Wave Dispersion in Multilayered Reinforced Nonlocal Plates under Nonlinearly Varying Initial Stress

1. Introduction

Nanomaterials have attracted noticeable interest in different engineering-related disciplines since last decade due to their promising thermo-electro-mechanical properties [1,2,3,4]. Initial stresses influence the energy efficiency and performance of many macroscale and small-scale electromechanical devices and machines since the mechanical response of the fundamental parts of these systems is changed in the presence of initial stresses [5,6]. Many factors such as imprecision in manufacturing processes and inappropriate operating conditions can cause initial stresses. In many real situations, completely eliminating initial stresses is difficult and costly. Therefore, to have better energy efficiency and performance, it is important to increase our level of knowledge of the influences of initial stresses on the mechanical response [7].

A notable amount of effort has been made in recent years in order to analyze the influences of initial stresses on the mechanical response of small-scale structures using elasticity models [8]. Since the mechanical response is highly scale-dependent at ultrasmall levels [9,10,11,12,13,14,15,16,17], modified elasticity theories incorporating size effects are often used for small-scale structures [18,19,20,21,22,23,24]. Wang and Cai [25] investigated the influences of initial stresses on the vibrational characteristics of multi-walled carbon nanotubes (CNTs) via a elasticity model. Furthermore, Song et al. [26] examined the initial stress effects on the wave propagation in CNTs via applying a nonlocal model of elasticity. Heireche et al. [27] also explored sound wave propagations in a single-walled CNT subject to initial longitudinal stresses. In another investigation, Güven [28] studied the effects of initial stresses on the vibrational characteristics of CNTs under a magnetic field. In addition, in a continuum-based study conducted by Selim et al. [29], a theoretical model was developed for analyzing initial stress effects on the wave propagation characteristics of CNTs. Shen et al. [30] developed a scale-dependent elasticity model for the vibration of nanoscale mechanical sensors using double-walled CNTs under initial longitudinal stresses.

Besides developing elasticity models for small-scale tubes such as CNTs under initial stresses, the mechanical response of small-scale plates with initial stresses has been analyzed via scale-dependent models. Asemi et al. [31] examined initial stress effects on the vibrational characteristics of a system of two piezoelectric nanoscale plates employing a nonlocal model of plates. In another study, size effects together with the influences of initial stresses on the flexural wave propagations in nanoscale plates [32] were explored using the nonlocal elasticity. Moreover, the effects of uniaxial initial stress on the nonlocal vibrational response of nanoplates was examined in the literature [33]. Karami et al. [34] also analyzed initial stress effects on the wave dispersion of graphene sheets mounted on an elastic matrix. Ebrahimi and Shafiei [35] developed a modified plate model for the vibrational response of graphene sheets with initial stresses employing Reddy’s higher-order theory of shear deformations. In another investigation by Mohammadi et al. [36], the influences of shear initial stresses on the vibration of small-scale plates embedded in an elastic matrix were explored.

Composite ultrasmall structures such as hybrid plates have been synthesized and applied in a variety of applications in recent years [2,3,37]. On the other hand, due to the excellent mechanical properties of shape memory alloy (SMA) wires, they have been used for many different applications ranging from reinforced smart concrete [38] to robotic neurosurgery [39]. Furthermore, different applications of SMA small-scale structures in micro/nanoscale electromechanical systems have recently been reported [40]. In this article, the influences of nonlinearly varying initial stress on the wave propagation in multilayered small-scale plates with SMA nanoscale wires are investigated. The mechanism of wave dispersion in typical small-scale plates is known to researchers. However, to the best of our knowledge, no study has been reported on wave dispersion in reinforced nonlocal plates under nonlinear initial stresses. For this study, a scale-dependent model of plates is proposed employing the nonlocal elasticity. The effects of SMA nanoscale wires on the wave propagation are modeled using Brinson’s model. The coupled differential equations for scale-dependent wave propagations in the small-scale plate are presented. The phase and group velocities of the small-scale system are obtained. The effects of non-uniformly distributed initial stresses in conjunction with size and SMA effects on the phase and group velocities are examined. The theoretical formulation and accurate analysis performed in this article would be useful in the design and manufacture of cutting-edge technology-based machines in the field, especially in developing devices and tools for analyzing wave dispersion and stress evaluation in ultrasmall structures.

