Wave propagation analysis of porous functionally graded piezoelectric nanoplates with a visco-Pasternak foundation

This study investigates the size-dependent wave propagation behaviors under the thermoelectric loads of porous functionally graded piezoelectric (FGP) nanoplates deposited in a viscoelastic foundation. It is assumed that (i) the material parameters of the nanoplates obey a power-law variation in thickness and (ii) the uniform porosity exists in the nanoplates. The combined effects of viscoelasticity and shear deformation are considered by using the Kelvin-Voigt viscoelastic model and the refined higher-order shear deformation theory. The scale effects of the nanoplates are captured by employing nonlocal strain gradient theory (NSGT). The motion equations are calculated in accordance with Hamilton’s principle. Finally, the dispersion characteristics of the nanoplates are numerically determined by using a harmonic solution. The results indicate that the nonlocal parameters (NLPs) and length scale parameters (LSPs) have exactly the opposite effects on the wave frequency. In addition, it is found that the effect of porosity volume fractions (PVFs) on the wave frequency depends on the gradient indices and damping coefficients. When these two values are small, the wave frequency increases with the volume fraction. By contrast, at larger gradient index and damping coefficient values, the wave frequency decreases as the volume fraction increases.


Introduction
terials in various applications such as sensors, miniature piezoelectric generators, and resonant ultrasonic inspection devices [3] , thus promoting the rapid development of nanoelectromechanical systems (NEMS). As the trend of application of nanostructures in modern engineering design has expanded significantly, the study of nanomechanics has become an important research topic. However, the experimental and atomic simulation studies revealed that nanostructures exhibit an obvious size-dependent behavior as the structure size decreases progressively [4] . Using the classical continuum theory, Alizada and Sofiyev [5] found that scale effects and vacancies considerably affect Young's modulus of nanomaterials. This implies that the classical continuum theory fails to accurately assess the mechanical responses of these nanoelements. Therefore, to ensure the accuracy and rationality of nanostructure design, a series of size-dependent nonclassical theories have been proposed to capture the scale effects of small-scale structures, such as the couple stress theory (CST) [6] , the strain gradient elasticity theory (SGET) [7] , and the nonlocal elasticity theory (NET) [8] . Among them, the NET developed by Eringen is often used to explain the effect of size reductions. According to the NET, the stress at each point in the solid depends on the strain at not only the reference point, but also all other points throughout the object. When the nonlocal scale parameter is set to zero, the NET can be reduced to the classical elasticity theory (CET). Although the NET is widely used to consider small-scale effects, it does not fully cover the mechanical behavior of nanostructures. Many researchers have observed the stiffness-hardening effect of structures in experiments [9][10] , which cannot be predicted by the NET. In fact, the NET successfully predicts the stiffness decrease phenomenon of structures observed in the experiment, while ignoring the influence of another increasing effect. Subsequently, Lim et al. [11] innovatively proposed a nonlocal strain gradient theory (NSGT) to capture the size-dependent properties of nanostructures more accurately and completely. This theory has been demonstrated to successfully describe two completely different features of small-scale structures by introducing two scale parameters, namely nonlocal parameters (NLPs) and length scale parameters (LSPs). In addition, comparison between the results obtained from different nonclassical theories and those obtained experimentally indicates that the results obtained from the NSGT framework are closer to the molecular dynamics (MD) simulation results [12][13] . Therefore, researchers have preferred to use the NSGT rather than the NET to explore the small-scale behaviors of nanostructures in subsequent studies. Sahmani and Aghdam [14] employed the NSGT for nonlinear vibration responses of micro-/nanotubules and effectively captured the softening and hardening effects. Farajpour et al. [15] analyzed the effect of scale parameters on the buckling response of nanoplates employing the NSGT. Ebrahimi et al. [16] explored the dispersion characteristics of temperature-dependent nanoplates composed of ceramics and metals with the NSGT. Subsequently, they also employed the higher-order plate theory for the flexural wave propagation analysis of functionally graded (FG) nanoplates under magnetic and electric fields, and obtained the variation curve of dispersion relations with different factors [17] . Ebrahimi and Dabbagh [18] explored wave propagation behaviors of piezoelectric nanoplates by considering surface effects and the NSGT. Furthermore, Abazid [19] investigated the hygrothermal effects of wave propagation in embedded piezoelectromagnetic nanoplates based on the NSGT. It can be seen that most of the above studies have focused on homogeneous material structures and rarely involved composite materials.
