Three-dimensional analysis of freely vibrating multilayered piezoelectric plates through adaptive global piecewise-smooth functions

Introduction

Smart materials have for the last few decades attracted the attention of several researchers all over the world. The development of modern computers has also enforced such attraction by first allowing to theoretically describe multiphysics problems and to subsequently test the relevant models. The smart materials we are dealing with here regard piezoelectric layers coupled to stressed and deformed elastic plates. Such a category of systems plays an important role in engineering for controlling vibrations and health monitoring. The piezoelectric layers are generally embedded into structural systems at specific locations; strains may be induced (actuation) on the structure by applying requested voltages at piezoelectric layers or voltages may be relieved (sensing) during the motion of a structure. This double-faced (electric and mechanical) behavior is technologically attractive, but it needs non-negligible challenges to be mathematically described; a mathematical description, which would allow to foresee and thus design the behavior of the system. The mathematical model can be based on three-dimensional or two-dimensional equations by involving higher or lower complexity, respectively. Of course, the best choice is addressed to models requiring a low analytical complexity along with the results as far as possible nearest to the exact ones. In this regard, approximate models play an important role for their capability to accurately describe the relevant physical phenomenon while the three-dimensional models offer irreplaceable benchmarks to compare the relevant approximating performance.

The first theoretical studies addressed to the analysis of piezoelectric crystal plates can be attributed to Mindlin (1952, 1961, 1972). Since then, investigations and models have been taken into account and designed from both theoretical and numerical points of view. With a look at the specific interests of this article (freely vibrating plates), the literature lacks a theoretical treatment of the problem until the beginning of the 90s, when Yang and Batra (1994) presented an extensive analysis of the eigenvalue problem associated with free vibrations of a finite piezoelectric body. Almost in the same period, Heyliger and Saravanos (1995) presented a three-dimensional analysis enriching the literature with numerical results; these latter regard both the analysis of single piezoelectric layers and multilayer plates. The extensive set of the numerical values provided by Heyliger and Saravanos (1995) was later on used as a benchmark; traces of numerical difficulties encountered in the relevant numerical evaluations can also be found (e.g. Cupial, 2005). In Chen et al. (1998), the natural frequencies of a single-layer rectangular piezoelectric plate were calculated using an approach based on the transfer matrix formulation. Chen et al. (1998) investigated simply supported plates; similarly, Batra and Liang (1997) and Liang and Batra (1997) analyzed such a multiphysics problem closely related to simply supported boundary conditions by assuming the electric potential to be a linear function of the transversal coordinate of the plate. Subsequently, Vel and Batra (2000, 2001) carried out static analyses of three-dimensional deformations of multilayered piezoelectric rectangular plates using the Eshelby–Stroh formalism. About 5 years later, Ballhause et al. (2005) developed a unified formulation to assess multilayered theories for piezoelectric plates; in particular, equivalent single-layer and layer-wise models were developed to carry out static and dynamic analyses; these analyses regarded simply supported boundary conditions which allowed the authors to obtain solutions within the approximation used in the expansions of displacement components and electric potential (up to the fourth order through the transversal direction of the plate). The basic principles of models described in Ballhause et al. (2005) were later extended in Carrera et al. (2010) to derive consistent differential electromechanical governing equations through the principle of virtual displacements and Reissner’s (1984) mixed variational theorem.

Based on the above-mentioned references, Messina and Carrera (2014) investigated an approximate methodology aimed at approaching the three-dimensional dynamics of freely vibrating piezoelectric plates using an expansion level allowing any degree of accuracy in a unique formulation for single- or multilayer piezoelectric plates. Such an objective was pursued by expanding the unknown essential functions (displacement and electric potential) on both classical smooth bases (on the middle plane) and global piecewise-smooth function (GPSF) series (Messina, 2002) (through the thickness of the plate). The main advantage of using GPSF series consists of treating even a multilayer plate as if it were made up of a single-layer plate without reducing the accuracy of the results with regard, for example, to a counterpart theory based on a layer-wise model (e.g. Carrera et al., 2010), where the mathematical description is made layer by layer and the compatibility and/or equilibrium conditions are explicitly imposed at each layer interface. On the other hand, the opportunity to treat a multilayer plate in a global manner was also suggested by Reddy (1987); however, this latter proposal was theoretically illustrated within the frame of pure mechanics and without recurring to global approximating functions such as GPSFs which effectively allow to treat the multilayer plate as if it were made up of a single-layer plate.

The approximate model introduced by Messina and Carrera (2014) was also successfully compared with an exact model. The exact model was developed in the same article where numerical instabilities were challenged and solved by appropriately scaling the electric potential. In Messina and Carrera (2014), the comparisons were carried out only from a numerical point of view by evaluating natural frequencies.

Messina and Carrera (2014) based the approximate model on the original GPSFs developed by Messina (2002) which needed a unique expansion level for the unknown essential functions over all the layers involved in the multi-stacking sequence. Such a unique expansion level thus constrained the investigators to possibly overfit the unknown essential function; for example, plates having very thin layers with respect to thick ones (i.e. sandwich configurations) or plates and shells actuated or sensed with very thin layers of piezoelectric materials could need lower order terms (e.g. linear or parabolic) rather than higher order terms as required in thicker parts of the stacks. This accurate selection would result in a significant computational saving. In this latter regard, Messina (2015) recently developed the adaptive global piecewise-smooth functions (A-GPSFs) in order to allow the analyst to make a highly flexible choice with respect to the material and geometric characteristics of the analyzed plates. In particular, such a subset (A-GPSFs) allows to adaptively select the number of functional components for each single layer of the plate by preserving, or checking in a self-contained way, both the accuracy of the results and the global modeling of the GPSFs through the whole stack of the plate. In Messina (2015), the A-GPSFs were both singled out from the relevant mother set (GPSF) and tested within the frame of simply supported cross-ply laminated plates through a Navier-type solution.

