## 4 Closed-Form Solutions for FG Nano-Rods
### Case III: CC FG Nano-Rod Subject to a Uniformly Distributed Axial Load
It is apparent that for \(\mu=0.2\) and \(\lambda<0.070156\) or \(\mu=0.6\) and \(\lambda<0.4\) the NSG model is stiffer than the local model and for \(\mu=0.2\) and \(\lambda>0.070156\) or \(\mu=0.6\) and \(\lambda>0.4\) the NSG model softens the nano-rod. The limit values of the non-dimensional maximum axial displacement \(\bar{u}(1/2)\) for \(\lambda\to 0^{+}\) are given by \(0.0915228\) for \(\mu=0.2\) and by \(0.0281989\) for \(\mu=0.6\). Figure 9: CC nano-rod subject to a uniformly distributed axial load. Plot of the non-dimensional axial displacement \(u\), limit non-dimensional axial displacement \(\bar{u}_{\infty}\) (black dotted line) and local non-dimensional axial displacement \(\bar{u}_{e}\) (black dot dashed line) vs. the nano-rod non-dimensional length \(\xi\) in terms of the nonlocal non-dimensional parameter \(\lambda\) in the set \(\{0.2,0.4,0.8,1.0\}\) and gradient non-dimensional parameter \(\mu\) in the set \(\{0.15,0.30\}\). The plots of the non-dimensional maximum axial displacement \(\bar{u}(1/2)\) for the NSG method versus the non-dimensional characteristic parameter \(\mu\) for \(\lambda=0.2\) and \(\lambda=0.6\) are provided in Figure 11b. It is immediate to note that for \(\lambda=0.2\) and \(\mu<0.374166\) or \(\lambda=0.6\) and \(\mu<0.812404\) the NSG model is stiffer than the local model and for \(\lambda=0.2\) and \(\mu>0.374166\) or \(\lambda=0.6\) and \(\mu>0.812404\) the NSG model softens the nano-rod. The limit values of the non-dimensional maximum axial displacement \(\bar{u}(1/2)\) for \(\mu\to 0^{+}\) are given by 0.265 for \(\lambda=0.2\) and by 0.785 for \(\lambda=0.6\). Finally, a standard numerical analysis of Equation (50) shows that the non-dimensional maximum axial displacement \(\bar{u}(1/2)\) for the proposed NSG method is attained at the midspan and is reported in Table 3 in terms of the non-dimensional nonlocal parameter \(\lambda\) and gradient parameter \(\mu\). It worth noting that the maximum displacement is attained at the midspan \(\xi=1/2\) of the FG nano-rod independent of the values of \(\lambda\) and \(\mu\). The non-dimensional maximum axial displacement of the classical (local) model is \(\bar{u}_{e}(1/2)=1/8=0.125\). Accordingly, the non-dimensional maximum axial displacements \(\bar{u}(1/2)\) in terms of the pairs \((\lambda,\mu)\), which are less than the non-dimensional maximum classical axial displacements \(\bar{u}_{e}(1/2)=0.125\) of the FG nano-rod, are reported in italic in Table 3.  
Figure 11: CC nano-rod subject to a uniformly distributed axial load. (**a**) The non-dimensional maximum axial displacement \(\bar{u}(1/2)\) for the NSG method vs. the non-dimensional characteristic parameter \(\lambda\) for \(\mu=0.2\) and \(\mu=0.6\). (**b**) The non-dimensional maximum axial displacement \(\bar{u}(1/2)\) for the NSG method vs. the non-dimensional characteristic parameters \(\mu\) for \(\lambda=0.2\) and \(\lambda=0.6\). Figure 10: CC nano-rod subject to a uniformly distributed axial load. 3D plot of the non-dimensional maximum axial displacement \(\bar{u}(1/2)\) vs. the non-dimensional characteristic parameters \(\lambda\) and \(\mu\). The non-dimensional local maximum axial displacement \(\bar{u}_{e}(1/2)=1/8=0.125\) is the horizontal plane. Hence, the italic values of the non-dimensional maximum axial displacement \(\bar{u}(1/2)\) of the NSG method are smaller than the one of the classical (local) model, thus Table 3 allows one to identify the corresponding pairs \((\lambda,\mu)\) having the effect of stiffen or soften the FG nano-rod with respect the classical (local) behavior.