- Original Article
- Open Access
Transient bending analysis of a functionally graded circular plate with integrated surface piezoelectric layers
- Ali Asghar Jafari^{1},
- Ali Akbar Jandaghian^{2} and
- Omid Rahmani^{2}Email author
https://doi.org/10.1186/s40712-014-0008-5
© Jafari et al.; licensee Springer 2014
- Received: 24 February 2014
- Accepted: 1 July 2014
- Published: 16 September 2014
Abstract
Background
Thin and piezoelectric materials are widely used as sensors or actuators in smart structures by embedding or surface-mounted them.
Methods
This paper report on the exact, explicit solution for the transient bending of a circular functionally graded (FG) plates integrated with two uniformly distributed actuator layers made of piezoelectric material based on the classical plate theory (CPT). The material properties of the FG substrate plate are assumed to be graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents and the distribution of electric potential field along the thickness direction of piezoelectric layers is simulated by a quadratic function. The form of the electric potential field in the piezoelectric layer is considered such that the Maxwell static electricity equation is satisfied. The governing equations are solved for clamped and simply supported edge boundary condition of the circular plate. The solutions are expressed by elementary Bessel functions and derived via exact inverse Laplace transform.
Results and Conclusions
It is seen that the power index (g) and thickness of piezo-layer have significant effect on the deflection amplitude and natural frequency of piezo-FG plate.
Keywords
- Functionally graded material
- Piezoelectric
- Transient bending
- Elasticity solution
Background
The active and controllable mechanical properties of piezoelectric materials are extensively recognized as one of the most important resources for the development of intelligent self-monitoring and self-adaptive lightweight structures (Rahmani and Noroozi Moghaddam 2014). Adaptive structures, incorporating piezoelectric patches for sensing and actuation, are now broadly used in the field of active and passive vibration and shape control, in medical instruments, in measuring apparatus, and in micro-electromechanical systems (Jandaghian et al. 2013). A metal substrate surface bonded or embedded by a piezoelectric layer has received considerable attention throughout the last decades for practical designs of sensors and actuators because of the electromechanically coupling characteristics (Jandaghian et al. 2014).
Recently, functionally graded (FG) materials which exhibit smooth variation of material properties (Rahmani and Pedram 2014) have been studied for developing smart FG structures (Rahmani et al. 2010). By utilizing piezoelectric materials as actuators or sensors, smart FG structures have been made with capabilities of self-controlling and self-monitoring. The design of devices including active piezoelectric materials requires, as an initial step, an efficient modeling of the electrical, mechanical, and coupling properties of the host structure, the piezoelectric elements, and their interactions. In the following, a review of recent findings and developments in modeling of smart FG structures will be presented. Pan and Han (2005) presented an exact solution for multilayered rectangular plate made of functionally graded, anisotropic, and linear magneto-electro-elastic materials. In this work, the influence of the exponential factor, the magneto-electro-elastic properties, and loading types on induced magneto-electric-elastic fields have been investigated. Batra (Batra and Liang 1997; Vel and Batra 2001) investigated the vibration behavior of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators with a piezoelectric plate subjected to transient thermal loading. Batra and Geng (2002) considered a FG viscoelastic layer but a homogeneous PZT constraining layer and performed the three-dimensional transient analysis of the problem with the finite element method (FEM). He et al. (2001) suggested a finite element formulation based on the classical laminated plate (CLP) theory for the shape and vibration control of the FG material plates with integrated piezoelectric sensors and actuators and used a constant velocity feedback control