Based on the above-mentioned research work, in this article the investigation of simply supported plates stacked along with piezoelectric layers is carried out using A-GPSFs. In order to compare the performance of this model with the existing ones, the simply supported plates are analyzed through a Navier-type solution. The results of this approximate model are also analyzed from a graphical point of view by comparing stress, displacement, and electric potential through the thickness of the multilayered plates to those achieved by the exact model developed in Messina and Carrera (2014). Convergence tests have also been carried out in order to show how the convergence to the exact eigenvalues based on the use of A-GPSFs can be achieved with minimum computational effort. The analysis and the comparisons herein carried out clearly show how the A-GPSFs are (1) an efficient mathematical tool to get the analytical results with an increasing arbitrary accuracy also within the frame of multiphysics problems and (2) still allow to treat a multilayer plate as if it were made up of a single layer; moreover, A-GPSFs (3) leave the analyst to choose, layer by layer, the degree of approximation by saving the relevant computational effort consistently with the required accuracy, and finally (4) contain previous theories as particular cases.

The variationally consistent displacement-based plate model

Let us consider a rectangular plate having a constant thickness h, an axial length L_x, and a transversal length L_y (Figure 1). The in-plane and normal coordinate length parameters are denoted by x, y, and z, respectively, where U, V, and W represent the corresponding displacement components while Φ denotes the electric potential. These variables (U, V, W, Φ) are the essential unknown functions, which variationally varied through the theorem of virtual displacements–potentials and lead to governing differential equations and consistent boundary conditions. The plate is made up of an arbitrary number, N_L, of the linearly elastic orthotropic and piezoelectric layers, with the material axes, which are coincident with the axes of the adopted coordinate system (Figure 1). As far as the electric potential Φ is concerned, the value is assumed always nil at the lateral surfaces (x, y = 0, L_x, L_y), while it is set nil or it is left free to assume any value at the top and bottom of the plate; these conditions are also symbolically recalled in Figure 1 with the relevant surfaces as grounded.

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Figure 1.

Nomenclature and system of coordinates for the multilayered piezoelectric plate.

This general laminated plate theory begins with the following displacement expansion

where Z^{u,v,w, φ} _i are the known functions expressed through the z coordinate while u, v, w, and φ are the functions defined in the middle plane of the plate (x, y). In this work, the number of expanding terms will be assumed equal for each displacement component but different for the potential (N_disp = N^u = N^v = N^w; N_pot = N^φ); u^T, v^T, w^T, and φ^T are the transpose vectors, and therefore, the vectors (u, v, w, φ, Z ^u , Z ^v , Z ^w , Z ^φ ) placed on the third member of equation (1) have N_disp × 1 and N_pot × 1 dimensions, respectively. This latter choice regards the numerical implementation, but each single essential variable can have different expansion levels.

At this stage, it should be noticed how equation (1) is the only assumption needed to build the variationally consistent plate model; such an assumption does not depend on whether the plate is made up of a single or multiple different layers. The capability of this model to converge at the three-dimensional exact results is entrusted both to the functional components contained in Z ^u,v,w,φ (herein each ith function Z^{u,v,w, φ} _i will be assumed to be coincident with an A-GPSF component) and to the theorem of virtual displacements–potentials.

The A-GPSFs, herein referred to as Z^{u,v,w, φ} _i , should be seen as globally defined through the whole thickness of the laminate, that is, −1/2 ≤ z/h = ζ ≤ 1/2, rather than at each layer level. The A-GPSFs are chosen in order to satisfy the essential boundary conditions at the top and bottom of the plate.

Based on the assumed displacement and potential field (1), the strain–displacement and electric field–potential equations are as follows (()′ = d()/dz; (), _x = ∂()/∂x; to carry out layer by layer)

with ε^T = (ε_x, ε_y, ε_z, γ_xy, γ_xz, γ_yz) and E^T = (E_x, E_y, E_z) represent the strain components of the symmetric strain tensor and the components of the electric field, respectively. Moreover, in accordance with the electromechanical behavior of the plate, the following constitutive equations are taken into account

In equation (3), C, e, and ∈ (elastic stiffness, piezoelectric, and permittivity matrices, respectively) couple the mechanical entities with those electric and correspond to the following explicit expressions

where σ^T = (σ_x, σ_y, σ_z, τ_yz, τ_xz, τ_xy) lists the stress components of the symmetric stress tensor while D^T = (D_x, D_y, D_z) contains the electric displacement field components.

Therefore, based on the theorem of virtual displacements and potentials (5) (for freely vibrating plates)

the governing equations of motion (6) and the consistent boundary conditions (7) and (8) can be obtained through suitable integrations by parts of equation (5) along with the congruence of the virtual operators δ applied to the 4 × N unknown functions (u, v, w, φ). Therefore, the following governing equations of motion

are retrieved associated with the following boundary conditions, variationally and consistently derived through equation (5), at x = 0, L_x

and at y = 0, L_y

where

At this stage, let us take into account the following in-plane displacement and electric potential field

where q_x = mπ/L_x and q_y = nπ/L_y with m and n integer numbers corresponding to the number of half waves in x- and y-directions, respectively; the vectors A ^u,v,w,φ are the unknown vectors needing to be determined; and ω_mn is the circular frequency (rad/s). Equation (11) is able to fulfill the following, what we could call, three-dimensional lateral simply supported boundary conditions

which, in this reduced two-dimensional model (6), correspond to the lateral essential and natural boundary conditions (7) and (8).