2. Prestressed Small-Scale Plates with SMA Nanoscale Wires

In this section, wave propagations in prestressed small-scale plates with SMA nanoscale wires are formulated using a scale-dependent plate model. Figure 1 illustrates a small-scale plate made of five layers reinforced by SMA nanoscale wires. Let us indicate the length, thickness and width of each layer by l_x, h and l_y, respectively. Furthermore, Poisson’s ratio, elasticity moduli, shear modulus and density of the small-scale plate are denoted by ${\nu }_{12}^{LP}$, $E_i^{LP}$, $G_{12}^{LP}$ and ${\rho }_{LP}$, respectively. For these properties, we have [41]

$$\begin{array}{l}{E}_1^{LP}\left(\zeta \right)=E_W\left[V_W^{SMA}\left(\zeta \right)\right]+E_P\left[1-V_W^{SMA}\left(\zeta \right)\right],\\ {\rho }_{LP}\left(\zeta \right)={\rho }_W\left[V_W^{SMA}\left(\zeta \right)\right]+{\rho }_P\left[1-V_W^{SMA}\left(\zeta \right)\right],\\ {\nu }_{12}^{LP}\left(\zeta \right)={\nu }_W\left[V_W^{SMA}\left(\zeta \right)\right]+{\nu }_P\left[1-V_W^{SMA}\left(\zeta \right)\right],\\ E_2^{LP}\left(\zeta \right)=\frac{E_W{E}_P}{\left\{E_P\left[V_W^{SMA}\left(\zeta \right)\right]+E_W\left[1-V_W^{SMA}\left(\zeta \right)\right]\right\}},\\ G_{12}^{LP}\left(\zeta \right)=\frac{G_W{G}_P}{\left\{G_P\left[V_W^{SMA}\left(\zeta \right)\right]+G_W\left[1-V_W^{SMA}\left(\zeta \right)\right]\right\}},\end{array}$$

(1)

where $\zeta$ and $V_W^{SMA}$ indicate the martensite fraction and volume fraction of SMA nanoscale wires, respectively; “LP”, “W” and “P” are used to refer to the laminated plate, SMA nanoscale wires and plate, respectively. Using one-dimensional Brinson’s model, for a SMA nanoscale wire, one obtains [42,43]

$$E_W(\zeta )=\frac{E_{mar}{E}_{aus}}{E_{mar}\left(1-\zeta \right)+E_{aus}\zeta },$$

(2)

in which “mar” and “aus” are used to refer to the martensite and austenite phases, respectively. The martensite fraction is determined by two dominant factors as follows

$$\zeta ={\zeta }_{tem}+{\zeta }_{str}.$$

(3)

where

$$\left\{\begin{array}{c}{\zeta }_{str}\\ {\zeta }_{tem}\end{array}\right\}=\left\{\begin{array}{c}{\zeta }_{str0}\\ {\zeta }_{tem0}\end{array}\right\}-\left(\frac{{\zeta }_0-\zeta }{{\zeta }_0}\right)\left\{\begin{array}{c}{\zeta }_{str0}\\ {\zeta }_{tem0}\end{array}\right\}.$$

(4)

Here “str”, “tem” and “0” denote the stress, temperature and initial value of a factor, respectively. Using one-dimensional Brinson’s model, we have the following relation for the martensite fraction

$$\begin{array}{l}\zeta =\frac{{\zeta }_0}{2}\cos \left[{\lambda }_c\left(C_{A}T-C_A{A}_{sta}-\sigma \right)\right]+\frac{{\zeta }_0}{2}\\ \mathrm{for}{C}_{A}T-C_A{A}_{fin}<\sigma <C_{A}T-C_A{A}_{sta}\mathrm{and}\mathrm{T}>A_{sta},\end{array}$$

(5)

where

$${\lambda }_c=\frac{\pi }{C_A\left(A_{fin}-A_{sta}\right)},$$

(6)

where T and $\sigma$ are, respectively, the temperature and stress; A_fin, A_sta and C_A are, respectively, the finish temperature of the austenite phase, start temperature of this phase and critical stress slope. Let us denote the mid-plane displacement in z, y and x directions by w, v and u, respectively. For the normal and shear strains, one can write

$$\begin{array}{l}{\epsilon }_{xx}={\epsilon }_{xx}^0-z{\kappa }_{xx},\\ {\epsilon }_{yy}={\epsilon }_{yy}^0-z{\kappa }_{yy},\\ {\gamma }_{xy}={\gamma }_{xy}^0-z{\kappa }_{xy},\end{array}$$

(7)

where

$$\left\{\begin{array}{c}{\epsilon }_{xx}^0\\ {\epsilon }_{yy}^0\\ {\gamma }_{xy}^0\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\end{array}\right\},\hspace{1em}\left\{\begin{array}{c}{\kappa }_{xx}\\ {\kappa }_{yy}\\ {\kappa }_{xy}\end{array}\right\}=\left\{\begin{array}{c}\frac{{\partial }^{2}w}{\partial x^2}\\ \frac{{\partial }^{2}w}{\partial y^2}\\ 2\frac{{\partial }^{2}w}{\partial y\partial x}\end{array}\right\}.$$

