Document Type : Research Article
Authors
^{1} Department of Civil Engineering, Pimpri Chinchwad College of Engineering & Research, Pune – 412101, India
^{2} Department of Structural Engineering, Veermata Jijabai Technological Institute, Mumbai – 400019, India
Abstract
Keywords
Functionally Graded Piezoelectric Plates in Cylindrical Bending by Semianalytical Approach
^{a} Department of Civil Engineering, Pimpri Chinchwad College of Engineering & Research, Pune – 412101, India
^{b} Department of Structural Engineering, Veermata Jijabai Technological Institute, Mumbai – 400019, India
KEYWORDS 

ABSTRACT 
Static analysis; Semianalytical method; Functionally graded piezoelectric material; 2D domain. 
A simply supported (SS) functionally graded piezoelectric material (FGPM) plate in a 2D domain has been analyzed for stress and displacement by a Semianalytical approach. Inplane variation in stresses and displacements is assumed to be trigonometric. The elasticity approach is used and no simplifying assumption is made on the stress and displacement fields in the throughthickness direction. The FGPM plate is subjected to a transverse electromechanical load whose intensity remains constant in the outofplane direction. Thus, the plate is under plane stress and plane strain conditions of elasticity. Exponential law or power law has been considered for smooth gradation of material properties in the throughthickness direction. The formulation is a set of firstordered ordinary differential equations (ODE), which has been solved using numerical integration. Exact outcomes in the literature have been used to correlate and validate the present model results. Additional investigation has been carried out on FGPM plates and beams and results are provided for future reference. 
Stress and displacement analysis of smart materials remain to be an active area of research to date. Smart materials are formed with an elastic substrate having embedded or attached patches of piezoelectric materials. By virtue of actuation, piezomaterials undergo deformation under the applied electric field and by virtue of sensing, produce an electric charge on deforming mechanically. This ability of interconversion of mechanical and electrical energy of piezomaterials is judiciously used to develop selfcontrolling, selfgoverning smart materials.
Smart materials find numerous applications in every walk of engineering, including aerospace and aeronautical industry, robotics, and medical instrumentation. These highend applications demand accurate and involved analysis and it is essential to have a robust, versatile, and computationally inexpensive analysis tool.
Researchers have proposed several analytical and numerical solutions based on exact and approximate theories. A functionally graded piezoelectric plate loaded with electromechanical loading in the 2D field has been analyzed by Lu et al. [1] with the help of elasticity solutions. Similarly, Lu et al. [2] have analyzed an allaround simply supported FGPM plate for the exact solution. Xiang and Shi [3] have presented a static analysis of the FGPM sandwich cantilever using Airy’s stress function. Mikaeeli and Behjat [4] have used the threedimensional elementfree Galerkin method to investigate the static behavior of thick functionally graded piezoelectric plates. Kulikov and Plotnikova [5] have used the sampling surfaces method for the exact analysis of thick and thin FG piezoelectric laminated plates with specified accuracy. Exact solutions for FGPM plates have also been provided by Lim and He [6], Reddy and Cheng [7], and Zhong and Shang [8].
Exact solutions, though invaluable, are often difficult to obtain due to the mathematical complexities involved in the solution techniques. Thus, efforts are made to put forth approximate models based upon equivalent singlelayer theory or layerwise theories. A large quantum of literature is found on approximate analysis of FGPM plates, including those from Almajid et al. [9], Joshi et al. [10], Taya et al. [11], Zhong and Yu [12], Wu et al. [13], Loja et al. [14], Li et al. [15], Chuaqui and Roque [16], Nourmohammadi and Behjat [17], Behjat and Khoshravan [18], Raissi H. [19], Raissi et al. [20, 21] among others.
In addition to the static analysis, extensive work has been carried out by the researchers on vibrations and wave propagation in Functionally Graded Material (FGM) plates and FGPM plates, a significant contribution coming from Song and Luo [22], Mazzotti et al. [23], Li et al. [24], Li and Han [25], Li et al. [26], Vinh and Tounsi [27], Tahir et al. [28], Rachid et al. [29], Habib et al. [30], Boufia et al. [31], Zaitoun et al. [32], Mudhaffar et al. [33], Kouider et al. [34], Merazka et al. [35], Hachemi et al. [36] and Bakoura et al. [37].
The present paper gives a semianalytical model for the stress and displacement analysis of a simply supported FGPM plate in cylindrical bending. Inplane displacement (u), transverse displacement (w), transverse normal stress (s_{z}), transverse shear stress (t_{xz}), electric potential (f) and transverse electric displacement (D_{z}) are considered as primary variables. An FGPM plate acted upon by electromechanical loading is formulated as a mixed twopoint boundary value problem (BVP) in the interval h/2 £ z £ h/2, with half of the variables specified at the edges
z = ± h/2. Inplane variation in primary variables is assumed to be trigonometric, keeping consistent with the relevant electroelastic boundary conditions (BCs). The Semianalytical model developed with algebraic manipulation of governing elasticity equations is a set of firstordered ordinary differential equations, which can be easily solved using numerical integration. The model is simple, mixed, versatile, accurate, and computationally inexpensive. However, this approach is suitable only for simply supported or clampedclamped BCs and not for any arbitrary BCs.
