## 4 Closed-Form Solutions for FG Nano-Rods
### Reduced Young's Modulus  
In applications, the evaluation of the Young's modulus of micro- and FG nano-rods is of great interest. Hence, we evaluate the reduced rigidity \(K_{r}\) from Equation (34) to get  
\[K_{r}=\frac{L}{L+2L_{c}}A_{E}. \tag{36}\]  
Note that the classical rigidity \(K_{r}=A_{E}\) can be recovered if \(L_{c}\to 0\). Figure 3: CF nano-rod with a concentrated load at the free end. 3D plot of the non-dimensional maximum axial displacement \(\bar{u}(1)\) vs. the non-dimensional characteristic parameters \(\lambda\) and \(\mu\). The non-dimensional local maximum axial displacement \(\bar{u}_{e}(1)=1\) is the horizontal plane. Figure 2: CF nano-rod with a concentrated load at the free end. Plot of the non-dimensional axial displacement \(\bar{u}\), limit non-dimensional axial displacement \(\bar{u}_{\infty}\) (black dotted line) and local non-dimensional axial displacement \(\bar{u}_{e}\) (black dot dashed line) in terms of the nonlocal non-dimensional parameter \(\lambda\) in the set \(\{0.2,0.4,0.8,1.0\}\) and: (**a**) gradient non-dimensional parameter \(\mu=0.15\); and (**b**) gradient non-dimensional parameter \(\mu\) in the set \(\{0.15,0.30\}\). Using the reduced rigidity in Equation (36), the axial displacement of the modified nonlocal strain gradient model can be calculated by adopting the classical (local) analysis for rods. Let us consider a constant Young's elastic modulus \(E(y)=E\) so that the elastic area is \(A_{E}=\int_{\Omega}E\left(y\right)dA=EA\). Hence, the reduced [PERSON]'s modulus \(E_{r}\) can be defined from Equation (36) as  
\[E_{r}=\frac{L}{L+2L_{c}}E \tag{37}\]  
depending on the nonlocal parameter \(L_{c}\). The upper bound of the nano-rod reduced [PERSON]'s modulus \(E_{r}\) for \(L\rightarrow\infty\) is provided by the local [PERSON]'s modulus \(E\). Analogously, for \(L_{c}\to 0\), we recover the local [PERSON]'s modulus \(E\) and for \(L_{c}\rightarrow+\infty\), the nano-rod reduced Young's modulus tends to vanish. To make a comparison, we considered the data presented in [40] for a SWCNT of armchair (10, 10). The diameter \(d\) of the SWCNT (n, m) can be calculated by  
\[d=\frac{a}{\pi}\sqrt{3\left(n^{2}+n\cdot m+m^{2}\right)}=1.356\ nm \tag{38}\]  
where the carbon-carbon bond length is \(a=0.142\) nm. The effective thickness of the considered SWCNT is \(t=0.34\) nm and the classical (local) [PERSON]'s modulus assumed in [40,44] is \(E=909.5\) GPa. The [PERSON]'s modulus predicted by the MD simulations is reported in [40,44]. Note that the strain gradient elastic theory with high-order boundary conditions was adopted in [40] and the necessity to impose higher-order (non-classical) boundary conditions has the effect that the classical boundary conditions of the classical theory (such as free and clamped boundary conditions) may no longer be meaningful for the modified nonlocal strain gradient rod. As a result, following [28,35], one needs to take into account further boundary conditions involving higher-order stress and strain distributions. Accordingly, four nonlocal nano-rods are considered in [40], depending on the considered higher-order boundary conditions, and for each of them the small-scale parameters are set to match the results of the MD simulations. On the contrary, the proposed NSG model has no higher-order boundary condition to add so that the classical definitions of external constraints must not be modified. Hence, a unique model of rod can be considered and a unique value of the nonlocal parameter has to be set to match the results of the MD simulations. In Figure 4 the results provided by Equation (37) are plotted together with the MD data versus the SWCNT length. Upper and lower bounds of the nonlocal parameter \(L_{c}=0.04272\) nm and \(L_{c}=0.06942\) nm are reported in Figure 4a to include the values provided by the MD simulations. A good agreement between the Young's modulus obtained by the NSG model and the MD results could be obtained by setting the nonlocal parameter \(L_{c}=0.0534\) nm, as shown in Figure 4b.