## 4 Discussion  
To tackle nerve neuropathies deriving from different kinds of pathologies (i.e., diabetes, scleroderma) [33], dietary deficiency, or exposure to ionizing radiation (e.g., neuromalacia) [34], an explicit understanding of the main features of the peripheral nerve response to stretch is needed. In particular, a computationally light and physically-based theoretical approach, linking in a deterministic and explicit way the main physical constituents of the connective tissue to the macroscopic response of the peripheral nervous tissue, could be strategic to make quantitative and reliable predictions. Since peripheral nerves are anatomically complex (i.e., they have branching points), but their overall response is closely related to the response of their main rectilinear tract (which is the main structural segment), in this work, they were modeled as continuum rectilinear transverse isotropic bodies. Indeed, although peripheral nerves are hierarchically organized, ultrastructural studies revealed that the main constituents of the connective tissue are a ground matrix of elastin and a longitudinally-oriented fibrous network of collagen [4]. As a consequence, the strain energy function, ruling the nerve behavior, was dominated by \(\Psi_{mat}\) and \(\Psi_{fib}\), that is by the energetic contributions of the matrix and fibrils, respectively. In particular, the energetic contribution of the elastic ground matrix was modeled through a first order Yeoh-like strain energy function [28], while the energetic contribution of the collagen fiber network was modeled through Equation (10), where, for the sake of simplicity, the function \(\omega(\lambda,D)\) was a convex energy function, traditionally used to model the behavior of soft tissues [29; 30]. The superimposition of these energetic contributions was able to closely reproduce the experimental evolution of the mean Cauchy stress. First, to qualitatively test the suitability of the provided approach, changes in numerical parameters \((K_{1},A,D)\) were investigated with respect to their effectiveness in reshaping both the \(\sigma_{mat}\) and \(\sigma_{fib}\) curves. In particular, with reference to \(\sigma_{mat}\), the more \(K_{1}\) increased, the steeper the stress curve was, due to the action of the ground matrix. Similarly, the steepness of \(\sigma_{fib}\) increased in a linear way by increasing the \(A\) value, since it resulted in a proportional magnification of the curve. On the contrary, \(D\) affected \(\sigma_{fib}\) in a highly non-linear way. In the same way, the dimensionless extrinsic curvature \(\hat{k}_{mat}\) showed that the contribution of the ground matrix to the nerve stiffening was limited with stretch, while \(\hat{k}_{fib}\) heavily affected the stiffening process. As a consequence, three parameters were enough to provide a variety of shapes, allowing experimental trends to be correctly reproduced by theoretical predictions, while in the literature, more parameters were needed to account for the collagen action through an iterative and probabilistic approach [22; 23]. In addition, the sensitivity of \(\sigma_{mat}\) and \(\sigma_{fib}\) and \(\hat{k}_{fib}\) towards changes in numerical parameters was explored. In particular, the sensitivity of the matrix stress function to the variation of \(K_{1}\) and the sensitivity of the fiber stress function to the variation of the \(A\) parameter (with respect to the unit values of the other constants) converged towards zero for increasing values of \(K_{1}\) and \(A\). On the contrary, the sensitivity of the fibril stress function (\(\sigma_{fib}\)) to the variation of the \(D\) parameter quickly reached the steady value of one. The higher the value of \(D\) was, the faster the steady state was reached. As a consequence, the parameter \(D\) affected \(\sigma_{fib}\) more extensively than the parameter \(A\). In contrast, \(A\) affected \(\sigma_{fib}\) exactly as \(K_{1}\) affected \(\sigma_{mat}\).