(8)

Taking into account a scale parameter in the form of ${\mu }_{nl}={\left(e_0{a}_c\right)}^2$ in which e₀ and a_c are a calibration constant and an internal characteristic length [44,45], the constitutive relation is

$$\begin{array}{l}\left\{\begin{array}{c}{\sigma }_{xx}^{(k)}\\ {\sigma }_{yy}^{(k)}\\ {\sigma }_{xy}^{(k)}\end{array}\right\}-{\mu }_{nl}{\nabla }^2\left\{\begin{array}{c}{\sigma }_{xx}^{(k)}\\ {\sigma }_{yy}^{(k)}\\ {\sigma }_{xy}^{(k)}\end{array}\right\}=\left[{\tilde{C}}^{(k)}(\zeta ,{\varphi }_k)\right]\left\{\begin{array}{c}{\epsilon }_{xx}^0\\ {\epsilon }_{yy}^0\\ {\gamma }_{xy}^0\end{array}\right\}\\ -z\left[{\tilde{C}}^{(k)}(\zeta ,{\varphi }_k)\right]\left\{\begin{array}{c}{\kappa }_{xx}\\ {\kappa }_{yy}\\ {\kappa }_{xy}\end{array}\right\}+V_{W,k}^{SMA}{\sigma }_{RS}^{(k)}\left\{\begin{array}{c}{\eta }_1\left({\varphi }_k\right)\\ {\eta }_2\left({\varphi }_k\right)\\ {\eta }_3\left({\varphi }_k\right)\end{array}\right\},\end{array}$$

(9)

where

$$\begin{array}{l}\left[{\tilde{C}}^{(k)}(\zeta ,{\varphi }_k)\right]=\left[\begin{array}{ccc}{\tilde{C}}_{11}^{(k)}(\zeta ,{\varphi }_k)& {\tilde{C}}_{12}^{(k)}(\zeta ,{\varphi }_k)& {\tilde{C}}_{16}^{(k)}(\zeta ,{\varphi }_k)\\ {\tilde{C}}_{12}^{(k)}(\zeta ,{\varphi }_k)& {\tilde{C}}_{22}^{(k)}(\zeta ,{\varphi }_k)& {\tilde{C}}_{26}^{(k)}(\zeta ,{\varphi }_k)\\ {\tilde{C}}_{16}^{(k)}(\zeta ,{\varphi }_k)& {\tilde{C}}_{26}^{(k)}(\zeta ,{\varphi }_k)& {\tilde{C}}_{66}^{(k)}(\zeta ,{\varphi }_k)\end{array}\right],\\ \left\{\begin{array}{c}{\eta }_1\left({\varphi }_k\right)\\ {\eta }_2\left({\varphi }_k\right)\\ {\eta }_3\left({\varphi }_k\right)\end{array}\right\}=\left\{\begin{array}{c}{\cos }^2\left({\varphi }_k\right)\\ {\sin }^2\left({\varphi }_k\right)\\ \cos \left({\varphi }_k\right)\sin \left({\varphi }_k\right)\end{array}\right\}.\end{array}$$

(10)

In Equations (9) and (10), ${\tilde{C}}_{ij}^{(k)}$, ${\sigma }_{RS}^{(k)}$ and ${\varphi }_k$ stand for the elasticity constants of the plate, the recovery stress and angle of SMA nanoscale wires, respectively; ${\nabla }^2$ represents the Laplace operator. Taking into account a reinforced small-scale plate made of n layers, the stress resultants are

$$\begin{array}{l}{N}_{xx}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{\sigma }_{xx}^{(l)}dz}},\hspace{1em}{N}_{yy}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{\sigma }_{yy}^{(l)}dz}},\\ N_{xy}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{\sigma }_{xy}^{(l)}dz}},\hspace{1em}{M}_{xx}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{\sigma }_{xx}^{(l)}zdz}},\\ M_{yy}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{\sigma }_{yy}^{(l)}zdz}},\hspace{1em}{M}_{xy}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{\sigma }_{xy}^{(l)}zdz}}.\end{array}$$

(11)

