# Postbuckling and Free Vibration of Multilayer Imperfect Nanobeams under a Pre-Stress Load
## 2 Problem Formulation
### Constitutive Equations  
According to [PERSON]'s nonlocal elasticity differential model, the constitutive equation that describes the stress-strain relation for a laminated composite material is given by  
\[\sigma_{x}-\mu\sigma_{x}^{\prime\prime}=\overline{Q}_{11}\varepsilon_{x}= \overline{Q}_{11}(\varepsilon_{0}+zk_{0}) \tag{5}\]  
where \(\overline{Q}_{11}\) is the reduced transformed stiffness from the material coordinates to the problem coordinates, and \(\mu\) is a size parameter such that \(\mu=\left(\varepsilon_{0}a\right)^{2}\), where \(\varepsilon_{0}\) is a constant and \(a\) is an internal-length parameter. The length parameter, \(\varepsilon_{0}a<2\) nm, for a single-wall carbon-nano tube [67]. Multiplying Equation (5) by the beam's cross-sectional area, \(A\), and integrating over the area yields the nonlocal resultant force, \(N\), as follows:  
\[N-\mu N^{\prime\prime}=\epsilon_{0}\int_{h}b\overline{Q}_{11}dz+k_{0}\int_{h}zb \overline{Q}_{11}dz=\epsilon_{0}A_{11}+k_{0}B_{11} \tag{6}\]  
where \(b\) is the beam's width, and \(A_{11}=\int_{h}b\overline{Q}_{11}dz\) and \(B_{11}=\int_{h}zb\overline{Q}_{11}dz\) are the axial and coupling stiffness constants of the beam's cross section, respectively. From the equilibrium equations, we have \(N^{\prime}=0\), and hence Equation (6) reduces to  
\[N=\epsilon_{0}A_{11}+k_{0}B_{11} \tag{7}\]  
which in light of Equation (3) reads as  
\[N=A_{11}\bigg{[}u^{\prime}+\frac{1}{2}\Big{(}{w^{\prime}}^{2}-{w^{\prime}_{0} }^{2}\Big{)}\bigg{]}-B_{11}\big{(}{w^{\prime\prime}}-{w^{\prime\prime}_{0}} \big{)} \tag{8}\]  
Differentiating Equation (8) twice with respect to \(x\), and noting that \(N^{\prime\prime}=0\), results in the following relation:  
\[\epsilon^{\prime\prime}_{0}=\frac{B_{11}}{A_{11}}\big{(}{w^{\prime\prime\prime }}-{w^{\prime\prime\prime}_{0}}\big{)} \tag{9}\]  
Similarly, the nonlocal resultant moment, \(M\), can be represented by  
\[M-\mu M^{\prime\prime}=\epsilon_{0}B_{11}+k_{0}D_{11} \tag{10}\]  
where \(D_{11}=\int_{h}b\;z^{2}\overline{Q}_{11}dz\) is the bending stiffness of the cross section. Substituting the term, \(M^{\prime\prime}\), from Equation (4b) into Equation (10) results in the moment stress resultant, \(M\), as a function of only the displacements as  
\[M=\mu\big{[}m\;\tilde{w}-\big{(}N-\overline{N}\big{)}{w^{\prime\prime}}\big{]} +\epsilon_{0}B_{11}+k_{0}D_{11} \tag{11}\]  
Differentiating Equation (11) twice with respect to \(x\) and substituting the result back into Equation (4b), we obtain  
\[-m\;\tilde{w}+\mu\;m\;\tilde{w}^{\prime\prime}+\big{(}N-\overline{N}\big{)}{ w^{\prime\prime}}-\mu\big{(}N-\overline{N}\big{)}{w^{\prime\prime\prime}}+B_{11} \epsilon^{\prime\prime}_{0}+D_{11}k^{\prime\prime}_{0}=0 \tag{12}\]  
In order to eliminate the induced axial force, \(N\), from Equation (12), we integrate Equation (8) once with respect to \(x\) and apply the non-moving end conditions, \(u(0)=0,\;u(L)=0\), to get  
\[N=\frac{A_{11}}{2L}\int_{0}^{L}\Big{(}{w^{\prime}}^{2}-{w^{\prime}_{0}}^{2} \Big{)}dx-\frac{B_{11}}{L}\int_{0}^{L}\big{(}{w^{\prime\prime}}-{w^{\prime \prime}_{0}}\big{)}dx \tag{13}\]  
Finally, by substituting \(N,\;\epsilon_{0}\), and \(k_{0}\) into Equation (12), we obtain the equation of motion of a nonlocal, initially curved laminated multilayer nanobeam in terms of only the displacement, \(w\), as follows:  
\[ \begin{array}{ll}m\;\tilde{w}-\mu\;m\;\tilde{w}^{\prime\prime}&+\Bigg{\{}D_ {11}+\mu\bigg{[}\frac{A_{11}}{2L}\int_{0}^{L}\Big{(}{w^{\prime}}^{2}-{w^{\prime }_{0}}^{2}\Big{)}dx-\frac{B_{11}}{L}\int_{0}^{L}\big{(}{w^{\prime\prime}}-{w^{ \prime\prime}_{0}}\big{)}dx-\overline{N}\bigg{]}\\ &-\frac{B_{11}^{2}}{A_{11}}\bigg{\}}{w^{i\prime}}-\bigg{[}\frac{A_{11}}{2L} \int_{0}^{L}\Big{(}{w^{\prime}}^{2}-{w^{\prime}_{0}}^{2}\Big{)}dx-\frac{B_{11} }{L}\int_{0}^{L}\big{(}{w^{\prime\prime}}-{w^{\prime\prime}_{0}}\big{)}dx- \overline{N}\bigg{]}{w^{\prime\prime}}\\ &-\bigg{(}D_{11}-\frac{B_{11}^{2}}{A_{11}}\bigg{)}{w^{i\prime}_{0}}=0\end{array}  \tag{14}\]For a symmetric laminate, \(B_{11}=0\).