# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 3 Mathematical Formulations of the Equations of Motion for  
an Orthotropic Nanoplate  
Based on [PERSON]'s plate theory [26; 27], the displacement components, \(u(x,y,t)\), \(v(x,y,t)\), and \(w(x,y,t)\), for an arbitrary point on the middle plane of the nanoplate along the \(x\), \(y\), and \(z\) axes can be written in the following forms:  
\[\overline{u}_{x}=u(x,y,t)-z\frac{\partial w(x,y,t)}{\partial x},\overline{v}_ {y}=v(x,y,t)-z\frac{\partial w(x,y,t)}{\partial y},\overline{w}_{z}=w(x,y,t). \tag{5}\]  
The strain-displacement relations for linear strain theory are  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]  
The nonlocal constitutive relation between the stress and strain of one orthotropic nanoplate, as expressed in the form of (4), based on nonlocal stresses, \(\sigma_{xx}\), \(\sigma_{yy}\), and \(\sigma_{xy}\), can be written as follows:  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]  
\[(1\ -\ \mu\Delta)\sigma_{yy}=\frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}} \varepsilon_{xx}+\frac{E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{8}\]  
\[(1\ -\ \mu\Delta)\sigma_{xy}=\ G_{xy}\gamma_{xy}. \tag{9}\]  
Based on [PERSON]'s second law for the infinitesimal element of the nanoplate, equilibrium equations can be obtained in the following form:  
\[\frac{\partial N_{xx}}{\partial x}+\frac{\partial N_{xy}}{\partial y}=\rho h \frac{\partial^{2}u}{\partial t^{2}}, \tag{10}\]  
\[\frac{\partial N_{yy}}{\partial y}+\frac{\partial N_{xy}}{\partial x}=\rho h \frac{\partial^{2}v}{\partial t^{2}}, \tag{11}\]  
\[\frac{\partial^{2}M_{xx}}{\partial x^{2}}+\frac{\partial^{2}M_{yy}}{\partial y ^{2}}+2\frac{\partial^{2}M_{xy}}{\partial x\partial y}+q(x,y,t)=\rho h\frac{ \partial^{2}w}{\partial t^{2}}, \tag{12}\]  
where \(N_{xx}\), \(N_{yy}\), and \(N_{xy}\) are the in-plane stress variables, and \(M_{xx}\), \(M_{yy}\) and \(M_{xy}\) are the moment variables, defined as  
\[(N_{xx},N_{yy},N_{xy})= \int_{-\frac{1}{2}}^{\frac{1}{2}}(\sigma_{xx},\ \sigma_{yy},\tau_{xy})dz, \tag{13}\]  
\[(M_{xx},M_{yy},M_{xy})= \int_{-\frac{1}{2}}^{\frac{1}{2}}(\sigma_{xx},\ \sigma_{yy},\tau_{xy})zdz, \tag{14}\]  
and \(q(x,y,t)\) is the external force caused by the transverse force \(f\):  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]0  
Using expression (6), (13) and (14) and taking into account Equations (7)-(9) yields  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]1  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]2\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]3  
and  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]4  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]5  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]6  
where \(A_{11},\ A_{12},\ A_{22},\) and \(A_{66}\) are extensional and \(D_{11},\ D_{12},\ D_{22},\) and \(D_{66}\) are the bending stiffnesses of the orthotropic elastic nanoplate, expressed as  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]7  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]8  
By using Equations (10)-(12), (16)-(18) and (19)-(21), the PDEs describing the displacements in the \(xOy\) plane and the transverse displacement in the \(z\)-axis direction of a nanoplate subjected to an external load, \(f(x,y,t)\), have the forms  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{ \partial x^{2}},\varepsilon_{yy}=\frac{\partial v}{\partial y}-z\frac{ \partial^{2}w}{\partial y^{2}},\gamma_{xy}=\frac{\partial u}{\partial y}+ \frac{\partial v}{\partial x}-2z\frac{\partial^{2}w}{\partial x\partial y}. \tag{6}\]9  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]0  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]1  
Mathematical Formulation of the Equations of Motion for an Orthotropic Doubly Curved Shallow Nano-Shell  
According to [PERSON]'s linear shell theory presented by [PERSON] [29], the strain components at an arbitrary point in the panel are related to the middle surface strains, \(e_{xx}\), \(e_{yy}\), and \(e_{xy}\), and changes in the curvature and torsion of the middle surface, \(k_{xx}\), \(k_{yy}\), and \(k_{xy}\), which are defined by the following relationships:  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]2  
where  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]3  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]4  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]5  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]6  
\[(1\ -\mu\Delta)\sigma_{xx}=\frac{E_{x}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{xx}+ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}\varepsilon_{yy}, \tag{7}\]7The nonlocal constitutive relation between stress and strain is defined by expression (4).