# Forced Dynamics of Elastically Connected Nano-Plates and Nano-Shells in Winkler-Type Elastic Medium
## 2 Brief Review of [PERSON]'s Constitutive Elasticity  
It is known that the effect of nano-structures' size is important for their mechanical behavior because their dimensions are small and comparable to molecular distances. There are various higher-order continuum theories involving material length scale parameters. Among them are a couple of stress elasticity theories, such as those by [PERSON] [30], [PERSON] [31], and [PERSON] [32], strain gradient elasticity theory by [PERSON] et al. [3], and [PERSON]'s nonlocal theory [4; 5; 6]. All these theories provide extensions of the classical equations of elasticity with additional higher-order strain gradient terms based on the assumption that the materials cannot be modeled as collections of points, but they have to be considered as atoms with a higher-order deformation mechanism at the micro- or nano-scale. Nonlocal strain gradient theory uses two independent short length scale parameters. Unlike the nonlocal elasticity theory that will be used in this paper, nonlocal strain gradient theory takes into account both the interatomic forces and higher-order microstructure deformation mechanism. This work provides an opportunity for the same model to be used by applying multiple methods. It is worth noting that the state of stress and strain depends on the external load, the shape of the profile, and the type of material, but the relationship between them is directly linearly proportional under conditions of ideal elasticity at stresses below the elastic limit. According to [PERSON]'s nonlocal elasticity theory [4; 5; 6], the state of stress at some point in the domain is defined as a function of the state of strain not only at that point but as a function of the state of strain at all points in the observed domain. This assumption accounts for long-range intermolecular interactions and leads to a size-dependent theory of elasticity, which aligns well with the atomic theory of lattice dynamics. Using the fundamental equations of the nonlocal elasticity theory from papers written by [PERSON] and co-workers [4; 5; 6], [PERSON]'s constitutive elastic relation from the stress and strain is given as  
\[\Big{(}1\ -\ \mu\
abla^{2}\Big{)}\sigma_{ij}=C_{ijkl}\varepsilon_{kl}, \tag{1}\]  
where \(\mu=\Big{(}e_{0}\widetilde{a}\Big{)}^{2}\) is the nonlocal parameter, \(\widetilde{a}\) describes the internal characteristic length, and \(e_{0}\) is a constant corresponding to each material that can be identified from atomistic simulations or by using the dispersive curve of the Born-Karman model of lattice dynamics. \(C_{ijkl}\) is the elastic modulus tensor for classical isotropic elasticity, and \(\sigma_{ij}\) and \(\varepsilon_{kl}\) are the stress and strain tensors, respectively. Using Equation (1), the constitutive relations can be written for one-dimensional structures in the following forms:  
\[\sigma_{xx}-\mu\frac{\partial^{2}\sigma_{xx}}{\partial x^{2}}=E\varepsilon_{ xx}, \tag{2}\]  
\[\tau_{xz}-\mu\frac{\partial^{2}\tau_{xz}}{\partial x^{2}}=G\gamma_{xz}, \tag{3}\]  
where \(E\) is [PERSON]'s modulus of elasticity and \(G\) is the shear modulus. By applying relation (1), the constitutive relation for orthotropic two-dimensional nano-structures can be expressed as follows:  
\[(1-\mu\Delta) \begin{Bmatrix}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{yy}\\ \sigma_{yy}\end{Bmatrix} = \begin{bmatrix}\frac{E_{x}}{1-\tau_{xy}\
u_{yx}}& \frac{\
u_{xy}E_{y}}{\Gamma-\
u_{xy}\
u_{yx}}&0\\ \frac{\
u_{xy}E_{y}}{1-\
u_{xy}\
u_{yx}}&\frac{E_{y}}{1-\tau_{xy}\
u_{yx}}&0\\ 0&0&G_{xy}\end{bmatrix}\begin{Bmatrix}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{xy}\end{Bmatrix} , \tag{4}\]  
where \(\
abla^{2}=\Delta=\frac{d^{2}}{dx^{2}}+\frac{d^{2}}{dy^{2}}\) is the Laplacian, \(\
u_{xy}\) and \(\
u_{yx}\) denote [PERSON]'s ratios (\(\
u_{xy}=\
u_{yx}\)), \(\sigma_{xx}\), \(\sigma_{yy}\), and \(\sigma_{xy}\) are the normal and shear stresses, \(\varepsilon_{xx}\), \(\varepsilon_{yy}\), and \(\gamma_{xy}\) are the normal and shear strains, and \(G_{xy}\) is the shear modulus.