An FGPM plate with dimensions a, b, and h in x, y, and z directions respectively, has been considered. The plate's midplane is assumed to be the reference xy plane and the transverse axis is in the zdirection (Figure 1). Edges at x = 0, a are diaphragm supported and grounded to zero potential. The top layer of the plate is loaded with transverse mechanical and electrical loading, which is independent of the ydirection. Condition of elasticity shall be considered as of plane strain or plane stress depending upon the dimension b being extremely long or extremely short.
The material properties of FGPM are assumed to vary in the depth direction as;
, , 
(1) 
Fig. 1. Simply Supported FGPM Plate
where, , e_{ij} and are the values at any arbitrary depth and,, are the available reference values. Poisson's ratios n_{ij} are invariants. Gradation f(z) is either exponential law or power law, which is popularly used in literature.
Coupled elastic and electric fields equations given by Tirsten [38] are;
{s} = [C^{E}]{ e } – [e]{E}, {D} = [e]^{T}{ e } + [g^{S}]{E} 
(2) 
2D elasticity equilibrium equations and straindisplacement equations are;
, 
(3) 
, , 
(4) 
Maxwell [39] has given a charge equilibrium equation in a 2D domain as;
(5) 
The vectors and matrices appearing in Eqs. (2) are given in the Appendix.
Equations (2)(5) consist of interdependent 11 unknowns, viz. u, w, e_{x}, e_{z}, g_{xz}, s_{x}, s_{z}, t_{xz}, D_{x}, D_{z} and f in 11 equations. After an algebraic simplification of the above set of equations, a set of partial differential equations (PDEs) involving only six chosen primary variables and the gradation function f(z) is obtained as;
(6) 
Kantorovich [40] approach is used to convert the obtained set of PDEs into a set of ordinary differential equations (ODEs). The inplane variation in displacement field and stress field is considered to be trigonometric, satisfying elastic and electric boundary conditions at x = 0, a, as;
(7) 
where, m = 1, 3, 5, …
The transverse mechanical load p(x,z) and electrostatic potential f(x, z) are represented using Fourier series to facilitate the application of an arbitrarily distributed load as;
(8) 
Substituting Eqs. (7), (8) and the derivatives into Eqs. (6), a set of firstorder ODEs containing primary dependent variables and the gradation function f(z) is obtained as;
(9) 
The electroelastic coefficients Q_{11}–Q_{66 }in Eqs. (9) are given in the Appendix.
The above Eqs. (9) show the mixed twopoint BVP in the realm h/2 £ z £ h/2, with stress components and transverse electric displacement (open circuit condition) or electric potential (closedcircuit condition) known at the upper and lower faces of the plate. The secondary variables are represented in the form of primary variables as;
(10) 
Numerical integration is used to obtain solutions to Eq.s (9). The BVP is converted into an initial value problem (IVP) [41] and the shooting approach is used to solve the ODEs in Eq. (9). The solution methodology has been discussed in detail elsewhere [42] and is not repeated here.
Numerical investigation using Semianalytical methodology is discussed below. A simply supported infinite FGPM plate and an FGPM/Homogeneous bilayered laminate under electromechanical loading have been investigated to validate the model. Additionally, a few examples of FGPM and hybrid beams of different piezo materials have been addressed.
Example 1
An infinitely long PZT4 based FGPM simply supported plate has been considered. Elastic and electric properties of the plate are considered to vary exponentially in the depth direction as;
,, 
(11) 
where β = 1, 0.5, 0, 0.5, 1 is the material grading constant. Reference values, and are expressed in Table 1. The plate is considered to be thick with an aspect ratio of a/h = 1. Firstly, the plate is loaded by a mechanical load; with upper and lower faces held at zero potential. Secondly, it is exposed to electric load; with no applied stresses at the upper and lower faces. Loading being independent of ydirection, the FGPM plate is in plane strain condition of elasticity. Comparison of present theory (PT) results of throughthickness variation in displacements and stresses evaluated at section x = 0.25a with analytical results given by Lu et al. [1] is illustrated for a sensory plate in Figure 2 and for an actuating plate in Figure 3. Present results match the exact result for both, the sensory and the actuating plates.
Table 1. Material Properties (^{a}Reference [1], ^{b}Reference [43] ^{c}Reference [44])
Material 
Properties 
PZT4^{a} 
C_{11 }= 139 (GPa), C_{13 }= 74.3, 
PVDF^{b} 
E_{1 }= 237 (GPa), E_{3 }= 10.5, 
Ni^{c} 
Y = 199.5 GPa, n = 0.3 
Al_{2}O_{3}^{c} 
Y = 393 GPa, n = 0.25 
PZT5A^{c} 
Y_{1} = 61 (GPa), Y_{2} = 61, Y_{3} = 53.2, 
It can be observed for the actuating plate in Figure 3(a) and (b) that change in material grading constant b does not affect inplane displacement (u) and transverse displacement (w) to any large extent. However, a considerable increase in the value of transverse normal stress (s_{z}) and transverse shear stress (t_{xz}) is observed (Figure 3(c) and (d)) with the increase in value of b. Thus, a grading stiff (b > 0) material may fail under large electric force. On the other hand, a grading soft (b < 0) material may effectively reduce the stresses under electric load. Figures 2 and 3 show that in sensory and actuating plates, the transverse displacement (w) does not vary linearly through the depth, as assumed in a few approximate 2D plate theories. Further, the grading soft material shows significant nonlinearity in w compared to the grading stiff material. Thus, for a grading soft FGPM plate, the assumption of linear variation in w may lead to a considerable error.