algorithm for the active control of the dynamic response of the plate through closed-loop control. Reddy and Cheng (2001) studied the bending of a FG rectangular plate with an attached piezoelectric actuator. They have employed the transfer matrix and asymptotic expansion techniques to obtain a three-dimensional asymptotic solution. Bhangale and Ganesan (2006) studied the static behavior of functionally graded, anisotropic, and linear magneto-electro-elastic plates, using a semi-analytical finite element plate model. In this study, the material constitution was assumed to vary in an exponential way across the thickness direction. A FE formulation was suggested by Liew et al. (2002) for modeling and controlling piezoelectric shell laminates under coupled displacement, temperature, and electric potential fields. The base shell was of FG material type that consists of combined ceramic-metal materials with different mixing ratios of the ceramic and metal constituents. Ray and Sachade (2006) presented a finite element model for the static analysis of FG plates with a layer of piezoelectric fiber reinforced composite material and investigated the effect of varying the fiber angle in the piezoelectric fiber reinforced composite layer on its actuating capability of the functionally graded plates. Ootao and Tanigawa (Ootao and Tanigawa 2000; Tanigawa 2001) studied the FGM simply supported rectangular plate bonded to a piezoelectric material. A study on the nonlinear vibration and dynamic response of a FG material plate with surface-bonded piezoelectric layers in thermal environments was developed by Huang and Shen (2006). The nonlinear formulations were based on the higher-order shear deformation plate theory (HSDT) including thermo-piezoelectric effects. In this study, they accounted to heat conduction and temperature-dependent material properties, and a variation through the plate thickness both for the temperature field and for the electrical field intensity was assumed. Wang and Noda (2001) investigated a smart FG composite structure composed of a metal layer, piezoelectric layer, and a FG layer in between.
A study on FG beams with surface-integrated piezoelectric actuators and sensors based on a state space formulation was carried out by Bian et al. (2006). In their study, a spring layer modeled the bonding adhesive between the host beam and the piezoelectric layers in order to consider its effect. The bonding conditions were simulated through the consideration of different spring layer parameters. Free axisymmetric vibration problem of piezoelectric coupled thin circular (Ebrahimi and Rastgo 2008) and thin annular (Ebrahimi and Rastgoo 2008) FGM plates has been carried out by Ebrahimi and Rastgoo. Also, Ebrahimi et al. suggested an analytical solution to the analysis of smart moderately thick shear deformable annular (Ebrahimi et al. 2009) and circular (Ebrahimi et al. 2008) FG plate based on Mindlin's plate theory. Wang et al. (2001) studied free vibration of a circular plate surface bonded by two piezoelectric layers, based on the Kirchhoff theory. They have shown that the mode shape of the electric potential obtained from free vibration analysis is generally to be nonuniform in the radial direction in contrast to what is commonly assumed. Zhang et al. present an exact, explicit solution to study the static and transient bending characteristics of a thin piezoelectric circular plate under electromechanical loading, grounded over the whole surface and built-in or simply supported at the edge (Zhang et al. 2004). A study on the performance of vertically reinforced 1-3 piezoelectric composite distributed actuator in the active constrained layer damping system bonded to a FG plate was carried out by Ray and Batra (2007). They modeled the deformations of each layer by the first-order shear deformation theory. Hosseini-Hashemi et al. suggested an analytical solution to the investigation and vibration of thick plates. They studied free vibration of piezoelectric coupled thick circular/annular FG plates with different boundary conditions based on Reddy's third-order shear deformation theory (TSDT) (Hosseini Hashemi et al. 2010), thick circular isotropic plate (Hosseini-Hashemi et al. 2010b), and smart Reddy plate (Hosseini-Hashemi et al. 2010a).
