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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Prestressed Small-Scale Plates with SMA Nanoscale Wires

_{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]

_{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

_{0}and a

_{c}are a calibration constant and an internal characteristic length [44,45], the constitutive relation is

_{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

_{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

_{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

_{g}) is determined as follows

## 3. Results and Discussion

^{3}, respectively [49]. It is found that the calculated results excellently match those determined in the literature.

^{3}, E

_{W}= 30 GPa, ${\varphi}_{k}$ = 0

^{0}, ${V}_{W}^{SMA}$ = 0.3 and ${\sigma}_{RS}$ = 0.2 GPa while for the small-scale plate, we have ${\nu}_{P}$ = 0.3, ${\rho}_{P}$ = 1600 kg/m

^{3}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).

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Wave propagation in a prestressed small-scale plate with shape memory alloy (SMA) nanoscale wires.

**Figure 2.**Group velocities calculated by the current modeling and those calculated in Ref. [49] for wave propagations in small-scale plates.

**Figure 3.**Effects of the initial stress ratio on the phase velocity of small-scale plates with SMA nanoscale wires.

**Figure 4.**Effects of the initial stress ratio on the group velocity of small-scale plates with SMA nanoscale wires.

**Figure 5.**Effects of the scale parameter together with initial stress influences on the phase velocity of small-scale plates with SMA nanoscale wires.

**Figure 6.**Effects of the scale parameter together with initial stress influences on the group velocity of small-scale plates with SMA nanoscale wires.

**Figure 7.**Effects of the volume fraction on the phase velocity of small-scale plates with SMA nanoscale wires.

**Figure 8.**Effects of the volume fraction on the group velocity of small-scale plates with SMA nanoscale wires.

**Figure 9.**Effects of the recovery stress on the phase velocity of small-scale plates with SMA nanoscale wires.

**Figure 10.**Effects of the recovery stress on the group velocity of small-scale plates with SMA nanoscale wires.

**Figure 11.**Effects of the wire orientation and initial stress on the phase velocity of small-scale plates with SMA nanoscale wires.

**Figure 12.**Effects of the wire orientation and initial stress on the group velocity of small-scale plates with SMA nanoscale wires.

**Figure 13.**Effects of the non-uniform coefficient on the phase velocity of small-scale plates with SMA nanoscale wires.

**Figure 14.**Effects of the non-uniform coefficient on the group velocity of small-scale plates with SMA nanoscale wires.

**Figure 15.**Effects of various profiles of initial stress on the phase velocity of small-scale plates with SMA nanoscale wires.

**Figure 16.**Effects of various profiles of initial stress on the group velocity of small-scale plates with SMA nanoscale wires.

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**MDPI and ACS Style**

Farajpour, M.R.; Shahidi, A.R.; Farajpour, A.
Wave Dispersion in Multilayered Reinforced Nonlocal Plates under Nonlinearly Varying Initial Stress. *Eng* **2020**, *1*, 31-47.
https://doi.org/10.3390/eng1010003

**AMA Style**

Farajpour MR, Shahidi AR, Farajpour A.
Wave Dispersion in Multilayered Reinforced Nonlocal Plates under Nonlinearly Varying Initial Stress. *Eng*. 2020; 1(1):31-47.
https://doi.org/10.3390/eng1010003

**Chicago/Turabian Style**

Farajpour, Mohammad Reza, Ali Reza Shahidi, and Ali Farajpour.
2020. "Wave Dispersion in Multilayered Reinforced Nonlocal Plates under Nonlinearly Varying Initial Stress" *Eng* 1, no. 1: 31-47.
https://doi.org/10.3390/eng1010003