## 2 Modified Nonlocal Strain Gradient Law for Rods  
Let us consider a functionally graded (FG) straight nano-rod of length \(L\), the \(x\)-coordinate is taken along the length of the nano-rod with the \(y\)-coordinate along the thickness and the \(z\)-coordinate along the width of the nano-rod. The local Young's modulus \(E\) of the FG nano-rod continuously changes in the thickness direction \(y\), so that the Young's elastic modulus at the point \(y\) is \(E(y)\) and the elastic area is \(A_{E}=\int_{\Omega}E\left(y\right)dA\), being \(\Omega\) the nano-rod cross-section. In the modified nonlocal strain gradient (NSG) model for FG nano-rods, the axial force \(N\) is defined in terms of elastic axial strain \(\varepsilon_{el}\) and of its derivative \(\partial_{x}\varepsilon_{el}\) is [27]  
\[ \begin{array}{l}N\left(x,\lambda_{0},\lambda_{1},l\right)=\left(\phi_{0} \ast\left(A_{E}\;\varepsilon_{el}\right)\right)\left(x,\lambda_{0}\right)-l^{2 }\partial_{x}\left(\phi_{1}\ast\left(A_{E}\;\partial_{x}\varepsilon_{el} \right)\right)\left(x,\lambda_{1}\right)\\ =\int_{0}^{L}\phi_{0}\left(x-\xi,\lambda_{0}\right)\;\left(A_{E}\; \varepsilon_{el}\right)\left(\xi\right)d\xi-l^{2}\partial_{x}\int_{0}^{L}\phi _{1}\left(x-\xi,\lambda_{1}\right)\;\left(A_{E}\;\partial_{\xi}\varepsilon_{el }\right)\left(\xi\right)d\xi.\end{array}  \tag{1}\]The smoothing kernels \(\phi_{0}\) and \(\phi_{1}\) depend on two non-dimensional nonlocal parameters \(\lambda_{0}>0\) and \(\lambda_{1}>0\). The scale parameter \(l>0\), characteristic of the strain gradient elasticity [21], was subsequently introduced in [27] to make dimensionally homogeneous the convolutions in Equation (1) and to describe the importance of higher-order strain gradient fields. Following [27; 38], we consider that the nonlocal parameters are coincident, i.e., \(\lambda:=\lambda_{0}=\lambda_{1}\), and the kernels \(\phi_{0}\) and \(\phi_{1}\) are coincident with the bi-exponential averaging function given by  
\[\phi(x,L_{c})=\frac{1}{2L_{c}}\exp\left(-\frac{\left|x\right|}{L_{c}}\right), \tag{2}\]  
being \(L_{c}=\lambda L\) the characteristic length of Eringen nonlocal elasticity. The kernel (Equation (2)) fulfills positivity, symmetry, normalization and impulsivity [42]. Introducing the following fields  
\[ \begin{split}& N_{0}\left(x,L_{c}\right)=\int_{0}^{L}\phi\left(x- \xi,L_{c}\right)\ \left(A_{E}\ \varepsilon_{el}\right)\left(\xi\right)d\xi\\ & N_{1}\left(x,L_{c},l\right)=l^{2}\int_{0}^{L}\phi\left(x-\xi,L _{c}\right)\ \left(A_{E}\ \partial_{\xi}\varepsilon_{el}\right)\left(\xi\right)d\xi,\end{split}  \tag{3}\]  
the modified nonlocal strain gradient elastic law (Equation (1)) can be rewritten as  
\[N\left(x,L_{c},l\right)=N_{0}\left(x,L_{c}\right)-\partial_{x}N_{1}\left(x,L_ {c},l\right). \tag{4}\]  
As proven in the next proposition, the modified nonlocal strain gradient integral relation (Equation (4)) for FG nano-rods is equivalent to a suitable differential law with constitutive boundary conditions. **Proposition 1** (Constitutive equivalence for FG nano-rods). : _The modified nonlocal strain gradient constitutive law (Equation (4)) equipped with the bi-exponential kernel (Equation (2))_  
\[N\left(x,L_{c},l\right)=N_{0}\left(x,L_{c}\right)-\partial_{x}N_{1}\left(x,L_{ c},l\right), \tag{5}\]  
_with \(x\in[0,L]\), is equivalent to the differential relation_  
\[\left(A_{E}\cdot\varepsilon_{el}\right)\left(x\right)-l^{2}\partial_{x}^{2} \left(A_{E}\cdot\varepsilon_{el}\right)\left(x\right)=N\left(x,L_{c},l\right) -L_{c}^{2}\ \partial_{x}^{2}N\left(x,L_{c},l\right) \tag{6}\]  
_subject to the following two constitutive boundary conditions (CBC)_  
\[\left\{