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18 pages, 10413 KB

Open AccessArticle

Analytical Analysis for a Uniformly Loaded Circular Plate of Functionally Graded Piezoelectric Materials Using Mian and Spencer Theory

by Huiduo Fan, Lulu Shen, Zaihu Zhu and Bo Yang

Materials 2026, 19(14), 2942; https://doi.org/10.3390/ma19142942 - 8 Jul 2026

Viewed by 283

Abstract

Mian and Spencer developed a procedure for constructing exact solutions to the equations of linear elasticity in thick isotropic functionally graded plates. In this study, the Mian and Spencer plate theory is extended from elastic material to piezoelectric material by incorporating intrinsic electro-elastic [...] Read more.

Mian and Spencer developed a procedure for constructing exact solutions to the equations of linear elasticity in thick isotropic functionally graded plates. In this study, the Mian and Spencer plate theory is extended from elastic material to piezoelectric material by incorporating intrinsic electro-elastic coupling. An electric potential function is constructed by analogy with the displacement components, and the corresponding functions are obtained through stepwise integration using the mechanical and electrical boundary conditions on the top and bottom surfaces of the plates. Exact 3D elasticity solutions are then derived for a uniformly loaded FGPM circular plate under simply supported and clamped boundary conditions, with the material coefficients allowed to vary arbitrarily along the thickness direction

z

. The analytical solutions are validated through comparison with results reported in the literature. Finally, a systematic parametric study is performed to investigate the effects of the material gradient index, boundary conditions, thickness-to-radius ratio, and piezoelectric effect on the electro-elastic effect of FGPM circular plates. Full article

(This article belongs to the Section Smart Materials)

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21 pages, 1400 KB

Open AccessArticle

Frictional Contact of Functionally Graded Piezoelectric Materials with Arbitrarily Varying Properties

by Xiuli Liu, Kaiwen Xiao, Changyao Zhang, Xinyu Zhou, Lingfeng Gao and Jing Liu

Mathematics 2026, 14(3), 450; https://doi.org/10.3390/math14030450 - 27 Jan 2026

Viewed by 585

Abstract

This study investigates the two-dimensional (2D) steady-state frictional contact behavior of functionally graded piezoelectric material (FGPM) coatings under a high-speed rigid cylindrical punch. An electromechanical coupled contact model considering inertial effects is established, while a layered model is employed to simulate arbitrarily varying [...] Read more.

This study investigates the two-dimensional (2D) steady-state frictional contact behavior of functionally graded piezoelectric material (FGPM) coatings under a high-speed rigid cylindrical punch. An electromechanical coupled contact model considering inertial effects is established, while a layered model is employed to simulate arbitrarily varying material parameters. Based on piezoelectric elasticity theory, the steady-state governing equations for the coupled system are derived. By utilizing the transfer matrix method and the Fourier integral transform, the boundary value problem is converted into a system of coupled Cauchy singular integral equations of the first and second kinds in the frequency domain. These equations are solved semi-analytically, using the least squares method combined with an iterative algorithm. Taking a power-law gradient distribution as a case study, the effects of the gradient index, relative sliding speed, and friction coefficient on the contact pressure, in-plane stress, and electric displacement are systematically analyzed. Furthermore, the contact responses of FGPM coatings with power-law, exponential, and sinusoidal gradient profiles are compared. The findings provide a theoretical foundation for the optimal design of FGPM coatings and for enhancing their operational reliability under high-speed service conditions. Full article

(This article belongs to the Special Issue Structural Optimization and Numerical Modeling of Materials and Engineering)

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19 pages, 7240 KB

Open AccessFeature PaperArticle

Finite Element Simulation of Thermal Sliding Friction and Wear in an FGPM-Coated Half-Plane

by Lingfeng Gao, Jing Liu, Jiajia Mao and Kaiwen Xiao

Mathematics 2025, 13(21), 3414; https://doi.org/10.3390/math13213414 - 27 Oct 2025

Cited by 1 | Viewed by 916

Abstract

This study investigates the thermoelastic frictional contact and wear behavior during reciprocating sliding of a conductive cylindrical punch on a functionally graded piezoelectric material (FGPM)-coated half-plane. The thermo-electro-elastic properties of the coating vary continuously along the thickness direction according to arbitrary gradient functions, [...] Read more.