In another study, Kargarnovin et al. (2007) investigated the active vibration control of FG material plates using piezoelectric sensor/actuator patches, using CLP theory. In their work, the effect of the feedback gain and the volume fraction on the plate frequency and displacement was studied. Es’haghi et al. (2011) presented an analytical solution for vibration study of piezoelectric coupled FG Mindlin plates which have open-circuit piezoelectric patches and have been used as sensors. Alibeigloo (2010) presented an analytical solution for FG beams integrated with orthotropic piezoelectric actuator and sensor, under an applied electric field and thermo-mechanical load. Studies on the effect of the grading index, thickness ratio, the electromechanical coupling, and thermo-mechanical surface boundary condition on the bending behavior of the structure were analyzed. Static analysis of functionally graded, transversely isotropic, and magneto-electro-elastic circular plate under uniform mechanical load was discussed by Li et al. (2008). In this investigation, they represented displacements and electric potential by appropriate polynomials in the radial coordinate. Shiyekar and Kant (2010) investigated the bidirectional flexure behavior of FG plates with integrated piezoelectric fiber reinforced composites. They considered the HSDT theory to analyze the performance of this plate subjected to electromechanical loadings. The electrostatic potential was assumed as layer-wise linear through the thickness of the piezoelectric layer. Shen (2005) presented a postbuckling analysis for a simply supported and functionally graded plate with piezoelectric actuators subjected to the combined action of mechanical, electrical, and thermal loads. The governing equations in his study were based on a thermo-piezoelectric higher-order shear deformation plate theory. Fakhari et al. (2011) suggested a FE formulation based on the HSDT plate theory to investigate the nonlinear natural frequencies, time, and frequency responses of FG plate with surface-bonded piezoelectric layers under thermal, electrical, and mechanical loads. Numerical results have been presented to study the effects of the volume fraction exponent, the applied voltage in piezoelectric layers, the thermal load, and the vibration amplitude on nonlinear natural frequencies, and time response of the plate with integrated piezoelectric layers was studied. Hashemi et al. (2012) developed an analytical solution for the free vibration of piezoelectric coupled FG thick circular/annular plates on the basis of Mindlin's first-order shear deformation plate theory (FSDT) and studied the effects of coupling between in-plane and transverse displacements on the frequency parameters. Komeili et al. (2011) carried out the bending of problem of beams made of FG piezoelectric materials under a combined thermo-electro-mechanical load. They considered different shear deformation theories to perform a parametric study focused on the evaluation of the effect of the material composition, voltage, end supports, shear deformation, and the slenderness ratio on the thermo-electro-mechanical performance. Loja et al. (2013) studied the static and free vibration behavior of functionally graded sandwich plate-type structures, using B-spline finite strip element models based on different shear deformation theories. Geometrical nonlinear static and free vibration analyses of FG piezoelectric plates using FEM were studied by Behjat and Khoshravan (2012). On their work, different sets of mechanical and electrical loadings were considered. The plate with FG piezoelectric material was considered to vary gradually through the thickness by a power law distribution. The electric potential was assumed to vary in a quadratic way through the thickness and was considered to be a nodal degree of freedom. Liew et al. (2003) presented a FE model for the static and dynamic piezothermoelastic analysis and control of FGM plates under temperature gradient environments using integrated piezoelectric sensor/actuator layers. They also applied a feedback control algorithm that couples the direct and inverse piezoelectric effects to provide active control of the integrated FGM plate in a closed-loop system.
To the best knowledge of authors and from carried out literature review, there are no studies on the transient motion of FG circular plate integrated with piezoelectric material. In this work, the transient bending of a thin, circular FGM plate integrated with piezoelectric layers under axisymmetric mechanical loading is presented for different boundary conditions. The piezoelectric layers are electrically grounded over the edge, and electrodes at the two surfaces of the piezoelectric layers are shortly connected. The properties of the substrate plates were graded in the thickness direction according to a volume fraction power law distribution. A consistent formulation based on the classical plate theory that satisfies the Maxwell static electricity equation is proposed for the piezoelectric layers. The distribution of electric potential field in the thickness direction of the piezoelectric layers is simulated by a quadratic function, and the solutions are presented in terms of a single, elementary Bessel function. In the following, the explicit time history of the solution is achieved by precise inverse Laplace transformation. Consequently, accurate results of system characteristics can simply be acquired applying straightforward numerical process.
