## 1 Introduction  
Peripheral nerves are crucial to connect sensory and motor organs to the central nervous system [1, 2]. Motor nerves carry information towards organs, muscles, or other peripheral effectors, and also, sensory nerves link sensory receptors to the brain [3]. These complex structures are hierarchically organized: internal axons, together with Schwann cells, are protected by a loose connective tissue (i.e., the endoneurium), which was found to be mainly composed by I and II type collagen, macrophages, and mast cells and filled by endoneurial fluid [2, 4, 5]. Endoneurial components are, in turn, enveloped in fascicles by the epineurial sheet, which is formed by layers of perineural cells, I and II type collagen, together with elastic fibers [2, 4, 5]. Finally, fascicles are grouped together by a further layer of connective tissue (i.e., the epineurium) mainly composed by I and III type collagen, elastic fibers, and mast and fat cells [2]. Despite this richness of internal substructures, the main constitutive components of the connective tissues of nerves are elastin and collagen, which were found to be mainly axially oriented [4].  
In addition, peripheral nerves are physical objects, which obey physical laws [6]. Indeed, they react to external mechanical stimuli trough a specific mechanical response, increasing their stiffness to keep axons' integrity and to preserve endoneurial structures from longitudinal overstretch [7]. The knowledge and the prediction of this kind of response are important in medicine [8, 9] and physical therapy [2; 10; 11], while they are strategic in modern and high-tech bioengineering fields, as in neural interfaces design [12; 13; 14].  
Although the anatomic path of nerves involves branching points, their mechanical response to longitudinal stretch was closely related to the response of the rectilinear nerve trunks [7]. As a consequence, and for the sake of simplicity, different kinds of models were implemented to predict the rectilinear nerve trunk response to external stimuli. Indeed, from one side, peripheral nerves were modeled as isotropic materials, providing a deterministic elastic [15; 16] or hyperelastic response [17; 18; 19; 20; 21]. From the other side, a stochastic iterative fibril-scale mechanical model was implemented to reproduce the straightening of wavy fibrils and to account for the effects of interfibrillar crosslinks on the overall properties of the tissue [22; 23]. Furthermore, recent studies aimed at investigating the relationship between macroscopic tensile loading of nerves and micro-scale deformations [24].  
Nevertheless, a deterministic and explicit continuum model to explore the influence of the ground matrix and collagen fibrils on the nerve stiffening is apparently not available in literature. As a consequence, in this work, such a theoretical framework is provided to quantify the action of both the ground matrix and collagen fibrils on the nerve stiffening.  
In particular, the text is organized as follows: First, a closed-form strain energy function (SEF) is proposed to account for the actions of the matrix and fibrils. Then, the change of the shape of the stress/stretch curve is studied for variations of relevant numerical parameters, while a further sensitivity study is performed. Finally, the proposed framework is used to investigate the stiffening phenomenon in different nerve specimens coming from different animal models [19; 25].