## 2 Methods  
The peripheral nerve was modeled as a three-dimensional continuum body (elliptic cylinder), which was deformed into a final configuration \(\Gamma\) starting from a reference configuration \(\Gamma_{r}\). The position of each point of this solid was identified through the vectors \(\mathbf{X}\) and \(\mathbf{x}\), respectively related to the reference and the deformed configuration. As shown in Figure 1 below. Figure 1: (**a**) The reference configuration \(\Gamma_{r}\). The nerve is modeled as a transverse isotropic cylindrical body (ground matrix = white areas, collagen fibrils dark lines). In the reference configuration (\(t\) = 0, non-stretched body), the coordinates of a solid point (red point) are identified through the vector \(\mathbf{X}\). The cross-section WW shows how the internal volume of the specimen is filled by both the ground matrix and collagen fibrils. Axons inside bundles are not shown. (**b**) After stretching, the axial length of the cylinder is changed in \(L_{f}\), and the current coordinates of the solid point are identified through the vector \(\mathbf{x}\). The deformation of the body was described through a time-dependent map \(\mathbf{x}=\beta(\mathbf{X},t)\), where \(\beta\) is an invertible and regular function. Unlike previous models [17; 18; 19; 20; 21; 22; 23], a unit vector field \(\mathbf{M}(\mathbf{X})\) was used to explicitly account for the presence of the collagen fibrils in the reference configuration \(\Gamma_{r}\)[26; 27]. For the sake of simplicity, the collagen fibrils were modeled as axially oriented with no dispersion [4]. The response of the nerve was described through a strain energy function \(\Psi\) (per unit of volume), which was affected by the initial fibrils' direction, as well as by the strain measure. Therefore, the mean Cauchy stress tensor within the nerve is written as:  
\[\sigma=J^{-1}\mathbf{F}\frac{\partial\Psi(\mathbf{C},\mathbf{M}\otimes \mathbf{M})}{\partial\mathbf{F}} \tag{1}\]  
where \(\mathbf{F}=Grad\mathbf{x}\) is the deformation gradient, \(J=\det(\mathbf{F})\), and \(\mathbf{C}=\mathbf{F}^{\mathbf{T}}\mathbf{F}\) is the right Cauchy-Green strain tensor. In particular, an invariant-based form of this energy is used:  
\[\Psi=\Psi(I_{1},I_{2},I_{3},I_{4},I_{5}) \tag{2}\]  
where, \(I_{1}=tr(\mathbf{C})\), \(I_{2}=\frac{1}{2}[I_{1}^{2}-tr(\mathbf{C^{2}})]\), \(I_{3}=det(\mathbf{C})\), \(I_{4}=\mathbf{M}\cdot(\mathbf{CM})\), \(I_{5}=\mathbf{M}\cdot(\mathbf{C^{2}M})\). The Cauchy stress tensor is, then, better specified as:  
\[\sigma=J^{-1}[2\Psi_{1}\mathbf{B}+2\Psi_{2}(I_{1}\mathbf{B}-\mathbf{B}^{2})+2 I_{3}\Psi_{3}\mathbf{I}+2\Psi_{4}\mathbf{m}\otimes\mathbf{m}+2\Psi_{5}( \mathbf{m}\otimes\mathbf{B}\mathbf{m}+\mathbf{B}\mathbf{m}\otimes\mathbf{m})] \tag{3}\]  
where \(\mathbf{B}=\mathbf{F}\mathbf{F}^{\mathbf{T}}\) is the left Cauchy-Green tensor, \(\Psi_{i}=\frac{\partial\Psi}{\partial i_{i}}\), \(i=1,2,3,4,5\), \(\mathbf{m}=\mathbf{F}\mathbf{M}\). As in previous literature [7; 19; 20; 21], the nerve was considered incompressible (i.e., the nerve volume remained the same, when the specimen underwent deformation), then \(I_{3}=det(\mathbf{C})=1\), and the pressure contribution within the Cauchy stress tensor was made explicit. Thus, the stress is written as:  
\[\sigma=-p\mathbf{I}+2\Psi_{1}\mathbf{B}+2\Psi_{2}(I_{1}\mathbf{B}-\mathbf{B}^ {2})+2\Psi_{4}\mathbf{m}\otimes\mathbf{m}+2\Psi_{5}(\mathbf{m}\otimes\mathbf{ B}\mathbf{m}+\mathbf{B}\mathbf{m}\otimes\mathbf{m}) \tag{4}\]  
where \(p\) is the hydrostatic pressure, while \(\mathbf{I}\) is the unit tensor. The physical behavior of the nerve was assumed to result from the contribution of the main structural components of its connective tissue, that is the elastin matrix and the collagen fibers.