Furthermore, the recovery stress resultants are as follows

$$\begin{array}{l}{N}_{xx}^{RS}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{V}_{W,l}^{SMA}{\sigma }_{RS}^{(l)}{\eta }_1\left({\varphi }_l\right)dz}},\\ N_{yy}^{RS}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{V}_{W,l}^{SMA}{\sigma }_{RS}^{(l)}{\eta }_2\left({\varphi }_l\right)dz}},\\ N_{xy}^{RS}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{V}_{W,l}^{SMA}{\sigma }_{RS}^{(l)}{\eta }_3\left({\varphi }_l\right)dz}},\\ M_{xx}^{RS}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{V}_{W,l}^{SMA}{\sigma }_{RS}^{(l)}{\eta }_1\left({\varphi }_l\right)zdz}},\\ M_{yy}^{RS}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{V}_{W,l}^{SMA}{\sigma }_{RS}^{(l)}{\eta }_2\left({\varphi }_l\right)zdz}},\\ M_{xy}^{RS}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{V}_{W,l}^{SMA}{\sigma }_{RS}^{(l)}{\eta }_3\left({\varphi }_l\right)zdz}}.\end{array}$$

(12)

Employing Equations (9) and (11) together with Equation (8), we obtain the following equations for the stress resultants

$$\left\{\begin{array}{c}{N}_{xx}\\ N_{yy}\\ N_{xy}\end{array}\right\}-{\mu }_{nl}{\nabla }^2\left\{\begin{array}{c}{N}_{xx}\\ N_{yy}\\ N_{xy}\end{array}\right\}=\left[\tilde{K}\right]\left\{\begin{array}{c}\frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\end{array}\right\}-\left[\tilde{Q}\right]\left\{\begin{array}{c}\frac{{\partial }^{2}w}{\partial x^2}\\ \frac{{\partial }^{2}w}{\partial y^2}\\ 2\frac{{\partial }^{2}w}{\partial y\partial x}\end{array}\right\}+\left\{\begin{array}{c}{N}_{xx}^{RS}\\ N_{yy}^{RS}\\ N_{xy}^{RS}\end{array}\right\},$$

(13)

$$\left\{\begin{array}{c}{M}_{xx}\\ M_{yy}\\ M_{xy}\end{array}\right\}-{\mu }_{nl}{\nabla }^2\left\{\begin{array}{c}{M}_{xx}\\ M_{yy}\\ M_{xy}\end{array}\right\}=\left[\tilde{Q}\right]\left\{\begin{array}{c}\frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\end{array}\right\}-\left[\tilde{S}\right]\left\{\begin{array}{c}\frac{{\partial }^{2}w}{\partial x^2}\\ \frac{{\partial }^{2}w}{\partial y^2}\\ 2\frac{{\partial }^{2}w}{\partial y\partial x}\end{array}\right\}+\left\{\begin{array}{c}{M}_{xx}^{RS}\\ M_{yy}^{RS}\\ M_{xy}^{RS}\end{array}\right\},$$

(14)

where

$$\left[\tilde{K}\right]=\left[\begin{array}{ccc}{\tilde{K}}_{11}& {\tilde{K}}_{12}& {\tilde{K}}_{16}\\ {\tilde{K}}_{12}& {\tilde{K}}_{22}& {\tilde{K}}_{26}\\ {\tilde{K}}_{16}& {\tilde{K}}_{26}& {\tilde{K}}_{66}\end{array}\right],\hspace{1em}\left[\tilde{Q}\right]=\left[\begin{array}{ccc}{\tilde{Q}}_{11}& {\tilde{Q}}_{12}& {\tilde{Q}}_{16}\\ {\tilde{Q}}_{12}& {\tilde{Q}}_{22}& {\tilde{Q}}_{26}\\ {\tilde{Q}}_{16}& {\tilde{Q}}_{26}& {\tilde{Q}}_{66}\end{array}\right],\hspace{1em}\left[\tilde{S}\right]=\left[\begin{array}{ccc}{\tilde{S}}_{11}& {\tilde{S}}_{12}& {\tilde{S}}_{16}\\ {\tilde{S}}_{12}& {\tilde{S}}_{22}& {\tilde{S}}_{26}\\ {\tilde{S}}_{16}& {\tilde{S}}_{26}& {\tilde{S}}_{66}\end{array}\right],$$

(15)

and

$$\left\{\begin{array}{c}{\tilde{K}}_{ij}\\ {\tilde{Q}}_{ij}\\ {\tilde{S}}_{ij}\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \sum _{l=1}^n{\tilde{C}}_{ij}^{(l)}\left(z_l-z_{l-1}\right)}\\ {\displaystyle \sum _{l=1}^n\frac{{\tilde{C}}_{ij}^{(l)}}{2}\left(z_l^2-z_{l-1}^2\right)}\\ {\displaystyle \sum _{l=1}^n\frac{{\tilde{C}}_{ij}^{(l)}}{3}\left(z_l^3-z_{l-1}^3\right)}\end{array}\right\}.$$