Methods
In this study, according to the ratio of the plate radius to its thickness, the Kirchhoff assumption for thin plates is applicable, whereby the shear deformation and rotary inertia can be omitted. In addition, the axisymmetric applied external load is assumed.
Equilibrium equations
\( {D}_1={\displaystyle \underset{-h}{\overset{h}{\int }}\frac{z^2E(z)}{1-{\nu}^2} dz},\kern0.5em {D}_2=\frac{2}{3}{h}_1\left(3{h}^2+3h{h}_1+{h}_1^2\right){C}_{11}^E \) and \( \overline{\rho}=\frac{1}{2h}{\displaystyle {\int}_{-h}^h{\rho}_1(z) dz}. \)
In the next section, the solution of two coupled partial differential Equations 36 and 38 will be discussed according to the following mechanical and electrical boundary conditions and symmetry constraints for the simply supported and fully grounded piezoelectric coupled FG plate.
Analytical solution procedure
The results of transient bending analysis are given below for two boundary conditions, corresponding to the simply supported and clamped circular plate.
Simply supported circular plate
where, in Equation 39, \( \dot{w}=\frac{\partial w}{\partial t} \) and \( \dot{\phi}=\frac{\partial \varphi }{\partial t} \).
It should be noted that the solutions w^{ * }(r) and φ^{*}(r) in Equations 47 and 48 satisfy Equations 43 and 44 and all initial and boundary conditions in the problem.
The final solutions for the bending moments M_{ r }(r,t) and M_{ θ }(r,t) are obtained introducing Equation 57 into Equations 32 and 33.
Clamped circular plate
The inverse Laplace transform is applied to w^{*}(r,s) and φ^{*}(r,s)to derive the final solution in the time domain. The inversions of A^{′}_{ j }(s) and B′_{ j }(s) in Equations 69 and 67 are simply derived using standard Laplace transform tables, e.g., (Spiegel 1965).
Note that for j = 1,2,…, β_{ j } is negative, and as a result, α_{ j }^{4} − β_{ j }^{ 4 } in Equation 70 is always greater than zero, making A^{′}_{ j } always bounded.
Results and discussion
Material properties of the piezoelectric coupled FGM plate (Ebrahimi and Rastgo 2008 )
Property | Value |
---|---|
Host FG plate | |
E_{c} = 205 (GPa) | ρ_{c} = 7,800 (kg/m^{3}) |
E_{m} = 200 (GPa) | ρ_{m} = 8,900 (kg/m^{3}) |
PZT | |
\( {C}_{11}^E = 132 \times {10}^9 \) | \( {C}_{12}^E = 71 \times {10}^9 \) |
\( {C}_{33}^E = 115 \times {10}^9 \) | \( {C}_{13}^E = 73 \times {10}^9 \) |
\( {\overline{C}}_{11}^E=85.7 \times {10}^9 \) | \( {\overline{C}}_{12}^E = 24.7 \times {10}^9 \) |
e_{31} = −4.1 (C/m^{2}) | e_{33} = 14.1 (C/m^{2}) |
\( {\overline{e}}_{31}=-13.05\ \left({\mathrm{C}/\mathrm{m}}^2\right) \) | |
Ξ_{11} = 7.124 × 10^{− 9} (F/m) | Ξ_{33} = 5.841 × 10^{− 9} (F/m) |
ρ_{2} = 7,500 (kg/m^{3}) |
The first three mode resonance frequencies (Hz) for piezo-FG plate for various values of power index
Power index (g) | Mode number | Present (Hz) | (Ebrahimi and Rastgo2008) | Difference (%) | (Wang et al.2001) |
---|---|---|---|---|---|
0 | 1 | 69.37 | 69.52 | 0.22 | 69.33 |
2 | 418.99 | 417.86 | 0.27 | 418.03 | |
3 | 1029.74 | 1039 | −0.9 | 1,038.2 | |
1 | 1 | 67.51 | 67.88 | −0.55 | - |
2 | 408.38 | 406.83 | 0.38 | - | |
3 | 1,003.65 | 1,001.51 | 0.21 | - | |
3 | 1 | 66.49 | 67.07 | −0.87 | - |
2 | 403.01 | 402.23 | 0.19 | - | |
3 | 990.45 | 983.5 | 0.19 | - | |
5 | 1 | 66.17 | 66.81 | −0.97 | - |
2 | 401.33 | 400.71 | 0.15 | - | |
3 | 986.33 | 982.2 | 0.42 | - | |
7 | 1 | 66.02 | 66.72 | −1.06 | - |
2 | 400.55 | 397.32 | 0.81 | - | |
3 | 984.42 | 980.4 | 0.41 | - | |