After substituting equations (10) in equation (6), the following eigenproblem ([K − ω²M]A = 0) is obtained

where each single K ^u,v,w,φ matrix is defined through the following sub matrices

and from which is evidently obvious the fact that each respective transpose matrix in the symmetric part of K is obtained by simply swapping the apices (e.g. K^uvT = K ^vu ). The eigenvalues (in the form of square circular frequencies ω_mn²) in problem (12a) could be thought as extracted through a suitable numerical routine once a couple (m, n) has been set a priori. However, here we followed the procedure to extract the mentioned eigenvalues by solving a reduced form of the eigenproblem (12a); the reduction only concerns the numerical evaluation being the relevant solutions equivalent to original problem (12a). Such a reduced form is based on decoupling A ^φ and A ^u,v,w . In particular, based on the last row of equation (12a), A ^φ can be set as depending on A ^u,v,w as in equation (13)

When A ^φ in equation (13) is substituted in equation (12), this latter can be rearranged in its reduced form as in equation (14)

from which eigenvalues (ω_mn²) and the respective eigenvectors (A ^u,v,w ) can be extracted through classical routines. Once eigenvectors (A ^u,v,w ) have been evaluated by solving problem equation (14), A ^φ can be recovered through equation (13).

The eigenvectors (A ^u,v,w,φ ) would make it possible, through equations (1) and (10), to attain the eigenfunctions (displacement components and electric potential) of the original problem. The same eigenvectors would also make it possible, through equations (1), (3), and (10), to get relevant state of stress and electric displacement associated with the specific eigenvalue and eigenvector. In any case, due to the specific lateral boundary conditions assumed in this work, only the trend through the whole thickness of the plate will be descriptive of the behavior of all the above-mentioned quantities (i.e. displacement, potential, stress). Indeed, the behavior through the middle plane only depends on circular functions (sin(q_xx), sin(q_yy), cos(q_xx), cos(q_yy)).

The resolution of the eigenproblem (14) requires the evaluation of matrices in equation (9) and the evaluation of the relevant integrals through the thickness of the plate. In this regard, the simplicity of this model is clearly due to the capability of the expansion through the z-coordinate (Z ^u,v,w,φ ) to properly model the z-part of both the displacement and electric potential field (1); these latter are described in the next section.

Modeling the expansion through the z-coordinate (Z ^u,v,w,φ ) using A-GPSFs

In section “The variationally consistent displacement-based plate model,” a theory has been developed without mentioning the approximating functions that, once introduced a posteriori into the model, can properly configure the displacement components through the whole thickness of the laminate. This objective is herein currently achieved through the A-GPSFs (Messina, 2015), which are adopted in order to fulfill the essential boundary conditions.

The classical Legendre polynomials (or equivalent bases) can be used when the plate consists of a single homogeneous layer; while, when a multilayer architecture is met, the classical Legendre polynomials are not able to efficiently expand the expected displacement and potential field (1) through the whole thickness. When a plate is made up of multiple layers, A-GPSFs are used in order to properly model the displacement and electric potential field; indeed, the continuity of the displacement and potential, intended as an internal essential boundary condition, is a priori fulfilled by A-GPSFs which, however, still allow to treat the plate as though it was made up of a single layer.

In order to clarify the above-mentioned assertions, let us consider Figure 2 where different sets of local functions are used to generate suitable and respective sets of A-GPSFs. In Figure 2, the case of a three-layer plate is taken into account in order to keep the analysis concise; however, such a simple case should not be considered a limitation for the applications of the A-GPSFs.

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Figure 2.

Local functions for the multilayered piezoelectric plate.

In Figure 2, each single case locally (domain by domain) shows functional components modeling the components in vectors Z ^u,v,w,φ ; the solid line represents the functional component, while the dashed line is the zero line. As previously stated in section “The variationally consistent displacement-based plate model,” the number of expanding terms will be assumed equal for each displacement component but different for the potential (N_disp = N^u = N^v = N^w; N_pot = N^φ); therefore, let us address our considerations, for example, to the electric potential. If we are analyzing the case where the potential can assume any value at the top and bottom of the plate, Case 1 shows suitable local bases, which, however, need an explicit introduction of the continuity at interfaces. Case 1 also shows a set of local functions along with the choice of an analyst who considers only linear terms (i.e. constant and linear for completeness) in the third layer (for example, due to a thinner thickness than others). Once the local trends are assumed to be [3,3,2], the A-GPSFs can be assembled in order to treat the multilayered plate as if it were made up of a single layer; for the case we are dealing with, the number of independent global piecewise-smooth functions corresponds to N_pot = 6 evaluated as N in equation (15) (Messina, 2015)

where ∑ N f corresponds to the total number of local functions (3 + 3 + 2 = 8) and N_L = 3 is the number of layers. The resultant A-GPSFs can be then extracted through characteristic graph-algorithms detailed in Messina (2002, 2015) and herein set as in Figure 3. In particular, Figure 3 illustrates in total seven paths through which a set of GPSFs (Messina, 2002) contain a redundant functional component (the red path (2,1)-(2,2)-(3,3) is coincident with the black path (2,1)-(2,2)-(2,3)). Six independent A-GPSFs can then be obtained by removing the red path.

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Figure 3.

Graph-algorithm to build a set of A-GPSFs (ref. Case 1, Figure 2).

Thus, the resulting set of A-GPSFs is obtained by joining the ends; this junction could be obtained by simply scaling a priori each local function in order to have an established algebraic constant at the ends. Once the junctions at the ends have been carried out, the related set of A-GPSFs is attained as illustrated in Figure 4.

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Figure 4.

Set of A-GPSFs for electric potential (ref. Case 1, Figures 2 and 3).

Whenever the case under investigation regards an electric potential grounded at the top and bottom of the plate, the local bases referred to both in Cases 2 and 3 (Figure 2) can be used. Case 2 ([3,3,2]) uses the same number of functional components previously illustrated for Case 1 (N_pot = 6), but the approximating functions (i.e. polynomials) have a higher order than those used in Case 1; indeed, the first and third layers in Case 2 use third-order and second-order polynomials, respectively. If the analyst wants to keep a linear approximation in the thinner layer (k = 3 in Figure 2) and a second order for the approximation in the other layers, he should recur to the use of sequence [2,3,1] (Case 3). This sequence would involve a set of A-GPSFs containing only N_pot = 4 functional components as illustrated in Figure 5.

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Figure 5.

Set of A-GPSFs for electric potential (ref. Case 3, Figure 2).