(16)

Applying Hamilton’s principle leads to motion equations as follows

$$\frac{\partial N_{xx}}{\partial x}+\frac{\partial N_{xy}}{\partial y}=m_{LP}\frac{{\partial }^{2}u}{\partial t^2},$$

(17)

$$\frac{\partial N_{xy}}{\partial x}+\frac{\partial N_{yy}}{\partial y}=m_{LP}\frac{{\partial }^{2}v}{\partial t^2},$$

(18)

$$\begin{array}{c}\frac{{\partial }^2{M}_{yy}}{\partial y^2}+2\frac{{\partial }^2{M}_{xy}}{\partial x\partial y}+\frac{{\partial }^2{M}_{xx}}{\partial x^2}+\frac{\partial }{\partial y}\left(N_{xy}\frac{\partial w}{\partial x}+N_{yy}\frac{\partial w}{\partial y}\right)\\ +\frac{\partial }{\partial x}\left(N_{xx}\frac{\partial w}{\partial x}+N_{xy}\frac{\partial w}{\partial y}\right)=m_{LP}\frac{{\partial }^{2}w}{\partial t^2},\end{array}$$

(19)

where m_LP is the mass per unit area of the small-scale plate. Substituting Equations (13) and (14) into Equations (17)–(19), the coupled equations for the wave propagation in the small-scale plate are obtained as

$$\begin{array}{l}{\tilde{K}}_{66}\frac{{\partial }^{2}u}{\partial y^2}+{\tilde{K}}_{11}\frac{{\partial }^{2}u}{\partial x^2}+2{\tilde{K}}_{16}\frac{{\partial }^{2}u}{\partial y\partial x}+{\tilde{K}}_{26}\frac{{\partial }^{2}v}{\partial y^2}+{\tilde{K}}_{16}\frac{{\partial }^{2}v}{\partial x^2}\\ +\left({\tilde{K}}_{66}+{\tilde{K}}_{12}\right)\frac{{\partial }^{2}v}{\partial y\partial x}-[{\tilde{Q}}_{26}\frac{{\partial }^{3}w}{\partial y^3}+\left({\tilde{Q}}_{12}+2{\tilde{Q}}_{66}\right)\frac{{\partial }^{3}w}{\partial y^2\partial x}\\ +3{\tilde{Q}}_{16}\frac{{\partial }^{3}w}{\partial y\partial x^2}+{\tilde{Q}}_{11}\frac{{\partial }^{3}w}{\partial x^3}]=m_{LP}\frac{{\partial }^{2}u}{\partial t^2}-m_{LP}{\mu }_{nl}{\nabla }^2\frac{{\partial }^{2}u}{\partial t^2},\end{array}$$

(20)

$$\begin{array}{l}{\tilde{K}}_{26}\frac{{\partial }^{2}u}{\partial y^2}+{\tilde{K}}_{16}\frac{{\partial }^{2}u}{\partial x^2}+\left({\tilde{K}}_{66}+{\tilde{K}}_{12}\right)\frac{{\partial }^{2}u}{\partial y\partial x}+{\tilde{K}}_{22}\frac{{\partial }^{2}v}{\partial y^2}\\ +{\tilde{K}}_{66}\frac{{\partial }^{2}v}{\partial x^2}+2{\tilde{K}}_{26}\frac{{\partial }^{2}v}{\partial y\partial x}-[{\tilde{Q}}_{22}\frac{{\partial }^{3}w}{\partial y^3}+3{\tilde{Q}}_{26}\frac{{\partial }^{3}w}{\partial y^2\partial x}\\ +\left({\tilde{Q}}_{12}+2{\tilde{Q}}_{66}\right)\frac{{\partial }^{3}w}{\partial y\partial x^2}+{\tilde{Q}}_{16}\frac{{\partial }^{3}w}{\partial x^3}]=m_{LP}\frac{{\partial }^{2}v}{\partial t^2}-m_{LP}{\mu }_{nl}{\nabla }^2\frac{{\partial }^{2}v}{\partial t^2},\end{array}$$