Fig. 2. Throughthickness variation in functionally gradient PZT4 sensory plate in
(a) inplane displacement, (b) transverse displacement, (c) transverse normal stress,
(d) transverse shear stress, (e) induced electric potential, (f) transverse electric displacement






Fig. 3. Throughthickness variation in functionally gradient PZT4 actuating plate in
(a) inplane displacement, (b) transverse displacement, (c) transverse normal stress,
(d) transverse shear stress, (e) applied electric potential, (f) transverse electric displacement
Example 2
A twolayered infinite simply supported FGPM/Homogeneous piezoelectric plate with overall thickness h in cylindrical bending is considered. The lower layer of thickness h_{1} is a homogeneous PZT4 piezoelectric material with constant material properties (given in Table 1) and the upper layer of thickness (h – h_{1}) is a PZT4 based FGPM. The elastic and electric properties in the upper FGPM layer vary as per the exponential law;
, , 
(12) 
where β = 1, 0.5, 0, 0.5, 1 is material gradient. The thickness of homogeneous layer h_{1 }is considered to be 0.2h and 0.8h. The laminate is subjected to two loading cases; mechanical singly sinusoidal load at the top surface with electric displacement D_{z }at top and bottom zero and electric singly sinusoidal load at the top surface with tractionfree faces. A comparison of throughthickness variation in stresses and displacements evaluated at a section x = 0.25a with exact solutions given by Lu et al. [1] is expressed in Figure 4 for the sensory plate and in Figure 5 for actuating plate.
Figure 4 shows that as the thickness of the FGPM layer decreases from 0.8h to 0.2h, the throughthickness variation curves of inplane displacement (u), transverse displacement (w), transverse normal stress (s_{z}) for grading stiff (b > 0) material and grading soft (b < 0) material converge to corresponding curves for homogeneous (b = 0) material. It can thus be observed that in the case of a sensory plate under mechanical loading, stresses and displacements may be restricted by using a suitable gradation factor, as well as by providing an appropriate thickness of the FGPM layer. However, in the case of actuating plates with an FGPM layer of 0.2h thickness (Figure 5), curves for the transverse normal stress (s_{z}) in grading stiff and grading soft materials do not show any convergence towards those with the homogeneous material, indicating that the gradation factor b plays a very vital role in actuating plate under electric load and that even a relatively thin layer of FGPM can have a distinct effect on stresses in the bilayered plate.
Example 3
A simply supported moderately thick (a/h = 10) beam made up of functionally graded PVDF is considered. The material properties are assumed to vary exponentially as;
, , 
(13) 
where the material grading constant β = 1, 0.5, 0, 0.5, 1. Reference values of ,, are taken from Helinger et al. [43] and given in Table 1. The sensory beam is subjected to sinusoidal mechanical load with unit intensity. The actuating beam is subjected to sinusoidal electric load, again with unit intensity. Values of the entities at salient points are given in Table 2 for the sensory beam and in Table 3 for actuating beam, which may be used as benchmark results.
Example 4
The beam in Example 3 is reinvestigated for its response to uniformly distributed mechanical and electric load applied at the top face. Fourier coefficients p_{0} and f_{0} in Eqs. 8 are taken as
where m = 1, 3, 5, … is varied. The solution is obtained by applying the RungeKutta Method of solving the IVP and convergence is observed to reach in about twenty iterations. Values of stresses and displacements at salient points are given in Table 4 for a sensory beam and in Table 5 for actuating beam for future reference.
Results indicate that inplane displacement (u) and transverse displacement (w) at the top surface in the case of grading stiff beam are considerably small as compared to grading soft beam. This is observed in actuating as well as sensory beam. However, as the stiff portion of the FGPM beam absorbs more stresses, the inplane stresses (s_{x}) at the top surface are much larger in grading stiff beam as compared to grading soft beam.






Fig. 4. Throughthickness variation in FGPM/homogeneous bilayer sensory plate in
(a) inplane displacement (h_{1}=0.2h), (b) inplane displacement (h_{1}=0.8h),
(c) transverse displacement (h_{1}=0.2h), (d) transverse displacement (h_{1}=0.8h),
(e) transverse normal stress (h_{1}=0.2h), (f) transverse normal stress (h_{1}=0.8h)





Fig. 5. Throughthickness variation in FGPM/homogeneous bilayer actuating plate in
(a) inplane displacement (h_{1}=0.2h), (b) inplane displacement (h_{1}=0.8h),
(c) transverse normal stress (h_{1}=0.2h), (d) transverse normal stress (h_{1}=0.8h),
(e) transverse shear stress (h_{1}=0.2h), (f) transverse shear stress (h_{1}=0.8h)
Table 2. Displacements and stresses in a simply supported sensory PVDF beam under sinusoidal mechanical loading
Entity 
z 
Gradation Constant b 