9 | 1 | 65.94 | 66.61 | −1.02 | - |
2 | 400.12 | 396.81 | 0.83 | - | |
3 | 983.36 | 979.02 | 0.44 | - | |
10 | 1 | 65.91 | 66.59 | −1.03 | - |
2 | 399.67 | 396.5 | 0.79 | - | |
3 | 982.98 | 978.1 | 0.5 | - |
Effect of piezoelectric layer on resonance frequencies
Thickness ratio | First mode | Second mode | Third mode |
---|---|---|---|
1/12 | 68.65 | 417.62 | 1,026.62 |
1/10 | 69.52 | 418.99 | 1,029.74 |
1/8 | 71.52 | 420.96 | 1,034.82 |
1/5 | 73.68 | 428.44 | 1,052.90 |
Conclusion
In this paper, the transient bending of a three-layer piezoelectric circular FG plate based on the Kirchhoff plate theory is studied. The material properties of the FG plate are assumed to vary through the thickness of the plate based on the power law distribution. Also the distribution of electric potential in the piezoelectric layers follows a quadratic function in short circuited form. The solution is presented in terms of elementary Bessel functions and analytical inverse Laplace transforms. The solution enables the efficient determination of the plate characteristics anywhere within the plate for small as well as large times with reliable precision. It can be observed that the power index (g) has significant effect on the deflection amplitude and natural frequency of the piezo-FG plate. It is shown that the thickness of the piezo-layer has significant effect on the deflection amplitude and natural frequency of the piezo-FG plate.
Declarations
Authors’ Affiliations
References
- Abramowitz, M, & Stegun, IA. (1964). Handbook of Mathematical Functions: With Formulars, Graphs, and Mathematical Tables (Vol. 55). New York: Courier Dover Publications.Google Scholar
- Alibeigloo, A. (2010). Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers. Composite Structures, 92(6), 1535–1543.View ArticleGoogle Scholar
- Batra, R, & Geng, T. (2002). Comparison of active constrained layer damping by using extension and shear mode piezoceramic actuators. Journal of Intelligent Material Systems and Structures, 13(6), 349–367.View ArticleGoogle Scholar
- Batra, R, & Liang, X. (1997). The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators. Computers & Structures, 63(2), 203–216.MATHView ArticleGoogle Scholar
- Behjat, B, & Khoshravan, M. (2012). Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates. Composite Structures, 94(3), 874–882.View ArticleGoogle Scholar
- Bhangale, RK, & Ganesan, N. (2006). Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates. International Journal of Solids and Structures, 43(10), 3230–3253.MATHView ArticleGoogle Scholar
- Bian, Z, Lim, C, & Chen, W. (2006). On functionally graded beams with integrated surface piezoelectric layers. Composite Structures, 72(3), 339–351.View ArticleGoogle Scholar
- Ebrahimi, F, & Rastgo, A. (2008). An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Structures, 46(12), 1402–1408.View ArticleGoogle Scholar
- Ebrahimi, F, & Rastgoo, A. (2008). Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers. Smart Materials and Structures, 17(1), 015044.Google Scholar
- Ebrahimi, F, Rastgoo, A, & Kargarnovin, M. (2008). Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers. Journal of Mechanical Science and Technology, 22(6), 1058–1072.View ArticleGoogle Scholar
- Ebrahimi, F, Rastgoo, A, & Atai, A. (2009). A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate. European Journal of Mechanics-A/Solids, 28(5), 962–973.MATHView ArticleGoogle Scholar
