# Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method
## 2 Theory and Formulation  
In what follows, the nonlocal strain gradient theory [36] is applied to account for both the nonlocal stress field and the strain gradient effects, by means of two small-scale parameters. This theory defines the stress field as  
\[\sigma_{ij}-l_{1}^{2}\sigma_{ij,mm}=C_{ijkl}(\varepsilon_{kl}-l_{2}^{2} \varepsilon_{kl,mm}) \tag{1}\]  
where \(\sigma_{ij}\) and \(\varepsilon_{ij}\) are the stress and strain tensors; \(C_{ijkl}\) refers to the elastic properties' matrix, while \(l_{1}\) and \(l_{2}\) denote the internal length scales to be determined experimentally or numerically by means of microscopic models, e.g., the MD simulations. A triclinic nanobeam of length \(L\), width \(b\), thickness \(h\) is shown in Figure 1. In the current study, it is assumed that the stress components depend on both the longitudinal and transverse shear strains, i.e.,  
\[\left\{ \begin{array}{l}\sigma_{xx}\\ \tau_{xz}\end{array} \right\}=\left[ \begin{array}{cc}c_{11}&c_{15}\\ c_{51}&c_{55}\end{array} \right]\left\{ \begin{array}{l}\varepsilon_{xx}\\ \gamma_{xz}\end{array} \right\} \tag{2}\]  
where \(\varepsilon_{xx}\), \(\gamma_{xz}\) denote the longitudinal and transverse shear strains, respectively, whereas \(\sigma_{xx}\), \(\tau_{xz}\) stand for the axial and shear stress, respectively. In a context where the Timoshenko beam theory has been largely applied to model isotropic structures, herein the theory is extended to handle triclinic beams. According to the proposed continuum model, the displacement field takes the following form  
\[u_{x}(x,z,t)=z\psi(x,t) \tag{3}\]  
\[u_{y}(x,z,t)=0 \tag{4}\]  
\[u_{z}(x,z,t)=w(x,t) \tag{5}\]  
where \(w\) refers to the transverse displacements; \(\psi\) stands for the rotation of the cross-section and \(t\) denotes the time. According to the small deformations assumption, the constitutive relations for the anisotropic nanobeam are defined as follows  
\[\varepsilon_{xx}=z\frac{\partial\psi(x,t)}{\partial x} \tag{6}\]  
\[\gamma_{xz}=\frac{\partial w(x,t)}{\partial x}+\psi(x,t) \tag{7}\]  
\[\sigma_{xx}=c_{11}(z\frac{\partial\psi(x,t)}{\partial x})+c_{15}(\frac{ \partial w(x,t)}{\partial x}+\psi(x,t)) \tag{8}\]  
\[\tau_{xz}=c_{51}(z\frac{\partial\psi(x,t)}{\partial x})+c_{55}(\frac{\partial w (x,t)}{\partial x}+\psi(x,t)) \tag{9}\]  
\(c_{ij}\) being the elastic components of the triclinic material, defined as [13]  
\[ \begin{array}{l}c_{11}=98.84\times 10^{9}\ \text{Pa}\\ c_{15}=c_{51}=1.05\times 10^{9}\ \text{Pa}\\ c_{55}=21.10\times 10^{9}\ \text{Pa}\end{array}  \tag{10}\]  
The governing Equations of the problem are determined through the Hamiltonian principle as follows  
\[\int_{0}^{t}\delta(U-T)dt=0 \tag{11}\]  
\(U\) and \(T\) being the strain and kinetic energy, respectively. More specifically, the strain energy has the following form  
\[ \begin{array}{l}\delta U=\int_{V}\sigma_{ij}\delta\varepsilon_{ij}dV=\int_{ V}\sigma_{xx}\delta\varepsilon_{xx}+\tau_{xz}\delta\gamma_{xz}=\int_{0}^{L}M \frac{\partial\psi(y)}{\partial x}+Q(\delta\psi+\frac{\partial\psi(w)}{ \partial x})dx\\ =\int_{0}^{L}\Bigl{[}(-\frac{\partial M}{\partial x}+Q)\delta\psi-\frac{ \partial\phi(x)}{\partial x}\delta w\Bigr{]}dx+[M\delta\psi]_{0}^{L}+[Q\delta w ]_{0}^{L}\end{array}  \tag{12}\]  
where  
\[M=\int_{A}z\sigma_{xx}dA \tag{13}\]  
\[Q=\kappa\int_{A}\tau_{xx}dA \tag{14}\]  
Figure 1: Geometry of the nanobeam. And \(\kappa\) is the shear correction factor which depends on the material properties.