1 
0.5 
0 
0.5 
1 

u×10^{10} 
0 
1.207 
1.023 
0.8647 
0.7288 
0.6118 
0.5h 
0.229 
0.088 
0.0029 
0.058 
0.088 

h 
1.649 
1.193 
0.8594 
0.6171 
0.4421 

w×10^{9} 
0 
0.9842 
0.7637 
0.594 
0.4632 
0.3621 
0.5h 
0.9874 
0.7667 
0.5968 
0.4657 
0.3643 

h 
0.9824 
0.7633 
0.5945 
0.4642 
0.3634 

f ×10^{3} 
0.5h 
2 
1.6 
1.3 
0.9966 
0.7375 
t_{xz} 
0.5h 
4.6944 
4.7403 
4.7557 
4.7403 
4.6942 
s_{z} 
0.5h 
0.5636 
0.5318 
0.5 
0.4681 
0.4362 
D_{z}×10^{10} 
0 
0.2002 
0.1516 
0.0998 
0.0482 
0.0001 
0.5h 
0.2314 
0.1811 
0.1276 
0.0741 
0.02376 

h 
0.2553 
0.207 
0.1554 
0.1037 
0.05505 

s_{x} 
0 
31.7027 
44.3007 
61.7555 
85.815 
118.761 
0.5h 
7.9285 
5.7398 
1.723 
4.9053 
15.1802 

h 
43.3258 
51.6678 
61.3678 
72.6612 
85.8278 
Table 3. Displacements and stresses in a simply supported actuating PVDF beam under sinusoidal electric loading
Entity 
z 
Gradation Constant b 

1 
0.5 
0 
0.5 
1 

u×10^{11} 
0 
0.1621 
0.2249 
0.2924 
0.36 
0.4231 
0.5h 
0.219 
0.2183 
0.218 
0.2183 
0.219 

h 
0.4341 
0.3716 
0.3041 
0.2365 
0.1733 

w×10^{10} 
0 
0.0001 
0.0482 
0.0998 
0.1516 
0.2002 
0.5h 
0.0954 
0.0623 
0.0263 
0.0097 
0.0431 

h 
0.2553 
0.207 
0.1554 
0.1037 
0.055 

f 
0.5h 
0.3733 
0.4328 
0.4943 
0.5558 
0.6155 
t_{xz} 
0.5h 
0.0005 
0.0004 
0 
0.0009 
0.0015 
s_{z}×10^{3} 
0.5h 
0.3912 
0.7791 
1.1 
1.3 
1.1 
D_{z}×10^{8} 
0 
0.6493 
0.8598 
1.115 
1.418 
1.765 
0.5h 
0.6557 
0.8689 
1.128 
1.435 
1.789 

h 
0.6715 
0.8939 
1.167 
1.496 
1.883 

s_{x} 
0 
0.5749 
0.3509 
0.3685 
2.0541 
5.4925 
0.5h 
0.4352 
0.3935 
0.1823 
0.3577 
1.4922 

h 
0.1054 
0.2315 
0.372 
0.4787 
0.4619 
Table 4. Displacements and stresses in a simply supported sensory PVDF beam under uniformly distributed mechanical loading

Entity 
Location 
Gradation Factor b 

1 
0.5 
0 
0.5 
1 

u ×10^{10} 
+ h/2 
2.1305211 
1.5619067 
1.0758951 
0.8027623 
0.5814465 
w×10^{10} 
+ h/2 
12.6232173 
9.80410553 
7.64594844 
5.96345347 
4.66761745 
f ×10^{3} 
0 
3.09304359 
2.4978856 
2.0161351 
1.53476254 
1.13796579 
t_{xz} 
0 
7.10130203 
7.1779797 
7.2132 
7.15415914 
7.18174222 
s_{z} 
0 
1.43362666 
1.35486238 
1.2848 
1.14052316 
1.18774109 
D_{z} ×10^{11} 
+ h/2 
6.008076 
4.783087 
6.45538 
3.183497 
2.780888 
s_{x} 
 h/2 
42.8965 
59.8896 
83.3995 
115.6573 
160.1848 
0 
10.745 
7.7635 
2.3014 
6.6394 
20.6428 

h/2 
57.168 
83.5723 
35.2159 
106.0003 
143.5741 
Table 5. Displacements and stresses in a simply supported actuating PVDF beam
under uniformly distributed electric loading
Entity 
Location 
Gradation Factor b 

1 
0.5 
0 
0.5 
1 

u ×10^{11} 
+ h/2 
1.584256 
0.9602038 
10.690703 
1.1152712 
1.06158 
w ×10^{11} 
+ h/2 
6.388092 
5.68934 
2.77707 
4.47325 
3.019088 
f 
0 
0.8052 
0.9329 
1.0635 
1.1968 
1.3266 
t_{xz} 
h/4 
0.8876 
1.2576 
1.8416 
1.9147 
2.4401 
0 
0.0091 
0.1522 
0.7494 
0.3334 
1.0751 

+h/4 
1.6587 
2.4391 
2.3543 
5.0003 
5.5395 

s_{z} 
0 
0.8566981 
1.3580919 
2.8218 
1.8717 
8.8592 
D_{z} ×10^{8} 
+ h/2 
2.90026 
4.21218 
6.05078 
8.87531 
12.85177 
s_{x} 
 h/2 
0.8807 
3.553 
9.6994 
19.713 
40.7047 
0 
1.7939 
2.2672 
2.3797 
3.1374 
2.0641 