It could be interesting to notice in Figure 5 how each single global function (or A-GPSF) fulfills the essential boundary conditions for the electric potential, which is required to assume nil values at the top and bottom of the plate. Similar counterpart considerations can be made for Figure 4 where the electric potential is left free to assume any values at the top and bottom of the plate.

The same choices previously described for the electric potential can be carried out layer by layer for the displacement components (Z ^u , Z ^v , Z ^w ) in order to assemble suitable sets of A-GPSFs and treat a multilayered plate as if it were made up of a single global layer through the analytical model described in section “The variationally consistent displacement-based plate model.”

Finally, it is interesting to notice that the local bases can be built through well-established recursive procedures (e.g. Messina, 2011; Messina and Carrera, 2014; Messina and Rollo, 2010) and once these are set in a specific subdomain, such local bases can be adapted in shorter or longer domains through trivial changes in variables; in other terms, the recursive above-mentioned procedures should be applied once and for all.

Numerical and graphical analysis

In this part of the article, the analytical model of section ““The variationally consistent displacement-based plate model” is tested in conjunction with the A-GPSFs described in section “Modeling the expansion through the z-coordinate (Z ^u,v,w,φ ) using A-GPSFs.” These tests consist of the numerical and graphical results compared both to those of other models and to the exact results; these latter are achieved through the model described in Messina and Carrera (2014), which herein is not represented for the sake of brevity.

All the numerical evaluations refer to a multilayered plate made up of five layers symmetrically stacked through the material scheme [m₁/m₂/m₂/m₂/m₁] (ref. Table 1) along with orientation angle and thickness (mm) schemes equal to [0°/0°/90°/0°/0°] and [1/ 8 3 / 8 3 / 8 3 /1], respectively. This multilayered plate was initially analyzed by Heyliger and Saravanos (1995) and subsequently adopted by Carrera et al. (2010) and Messina and Carrera (2014) for comparisons; here, the same plate is reconsidered for completeness with respect to the previous analyses.

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Table 1.

Elastic and electric material properties of the multilayer plate.

Properties	m1	m2
E1 (GPa)	81.3	132.38
E2 (GPa)	81.3	10.756
E3 (GPa)	64.5	10.756
G23 (GPa)	25.6	3.606
G13 (GPa)	25.6	5.6537
G12 (GPa)	30.6	5.6537
ν 23	0.432	0.49
ν 13	0.432	0.24
ν 12	0.329	0.24
e15 (C/m2)	12.72	0.0
e24 (C/m2)	12.72	0.0
e31 (C/m2)	−5.2	0.0
e32 (C/m2)	−5.2	0.0
e33 (C/m2)	15.08	0.0
∈1 (nF/m)	13.06	0.0309897
∈2 (nF/m)	13.06	0.026563
∈3 (nF/m)	11.51	0.026563
ρ (kg/m3)	1.0	1.0

The numerical evaluations of this work have been summarized in Tables 2 to 5. Tables 2 and 3 refer to the switch closed in Figure 1, thus recalling bottom and top grounded (nil electric potential). The underlined values illustrated in Tables 2 to 5 recall exact eigenvalues. These provide an immediate comparison to the convergence tests, stopped when the same exact results were achieved.

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Table 2.

First six eigenvalues (ω/100) for square plate with m = n = 1, L_x/h = 4 and Φ (±h/2) = 0.

No.	Exact 3D analysis; a W = 109	LD1 b	LD2 b	LD3 b	LD4 b
1	57074.0	57252.5	57081.9	57074.0	57074.0
2	191,301	194,840	191,311	191,301	191,301
3	250,768	255,646	250,786	250,768	250,768
		A-GPSFs and GPSFs
		Disp. [2;2;2;2;2]; Pot. [1;2;2;2;1]; Ndisp = 6; Npot = 4; size: [22 × 22]	Disp. [3;3;3;3;3]; Pot. [2;3;3;3;2]; Ndisp = 11; Npot = 9; size: [42 × 42]	Disp. [4;4;4;4;4]; Pot. [3;4;4;4;3]; Ndisp = 16; Npot = 14; size: [62 × 62]	Disp. [5;5;5;5;5]; Pot. [4;5;5;5;4]; Ndisp = 21; Npot = 19; size: [82 × 82]	Disp. [6;6;6;6;6]; Pot. [5;6;6;6;5]; Ndisp = 26; Npot = 24; size: [102 × 102]
4	57074.0	57252.5	57081.9	57074.0	57074.0	57074.0
5	191,301	194,840	191,311	191,301	191,301	191,301
6	250,768	255,646	250,786	250,768	250,768	250,768
7	274,940	282,168	275,060	274,942	274,940	274,940
8	362,489	368,458	362,882	362,492	362,489	362,489
9	381,036	389,524	381,322	381,037	381,036	381,036
		Disp. [4;4;4;4;4]; Pot. [2;3;3;3;2]; Ndisp = 16; Npot = 9; size: [57 × 57]	Disp. [3;5;5;5;3]; Pot. [2;3;3;3;2]; Ndisp = 17; Npot = 9; size: [60 × 60]	Disp. [4;5;5;5;4]; Pot. [2;3;3;3;2]; Ndisp = 19; Npot = 9; size: [66 × 66]	Disp. [4;5;5;5;4]; Pot. [1;2;1;2;1]; Ndisp = 19; Npot = 3; size: [60 × 60]	GPSFs any cbc; c [6; 6; 6; 6; 6]; size: [3744 × 3744]
10	57074.0	57074.0	57074.1	57074.0	57039.0	57074.0
11	191,301	191,301	191,301	191,301	191,301	191,301
12	250,768	250,768	250,768	250,768	250,764	250,768
13	274,940	274,942	274,940	274,940	274,942	274,940
14	362,489	362,492	362,489	362,489	362,491	362,489
15	381,036	381,037	381,036	381,036	381,004	381,036
		Disp. [3;3;3;3;3]; Pot. [3;3;3;3;3]; Ndisp = 11; Npot = 11; size: [44 × 44]	Disp. [4;4;4;4;4]; Pot. [3;3;3;3;3]; Ndisp = 16; Npot = 11; size: [59 × 59]	Disp. [4;4;4;4;4]; Pot. [4;4;4;4;4]; Ndisp = 16; Npot = 16; size: [64 × 64]	Disp. [5;5;5;5;5]; Pot. [3;3;3;3;3]; Ndisp = 21; Npot = 11; size: [74 × 74]
16	57074.0	57081.9	57074.0	57074.0	57074.0
17	191,301	191,311	191,301	191,301	191,301
18	250,768	250,786	250,768	250,768	250,768
19	274,940	275,060	274,942	274,942	274,940
20	362,489	362,882	362,492	362,492	362,489
21	381,036	381,322	381,037	381,037	381,036