- Es’haghi, M, Hashemi, SH, & Fadaee, M. (2011). Vibration analysis of piezoelectric FGM sensors using an accurate method. International Journal of Mechanical Sciences, 53(8), 585–594.View ArticleGoogle Scholar
- Fakhari, V, Ohadi, A, & Yousefian, P. (2011). Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment. Composite Structures, 93(9), 2310–2321.View ArticleGoogle Scholar
- Hashemi, SH, Khorshidi, K, Es’haghi, M, Fadaee, M, & Karimi, M. (2012). On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates. Applied Mathematical Modelling, 36(3), 1132–1147.MATHMathSciNetView ArticleGoogle Scholar
- He, X, Ng, T, Sivashanker, S, & Liew, K. (2001). Active control of FGM plates with integrated piezoelectric sensors and actuators. International Journal of Solids and Structures, 38(9), 1641–1655.MATHView ArticleGoogle Scholar
- Hosseini Hashemi, S, Es’haghi, M, & Karimi, M. (2010). Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers. International Journal of Mechanical Sciences, 52(3), 410–428.View ArticleGoogle Scholar
- Hosseini-Hashemi, S, Es’haghi, M, & Rokni Damavandi Taher, H. (2010a). An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory. Composite Structures, 92(6), 1333–1351.View ArticleGoogle Scholar
- Hosseini-Hashemi, S, Es’haghi, M, Rokni Damavandi Taher, H, & Fadaie, M. (2010b). Exact closed-form frequency equations for thick circular plates using a third-order shear deformation theory. Journal of Sound and Vibration, 329(16), 3382–3396.View ArticleGoogle Scholar
- Huang, X-L, & Shen, H-S. (2006). Vibration and dynamic response of functionally graded plates with piezoelectric actuators in thermal environments. Journal of Sound and Vibration, 289(1), 25–53.View ArticleGoogle Scholar
- Jandaghian, A, Jafari, A, & Rahmani, O. (2013). Exact solution for Transient bending of a circular plate integrated with piezoelectric layers. Applied Mathematical Modelling, 37(12), 7154–7163.MathSciNetView ArticleGoogle Scholar
- Jandaghian, A, Jafari, A, & Rahmani, O. (2014). Vibrational response of functionally graded circular plate integrated with piezoelectric layers: An exact solution. Engineering Solid Mechanics, 2(2), 119–130.View ArticleGoogle Scholar
- Kargarnovin, M, Najafizadeh, M, & Viliani, N. (2007). Vibration control of a functionally graded material plate patched with piezoelectric actuators and sensors under a constant electric charge. Smart Materials and Structures, 16(4), 1252.Google Scholar
- Komeili, A, Akbarzadeh, AH, Doroushi, A, & Eslami, MR. (2011). Static analysis of functionally graded piezoelectric beams under thermo-electro-mechanical loads. Advances in Mechanical Engineering, 2011, 1–10.Google Scholar
- Li, X, Ding, H, & Chen, W. (2008). Three-dimensional analytical solution for functionally graded magneto–electro-elastic circular plates subjected to uniform load. Composite Structures, 83(4), 381–390.View ArticleGoogle Scholar
- Liew, K, He, X, Ng, T, & Kitipornchai, S. (2002). Active control of FGM shells subjected to a temperature gradient via piezoelectric sensor/actuator patches. International Journal for Numerical Methods in Engineering, 55(6), 653–668.MATHView ArticleGoogle Scholar
- Liew, K, He, X, Ng, T, & Kitipornchai, S. (2003). Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators. Computational Mechanics, 31(3–4), 350–358.MATHGoogle Scholar
- Loja, M, Soares, M, & Barbosa, J. (2013). Analysis of functionally graded sandwich plate structures with piezoelectric skins, using B-spline finite strip method. Composite Structures, 96, 606–615.View ArticleGoogle Scholar