+ h/2 
34.5471 
14.6918 
4.3232 
47.4725 
53.3201 
Example 5
A simply supported hybrid beam of thickness h consisting of Ni and Al_{2}O_{3} bimaterial functionally gradient elastic substrate with a piezoelectric layer of PZT5A of thickness 0.1h bonded to its ceramicrich top surface is considered. The FGM substrate comprises a 100% metal layer of thickness 0.1h_{e} and nine perfectly bonded isotropic layers of equal thickness (Figure 6) with different material properties computed at the midsurface of the respective layers. The volume fractions V_{c} and V_{m} of the ceramic and metal are assumed to vary along the thickness using the power law as;
(14) 
M = 0.25, 4 is the inhomogeneity parameter. Effective Young's modulus for FGM is obtained as;
(15) 
where q = 4.5 GPa for NiAl_{2}O_{3}. Properties of Ni, Al_{2}O_{3,} and PZT5A are given in Table 1. For FGM substrate, piezoelectric stress constants e_{ij} in Eq. (9) are made zero.
Fig. 6. Hybrid Beam Configuration
Three aspect ratios S = a/h = 5, 10, 40 are investigated. The hybrid beam is subjected to the following two loading cases;
(16) 
(17) 
whereY_{0} = 199.5 GPa, d_{0} = 374×10^{12} CN^{1}.
Comparison of results obtained by the present model with exact theory and a few approximate theories like Zigzag theory (ZIGT), Consistent thirdorder theory (CTOT), Firstorder shear deformation theory (FSDT) given by Kapuria et al. [44] is shown in Table 6 for loading case 1 and Table 7 for loading case 2. It can be seen that the present model is performing quite efficiently.
Table 6. Exact 2D results and percentage error in Present theory, ZIGT, CTOT, FSDT results
in hybrid beam for M = 0.25 (^{a}Reference [44])
Entity 
S 
Load Case 1 
Load Case 2 

Exact^{a} 
Present 
ZIGT^{a} 
CTOT^{a} 
FSDT^{a} 
Exact^{a} 
Present 
ZIGT^{a} 
CTOT^{a} 
FSDT^{a} 

w 
5 
10.897 
0.68 
1.71 
1.3 
1.74 
0.77736 
0.244 
0.26 
0.51 
1.21 
10 
10.448 
0.38 
0.45 
0.34 
0.46 
0.77509 
0.054 
0.07 
0.13 
0.3 

40 
10.308 
0.08 
0.03 
0.02 
0.03 
0.77438 
0.006 
0 
0.01 
0.02 

10s_{x}^{e} 
5 
8.2616 
0 
1.03 
0.96 
0.62 
0.47717 
0.886 
0.87 
8.02 
1.98 
10 
8.2241 
0.18 
0.25 
0.31 
0.17 
0.47008 
0.206 
0.22 
2.03 
0.5 

40 
8.2112 
0.01 
0.02 
0.01 
0.01 
0.46786 
0.016 
0.01 
0.13 
0.03 

s_{x}^{p} 
5 
0.2048 
0.19 
5.16 
2.2 
3.16 
0.12517 
0.124 
0.14 
0.89 
0.27 
10 
0.20014 
0.44 
1.41 
0.65 
0.9 
0.12549 
0.004 
0.02 
0.24 
0.06 

40 
0.19876 
0.21 
0.24 
0.19 
0.21 
0.12559 
0.016 
0.02 
0.03 
0.02 

t_{xz} 
5 
52.463 
0.07 
0.26 
0.14 
0.36 
38.779 
0.314 
0.33 
1.07 
0.72 
10 
52.607 
0.04 
0.06 
0.03 
0.09 
38.99 
0.064 
0.08 
0.27 
0.18 

40 
52.65 
0.03 
0 
0 
0.01 
39.056 
0.006 
0.01 
0.02 
0.01 

f/D_{z} 
5 
55.039 
1.03 
1.08 
8.39 
8.06 
0.77318 
0.066 
0.05 
0.07 
0.12 
10 
88.986 
0.69 
1.66 
13 
12.5 
0.77204 
0.006 
0 
0.03 
0.02 

40 
99.635 
0.12 
0.09 
0.73 
0.4 
0.77169 
0.004 
0.02 
0.02 
0.02 
Table 7. Exact 2D results and percentage error in Present theory, ZIGT and CTOT results
in hybrid beam for M = 4 (^{a}Reference [44])
Entity 
S 
Load Case 1 
Load Case 2 

Exact^{a} 
Present 
ZIGT^{a} 
CTOT^{a} 
Exact^{a} 
Present 
ZIGT^{a} 
CTOT^{a} 

w 
5 
13.576 
0.786 
1.52 
1.07 
0.91689 
0.178 
0.18 
1 
10 
12.791 
0.134 
0.4 
0.27 
0.90965 
0.048 
0.05 
0.25 

40 
12.544 
0.04 
0.03 
0.02 
0.90738 
0.002 
0 
0.02 

10s_{x}^{e} 
5 
8.885 
0.024 
0.71 
0.53 
0.56323 
0.112 
0.51 
6.02 
10 
8.8007 
0.14 
0.17 
0.12 
0.55734 
0.102 
0.13 
1.52 