Recently, with the rapid development of the NEMS, the design and the application of small-scale composite structures made of functionally graded materials (FGMs) have gradually emerged as a possibility. The FGMs are multiphase materials consisting of two or more single materials with material properties that can be smoothly and continuously controlled in the expected direction to obtain the best response produced when subject to different external loads. Owing to its special material composition, there is a significant increase in the trend of using FGMs as ultra-high-temperature materials [20] . In addition, FG piezoelectric (FGP) materials are an important part of FGMs, which compensate for the defects of piezoelectric materials, such as low resistance, high temperature creep, and stress concentration. In the past decades, the dynamic property analysis of FGP structures has attracted ample scholarly attention. Jadhav and Bajoria [21] evaluated the dynamic properties of FGP plates considering a thermoelectric field on the basis of the higher-order deformation theory. Jandaghian and Rahmani [22] presented the dynamic characteristics of Kirchhoff FGP nanoplates employing the NET. Afterwards, they analyzed the scale-dependent vibration characteristics of FGP nanoplates under thermoelectric loads [23] . Sharifi et al. [24] extended the NSGT to the analysis of the free vibration behavior of Kirchhoff FGP nanoplates. In addition, some relevant studies on the thermoelectromechanical vibration properties of FGP nanobeams and nanoshells were conducted [25][26] . It is worth mentioning that the above studies on FGP plates are limited to perfect material structures and ignore the effect of particles or porosity on the mechanical characteristics. In fact, the inadequacy of the preparation techniques results in excessive pore formation in FGMs during their preparation, which may affect or alter the elastic modulus, bending strength, thermal conductivity, and other properties of the material. Therefore, it is essential to consider the effects of porosity. Joubaneh et al. [27] performed thermal buckling analysis on porous cylindrical plates. Barati et al. [28] employed the refined higher-order plate theory for analyzing the effect of porosity on the free vibration response of FGP plates. In addition, they analyzed the vibration characteristics for porous FGP plates under electromechanical loads [29] . Wang and Zu [30] studied nonlinear vibration responses of porous FGP plates and detected the nonlinear broadband vibration phenomenon. Nguyen et al. [31] performed an isogeometric Bézier finite element method to explore the electromechanical vibration responses of porous FGP plates. Studies on the vibration and buckling of FGP nanoplates are plenty; however, the analysis of wave propagation in such structures is still relatively insufficient. Moreover, the above-mentioned studies have not considered the viscoelastic effect of structures.
As a matter of fact, in some nanostructures, such as flexible electronic sensors, nanorobots are often embedded in various types of viscoelastic polymeric structures and coatings. Because the dynamic properties of nanostructures are often affected by viscoelasticity, researchers usually establish viscoelastic models to study the mechanism of their action in micro-/nanostructures. For instance, Ebrahimi and Barati [32] developed a higher-order FG nanobeam embedded in a viscoelastic foundation to study the viscoelastic effect on vibration behaviors of nanobeams. Furthermore, they presented the effects of different factors, including viscoelasticity, on the size-dependent vibration properties of piezoelectric polymeric nanoplates with the NSGT [33] . Arefi and Zenkour [34] presented the vibration properties of viscoelastic sandwich nanoplates under thermoelectric loads, and found that the foundation parameters have an obvious effect on the vibration response of nanoplates. Liu et al. [35] developed FG magnetoelectro nanobeams containing visco-Pasternak foundation based on the Timoshenko beams theory to analyze the vibration properties of nanobeams employing the NET. Zenkour and Sobhy [36] explored the viscoelastic effects on vibration characteristics of piezoelectric nanoplates based on the NET and the Kelvin-Voigt model. Arani et al. [37] developed viscoelastic sandwich nanoplates to analyze the scale effects on dispersion relations of wave propagation.