(21)

$$\begin{array}{l}{\tilde{Q}}_{26}\frac{{\partial }^{3}u}{\partial y^3}+3{\tilde{Q}}_{16}\frac{{\partial }^{3}u}{\partial y\partial x^2}+\left({\tilde{Q}}_{12}+2{\tilde{Q}}_{66}\right)\frac{{\partial }^{3}u}{\partial y^2\partial x}+{\tilde{Q}}_{11}\frac{{\partial }^{3}u}{\partial x^3}\\ +{\tilde{Q}}_{22}\frac{{\partial }^{3}v}{\partial y^3}+\left({\tilde{Q}}_{12}+2{\tilde{Q}}_{66}\right)\frac{{\partial }^{3}v}{\partial y\partial x^2}+3{\tilde{Q}}_{26}\frac{{\partial }^{3}v}{\partial y^2\partial x}+{\tilde{Q}}_{16}\frac{{\partial }^{3}v}{\partial x^3}\\ -[{\tilde{S}}_{22}\frac{{\partial }^{4}w}{\partial y^4}+2\left({\tilde{S}}_{12}+2{\tilde{S}}_{66}\right)\frac{{\partial }^{4}w}{\partial y^2\partial x^2}+4{\tilde{S}}_{26}\frac{{\partial }^{4}w}{\partial y^3\partial x}+{\tilde{S}}_{11}\frac{{\partial }^{4}w}{\partial x^4}\\ +4{\tilde{S}}_{16}\frac{{\partial }^{4}w}{\partial x^3\partial y}]+\left(N_{yy}^{RS}+N_{yy}^{PS}\right)\frac{{\partial }^{2}w}{\partial y^2}+\left(N_{xx}^{RS}+N_{xx}^{PS}\right)\frac{{\partial }^{2}w}{\partial x^2}+2N_{xy}^{RS}\frac{{\partial }^{2}w}{\partial x\partial y}\\ -{\mu }_{nl}[\left(N_{yy}^{RS}+N_{yy}^{PS}\right)\frac{{\partial }^{4}w}{\partial y^4}+\left(N_{xx}^{RS}+N_{xx}^{PS}\right)\frac{{\partial }^{4}w}{\partial x^4}+2N_{xy}^{RS}\frac{{\partial }^{4}w}{\partial x^3\partial y}\\ +2N_{xy}^{RS}\frac{{\partial }^{4}w}{\partial x\partial y^3}+\left(N_{xx}^{RS}+N_{yy}^{RS}+N_{xx}^{PS}+N_{yy}^{PS}\right)\frac{{\partial }^{4}w}{\partial y^2\partial x^2}]\\ =m_{LP}\frac{{\partial }^{2}w}{\partial t^2}-m_{LP}{\mu }_{nl}{\nabla }^2\frac{{\partial }^{2}w}{\partial t^2},\end{array}$$

(22)

where $N_{ij}^{PS}$ is the in-plane load induced by initial stresses, which is obtained by

$$N_{xx}^{PS}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{\sigma }_{xx(l)}^{PS}dz}},\hspace{1em}{N}_{yy}^{PS}={\displaystyle \sum _{l=1}^n{\displaystyle \underset{z_{l-1}}{\overset{z_l}{\int }}{\sigma }_{yy(l)}^{PS}dz}},$$

(23)

where ${\sigma }_{ij(l)}^{PS}$ denotes initial stresses. Assuming non-uniformly distributed initial stresses along the thickness of the small-scale plate, we have

$$\begin{array}{l}{\overline{\sigma }}_{xx(k)}^{PS}=a{\overline{\sigma }}_{xx(1)}^{PS}{\left({\overline{z}}_k-{\overline{z}}_1\right)}^b+{\overline{\sigma }}_{xx(1)}^{PS},\\ {\overline{\sigma }}_{yy(k)}^{PS}=a{\overline{\sigma }}_{yy(1)}^{PS}{\left({\overline{z}}_k-{\overline{z}}_1\right)}^b+{\overline{\sigma }}_{yy(1)}^{PS},\end{array}$$

(24)

in which ${\overline{z}}_j=z_j/h$ and ${\overline{\sigma }}_{ij(k)}^{PS}={\sigma }_{ij(k)}^{PS}/C_{11}^{(k)}$. Moreover, a and b are constants, which determine the variation of initial stress. In order to determine the phase and group velocities of the small-scale plate with SMA nanoscale wires, the displacements along the length, width and thickness directions are expressed as [46]

$$\begin{array}{c}u(x,y,t)=\widehat{U}\exp \left(-i\omega t+i{k}_{x}x+i{k}_{y}y\right),\\ v(x,y,t)=\widehat{V}\exp \left(-i\omega t+i{k}_{x}x+i{k}_{y}y\right),\\ w(x,y,t)=\widehat{W}\exp \left(-i\omega t+i{k}_{x}x+i{k}_{y}y\right),\end{array}$$