40 
8.7728 
0.024 
0.01 
0.01 
0.5555 
0.012 
0.01 
0.1 

s_{x}^{p} 
5 
0.24047 
1.674 
3.94 
1.8 
0.12256 
0.098 
0.1 
0.82 
10 
0.23491 
1.034 
1.1 
0.55 
0.12284 
0.008 
0.01 
0.22 

40 
0.23327 
0.164 
0.23 
0.19 
0.12293 
0.02 
0.02 
0.04 

t_{xz} 
5 
49.19 
0.024 
0.11 
0.09 
37.892 
0.298 
0.3 
1 
10 
49.35 
0.004 
0.03 
0.02 
38.089 
0.078 
0.08 
0.25 

40 
49.399 
0 
0 
0 
38.151 
0.008 
0.01 
0.02 

f /D_{z} 
5 
723.27 
0.154 
7.42 
65.5 
0.77447 
0.005 
0.05 
0.07 
10 
1060.7 
1.174 
1.24 
11.2 
0.77335 
0 
0 
0.03 

40 
1166.6 
0.004 
0.07 
0.63 
0.77301 
0.018 
0.02 
0.02 
A Semianalytical formulation for electromechanical analysis of simply supported FGPM laminate in cylindrical bending has been developed. The formulation is based upon elasticity theory with no simplifying assumption on stress and displacement fields. Solutions are obtained using numerical integration. The model is computationally inexpensive and versatile. By appropriate substitution of material property coefficients and the gradation law, it may be used for homogeneous, grading stiff, grading soft, and sandwich plates. The approach may be extended for clampedclamped BCs but not for arbitrary BCs. Results obtained by the present formulation are in very good agreement with the exact results. A few other results and observations have been noted for future reference.
Nomenclature
a,b,h 
Length, breadth, and depth of plate 
C^{E} 
Elasticity coefficients at constant electric field 
E_{1},E_{3} 
Elastic moduli in principal directions 
g^{S} 
Dielectric constants at constant strain 
n_{ij} 
Generalized Poisson’s ratios 
u,w 
x and z direction displacements 
s_{x},s_{z} 
Normal stress in x and z direction 
t_{xz} 
Shear stress in xz plane 
e_{x},e_{z} 
Normal strain in x and z direction 
g_{xz} 
Shear strain in xz plane 
C_{ij}, 
Material stiffness coefficients 
E_{x},E_{z} 
Electric field intensities in x and z directions 
e_{ij}, 
Piezoelectric constants 
g_{ii}, 
Dielectric constants 
D_{x},D_{z} 
Electric displacements in x and z directions 
B_{x},B_{z} 
Body force intensities in x and z directions 
Appendix
The vectors and matrices in Eqs. (2) are as follows;
Stress vector; 
(18) 
Strain vector; 
(19) 
Stiffness matrix at the constant electric field;
(20) 
in which the reduced material coefficients C_{ij }in plane stress condition of elasticity are;
, , , 
(21) 
and inplane strain condition of elasticity;
,,; , 
(22) 
Piezoelectric stress constants matrix due to Cady [45] and dielectric constant matrix at constant strain due to Tzau and Pandita [46] are respectively;
(23) 

(24) 
The electric field intensity vector and electric displacement vector are respectively;
(25) 