This study investigates the thermoelastic frictional contact and wear behavior during reciprocating sliding of a conductive cylindrical punch on a functionally graded piezoelectric material (FGPM)-coated half-plane. The thermo-electro-elastic properties of the coating vary continuously along the thickness direction according to arbitrary gradient functions, with thermal parameters being temperature-dependence. A theoretical framework for the coupled thermo-electro-elastic frictional contact problem is developed and solved using the finite element method. A sequential coupling approach is employed to integrate thermoelastic frictional contact with piezoelectric effects. Furthermore, wear on the coating surface is modeled using an improved Archard formulation, accounting for its impact on thermal sliding frictional contact characteristics. Numerical simulations examine the influence of wear, cycle number, friction coefficient, gradient index and gradient form on the coupled thermo-electro-elastic response of the FGPM coating structure. The numerical results demonstrate the gradient index and gradient form can effectively mitigate thermo-electrical contact-induced damage and reduce friction and wear in piezoelectric materials. Full article

(This article belongs to the Special Issue Finite Element Method and Applications in Mathematical Modeling and Engineering: Modeling, Analysis, and Simulations)

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22 pages, 2243 KB

Open AccessArticle

Thermal Friction Contact Analysis of Graded Piezoelectric Coatings Under Conductive Punch Loading

by Xinyu Zhou, Jing Liu and Jiajia Mao

Coatings 2025, 15(2), 222; https://doi.org/10.3390/coatings15020222 - 13 Feb 2025

Cited by 7 | Viewed by 1500

Abstract

In this paper, we investigate the thermal friction sliding contact of a functionally graded piezoelectric material (FGPM)-coated half-plane subjected to a rigid conductive cylindrical punch. This study considers the effect of the thermal convection term in heat conduction. The thermo-electro-elastic material parameters of [...] Read more.

In this paper, we investigate the thermal friction sliding contact of a functionally graded piezoelectric material (FGPM)-coated half-plane subjected to a rigid conductive cylindrical punch. This study considers the effect of the thermal convection term in heat conduction. The thermo-electro-elastic material parameters of the coating vary exponentially along its thickness direction. Utilizing thermoelastic theory and Fourier integral transforms, the problem is formulated into Cauchy singular integral equations of the first and second kinds with surface stress, contact width, and electric displacement as the unknown variables. The numerical solutions for the contact stress, electric displacement, and temperature field of the graded coating surface are obtained using the least-squares method and iterative techniques. It can be observed that the thermo-electro-elastic contact behavior of the coating surface undergoes significant changes as the graded index varies from −0.5 to 0.5, the friction coefficient ranges from 0.1 to 0.5, and the sliding velocity changes from 0.01 m/s to 0.05 m/s. The results indicate that adjusting the graded index of the coating, the sliding speed of the punch, and the friction coefficient can improve the thermo-electro-elastic contact damage of the material’s surface. Full article

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14 pages, 1979 KB

Open AccessArticle

Active Vibration Control of Timoshenko Sigmoid Functionally Graded Porous Composite Beam with Distributed Piezoelectric Sensor/Actuator in a Thermal Environment

by Khalid El Harti, Rachid Saadani and Miloud Rahmoune

Designs 2023, 7(1), 2; https://doi.org/10.3390/designs7010002 - 22 Dec 2022

Cited by 10 | Viewed by 2629

Abstract

This work presents the study of the dynamics and active control of a cantilever sigmoid FGM beam with porosities in a thermal environment. During this study, we considered the Timoshenko beam’s theory combined with the finite element method (FEM). This work also presents [...] Read more.