3D: three-dimensional; A-GPSFs: adaptive global piecewise-smooth functions; GPSFs: global piecewise-smooth functions.

a

Messina and Carrera (2014).

b

Carrera et al. (2010).

c

Classical boundary conditions; Messina and Carrera (2014).

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Table 3.

First six eigenvalues (ω/100) for square plate with m = n = 1, L_x/h = 50 and Φ (±h/2) = 0.

No.	Exact 3D analysis; a W = 109	LD1 b	LD2 b	LD3 b	LD4 b
1	618.105	619.023	618.106	618.105	618.106
2	15681.5	15683.4	15681.6	15681.6	15681.6
3	21492.6	21494.4	21492.6	21492.6	21492.6
		A-GPSFs and GPSFs
		Disp. [2;2;2;2;2]; Pot. [1;2;2;2;1]; Ndisp = 6; Npot = 4; size: [22 × 22]	Disp. [3;3;3;3;3]; Pot. [2;3;3;3;2]; Ndisp = 11; Npot = 9; size: [42 × 42]	Disp. [4;4;4;4;4]; Pot. [3;4;4;4;3]; Ndisp = 16; Npot = 14; size: [62 × 62]	Disp. [5;5;5;5;5]; Pot. [4;5;5;5;4]; Ndisp = 21; Npot = 19; size: [82 × 82]	Disp. [6;6;6;6;6]; Pot. [5;6;6;6;5]; Ndisp = 26; Npot = 24; size: [102 × 102]
4	618.105	619.023	618.106	618.105	618.105	618.105
5	15681.5	15683.4	15681.5	15681.5	15681.5	15681.5
6	21492.6	21494.4	21492.6	21492.6	21492.6	21492.6
7	209,704	212,811	209,788	209,704	209,704	209,704
8	210,522	214,690	210,605	210,523	210,522	210,522
9	378,104	384,953	378,278	378,105	378,104	378,104
		Disp. [4;4;4;4;4]; Pot. [2;3;3;3;2]; Ndisp = 16; Npot = 9; size: [57 × 57]	Disp. [3;5;5;5;3]; Pot. [2;3;3;3;2]; Ndisp = 17; Npot = 9; size: [60 × 60]	Disp. [4;5;5;5;4]; Pot. [2;3;3;3;2]; Ndisp = 19; Npot = 9; size: [66 × 66]	Disp. [4;5;5;5;4]; Pot. [1;2;1;2;1]; Ndisp = 19; Npot = 3; size: [60 × 60]	GPSFs any cbc; a [6; 6; 6; 6; 6]; size: [3744 × 3744]
10	618.105	618.105	618.105	618.105	617.967	618.105
11	15681.5	15681.5	15681.5	15681.5	15681.5	15681.6
12	21492.6	21492.6	21492.6	21492.6	21492.5	21492.6
13	209,704	209,704	209,704	209,704	209,704	209,704
14	210,522	210,523	210,522	210,522	210,523	210,522
15	378,104	378,105	378,104	378,104	378,094	378,104
		Disp. [3;3;3;3;3]; Pot. [3;3;3;3;3]; Ndisp = 11; Npot = 11; size: [44 × 44]	Disp. [4;4;4;4;4]; Pot. [3;3;3;3;3]; Ndisp = 16; Npot = 11; size: [59 × 59]	Disp. [4;4;4;4;4]; Pot. [4;4;4;4;4]; Ndisp = 16; Npot = 16; size: [64 × 64]	Disp. [5;5;5;5;5]; Pot. [3;3;3;3;3]; Ndisp = 21; Npot = 11; size: [74 × 74]
16	618.105	618.106	618.105	618.105	618.105
17	15681.5	15681.5	15681.5	15681.5	15681.5
18	21492.6	21492.6	21492.6	21492.6	21492.6
19	209,704	209,788	209,704	209,704	209,704
20	210,522	210,605	210,523	210,523	210,522
21	378,104	378,278	378,105	378,105	378,104

3D: three-dimensional; A-GPSFs: adaptive global piecewise-smooth functions; GPSFs: global piecewise-smooth functions.

a

Messina and Carrera (2014).

b

Carrera et al. (2010).

c

Classical boundary conditions; Messina and Carrera (2014).

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Table 4.

First six eigenvalues (ω/100) for square plate with m = n = 1, L_x/h = 4 and Φ (±h/2) any.