Obviously, vibration, bending, and buckling properties of FGP nanostructures have been extensively studied by using the nonclassical continuum theory. However, the present study emphasizes certain aspects of the analysis of wave propagation in viscoelastic porous FGP nanoplates by using the NSGT. In this study, the NSGT containing NLPs and LSPs is used to investigate size-dependent wave propagation characteristics of FGP nanoplates with a viscoelastic foundation and subject to thermoelectric loads. The governing equations are obtained by applying Hamilton's principle. Then, the wave propagation frequency is obtained by substituting the analytical solution into the nonlocal motion equation. Finally, the effects of scale parameters, gradient index, and viscoelastic foundation parameters are investigated in detail.

Theory and formulation 2.1 FGMs
A porous FGP nanoplate model with thickness h and composed of PZT-4 and PZT-5H is shown in Fig. 1. The nanoplates are deposited on a two-layer viscoelastic foundation. The first layer is considered as a Kelvin-Voigt viscoelastic model, composed of a set of springs and dashpots that are connected in parallel. The second layer is considered as a Pasternak shear layer. An external electric voltage V 0 and a temperature variation ∆T are applied to the nanoplates. Before calculation, some limitations and assumptions of this research are stated as follows: (i) The material composition and properties of the FGP nanoplates are considered to show continuous gradient changes, and there is no obvious interface inside the material.
(ii) There are no debonding between porous FGP nanoplates and the viscoelastic foundation.
(iii) The effect of bending and shear deformation is considered by ignoring the gradient of strain in the thickness direction, that is, ε xz,z = 0. The equivalent material properties of FGP nanoplates are denoted based on the power-law function as follows [29] : where N is the gradient index. By considering porosity in Eq. (1), the elastic, piezoelectric, and dielectric constants and mass density are, respectively, denoted as follows [30] : where α is the porosity volume fraction (PVF). The electric potential Φ(x, y, z, t) distribution is assumed to be a combination of linear and cosine variations to satisfy the Maxwell equation [8] , i.e., According to the relationships between the electric potential and the electrostatic field, the electric field components E x , E y , and E z are obtained as

Governing equations
Adopting the higher-order shear deformation theory, the displacement components of the FGP nanoplate are formulated as follows [16] : where u 0 and v 0 are the displacements along the x-axis and y-axis, respectively. w b and w s are the deflections caused by bending and shear, respectively. f (z) represents a shape function, and is given as f (z) = 4z 3 /(3h 2 ). The nonzero strain can be expressed from Eqs. where The strain energy of the FGP nanoplate is expressed as Substituting Eqs. (7)- (9) and Eq. (13) into Eq. (15), we can obtain the following relationship: where The kinetic energy is written as follows: where the mass inertia is denoted by The work performed by the external force can be marked as where N 0 x and N 0 y are the normal forces caused by temperature and electric voltage. In addition, R v denotes the viscoelastic foundation. In the above relations [35] , where k w and k p are the Winkler and Pasternak moduli, and c d is the medium damping of the viscoelastic foundation. Using Hamilton's principle, and substituting Eqs. (16), (18), and (20) into Eq. (22), we can drive the governing equations of viscoelastic porous FGP nanoplates as follows:

Motion equations