(26) 
The electroelastic coefficients Q_{11} – Q_{66 }in Eqs. (9) are as below;
, , , , , , , , , , , , , , 
(27) 
References
[1] Lu P., Lee H.P. and Lu C., 2005. An exact solution for functionally graded piezoelectric laminates in cylindrical bending. International Journal of Mechanical Sciences 47(3), pp.437458. DOI: 10.1016/j.ijmecsci.2005.01.012
[2] Lu P., Lee H.P. and Lu C., 2006. Exact solution for simply supported functionally graded piezoelectric laminates by Strohlike formalism. Composite Structures 72(3), pp.352363. DOI: 10.1016/j.compstruct.2005.01.012
[3] Xiang H.J. and Shi Z.F., 2009. Static analysis for functionally graded piezoelectric actuators or sensors under a combined electrothermal load. European Journal of MechanicsA/Solids 28(2), pp.338346. DOI: 10.1016/j.euromechsol.2008.06.007
[4] Mikaeeli S. and Behjat B., 2016. Threedimensional analysis of thick functionally graded piezoelectric plate using EFG method. Composite Structures 154, pp.591599. DOI: 10.1016/j.compstruct.2016.07.067
[5] Kulikov G. M. and Plotnikova S. V., 2013. A new approach to threedimensional exact solutions for functionally graded piezoelectric laminated plates. Composite Structures, 106, pp.33–46. DOI: 10.1016/j.compstruct.2013.05.037
[6] Lim C.W. and He L.H., 2001. Exact solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting. International Journal of Mechanical Sciences 43, pp.24792492. DOI: 10.1016/S00207403(01)000595
[7] Reddy J.N. and Cheng Z.Q., 2001. Threedimensional thermomechanical deformations of functionally graded rectangular plates. European Journal of Mechanics  A/Solids, 20(5), pp.841855. DOI:10.1016/S09977538(01)011743
[8] Zhong Z. and Shang E.T., 2005. Exact analysis of simply supported functionally graded piezothermoelectric plates. Journal of Intelligent Material Systems and Structures 16(78), pp.643651. DOI: 10.1177/1045389X05050530.
[9] Almajid A., Taya M. and Hudnut S., 2001. Analysis of out ofplane displacement and stress field in a piezocomposite plate with functionally graded microstructure. International Journal of Solids and Structures, 38(19), pp.3377–3391. DOI:10.1016/S00207683(00)00264X.
[10] Joshi S., Mukherjee A. and Schmauder S., 2003. Numerical characterization of functionally graded active materials under electrical and thermal fields. Smart Materials and Structures, 12(4), pp.571–579. DOI:10.1088/09641726/12/4/309.
[11] Taya M., Almajid A., Dunn M. and Takahashi H., 2003. Design of bimorph piezocomposite actuators with functionally graded microstructure. Sensors and Actuators–A/ Physical, 107(3), pp.248–260. DOI: 10.1016/S09244247(03)003819.
[12] Zhong Z. and Yu T., 2007. Electroelastic analysis of functionally graded piezoelectric material beams. Journal of Intelligent Material Systems and Structures, 19(6), pp.707–713. DOI: 10.1177/1045389X07079453.
[13] Wu D., Gao W., Hui D., Gao K. and Li K., 2018. Stochastic static analysis of EulerBernoulli type functionally graded structures. Composites Part B/Engineering, 134, pp.69–80. DOI: 10.1016/j.compositesb.2017.09.050.
[14] Loja M. A. R., Mota Soares C. M. and Barbosa J. I., 2013. Analysis of functionally graded sandwich plate structures with piezoelectric skins, using Bspline finite strip method. Composite Structures, 96, pp.606–615. DOI: 10.1016/j.compstruct.2012.08.010.
[15] Li Y. S., Feng W. J. and Cai Z. Y. 2014. Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory. Composite Structures, 115, pp.41–50. DOI: 10.1016/j.compstruct.2014.04.005.
[16] Chuaqui T. R. C. and Roque C. M. C., 2017. Analysis of functionally graded piezoelectric Timoshenko smart beams using a multiquadric radial basis function method. Composite Structures, 176, pp.640–653. DOI: 10.1016/j.compstruct.2017.05.062.
[17] Nourmohammadi H. and Behjat B., 2019. Static analysis of functionally graded piezoelectric plates under Electrothermomechanical loading using a meshfree method based on RPIM. Journal of Stress Analysis, 4(2), pp.93–106.
DOI: 10.22084/JRSTAN.2020.20850.1125.
[18] Behjat B. and Khoshravan M. R., 2012. Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates. Composite Structures, 94(3),pp.874–882. DOI: 10.1016/j.compstruct.2011.08.024
[19] Raissi H., 2020. Stress analysis in adhesive layers of a fivelayer circular sandwich plate subjected to temperature gradient based on layerwise theory. Mechanics Based Design of Structures and Machines, pp.1–27. DOI: 10.1080/15397734.2020.1776619.
[20] Raissi H., Shishehsaz M. and Moradi S., 2019. Stress distribution in a fivelayer sandwich plate with FG face sheets using layerwise method. Mechanics of Advanced Materials and Structures, 26(14), pp.1234–1244. DOI: 10.1080/15376494.2018.1432796.
[21] Raissi H., Shishehsaz M. and Moradi S., 2020. Stress analysis of the fivelayer circular sandwich plate subjected to uniform distributed load by layerwise theory along with second order shear deformation theory. Australian Journal of Mechanical Engineering, pp.1–12. DOI: 10.1080/14484846.2020.1733170.
[22] Song D and Luo N, 2012, Wave propagation and transient response of a functionally graded material plate under a point impact load in thermal environment. Applied Mathematical Modelling, 36, pp.444462. DOI: 10.1016/j.apm.2011.07.023
[23] Mazzotti M. Bartoli I. Miniaci M. Marzani A. 2016. Wave dispersion in thinwalled orthotropic waveguides using the first order shear deformation theory. ThinWalled Structures, 103, pp.128140. DOI: 10.1016/j.tws.2016.02.014