(25)

in which k_j (j = x,y) represents wave numbers; $\widehat{U}$, $\widehat{V}$ and $\widehat{W}$ indicate wave amplitude coefficients; $\omega$ is the frequency of the small-scale plate. Substituting Equation (25) into Equations (20)–(22) leads to a matrix equation as follows

$$\left[\tilde{\Pi }\right]\left\{\widehat{\Delta }\right\}-{\omega }^2\left[\tilde{\Lambda }\right]\left\{\widehat{\Delta }\right\}=0,$$

(26)

where

$$\left[\tilde{\Pi }\right]=\left[\begin{array}{ccc}{\tilde{\mathsf{\Pi }}}_{11}& {\tilde{\mathsf{\Pi }}}_{12}& {\tilde{\mathsf{\Pi }}}_{13}\\ {\tilde{\mathsf{\Pi }}}_{21}& {\tilde{\mathsf{\Pi }}}_{22}& {\tilde{\mathsf{\Pi }}}_{23}\\ {\tilde{\mathsf{\Pi }}}_{31}& {\tilde{\mathsf{\Pi }}}_{32}& {\tilde{\mathsf{\Pi }}}_{33}\end{array}\right],\hspace{1em}\left[\tilde{\Lambda }\right]=\left[\begin{array}{ccc}{\tilde{\mathsf{\Lambda }}}_{11}& {\tilde{\mathsf{\Lambda }}}_{12}& {\tilde{\mathsf{\Lambda }}}_{13}\\ {\tilde{\mathsf{\Lambda }}}_{21}& {\tilde{\mathsf{\Lambda }}}_{22}& {\tilde{\mathsf{\Lambda }}}_{23}\\ {\tilde{\mathsf{\Lambda }}}_{31}& {\tilde{\mathsf{\Lambda }}}_{32}& {\tilde{\mathsf{\Lambda }}}_{33}\end{array}\right],\hspace{1em}\left\{\widehat{\Delta }\right\}=\left\{\begin{array}{c}\widehat{U}\\ \widehat{V}\\ \widehat{W}\end{array}\right\}.$$

(27)

The frequency of the small-scale plate with SMA nanoscale wires is calculated from the following relation

$$\left|\left[\tilde{\Pi }\right]-{\omega }^2\left[\tilde{\Lambda }\right]\right|=0,$$

(28)

where “$\left|\right|$” is used to denote the determinant. Assuming k = k_x = k_y, the general wave number is obtained as $K=\sqrt{k_x^2+k_y^2}=\sqrt{2}k$. The phase velocity (c_p) is defined as

$$c_p=\frac{\omega }{K}.$$

(29)

In addition, the group velocity (c_g) is determined as follows

$$c_g=\frac{d\omega }{dK}.$$

(30)

An ultrasonic wave dispersion in a reinforced multilayered small-scale plate depends on the stress, any components and their distributions, as well as the drive frequency. It is important to notice that there are many studies on stress evaluation by the ultrasonic velocity [47,48]. In other words, a wave propagation analysis can be used as a tool to measure the stress in a structural element.

3. Results and Discussion

Figure 2 compares the calculated group velocities with those determined in the literature for wave propagations in a small-scale plate without SMA nanoscale wires [49]. The nonlocal parameter, thickness, Poisson’s ratio, elasticity modulus and mass per volume of the small-scale plate are set to 1 nm, 0.34 nm, 0.25, 1.06 TPa and 2250 kg/m³, respectively [49]. It is found that the calculated results excellently match those determined in the literature.

The effects of initial stress ratio together with wave number effects on the phase velocity and group velocity of small-scale plates with SMA nanoscale wires are indicated in Figure 3 and Figure 4, respectively. The initial stress ratio is defined as ${\overline{\sigma }}_{xx(1)}^{PS}={\sigma }_{xx(1)}^{PS}/C_{11}^{(1)}$. A biaxial initial stress condition with a tension ratio of 1:1 is taken into account. It is assumed that all five layers have the same initial stress and tension ratio. The side length and thickness of each square layer are, respectively, set to 150 and 3 nm. The ratio of the nonlocal parameter to the side length (scale parameter) is 0.02. The physical properties of SMA nanoscale wires are taken as ${\nu }_W$ = 0.3, ${\rho }_W$ = 6450 kg/m³, E_W = 30 GPa, ${\varphi }_k$ = 0⁰, $V_W^{SMA}$ = 0.3 and ${\sigma }_{RS}$ = 0.2 GPa while for the small-scale plate, we have ${\nu }_P$ = 0.3, ${\rho }_P$ = 1600 kg/m³ and E_P = 3.44 GPa [50]. These physical properties are considered for each figure unless otherwise stated. From Figure 3 and Figure 4, it is found that increasing the tensile initial ratio leads to an increase in both phase and group velocities. The reason behind this phenomenon is that as the tensile biaxial initial stress is increased, the stiffness of the small-scale plate improves, and consequently the frequency increases. This leads to a noticeable increase in the phase velocity according to Equation (29).