According to the higher-order NSGT, the total stress considering the nonlocal elastic and strain gradient stress fields is given as [11] where σ (0) ij and σ (1) ij represent classical and higher-order stresses, respectively, where C ijkl denote elastic coefficients, e 0 a and e 1 a represent the NLPs, and l is the LSP. α 0 and α 1 denote the nonlocal kernel functions, and when they satisfy the condition proposed by Eringen, Eqs. (29) and (30) can be denoted as follows: By assuming e 1 = e 0 = e, Eq. (31) can be simplified as Considering temperature effects and neglecting the transverse normal stress, we can get the constitutive equation of FGP nanoplates in terms of NSGT as follows: where ∇ 2 is the Laplacian operator. ∇ 2 = ∂ ∂x 2 + ∂ ∂y 2 , µ = (ea) 2 , and η = l 2 . σ ij , ε ij , D i , E i , and ∆T are the stress, strain, electric displacement, electric field, and temperature change, respectively. c ij , e ij , s ij , β i , and p i are reduced constants of the FGP nanoplate, which are defined as follows [8] : Now, by integrating Eqs. (33)-(40) across the cross-section of FGP nanoplates, the forcestrain and moment-strain relationships can be obtained as follows: where Now, substituting Eqs. (42)-(47) into Eqs. (23)- (27), we can get the nonlocal motion equations of the FGP nanoplates as follows: ∂φ ∂y 3 Solution procedure To solve motion equations, the displacement function of wave propagation can be assumed as follows:  where U , V , W b , W s , and Φ denote the wave amplitudes. k 1 and k 2 stand for the wave number. The angular frequency is denoted by ω. Now, the following equation can be obtained by substituting Eq. (55) into Eqs. (50)-(54): The angular frequency is determined by resolving the following determinant: Setting k 1 = k 2 = k, we can obtain the frequency of wave propagation in porous FGP nanoplates as follows:

Results and discussion
In this section, wave propagation properties of viscoelastic porous FGP nanoplates under thermoelectric loads are analyzed. The thickness of the FGP nanoplates is h = 10 nm. Material properties are given in Table 1 [22] . Table 1 Material properties of FGP nanoplates [22] Property PZT-4 PZT-5H Elastic constants/GPa The NSGT is employed to accurately capture small-scale effects. In addition, some simple explanations are given. The comparative study between the results of the present model neglecting the visco-Pasternak foundation and those reported by Ma et al. [8] is illustrated in Fig. 2. This figure indicates the dispersion relationship for different values of scale parameters with ρ = 7 500 kg · m −3 , η = 0 nm 2 , ∆T = 0 K, and V 0 = 0 V. Clearly, the current results are consistent with those of Ma et al. [8] ; thus, the validity of the present study can be confirmed. It is worth mentioning that in the next analysis, if not specifically stated, the following parameters will be utilized: h = 10 nm, k = 5 × 10 8 m −1 , µ = 1 nm 2 , η = 3 nm 2 , N = 2, ∆T = 0 K, Ref. [8]: Present results: Fig. 2 Comparative study on dispersion characteristics of nanoplates (color online) Figure 3 presents the wave frequency as a function of wave number for various scale parameters. Obviously, in all cases, the frequency increases as the wave number increases. When the nonlocal elastic stress field is not considered (µ = 0 nm 2 ), the frequency approaches infinity with the increase in the wave number. As shown in Fig. 3(a), when NLPs are considered, and the strain gradient stress field is ignored (µ = 0 nm 2 , η = 0 nm 2 ), the wave frequency follows an approximately linear increase at lower wave numbers, and then gradually increases and eventually converges to a stable value as the wave number increases. However, when both NLP and LSP are considered (µ = 0 nm 2 , η = 0 nm 2 ), the above pattern of variation will be broken, and the frequency increases linearly with an increase in the wave number. In addition, Figs. 3(a)-3(d) indicate that the introduction of NLPs can markedly diminish the wave frequency, implying that the NET predicts the structural stiffness-softening effect. In contrast to the nonlocal elastic stress field, introducing LSPs increases the frequency, indicating that the NSGT predicts the structural stiffness-hardening effect. Furthermore, it is worth mentioning that in the case of NSGT, when NLPs are equal to LSPs (µ = η = 0 nm 2 ), the wave frequency is always equal for a given wave number. This indicates that increasing or decreasing both NLP and LSP simultaneously does not change the stiffness of the FGP nanoplate.
