## 4 Closed-Form Solutions for FG Nano-Rods
### Reduced Young's Modulus
The NSG model provides values of the reduced Young's modulus \(E_{r}\) tending to the classical (local) value \(E\) for increasing values of the SWCNT length \(L\). On the contrary, the values of the MD simulations appear to be constant for values of the SWCNT's length greater than \(27\) nm. The small-scale effect on displacements can be clearly observed in Figure 5 where the non-dimensional maximum displacement \(u(L)/L\), pertaining to the NSG model, is plotted versus the SWCNT's length \(L\) for an applied force \(F=1\) nN and the nonlocal parameter \(L_{c}\) ranging in the set \(\{0.04272,0.0534,0.06942\}\) nm. As can be seen, the small-scale effect on the displacement can be observed when the length of SWCNT is small and the small-scale effect increases for increasing values of the nonlocal parameter \(L_{c}\). If \(L\rightarrow+\infty\), the non-dimensional maximum displacement \(u(L)/L\) tends to the corresponding local one \(1/A_{E}=7.59117\times 10^{-4}\). In [45], a continuum mechanics model has been proposed to predict the effective wall thickness of a SWCNT and to calculate its Young's modulus. The deformation of a central long SWCNT in a bundle of SWCNTs, subjected to an external pressure, has been considered in plane-strain and has been modeled as a thin ring with a mean radius \(R\), thickness \(t\) in the radial direction and a unit width in the axial direction. Hence, it has been obtained that the radius of the nanotube is \(R=0.7066\) nm and the predicted thickness is \(t=0.0617\) nm so that the related Young's modulus is \(E=4880\) GPa. Considering the nonlocal parameter (dependent on the longitudinal atom spacing in armchair CNTs [46, 47]) \(L_{c}=0.0534\) nm as previously calibrated by means of the MD data, the variation law of Young's modulus obtained by the proposed NSG model for the SWCNT investigated in [45] is reported in Figure 6, with radius \(R=0.7066\) nm, thickness \(t=0.0617\) nm and Young's modulus \(E=4880\) GPa. The NSG model provides values of the reduced Young's modulus \(E_{r}\) tending to the value \(E=4880\) GPa for increasing values of the SWCNT length \(L\). Figure 4: SWCNT of armchair (10, 10) having diameter \(d=1.356\)\(nm\), effective thickness \(t=0.34\) nm and classical (local) Young’s modulus \(E=909.5\) GPa simulated by a CF nano-rod with a concentrated load at the free end. (**a**) Plot the variation of the [PERSON]’s modulus obtained by the NSG method together with the MD data versus the SWCNT length. The upper bound is given for the nonlocal parameter \(L_{c}=0.04272\) nm and the lower bound is given for the nonlocal parameter \(L_{c}=0.06942\) nm. (**b**) Variation of the [PERSON]’s modulus obtained by the NSG model and the MD data for the nonlocal parameter \(L_{c}=0.0534\) nm. Figure 5: SWCNT of armchair (10, 10) having diameter \(d=1.356\) nm, effective thickness \(t=0.34\) nm and classical (local) [PERSON]’s modulus \(E=909.5\) GPa simulated by a CF nano-rod with a concentrated load \(F=1\) nN at the free end. Plot of the non-dimensional maximum displacement \(u(L)/L\) of the NSG model vs. the SWCNT’s length \(L\) for the nonlocal parameter \(L_{c}\) ranging in the set \(\{0.04272,0.0534,0.06942\}\) nm. The non-dimensional local maximum displacement (black dotted line) is \(1/A_{E}=7.59117\times 10^{-4}\). It is apparent that, for a nanotube with L = 4 nm, the reduced Young's modulus is \(E_{r}=4753\) GPa, achieving 97.4% of the value of \(E\). Hence, we can conclude that, for the SWCNT considered in [45], the variation of the [PERSON]'s modulus \(E_{r}\) is really small if the length of the SWCNT is greater that 4 nm. The small-scale effect on displacements is observed in Figure 7, in which the non-dimensional maximum displacement \(u(L)/L\) of the NSG model is plotted versus the SWCNT's length L for an applied force \(F=1\) nN and the nonlocal parameter \(L_{c}\) ranging in the set \((0.04272,0.0534,0.06942)\) nm.