# A Quantitative Investigation on the Peripheral Nerve Response within the Small Strain Range
## 3 Results  
Experimental data were shown (white circles) for five consecutive extensions together with the theoretical prediction coming from Equation (19), which was plotted through a bold line (Figure 1a). The range of variation of the Cauchy stress was between 0 (reference configuration) and 1.183 kPa when the strain ranged between \(\epsilon=0\) and \(\epsilon=0.05\). In this range the predicted stiffness was \(E=21.188\) kPa, and the axial component of Equation (19) was able to closely reproduce the experimental evolution of stress (\(R^{2}=0.997\)). The distribution of errors, which were accounted for as the difference between experimental data and the theoretical prediction of the mean Cauchy stress, was shown in Figure 1b. The distribution of errors had a peak of probability (around 8.5%) for values ranging between \(-0.020\) kPa and \(-0.015\) kPa, while the maximum error was 0.104 kPa (around 0.5%), which represented a deviation from linearity of about 8.8%. The theoretical evolution of the axial displacement was studied for a fully incompressible nervous specimen with a length of 55 mm and cross sectional eccentricity of 0.95. Therefore, the transversal area was approximated with a \"tape like\" structure (1.398 mm \(\times\) 4.000 mm). The axial field of displacements was linearly increasing with the \(z\) coordinate, thus the boundary conditions to the limit sections, (i.e., \(z=0\), \(z=55\) mm) were automatically satisfied. Since the solution was expressed only as a function of the \(z\) coordinate, it was invariant with respect to \(\times\) and \(y\), and the displacement field had the form displayed in Figure 2a. The lateral contraction was also investigated and, for each axis (\(x\) and \(y\)), the displacement field was expressed as a function of only one variable (respectively \(x\) and \(y\)). In both cases, the field of displacements was symmetric with respect to the central axis, as well as invariant with respect to the \(z\) coordinate, as shown in Figure 2b,c. Figure 1: **(a)** Experimental data for five consecutive extensions together with the theoretical prediction (bold line), which was able to closely reproduce the experimental evolution of stress (\(R^{2}=0.997\)). **(b)** The distribution of errors had a peak of probability (around 8.5%) for values ranging between \(-0.020\) kPa and \(-0.015\) kPa, while the maximum error was 0.104 kPa (around 0.5%), which represented a deviation from linearity of about 8.8%. Moreover, the compressibility of the material was varied and, in Figure 3a, the evolution of the Poisson ratio was shown for increasing values of the \(\omega\) parameter. In particular, the Poisson ratio assumed the values of \(\
u\simeq 0.499\) for \(\omega=1\times 10^{-12}\), \(\
u\simeq 0.476\), for \(\omega=0.1\) and \(\
u=0.334\) for \(\omega=1\). Similarly, the variation of the bulk modulus, which was related to the compressibility parameter \(\omega\) through Equation (27), was shown in Figure 3b. For \(E=21.188\) kPa, together with the previous values of compressibility, the bulk modulus assumed the following values \(1.413\times 10^{13},148.316,21.188\) kPa. Figure 2: (**a**) The theoretical evolution of the axial displacement for a fully incompressible nervous specimen with a length of 55 mm and a section of 1.398 mm \(\times\) 4.000 mm; the axial field of displacements was linearly increasing with the \(z\) coordinate. (**b**) Lateral contraction; the displacement field was expressed as a function of \(x\). (**c**) Lateral contraction: the displacement field was expressed as a function of \(y\). In both cases, the field of displacements was symmetric with respect to the central axis as well as invariant with respect to the \(z\) coordinates. In a similar way, the evolution of the shear modulus, related to \(\omega\) through Equation (28), was shown in Figure 3c. In this case, the shear modulus values for \(\omega=1\times 10^{-12},0.1.1\) were respectively \(G=7.063,7.177,7.945\) kPa.