of \(D\) (see legend) and \(A=1\). In addition, Equation (18), accounting for the stiffening due to the matrix action \(\hat{\kappa}_{mat}(\lambda)\), monotonically increased for \(1\leq\lambda\leq 1.54\), while after a maximum \(\hat{\kappa}_{mat}(1.54)=0.133\), it monotonically decreased for \(1.54\leq\lambda\leq 2\) (Figure 2d). For \(\lambda\to 1\), its slope was \(\partial_{\lambda}\hat{\kappa}_{mat}(1)=2^{-1/2}\). Similarly, Equation (22), accounting for the stiffening due to the action of fibrils, for \(D=1\) (i.e., \(\tilde{\kappa}_{fib}(\lambda,1)\)), had a fixed bell-like shape (Figure 2e), with a maximum \(\tilde{\kappa}_{fib}(1.06,1)=11.317\). Its full width at half maximum was \(\text{FWHM}(\tilde{\kappa}_{fib})\simeq 0.082\), while for \(\lambda\to 1\), its slope was \(\partial_{\lambda}\tilde{\kappa}_{fib}(1)=192\). On the contrary, the same Equation (22) for different values of \(D=1+\Delta D\), \(\Delta D\in Q\) (i.e., \(\tilde{\kappa}_{fib}(\lambda,D)\)), had a variable bell-like shape (Figure 2f). The larger the value of \(D\) was, the more the value of the peak increased, while the more the value of the \(\text{FWHM}(\tilde{\kappa}_{fib})\) decreased. In other words, for \(D\to\infty\), \(max\{\tilde{\kappa}_{fib}(\lambda,D)\}\to\infty\), while \(\text{FWHM}(\tilde{\kappa}_{fib})\to 0\). However, these curves had the following characteristic: \(\lim_{\lambda_{max}\to\infty}\int_{1}^{\lambda_{max}}\tilde{\kappa}_{fib}( \lambda,D)d\lambda=1\), \(\forall D\in\mathbb{R}\). Furthermore, for \(\lambda\to 1\), its slope quadratically increased with \(D\), since it was \(\partial_{\lambda}\tilde{\kappa}_{fib}(D)=192D^{2}\). To better quantify the influence of each parameter, a sensitivity study was also performed on each contribution. More specifically, the sensitivity of \(\sigma_{mat}\) with respect to changes in \(K_{1}\) was tested in Figure 3a.