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

: This paper deals with the e ﬀ ects of initial stress on wave propagations in small-scale plates with shape memory alloy (SMA) nanoscale wires. The initial stress is exerted on the small-scale plate along both in-plane directions. A scale-dependent model of plates is developed for taking into consideration size inﬂuences on the wave propagation. In addition, in order to take into account the e ﬀ ects of SMA nanoscale wires, the one-dimensional Brinson’s model is applied. A set of coupled di ﬀ erential equations is obtained for the non-uniformly prestressed small-scale plate with SMA nanoscale wires. An exact solution is obtained for the phase and group velocities of the prestressed small-scale system. The inﬂuences of non-uniformly distributed initial stresses as well as scale and SMA e ﬀ ects on the phase and group velocities are explored and discussed. It is found that initial stresses as well as the orientation and volume fraction of SMA nanoscale wires can be used as a controlling factor for the wave propagation characteristics of small-scale


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].

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 ν LP 12 , E LP i , G LP 12 and ρ LP , respectively. For these properties, we have [41] E LP , G LP 12 (ζ) = Eng 2020, 1 33 where ζ and V SMA W 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 (ζ) = E mar E aus E mar (1 − ζ) + E aus ζ , (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 where 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 where where T and σ 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 where Taking into account a scale parameter in the form of µ nl = (e 0 a c ) 2 in which e 0 and a c are a calibration constant and an internal characteristic length [44,45], the constitutive relation is Eng 2020, 1 In Equations (9) and (10), RS and φ k stand for the elasticity constants of the plate, the recovery stress and angle of SMA nanoscale wires, respectively; ∇ 2 represents the Laplace operator. Taking into account a reinforced small-scale plate made of n layers, the stress resultants are Furthermore, the recovery stress resultants are as follows Employing Equations (9) and (11) together with Equation (8), we obtain the following equations for the stress resultants where Eng 2020, 1 35 and Applying Hamilton's principle leads to motion equations as follows 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 K 66 K 26 where N PS ij is the in-plane load induced by initial stresses, which is obtained by Eng 2020, 1 36 where σ PS ij(l) denotes initial stresses. Assuming non-uniformly distributed initial stresses along the thickness of the small-scale plate, we have in which 11 . 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] u in which k j (j = x,y) represents wave numbers;Û,V andŴ indicate wave amplitude coefficients; ω is the frequency of the small-scale plate. Substituting Equation (25) into Equations (20)- (22) leads to a matrix equation as follows where The frequency of the small-scale plate with SMA nanoscale wires is calculated from the following relation where "||" is used to denote the determinant. Assuming k = k x = k y , the general wave number is The phase velocity (c p ) is defined as In addition, the group velocity (c g ) is determined as follows 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.  (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.   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/m 3 , respectively [49]. It is found that the calculated results excellently match those determined in the literature.  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/m 3 , 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 Figures 3 and 4, respectively. The initial stress ratio is defined as

Present results
Literature for small-scale plates 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 Figures 3 and 4, respectively.
The initial stress ratio is defined as σ PS xx(1) = σ PS xx(1) /C 11 . 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 Eng 2020, 1 38 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 ν W = 0.3, ρ W = 6450 kg/m 3 , E W = 30 GPa, φ k = 0 0 , V SMA W = 0.3 and σ RS = 0.2 GPa while for the small-scale plate, we have ν P = 0.3, ρ P = 1600 kg/m 3 and E P = 3.44 GPa [50]. These physical properties are considered for each figure unless otherwise stated. From Figures 3 and 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 Figures 5 and 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 Figures 7 and 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. 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 Figures 7 and 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 Figures 9 and 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 Figures 11 and 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 k φ = 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 Figures 11 and 12. 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 Figures 9  and 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 Figures 11 and 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 φ 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 Figures 11 and 12.         Figures 13 and 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 Eng 2020, 1 43 initial stress involving uniform, linear and quadratic are compared in Figures 15 and 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. 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 Figures 15 and 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.    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 Figures 15 and 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.

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

SMA orientation
Uniform initial stress Linear initial stress Quadratic initial stress

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

SMA orientation
Uniform initial stress Linear initial stress Quadratic initial stress

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.