		A-GPSFs and GPSFs
No.	Exact 3D analysis; a W = 109	Disp. [2;2;2;2;2]; Pot. [2;2;2;2;2]; Ndisp = 6; Npot = 6; size: [24 × 24]	Disp. [3;3;3;3;3]; Pot. [3;3;3;3;3]; Ndisp = 11; Npot = 11; size: [44 × 44]	Disp. [4;4;4;4;4]; Pot. [4;4;4;4;4]; Ndisp = 16; Npot = 16; size: [64 × 64]	Disp. [5;5;5;5;5]; Pot. [5;5;5;5;5]; Ndisp = 21; Npot = 21; size: [84 × 84]	Disp. [6;6;6;6;6]; Pot. [6;6;6;6;6]; Ndisp = 26; Npot = 26; size: [104 × 104]
1	57088.7	57270.2	57096.6	57088.7	57088.7	57088.7
2	191,304	194,842	191,314	191,304	191,304	191,304
3	250,768	255,647	250,787	250,769	250,768	250,768
4	274,940	282,168	275,060	274,942	274,940	274,940
5	362,520	368,502	362,913	362,523	362,520	362,520
6	381,048	389,534	381,335	381,050	381,049	381,048
		Disp. [5;5;5;5;5]; Pot. [3;4;4;4;3]; Ndisp = 21; Npot = 14; size: [77 × 77]	Disp. [4;6;6;6;4]; Pot. [3;4;4;4;3]; Ndisp = 22; Npot = 14; size: [80 × 80]	Disp. [5;6;6;6;5]; Pot. [3;4;4;4;3]; Ndisp = 24; Npot = 14; size: [86 × 86]	Disp. [4;6;6;6;4]; Pot. [2;2;1;2;2]; Ndisp = 22; Npot = 5; size: [71 × 71]	GPSFs any cbc; b [6; 6; 6; 6; 6]; size: [3744 × 3744];
7	57088.7	57088.7	57088.7	57088.7	57054.7	57088.7
8	191,304	191,304	191,304	191,304	191,303	191,304
9	250,768	250,768	250,768	250,768	250,764	250,768
10	274,940	274,940	274,940	274,940	274,940	274,940
11	362,520	362,520	362,520	362,520	362,519	362,520
12	381,048	381,049	381,048	381,048	381,014	381,048

A-GPSFs: adaptive global piecewise-smooth functions; GPSFs: global piecewise-smooth functions; 3D: three-dimensional.

a

Messina and Carrera (2014).

b

Classical boundary conditions; Messina and Carrera (2014).

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Table 5.

First six eigenvalues (ω/100) for square plate with m = n = 1, L_x/h = 50 and Φ (±h/2) any.

		A-GPSFs and GPSFs
No.	Exact 3D analysis; a W = 109	Disp. [2;2;2;2;2]; Pot. [2;2;2;2;2]; Ndisp = 6; Npot = 6; size: [24 × 24]	Disp. [3;3;3;3;3]; Pot. [3;3;3;3;3]; Ndisp = 11; Npot = 11; size: [44 × 44]	Disp. [4;4;4;4;4]; Pot. [4;4;4;4;4]; Ndisp = 16; Npot = 16; size: [64 × 64]	Disp. [5;5;5;5;5]; Pot. [5;5;5;5;5]; Ndisp = 21; Npot = 21; size: [84 × 84]	Disp. [6;6;6;6;6]; Pot. [6;6;6;6;6]; Ndisp = 26; Npot = 26; size: [104 × 104]
1	618.108	619.026	618.109	618.108	618.108	618.108
2	15681.6	15683.4	15681.6	15681.6	15681.6	15681.6
3	21492.8	21494.7	21492.8	21492.8	21492.8	21492.8
4	209,707	212,827	209,792	209,707	209,707	209,707
5	210,573	214,736	210,656	210,573	210,573	210,573
6	378,105	384,953	378,279	378,106	378,105	378,105
		Disp. [4;4;4;4;4]; Pot. [2;3;3;3;2]; Ndisp = 16; Npot = 9; size: [57 × 57]	Disp. [4;4;4;4;4]; Pot. [3;3;3;3;3]; Ndisp = 16; Npot = 11; size: [59 × 59]	Disp. [4;5;5;5;4]; Pot. [3;3;3;3;3]; Ndisp = 19; Npot = 11; size: [68 × 68]	Disp. [4;5;5;5;4]; Pot. [2;2;1;2;2]; Ndisp = 19; Npot = 5; size: [62 × 62]	GPSFs any cbc; b [6; 6; 6; 6; 6]; size: [3744 × 3744];
7	618.108	617.969	618.108	618.108	617.969	618.108
8	15681.6	15681.6	15681.6	15681.6	15681.6	15681.6
9	21492.8	21492.8	21492.8	21492.8	21492.8	21492.8
10	209,707	209,707	209,707	209,707	209,707	209,707
11	210,573	210,573	210,573	210,573	210,573	210,573
12	378,105	378,095	378,106	378,105	378,094	378,105

A-GPSFs: adaptive global piecewise-smooth functions; GPSFs: global piecewise-smooth functions; 3D: three-dimensional.

a

Messina and Carrera (2014).

b

Classical boundary conditions; Messina and Carrera (2014).

Tables 2 and 3 refer to two different length-to-thickness ratios (L_x/h = 4 and L_x/h = 50, respectively). Tables 4 and 5 similarly refer to the same different length-to-thickness ratios but take into account the case of free potential at the bottom and top of the plate (switch open in Figure 1).

Tables 2 and 3 present a more extensive numerical investigation because they compare the performance of this model based on A-GPSFs to both the exact model (Messina and Carrera, 2014) and the model of Carrera et al. (2010). Conversely, Tables 4 and 5 could only be compared to the exact results.