[24] Li C.L., Han Q., Wang Z., Wu X., 2020, Analysis of wave propagation in functionally graded piezoelectric composite plates reinforced with graphene platelets. Applied Mathematical Modelling, 81, pp.487–505. DOI: 10.1016/j.apm.2020.01.016
[25] Li C.L. and Han Q., 2020. Semianalytical wave characteristics analysis of graphenereinforced piezoelectric polymer nanocomposite cylindrical shells. International Journal of Mechanical Sciences, 186:105890. DOI: 10.1016/j.ijmecsci.2020.105890
[26] Li C.L., Han Q., Y. J. Liu D. L. Xiao, 2017. Guided wave propagation in rotating functionally graded annular plates. Acta Mechanica, 228, pp.10831095. DOI: 10.1007/s0070701617529.
[27] Vinh P.V. and Tounsi A., 2022. Free vibration analysis of functionally graded doubly curved nanoshells using nonlocal firstorder shear deformation theory with variable nonlocal parameters. ThinWalled Structures,174. DOI: 10.1016/j.tws.2022.109084.
[28] Tahir S.I., Tounsi A., Chikh A., AlOsta M., AlDulaijan S.U. and AlZahrani M.M., The effect of threevariable viscoelastic foundation on the wave propagation in functionally graded sandwich plates via a simple quasi3D HSDT. Steel and Composite Structures 42(4), pp.501511. DOI:10.12989/scs.2022.42.4.501.
[29] Rachid A., Ouinas D., Lousdad A., Zaoui F.Z., Achour B., Gasmi H., Butt T.A., Tounsi A., 2022. Mechanical behavior and free vibration analysis of FG doubly curved shells on elastic foundation via a new modified displacements field model of 2D and quasi3D HSDTs. ThinWalled Structures, 172. DOI: 10.1016/j.tws.2021.108783.
[30] Habib H., Abdelbaki C., Bousahla A.A., Bourada F., 2022. Effect of the variable viscoPasternak foundations on the bending and dynamic behaviors of FG plates using integral HSDT model. Geomechanics and Engineering 28(1), pp.4964. DOI: 10.12989/gae.2022.28.1.049.
[31] Bouafia K., Selim M. M., Bourada F., Bousahla A. A., Bourada M., Tounsi A., Bedia E. A. A., Tounsi A., 2021. Bending and free vibration characteristics of various compositions of FG plates on elastic foundation via quasi 3D HSDT model. Steel and Composite Structures, 41 (4), pp. 487503. DOI: 10.12989/scs.2021.41.4.487.
[32] Zaitoun M. W., Chikh A., Tounsi A., AlOsta M. A., Sharif A., AlDulaijan S. U., AlZahrani M. M., 2022. Influence of the viscoPasternak foundation parameters on the buckling behavior of a sandwich functional graded ceramic–metal plate in a hygrothermal environment. ThinWalled Structures, 170. DOI: 10.1016/j.tws.2021.108549.
[33] Mudhaffar I. M.,^{ }Tounsi A., Chikh A.,^{ }AlOsta M. A.,^{ }AlZahrani M. M.^{ }AlDulaijan S. U., 2021. Hygrothermomechanical bending behavior of advanced functionally graded ceramic metal plate resting on a viscoelastic foundation. Structures, 33, pp.21772189. DOI: 10.1016/j.istruc.2021.05.090.
[34] Kouider D, Kaci A, Selim M. M., Bousahla A. A., Bourada F., Tounsi, A., Tounsi A., Hussain M., 2021, An original fourvariable quasi3D shear deformation theory for the static and free vibration analysis of new type of sandwich plates with both FG face sheets and FGM hard core. Steel and Composite Structures, 41 (2), pp.167191. DOI: 10.12989/scs.2021.41.2.167
[35] Merazka B., Bouhadra A., Menasria A., Selim M. M., Bousahla A. A., Bourada F., Tounsi A., Benrahou K.H., Tounsi A. and AlZahrani M. M., 2021. Hygrothermomechanical bending response of FG plates resting on elastic foundations. Steel and Composite Structures, 39 (5), pp.631643.
DOI: 10.12989/scs.2021.39.5.631.
[36] Hachemi H., Bousahla A. A., Kaci A., Bourada F., Tounsi A., Benrahou K. H., Tounsi A., AlZahrani M. M. Mahmoud S. R., 2021. Bending analysis of functionally graded plates using a new refined quasi3D shear deformation theory and the concept of the neutral surface position. Steel and Composite Structures, 39 (1), pp.5164. DOI: 10.12989/scs.2021.39.1.051
[37] Bakoura A., Bourada F., Bousahla A. A., Tounsi A., Benrahou K. H., Tounsi A., AlZahrani M. M., Mahmoud S. R., 2021. Buckling analysis of functionally graded plates using HSDT in conjunction with the stress function method. Computers and Concrete, 27 (1), pp. 7383.DOI: 10.12989/cac.2021.27.1.073.
[38] Tiersten H.F., 1969. Linear piezoelectric plate vibrations. Plenum Press New York.
[39] Maxwell J.C., 1865. A dynamical theory of the electromagnetic field. Royal Society Transactions 155, pp.459512. DOI: 10.1098/rstl.1865.0008
[40] Kantorovich L.V. and Krylov V.I., 1964. Approximate methods of higher analysis. Interscience Publishers, Inc. New York.
[41] Kant T. and Ramesh C.K., 1981. Numerical integration of linear boundary value problems in solid mechanics by segmentation method. International Journal of Numerical Methods in Engineering 17, pp.12331256. DOI: 10.1002/nme.1620170808
[42] Sawarkar S.S., Pendhari S.S. and Desai Y.M. 2016. Semianalytical solutions for static analysis of piezoelectric laminates. Composite Structures 153, pp.242252. DOI: 10.1016/j.compstruct.2016.05.106.
[43] Heylinger P.R., Ramirez G. and Pei K.C., 1994. Discrete layer piezoelectric plate and shell models for active tipclearance control. NASA Contractor Report, 195383.
[44] Kapuria S., Bhattacharyya M. and Kumar A.N., 2006. Assessment of coupled 1D models for hybrid piezoelectric layered functionally graded beams. Composite Structures 72(4), pp.455468. DOI: 10.1016/j.compstruct.2005.01.015
[45] Cady W.G., 1946. Piezoelectricity I & II. Dover Publications New York.
[46] Tzau H.S. and Pandita S., 1987. A multipurpose dynamic and tactile sensor for robot manipulators. Journal of Robotic Systems 4, pp.719741.