## 4 Closed-Form Solutions for FG Nano-Rods
### Case II: CF FG Nano-Rod Subject to a Uniformly Distributed Axial Load  
Let us now consider a FG nano-rod with a clamped end at \(x=0\) and a free end at \(x=L\) subject to a uniformly distributed axial load \(q(x)=p\). Following the steps reported in Section 3, the axial force \(N\) can be evaluated by means of the equilibrium equation so that Equation (24), with the boundary condition \(N(L)=0\), yields  
Figure 6: Plot the variation of the [PERSON]’s modulus obtained by the NSG method versus the SWCNT length for the nonlocal parameter \(L_{c}=0.0534\) nm simulated by a CF nano-rod with a concentrated load at the free end. The data of the SWCNT are: diameter \(d=1.4132\) nm, effective thickness \(t=0.0617\) nm and [PERSON]’s modulus \(E=4880\) GPa. Figure 7: 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 \(1/AE=7.48069\times 10^{-4}\). The data of the SWCNT are: diameter \(d=1.4132\) nm, effective thickness \(t=0.0617\) nm and [PERSON]’s modulus \(E=4880\) GPa. \(p(L-x)\). Hence, the closed-form solution of the NSG model for the considered FG nano-rod is obtained by the following nonlocal differential equation  
\[A_{E}\partial_{x}u(x)-l^{2}A_{E}\partial_{x}^{3}u(x)=p(L-x) \tag{39}\]  
under the CBC in Equation (26)  
\[\left\{ \begin{array}{c}\frac{pL}{L_{c}}+\frac{l^{2}}{L_{c}^{2}}A_{E}\left. \partial_{x}^{2}u\left(x\right)\right|_{x=0}=-p\\ \frac{l^{2}}{L_{c}^{2}}A_{E}\left.\partial_{x}^{2}u\left(x\right)\right|_{x=L} =-p\end{array} \right. \tag{40}\]  
and the classical boundary condition in Equation (27)  
\[u(0)=0. \tag{41}\]  
Hence, the axial displacement is  
\[ \begin{array}{c}u(x)=u_{e}(x)+\frac{p}{A_{E}\left(\mathrm{e}^{\frac{2L}{T}} -1\right)}\mathrm{e}^{-\frac{\xi}{T}}\left(\mathrm{e}^{\frac{\xi}{T}}-1\right) \cdot\\ \left[\mathrm{e}^{\frac{L}{T}}\left(l^{2}-L_{c}^{2}\right)+\mathrm{e}^{\frac{L \times x}{T}}\left(l^{2}-L_{c}^{2}\right)+\right.\\ \left.+\mathrm{e}^{\frac{2L}{T}}\left(-l^{2}+LL_{c}+L_{c}^{2}\right)+\mathrm{e }^{\frac{\xi}{T}}\left(-l^{2}+LL_{c}+L_{c}^{2}\right)\right]\end{array}  \tag{42}\]  
where \(u_{e}\) is the axial displacement of the local model  
\[u_{e}(x)=\frac{px}{2A_{E}}\left(2L-x\right). \tag{43}\]  
The maximum axial displacement of the \(CF\) FG nano-rod is attained at the free end \(x=L\) and is given by  
\[u(L)=u_{e}(L)+\frac{pLL_{c}}{A_{E}}=\frac{pL\left(L+2L_{c}\right)}{2A_{E}}. \tag{44}\]  
It is apparent from Equation (42) that the axial displacement \(u\) depends on the nonlocal parameters \(l\) and \(L_{c}\). On the contrary, the maximum axial displacement \(u(L)\) (see Equation (44)) is independent of the nonlocal gradient parameter \(l\). The limit maximum axial FG nano-rod displacement for the nonlocal parameter \(L_{c}\to 0\) is given by the classical (local) displacement \(u_{e}(L)=\frac{pL^{2}}{2A_{E}}\) of the FG nano-rod. The non-dimensional axial displacement \(\tilde{u}\) of the FG nano-rod versus the FG nano-rod non-dimensional length \(\xi\) is reported in Figure 8 in terms of the nonlocal non-dimensional parameter \(\lambda\) for \(\mu=0.15\) (thick lines) and \(\mu=0.30\) (dotted lines). The non-dimensional maximum axial displacement \(\tilde{u}(1)\) increases for increasing values of the nonlocal parameter \(\lambda\) and is independent of the non-dimensional gradient parameter \(\mu\). The limit FG nano-rod non-dimensional axial displacement for the non-dimensional nonlocal parameter \(\lambda\) tending to vanish is given by  
\[\tilde{u}_{\mathrm{lim}}(\xi)=\frac{\left(-1+\mathrm{e}^{\frac{\xi}{T}} \right)\left(\mathrm{e}^{\frac{\xi}{T}}-\mathrm{e}^{\frac{1}{T}}\right)}{1+ \mathrm{e}^{\frac{1}{T}}}\mu^{2}\mathrm{e}^{-\frac{\xi}{T}}-\frac{1}{2}\left( \xi-2\right)\xi \tag{45}\]  
and is reported with magenta thick line for \(\mu=0.15\) and dotted line for \(\mu=0.30\). Note that \(\tilde{u}_{\mathrm{lim}}(1)=\tilde{u}_{e}(1)=0.5\).