The first three numerical rows of Table 2 recall the results achieved by Carrera et al. (2010) who approached the exact results using the so-called LD1–4 models; such models are essentially based on assumed displacements at a layer-by-layer level; the number associated with the acronym “LD” refers to the maximum order of first Legendre polynomials adopted in each layer. The second block of numerical rows in Table 2 (i.e. rows 4–9) aims at proving how LD1–4 can be achieved as a particular case of this model based on A-GPSFs; indeed, the columns placed in the respective positions of LD1–4 show exactly the same eigenvalues, with all their six significant figures. Such equivalence clearly comes from the captions of the rows 4–9 in Table 2: these captions show the arrangement of local bases which originated the A-GPSFs. For example, “Disp.[2,2,2,2,2]; Pot.[1,2,2,2,1]” means that for each of the five domains (layers), only the first two functional components were taken into account in order to generate six A-GPSFs (N_disp = 2 + 2 + 2 + 2 + 2 −5 + 1 = 6) in Z ^u,v,w , while for modeling the electric potential through the thickness of the plate only the first linear term was assumed in the outer thinner layers, thus generating four A-GPSFs (N_pot = 1 + 2 + 2 + 2 + 1 − 5 + 1 = 4) in Z ^φ ; both these choices provide a displacement-based matrix (K ^uu , K ^vv , K ^ww ) in equation (12) having dimensions 3·6 × 3·6 and an electric potential–based matrix (K ^φφ ) having dimensions 4 × 4. Therefore, the resultant eigenproblem (12) has a dimension of 22 × 22. Based on the above description, the best performance is obtained with the local set “Disp.[5,5,5,5,5]; Pot.[4,5,5,5,4]” which originated a set of A-GPSFs achieving the first six significant digits of the first six exact eigenvalues. An increasing computational effort through the local set “Disp.[6,6,6,6,6]; Pot.[5,6,6,6,5]” just stably confirmed the exact values. Rows 10–15 in Table 2 show the advantage of using the flexibility offered by A-GPSFs in order to additionally reduce the computational effort. Based on the analysis of Table 2, rows 10–15 show the set “Disp.[4,5,5,5,4]; Pot.[2,3,3,3,2]” having a size of 66 × 66 which is clearly more convenient than 82 × 82. In order to provide an additional idea on the advantages which the A-GPSFs can offer an analyst, let us compare the results achieved at rows 10–15 to the results illustrated at rows 16–21. These latter rows show the performance of the same model presented in section “The variationally consistent displacement-based plate model” when the A-GPSFs are used as the original GPSFs (Messina, 2002). Rows 16–21 clearly show that the GPSFs are able to get the exact results but with greater computational effort (74 × 74 against 66 × 66).

Before closing the perusal of Table 2, two cases need to be compared: “Disp.[4,5,5,5,4]; Pot.[2,3,3,3,2]” versus “Disp.[4,5,5,5,4]; Pot.[1,2,1,2,1]”; this latter case uses a set of A-GPSFs having the same functional components related to the displacement of the former one while using a poorer set of local bases for the electric potential; this poorer base evidently generates less accurate eigenvalues than their exact counterparts (only the second eigenvalue is coincident with the exact eigenvalue). These less accurate eigenvalues are partially higher and lower than their exact counterparts. This means that in the multiphysics problem we are dealing with, in general, neither an approach from above nor from below can be expected during a convergence test based on the approximation process.

At this stage of the numerical perusal of Table 2, a relevant graphical analysis deserves to be shown. In this regard, Figure 6 illustrates the exact distribution through the thickness of displacement components, electric potential, and stress components. Such graphical results are essentially eigenfunctions defined regardless of an arbitrary coefficient of proportionality; thus, in order to make such graphical results comparable to each other, all the values were normalized through the absolute maximum value of the electric potential (whose maximum value is set at 1). Along with Figure 6, also Figures 7 to 9 regard exact representations, and in this regard such figures should be considered particularly valuable because produced for the first time through the exact method detailed in Messina and Carrera (2014) which is based on a virtual electric potential aimed at reducing numerical instabilities in the multiphysics problem we are dealing with.

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Figure 6.

Exact (Messina and Carrera, 2014) displacement, electric potential, and stress distribution through z regarding the first eigenfunction (ω/100 = 57074.0) for m = n = 1; ref. Table 2.

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Figure 7.

Exact (Messina and Carrera, 2014) displacement, electric potential, and stress distribution through z regarding the first eigenfunction (ω/100 = 618.105) for m = n = 1; ref. Table 3.

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Figure 8.

Exact (Messina and Carrera, 2014) displacement, electric potential, and stress distribution through z regarding the first eigenfunction (ω/100 = 57088.7) for m = n = 1; ref. Table 4.

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Figure 9.

Exact (Messina and Carrera, 2014) displacement, electric potential, and stress distribution through z regarding the first eigenfunction (ω/100 = 618.108) for m = n = 1; ref. Table 5.

Figure 6 correlates the mentioned quantities (displacement, potential, and stress) to eigenvalue ω/100 = 57074.0 in Table 2. In particular, the graph at the top of Figure 6 contains an inner zoom around the zero line showing an almost imperceptible oscillating U(z) through a magnitude which is three orders lower than other displacement components. This same Figure 6 shows in-plane stress components which are discontinuous, layer by layer, while the out-of-plane stress components are continuous and fulfill the nil boundary conditions at the bottom and top of the plate. The same electric potential results grounded at the bottom and top of the plate. The graphical counterparts of Figure 6, based on A-GPSFs model, are Figures 10 to 12. In particular, Figure 10 regards the best computational case previously described (Disp.[4,5,5,5,4], Pot.[2,3,3,3,2], size: 66 × 66) as presented in Table 2. Evidently, Figure 10 is in excellent agreement with Figure 6, even showing the above-mentioned almost imperceptible oscillation of U(z), and thus we could conclude on the excellent behavior of the A-GPSFs which are clearly able to obtain the exact results both from a numerical (eigenvalues) and graphical (displacement, potential, and stress) point of view. Figure 11 is a clear warning on the need to test the convergence eigenvalues through an adequate number of significant digits when an accurate stress distribution is requested (even retaining a number of significant figures higher than those normally interesting in engineering practice (e.g. Williams et al. 1997)). Indeed, Figure 11 referring to the case of Disp.[3,3,3,3,3] and Pot.[2,3,3,3,2] (size: 42 × 42) of Table 2 is related to the eigenvalues with three/four significant figures coincident with the exact counterparts; however, even though the correspondence of such three/four significant digits exists, the distribution of a part of the stress components is relatively inaccurate; in Figure 11, the out-of-plane stress components do not fulfill neither the inner (i.e. continuity of the stress) nor the outer boundary conditions and, moreover, its relevant distribution is relatively different with respect to its exact counterpart of Figures 6 and 10. Such particular sensitivity of the stress with regard to the eigenvalues should be attributed to the fact that stress components depend on the derivative of the essential variables (displacement and potential) and generally require a slightly increasing number of functional components in order to achieve a description which is closer to the exact trend. It is finally interesting to notice, in Figure 11, how the model based on A-GPSFs seems to fail when describing certain details (i.e. the imperceptible oscillation U(z)); however, we should positively appreciate that the model is able to approximate such imperceptible details in the mean when lower order of approximating bases are used. Indeed, among the first three components of Legendre polynomials (i.e. Figure 2), the higher unsymmetrical functional components are linear and, therefore, any possibility to get a better approximation than linear does not exist at all.