- Ootao, Y, & Tanigawa, Y. (2000). Three-dimensional transient piezothermoelasticity in functionally graded rectangular plate bonded to a piezoelectric plate. International Journal of Solids and Structures, 37(32), 4377–4401.MATHView ArticleGoogle Scholar
- Pan, E, & Han, F. (2005). Exact solution for functionally graded and layered magneto-electro-elastic plates. International Journal of Engineering Science, 43(3), 321–339.View ArticleGoogle Scholar
- Rahmani, O, & Noroozi Moghaddam, MH. (2014). On the Vibrational Behavior of Piezoelectric Nano-Beams. Advanced Materials Research, 829, 790–794.View ArticleGoogle Scholar
- Rahmani, O, & Pedram, O. (2014). Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science, 77, 55–70.MathSciNetView ArticleGoogle Scholar
- Rahmani, O, Malekzadeh, K, & Khalili, SMR. (2010). Analytical Solution for Free Vibration of Sandwich Structures with a Functionally Graded Syntactic Foam Core. Materials Science Forum, 636, 1143–1149.View ArticleGoogle Scholar
- Rao, SS. (2007). Vibration of continuous systems. New Jersey: John Wiley & Sons.Google Scholar
- Ray, M, & Batra, R. (2007). Vertically reinforced 1-3 piezoelectric composites for active damping of functionally graded plates. AIAA Journal, 45(7), 1779–1784.View ArticleGoogle Scholar
- Ray, M, & Sachade, H. (2006). Finite element analysis of smart functionally graded plates. International Journal of Solids and Structures, 43(18), 5468–5484.MATHView ArticleGoogle Scholar
- Reddy, J, & Cheng, Z-Q. (2001). Three-dimensional solutions of smart functionally graded plates. Journal of Applied Mechanics, 68(2), 234–241.MATHView ArticleGoogle Scholar
- Shen, H-S. (2005). Postbuckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings. International Journal of Solids and Structures, 42(23), 6101–6121.MATHView ArticleGoogle Scholar
- Shiyekar, S, & Kant, T. (2010). An electromechanical higher order model for piezoelectric functionally graded plates. International Journal of Mechanics and Materials in Design, 6(2), 163–174.View ArticleGoogle Scholar
- Sneddon, IN. (1995). Fourier transforms. New York: Courier Dover Publications.Google Scholar
- Spiegel, MR. (1965). Laplace transforms. New York: McGraw-Hill.Google Scholar
- Tanigawa, Y. (2001). Control of the transient thermoelastic displacement of a functionally graded rectangular plate bonded to a piezoelectric plate due to nonuniform heating. Acta Mechanica, 148(1–4), 17–33.MATHGoogle Scholar
- Timoshenko, S, Woinowsky-Krieger, S, & Woinowsky, S. (1959). Theory of plates and shells, vol 2. New York: McGraw-hill.Google Scholar
- Vel, SS, & Batra, R. (2001). Exact solution for rectangular sandwich plates with embedded piezoelectric shear actuators. AIAA Journal, 39(7), 1363–1373.View ArticleGoogle Scholar
- Wang, B, & Noda, N. (2001). Design of a smart functionally graded thermopiezoelectric composite structure. Smart Materials and Structures, 10(2), 189.Google Scholar
- Wang, Q, Quek, S, Sun, C, & Liu, X. (2001). Analysis of piezoelectric coupled circular plate. Smart Materials and Structures, 10(2), 229.Google Scholar
- Zhang, X, Veidt, M, & Kitipornchai, S. (2004). Transient bending of a piezoelectric circular plate. International Journal of Mechanical Sciences, 46(12), 1845–1859.MATHView ArticleGoogle Scholar
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