Scale effects together with initial stress effects on the phase and group velocities of small-scale plates with SMA nanoscale wires are, respectively, indicated in Figure 5 and Figure 6. The initial stress ratio is set to 0.5 for the cases with initial stress. It is found that greater scale parameters lead to lower phase and group velocities. It is rooted in the fact that greater nonlocal scale parameters reduce the stiffness, and thus are associated with lower frequencies. Therefore, the phase velocity of reinforced multilayered small-scale plates is significantly decreased as the scale parameter of stress nonlocality increases. The effects of the volume fraction of SMA nanoscale wires on the phase and group velocities are also indicated in Figure 7 and Figure 8, respectively. The initial stress ratio and scale parameter are 0.5 and 0.02, respectively. It is concluded that both group and phase velocities are sensitive to the volume fraction. Increasing this parameter slightly increases these velocities. It implies that the volume fraction of SMA nanoscale wires can be utilized as a controlling factor for wave propagation characteristics of small-scale plates.

The recovery stress effects in conjunction with the influences of the wave number on the phase and group velocities of small-scale plates with SMA nanoscale wires are, respectively, indicated in Figure 9 and Figure 10. The initial stress ratio, scale parameter and volume fraction are 0.5, 0.02 and 0.3, respectively. Greater recovery stresses increase both phase and group velocities. It implies that the recovery stress can also be used as a controlling factor for phase and group velocities of small-scale plates. In fact, enhanced recovery stresses are linked to a slight increase in the total equivalent structural stiffness of reinforced multilayered nonlocal plates, leading to slightly increased frequencies, and thus a small increase in the phase velocity. In addition, the influences of initial stress and the orientation of SMA nanoscale wires on the phase and group velocities are highlighted in Figure 11 and Figure 12, respectively. The recovery stress and wave number are set to 0.2 GPa and 1.0 1/nm. The maximum phase and group velocities are obtained for ${\varphi }_k$ = 45° for k = 1, 2, …, 5. Initial stresses are assumed to be tensile in this case, and thus they act as an improving agent for the total equivalent stiffness of reinforced multilayered nonlocal plates. This initial condition consequently yields enhanced frequencies and improved phase velocities as one can observe from Figure 11 and Figure 12.

Figure 13 and Figure 14 are plotted in order to show the effects of a (i.e., the non-uniform coefficient) on the phase and group velocities of non-uniformly prestressed small-scale plates with SMA nanoscale wires, respectively. The initial stress ratio of the first layer, scale parameter and volume fraction are 0.5, 0.02 and 0.3, respectively. Moreover, the wave number is set to 1.0 1/nm. It is found that both phase and group velocities are highly dependent on the non-uniform coefficient. Greater values of this coefficient lead to greater phase and group velocities. Moreover, the various profiles of the biaxial initial stress involving uniform, linear and quadratic are compared in Figure 15 and Figure 16 for the phase and group velocities, respectively. The quadratic tensile profile leads to the greatest phase and group velocities whereas the uniform tensile profile results in the lowest phase and group velocities.

4. Conclusions

The effects of non-uniformly distributed initial stresses on the wave propagation in small-scale plates with SMA nanoscale wires have been investigated. A scale-dependent model of plates was proposed utilizing the nonlocal elasticity in conjunction with Brinson’s model. The differential equations for scale-dependent wave propagations in the small-scale system were derived. The influences of non-uniformly distributed initial stresses together with SMA effects on the phase and group velocities were explored. It was observed that a greater tensile initial stress leads to an increase in both the phase and group velocities of the small-scale plate with SMA nanoscale wires since tensile prestresses improve the total equivalent structural stiffness of reinforced multilayered small-scale plates. However, greater scale parameters of stress nonlocality result in lower phase and group velocities as this parameter has a decreasing influence on the equivalent structural stiffness. The volume fraction and orientation of SMA nanoscale wires can be utilized as a controlling factor for the wave propagation characteristics of prestressed small-scale plates. It was also found that both group and phase velocities increase when the non-uniform coefficient increases. The group and phase velocities of the small-scale plate with initial stress of quadratic tensile profile are higher than those of linear and uniform profiles.