## 4 Closed-Form Solutions for FG Nano-Rods
### Case I: CF FG Nano-Rod with a Concentrated Load at the Free End  
Let us consider a FG nano-rod of length \(L\) with a clamped end at \(x=0\) and a free end at \(x=L\) subject to a concentrated load \(\mathcal{F}\) at the free end. Following the steps in Section 3, the axial force \(N\) can be evaluated by means of the equilibrium equation so that Equation (24), with the boundary condition \(N(L)=\mathcal{F}\), yields \(N(x)=\mathcal{F}\). For simplicity, we drop the dependence on the nonlocal characteristic parameters \((L_{c},l)\). Then, the analytical solution of the NSG model for the FG nano-rod is obtained from the following nonlocal differential equation  
\[A_{E}\partial_{x}u(x)-l^{2}A_{E}\partial_{x}^{3}u(x)=\mathcal{F} \tag{29}\]  
under the CBC (Equation (26))  
\[\left\{ \begin{array}{c}\frac{\mathcal{F}}{L_{c}}+\frac{l^{2}}{L_{c}^{2}}A_{ E}\ \ \partial_{x}^{2}u\left(x\right)\big{|}_{x=0}=0\\ -\frac{\mathcal{F}}{L_{c}}+\frac{l^{2}}{L_{c}^{2}}A_{E}\ \ \partial_{x}^{2}u\left(x \right)\big{|}_{x=L}=0\end{array} \right. \tag{30}\]  
and the kinematic boundary condition in Equation (27)  
\[u(0)=0. \tag{31}\]  
Hence, the solution of the differential equation (Equation (29)) with the boundary conditions in Equations (30) and (31) provides the axial displacement  
\[u(x)=u_{e}(x)+\frac{\mathcal{F}L_{c}}{A_{E}\left(\mathrm{e}^{\frac{L}{l}}-1 \right)}\mathrm{e}^{-\frac{x}{l}}\left(\mathrm{e}^{\frac{x}{l}}-1\right) \left(\mathrm{e}^{\frac{L}{l}}+\mathrm{e}^{\frac{x}{l}}\right) \tag{32}\]  
where \(u_{e}\) is the rod axial displacement of the local model  
\[u_{e}(x)=\frac{\mathcal{F}x}{A_{E}}. \tag{33}\]  
The maximum displacement takes place at \(x=L\) and can be obtained from Equation (32) setting \(x=L\)  
\[u(L)=u_{e}(L)+\frac{2\mathcal{F}L_{c}}{A_{E}}=\frac{\mathcal{F}\left(L+2L_{c} \right)}{A_{E}}. \tag{34}\]  
The classical (local) displacement \(u_{e}\) of the FG nano-rod is provided by \(L_{c}\to 0\). Moreover, the limit nano-rod displacement \(u_{\infty}\) for \(l\rightarrow+\infty\) is given by  
\[u_{\infty}(x)=u_{e}(x)+\frac{2\mathcal{F}L_{c}}{A_{E}L}x=\frac{\mathcal{F} \left(L+2L_{c}\right)}{A_{E}L}x. \tag{35}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Non-Dimensional Axial Displacement** & **Applied Load** \\ \hline \(\bar{u}=u\frac{A_{E}}{pL^{2}}\) & Uniform axial load \(p\) \\ \(\bar{u}=u\frac{A_{E}}{\mathcal{F}L}\) & Axial force \(\mathcal{F}\) at \(x=L\) \\ \(\bar{u}=\frac{u}{\delta}\) & Imposed axial displacement \(\delta\) at \(x=L\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Non-dimensional axial displacement \(\bar{u}\) in terms of the applied load. It is important to note that the NSG model for FG nano-rods exerts a softening effect, with respect to the local behavior, in terms of the nonlocal parameter \(L_{c}\). It is of interest that the maximum displacement \(u(L)\) of the NSG rod model does not depend on the gradient parameter \(l\) and the maximum axial displacement tends to the one of the classical (local) rod if \(L_{c}\to 0\). The effects of the non-dimensional characteristic parameters \(\lambda\) and \(\mu\) on the elastic response of nano-rods are examined in Figures 1 and 2. Figure 1a,b show the non-dimensional axial displacement \(\tilde{u}\) in terms of the gradient non-dimensional parameter \(\mu\) for \(\lambda=0.4\) and \(\lambda=0.8\), respectively. The local non-dimensional axial displacement \(\tilde{u}_{e}\) is recovered by \(\lambda\to 0^{+}\) and is reported with the dot dashed line. The limit non-dimensional axial displacement \(\tilde{u}_{\infty}\) follows from Equation (35) and is reported with the dotted line in terms of the non-dimensional nonlocal parameter \(\lambda\). The comparison between nonlocal FG nano-rods and classical (local) FG rods in Figure 1 highlights the increment of the axial displacement \(\tilde{u}\) due to the behavior of the nonlocal model. The parameter \(\lambda\) has the effect of increasing the axial displacement, i.e.