This work presents the study of the dynamics and active control of a cantilever sigmoid FGM beam with porosities in a thermal environment. During this study, we considered the Timoshenko beam’s theory combined with the finite element method (FEM). This work also presents a comparative study with an experimental study for the vibration of a functionally graded piezoelectric beam (FGPM) to validate the numerical model. Linear quadratic Gaussian (LQG) optimal control with a Kalman filter was used for the vibration control using piezoelectric sensors and actuators as symmetrical layers to eliminate membrane effects. The controlled and uncontrolled responses are presented, considering the influence of thermal effect, the porosity of the FGM material, and the location of the sensor pair on the smart structure. The results indicate that the porosity effect of the FGM material, as well as the application of the thermal effect, involves an increase in vibration frequencies, in contrast to the increase in the power law index. The study also shows that the thermal and porosity effects result in an increase in vibration amplitudes. Full article

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19 pages, 6833 KB

Open AccessArticle

Functionally Graded Piezoelectric Energy Harvester: A Numerical Study

by Anuruddh Kumar, Mohamed Nainar Mohamed Ansari, Sobhy M. Ibrahim, Paramanandam Thomas and Rahul Vaish

Electronics 2022, 11(16), 2595; https://doi.org/10.3390/electronics11162595 - 19 Aug 2022

Cited by 7 | Viewed by 2886

Abstract

The performance of linear energy harvesters is primarily confined to a very narrow operating frequency bandwidth around its natural frequency. Even a slight deviation of the excitation frequency from the fundamental frequency of the system tremendously reduces the harvester’s performance. In order to [...] Read more.

The performance of linear energy harvesters is primarily confined to a very narrow operating frequency bandwidth around its natural frequency. Even a slight deviation of the excitation frequency from the fundamental frequency of the system tremendously reduces the harvester’s performance. In order to minimize this shortcoming, the presented study considers the piezoelectric energy harvester with magnets introducing non-linearity in the system. The simple harmonic balance method is used to solve the non-linearity and for computing the voltage output and power in the frequency domain. In addition, the study also incorporates the functionally graded piezoelectric materials because of their superior properties. The distance between magnets (d₀) has been varied from 0.4 mm to 10 mm along with grading index (n) in the range of 0 to ∞. Finally, voltage and power across the resistance were computed. The effective harvesting frequency range for d₀ = 0.4 mm and n = 1 is observed in the range of 20 Hz to 85 Hz, while it was only between 35 Hz and 65 Hz for d₀ = 10 mm, yielding a 216% increase in the frequency bandwidth. Under different case studies, the peak output power varied from 2 mW (d₀ = 0.4 mm and n = ∞) to 6 mW (d₀ = 10 mm and n = 0). Full article

(This article belongs to the Special Issue Latest Advances in Energy Harvesting Technologies and Applications)

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23 pages, 10501 KB

Open AccessArticle

Functionally Graded Piezoelectric Medium Exposed to a Movable Heat Flow Based on a Heat Equation with a Memory-Dependent Derivative

by Ahmed E. Abouelregal, Hijaz Ahmad and Shao-Wen Yao

Materials 2020, 13(18), 3953; https://doi.org/10.3390/ma13183953 - 7 Sep 2020

Cited by 48 | Viewed by 3160

Abstract

The current work deals with the study of a thermo-piezoelectric modified model in the context of generalized heat conduction with a memory-dependent derivative. The investigations of the limited-length piezoelectric functionally graded (FGPM) rod have been considered based on the presented model. It is [...] Read more.

The current work deals with the study of a thermo-piezoelectric modified model in the context of generalized heat conduction with a memory-dependent derivative. The investigations of the limited-length piezoelectric functionally graded (FGPM) rod have been considered based on the presented model. It is assumed that the specific heat and density are constant for simplicity while the other physical properties of the FGPM rod are assumed to vary exponentially through the length. The FGPM rod is subject to a moving heat source along the axial direction and is fixed to zero voltage at both ends. Using the Laplace transform, the governing partial differential equations have been converted to the space-domain, and then solved analytically to obtain the distributions of the field quantities. Numerical computations are shown graphically to verify the effect of memory presence, graded material properties, time-delay, Kernel function, and the thermo-piezoelectric response on the physical fields. Full article

(This article belongs to the Special Issue Materials Science and Applications of Phase Change Memory Materials)

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15 pages, 2009 KB

Open AccessArticle

The Complex Rayleigh Waves in a Functionally Graded Piezoelectric Half-Space: An Improvement of the Laguerre Polynomial Approach

by Ke Li, Shuangxi Jing, Jiangong Yu, Xiaoming Zhang and Bo Zhang

Materials 2020, 13(10), 2320; https://doi.org/10.3390/ma13102320 - 18 May 2020

Cited by 17 | Viewed by 2829

Abstract

The research on the propagation of surface waves has received considerable attention in order to improve the efficiency and natural life of the surface acoustic wave devices, but the investigation on complex Rayleigh waves in functionally graded piezoelectric material (FGPM) is quite limited. [...] Read more.