The variation of wave frequency with gradient index at different PVFs is shown in Fig. 4. The three figures clearly indicate that an increasing gradient index decreases the wave frequency. This is because larger N will increase the proportion of the PZT-5H phase, thus considerably reducing the stiffness of nanoplates. Moreover, the effect of PVFs on the frequency is opposite at small and large gradient indices. Taking Fig. 4(a) as an example, the frequency increases with increasing PVFs before N ≈ 2.37; however, as the gradient index increases, increasing PVFs instead decreases the frequency. It can be seen from Eq. (2) that the gradient index will directly change the FGM properties, consequently affecting the stiffness. When N 2.37, the larger PVFs, the greater the stiffness, whereas when N > 2.37, the stiffness decreases with increasing PVFs. In addition, comparisons between Figs. 4(b)-4(c) and Fig. 4(a) reveal that although the overall shape of the curves is similar in the three situations, they can be clearly distinguished. The position of the intersection is affected by wave number k and thickness h of the nanoplates. Both larger thicknesses and wave numbers move the intersection backwards. Figure 5 presents the relationship between wave frequency and temperature change at various PVFs. It is seen that the wave frequency is not sensitive to temperature change. In fact, increasing temperature slightly reduces the stiffness of the nanoplates, which in turn reduces the frequency. Consistent with the previous results, the effect of porosity on the wave frequency is different at N = 1 and N = 5. In Fig. 4(a), the frequency increases with increasing PVFs, whereas in Fig. 4(b), increasing PVFs decreases the frequency.  Figure 6 presents the variation of wave frequency with applied voltage at different PVFs. As shown in Fig. 6, an increase in voltage will result in a decrease in the frequency. This is because the stiffness of the nanoplate gradually decreases as the positive voltage increases. In other words, applying a positive voltage induces an axial compressive force, whereas applying a negative voltage to the plate generates a tensile force. Therefore, the electric field can be properly used to control the wave propagation in FGP nanoplates. As expected, the effect of PVFs on the frequency is consistent with that shown in the previous graphs at N = 1 and N = 5. The variation of wave frequency with respect to viscoelastic foundation parameters for various PVFs is shown in Fig. 7. As shown in Figs. 7(a) and 7(b), the wave frequency increases with increasing Winkler and Pasternak coefficients. It is attributed to the increase in stiffness of the FGP nanoplate with the increase in the Winkler and Pasternak moduli. In addition, as the damping coefficient increases, the wave frequency decreases; this is because the damping medium has a retarding effect on the wave propagation. The larger the damping coefficient, the smaller the stiffness of FGP nanoplates. Similar to the previous results, the effect of PVFs on frequency also varies for different damping coefficients. When the damping coefficient is less than 46 × 10 6 N · s · m −3 , a larger PVF leads to a higher wave frequency, whereas when the damping coefficient is greater than 46 × 10 6 N · s · m −3 , increasing PVFs decreases the wave frequency. In addition, we can observe that the smaller the damping coefficient is, the more susceptible the wave frequency is to change in PVFs.

Conclusions
Based on the refined higher-order shear deformation plate theory and the Kelvin-Voigt viscoelastic model, this study investigates the wave propagation characteristics of porous FGP nanoplates. The small-scale effects of FGP nanoplates are investigated by using the NSGT. Finally, the effects of the scale parameters, voltage, temperature, PVFs, gradient index, and viscoelastic foundation parameters on the wave frequency are discussed. The following important conclusions are drawn.
(I) The nonlocal elastic stress field shows a stiffness-softening effect, which becomes more evident with increasing NLPs. The strain gradient stress field causes the nanoplates to exhibit a stiffness-hardening effect, which enhances with increasing LSPs.
(II) A larger gradient index decreases the stiffness and leads to a smaller wave frequency. In addition, imposing a positive voltage weakens the stiffness, whereas an increase in the negative voltage contributes to an increase in the stiffness. The frequency of wave propagation in FGP nanoplates is highly insensitive to temperature changes.
(III) The effect of PVFs on the wave frequency depends on the gradient index because it changes the material properties. The wave frequency increases with the PVFs at a lower gradient index. However, as the gradient index increases, increasing the PVFs decreases the wave frequency.
(IV) Larger Winkler and Pasternak moduli increase the stiffness of the nanoplate, thus increasing the wave frequency; however, increasing the damping coefficient reduces the wave frequency. In addition, the effect of PVFs on the wave frequency is affected by the value of the damping coefficients.
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