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Figure 10.

Displacement, electric potential, and stress distribution through z based on A-GPSFs (Disp.[4,5,5,5,4], Pot.[2,3,3,3,2]). First eigenfunction (ω/100 = 57074.0) for m = n = 1; ref. Table 2.

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Figure 11.

Displacement, electric potential, and stress distribution through z based on A-GPSFs (Disp.[3,3,3,3,3], Pot.[2,3,3,3,2]). First eigenfunction (ω/100 = 57081.9) for m = n = 1; ref. Table 2.

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Figure 12.

Displacement, electric potential, and stress distribution through z based on A-GPSFs (Disp.[4,5,5,5,4], Pot.[1,2,1,2,1]). First eigenfunction (ω/100 = 57039.0) for m = n = 1; ref. Table 2.

Based on the above-mentioned analysis, Figure 12 is finally worth mentioning. With respect to Figure 11, a higher number of functional components are used to model displacements along with an extremely reduced number of A-GPSFs aimed at modeling the electric potential through the thickness of the plate. In this case, the size decreases to 60, and the A-GPSFs are still able to capture an average of four/five significant digits of the exact eigenvalues along with the price of an electric potential which approximates its exact counterpart in the mean (Figure 6) through linear trends.

In addition to all the above-mentioned specific observations, made figure by figure (Figures 6, 10 to 12), we should finally realize that the A-GPSFs along with the theorem of virtual displacement are able to inherently achieve the fulfillment of the inner and outer natural boundary conditions without explicitly imposing the same conditions on the model and, moreover, this couple of theoretical tools is also able to treat the plate as if it were made up of a single layer by leaving the analyst to freely choose the accuracy in all layers for the multiphysics entities involved.

Table 3 has been designed in line with the aims of Table 2 but by taking into account a different length-to-thickness ratio (L_x/h = 50). Still in this case A-GPSFs are able to obtain the results coming from LD1–4 by Carrera et al. (2010) as particular cases (i.e. compare rows 1–3 with rows 4–6 in Table 3). In this case, A-GPSFs seem to behave slightly better than Carrera et al. (2010) models when compared to the exact results although the detected slight discrepancies could be attributed to printing errors rather than to any substantial significant difference between this model and LD1–4 model.

Table 3, compared to Table 2, shows the capability of this model to converge at the exact results faster when lower length-to-thickness ratio is taken into account.

Figure 7 concerns displacement, potential, and stress correlated to the eigenvalue ω/100 = 618.105 in Table 3. Here, an almost imperceptible behavior is detected for the electric potential which, rather than linear in the middle layer, has, through its detailed description, a second-order trend.

Tables 4 and 5 refer to the analysis with regard to an electric potential assuming free values at the bottom and top of the plate. The first six rows (1–6) in both Tables 4 and 5 illustrate the performance of A-GPSFs used as the original GPSFs; there is clear evidence of the better performance of the A-GPSFs. For example, in Table 4, the convergence to the exact results is achieved with N_disp = 22 and N_pot = 14 functional components of A-GPSFs involving a size of the eigenproblem 80 × 80 versus N_disp = 26 and N_pot = 26 functional components of GPSFs along with a size of the eigenproblem 104 × 104. The convergence still appears faster with higher length-to-thickness ratio (Table 5), and finally a slower convergence between Tables 2 to 5 can be observed. This latter observation justifies the need for theories adaptable to different boundary conditions along with a proper flexibility in order to choose the related approximation along with the minimum computational effort each time. The relevant graphical analysis, exact and based on A-GPSFs, of Tables 4 and 5 is recapped in Figures 8, 9, 13 and 14, respectively.

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Figure 13.

Displacement, electric potential, and stress distribution through z based on A-GPSFs (Disp.[4,6,6,6,4], Pot.[3,4,4,4,3]). First eigenfunction (ω/100 = 57088.7) for m = n = 1; ref. Table 4.

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Figure 14.

Displacement, electric potential, and stress distribution through z based on A-GPSFs (Disp.[4,6,6,6,4], Pot.[2,2,1,2,2]). First eigenfunction (ω/100 = 57054.7) for m = n = 1; ref. Table 4.

Conclusion

In this study, certain expansions in series recently introduced (A-GPSFs) have allowed to model and analyze the free vibrations of multilayered piezoelectric plates by saving computational efforts along with arbitrary accuracy. The A-GPSFs still allow to treat a multilayer plate as if it were made up of a single layer and leave the analyst to choose, layer by layer, the degree of approximation by saving the relevant computational effort consistently with the required accuracy. Any related comparable theory can be contained as a particular case.

The analytical model has been consistently derived from the classical theorem of virtual displacements along with its related boundary conditions. The plates are mechanically simply supported at the lateral edges and the electric potential is nil. The analysis herein carried out has demonstrated the capability of the A-GPSFs to deal with several areas of mechanical sciences and in particular with plates stacked within the frame of multiphysics problems such as laminated and piezoelectric layers. Three-dimensional results have been provided from a graphical point of view, and a part from the value of such ascertained exact results has corroborated the need to test approximate theories through comparisons of eigenvalues with a sufficiently high number of significant digits, even higher than those normally needed in engineering measurements. Finally, convergence tests have proved how theories based on assumed displacements and electric potential do not generally provide models converging neither from above nor from below. Further analysis aimed at analyzing convergence depending on boundary conditions and stacking layers would be worthy of attention.