## 4 Closed-Form Solutions for FG Nano-Rods
### Case III: CC FG Nano-Rod Subject to a Uniformly Distributed Axial Load  
Let us consider a fully clamped FG nano-rod subject to a uniformly distributed axial load \(q(x)=p\). Following the steps reported in Section 3, the axial force is \(N\left(x\right)=-px+a_{1}\) so that the analytical solution of the FG nano-rod is provided as  
\[A_{E}\partial_{x}u(x)-l^{2}A_{E}\partial_{x}^{3}u(x)=-px+a_{1} \tag{47}\]  
under the CBC in Equation (26)  
\[\left\{ \begin{array}{c}\frac{a_{1}}{L_{c}}+\frac{l^{2}}{L_{c}^{2}}A_{E}\ \left.\partial_{x}^{2}u\left(x\right)\right|_{x=0}=-p\\ \frac{1}{L_{c}}\left(px-a_{1}\right)+\frac{l^{2}}{L_{c}^{2}}A_{E}\ \left. \partial_{x}^{2}u\left(x\right)\right|_{x=L}=-p\end{array} \right. \tag{48}\]  
and the classical boundary condition in Equation (27)  
\[\left\{ \begin{array}{c}u(0)=0\\ u\left(L\right)=0.\end{array} \right. \tag{49}\]  
Hence, the axial displacement is  
\[ \begin{array}{c}u(x)=u_{e}(x)+\frac{p}{2A_{E}\left(\mathrm{e}^{\frac{L}{l}}+ 1\right)}\mathrm{e}^{-\frac{x}{l}}\left(\mathrm{e}^{\frac{x}{l}}-1\right) \cdot\\ \left[\left(\mathrm{e}^{\frac{x}{l}}-\mathrm{e}^{\frac{L}{l}}\right)\left(2l^ {2}-L_{c}-2L_{c}^{2}\right)\right]\end{array}  \tag{50}\]  
Figure 8: CF nano-rod with a uniformly distributed axial load. Plot of the non-dimensional axial displacement \(\bar{u}_{\infty}\), limit non-dimensional axial displacement \(\bar{u}_{\infty}\) (black dotted line) and local non-dimensional axial displacement \(\bar{u}_{e}\) (black dot dashed line) vs. the nano-rod non-dimensional length \(\xi\) in terms of the nonlocal non-dimensional parameter \(\lambda\) in the set \(\left\{0.2,0.4,0.8,1.0\right\}\) and gradient non-dimensional parameter \(\mu\) in the set \(\left\{0.15,0.30\right\}\). where \(u_{e}\) is the axial displacement of the local model  
\[u_{e}(x)=\frac{px}{2A_{E}}\left(L-x\right). \tag{51}\]  
Note that we have \(a_{1}=pL/2\) so that the axial force of the NSG method is \(N(x)=p(L-2x)/2\) and coincides to the local one. The non-dimensional axial displacement \(\bar{u}\) of the FG nano-rod versus the FG nano-rod non-dimensional length \(\xi\) is reported in Figure 9 in terms of the nonlocal non-dimensional parameter \(\lambda\) for \(\mu=0.15\) (thick lines) and \(\mu=0.30\) (dotted lines). The non-dimensional midspan axial displacement \(\bar{u}(1/2)\) increases for increasing values of the nonlocal parameter \(\lambda\) for a given \(\mu\). The limit FG nano-rod non-dimensional axial displacement for the non-dimensional nonlocal parameter \(\lambda\) tending to vanish follows from Equation (50)  
\[\bar{u}_{\text{lim}}(\xi)=\frac{\left(-1+\mathrm{e}^{\frac{\xi}{\beta}} \right)\left(\mathrm{e}^{\frac{\xi}{\beta}}-\mathrm{e}^{\frac{1}{\beta}} \right)}{1+\mathrm{e}^{\frac{1}{\beta}}}\mu^{2}\mathrm{e}^{-\frac{\xi}{\beta} }-\frac{1}{2}\left(\xi-1\right)\xi \tag{52}\]  
and is reported with magenta thick line for \(\mu=0.15\) and dotted line for \(\mu=0.30\). The limit non-dimensional axial displacement \(\bar{u}_{\infty}\) is obtained from Equation (50) for \(\mu\to\infty\) and is the vanishing one, i.e., \(\bar{u}_{\infty}(\xi)=0\). The corresponding plot is reported with the black dotted line in Figure 9.  
The 3D plot of the non-dimensional maximum axial displacement \(\bar{u}(1/2)\) for the proposed NSG method versus the non-dimensional characteristic parameters \(\lambda\) and \(\mu\) is reported in Figure 10. It is apparent that the NSG method stiffness or soften the nano-rod depending on the values of the non-dimensional nonlocal and gradient parameters \((\lambda,\mu)\). The cuts of the 3D plot for given values of the non-dimensional parameters \(\lambda\) and \(\mu\) are provided in Figure 11. In particular, the plots of the non-dimensional maximum axial displacement \(\bar{u}(1/2)\) for the NSG method versus the non-dimensional characteristic parameter \(\lambda\) for \(\mu=0.2\) and \(\mu=0.6\) are provided in Figure 11a.