The research on the propagation of surface waves has received considerable attention in order to improve the efficiency and natural life of the surface acoustic wave devices, but the investigation on complex Rayleigh waves in functionally graded piezoelectric material (FGPM) is quite limited. In this paper, an improved Laguerre orthogonal function technique is presented to solve the problem of the complex Rayleigh waves in an FGPM half-space, which can obtain not only the solution of purely real values but also that of purely imaginary and complex values. The three-dimensional dispersion curves are generated in complex space to explore the influence of the gradient coefficients. The displacement amplitude distributions are plotted to investigate the conversion process from complex wave mode to propagating wave mode. Finally, the curves of phase velocity to the ratio of wave loss decrements are illustrated, which offers extra convenience for finding the high phase velocity points where the complex wave loss is near zero. Full article

(This article belongs to the Section Materials Physics)

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20 pages, 2711 KB

Open AccessArticle

Thermo-Electro-Mechanical Vibrations of Porous Functionally Graded Piezoelectric Nanoshells

by Yun Fei Liu and Yan Qing Wang

Nanomaterials 2019, 9(2), 301; https://doi.org/10.3390/nano9020301 - 20 Feb 2019

Cited by 35 | Viewed by 5312

Abstract

In this work, we aim to study free vibration of functionally graded piezoelectric material (FGPM) cylindrical nanoshells with nano-voids. The present model incorporates the small scale effect and thermo-electro-mechanical loading. Two types of porosity distribution, namely, even and uneven distributions, are considered. Based [...] Read more.

In this work, we aim to study free vibration of functionally graded piezoelectric material (FGPM) cylindrical nanoshells with nano-voids. The present model incorporates the small scale effect and thermo-electro-mechanical loading. Two types of porosity distribution, namely, even and uneven distributions, are considered. Based on Love’s shell theory and the nonlocal elasticity theory, governing equations and corresponding boundary conditions are established through Hamilton’s principle. Then, natural frequencies of FGPM nanoshells with nano-voids under different boundary conditions are analyzed by employing the Navier method and the Galerkin method. The present results are verified by the comparison with the published ones. Finally, an extensive parametric study is conducted to examine the effects of the external electric potential, the nonlocal parameter, the volume fraction of nano-voids, the temperature rise on the vibration of porous FGPM cylindrical nanoshells. Full article

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20 pages, 501 KB

Open AccessArticle

Thermo-Electro-Mechanical Analysis of a Curved Functionally Graded Piezoelectric Actuator with Sandwich Structure

by Zhi Yan, Mostafa Zaman and Liying Jiang

Materials 2011, 4(12), 2151-2170; https://doi.org/10.3390/ma4122151 - 12 Dec 2011

Cited by 15 | Viewed by 8069

Abstract

In this work, the problem of a curved functionally graded piezoelectric (FGP) actuator with sandwich structure under electrical and thermal loads is investigated. The middle layer in the sandwich structure is functionally graded with the piezoelectric coefficient g₃₁ varying continuously along the [...] Read more.

In this work, the problem of a curved functionally graded piezoelectric (FGP) actuator with sandwich structure under electrical and thermal loads is investigated. The middle layer in the sandwich structure is functionally graded with the piezoelectric coefficient g₃₁ varying continuously along the radial direction of the curved actuator. Based on the theory of linear piezoelectricity, analytical solutions are obtained by using Airy stress function to examine the effects of material gradient and heat conduction on the performance of the curved actuator. It is found that the material gradient and thermal load have significant influence on the electroelastic fields and the mechanical response of the curved FGP actuator. Without the sacrifice of actuation deflection, smaller internal stresses are generated by using the sandwich actuator with functionally graded piezoelectric layer instead of the conventional bimorph actuator. This work is very helpful for the design and application of curved piezoelectric actuators under thermal environment. Full article

(This article belongs to the Special Issue Advances in Functionally Graded Materials)

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