Finite element modeling of active control of functionally graded shells in frequency domain via piezoelectric sensors and actuators

 A flat-shell element is presented for the active control of functionally graded material (FGM) shells through integrated piezoelectric sensor/actuator layers. The finite element formulation based on first-order shear deformation theory (FSDT) can be applied to shells ranging from relatively thin to moderately thick dimensions. A constant gain displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied to provide active control of the integrated FGM shell in a self-monitoring and self-controlling system. Frequency response characteristics of the FGM shell containing the piezoelectric sensors/actuators are analyzed in the frequency domain. The effects of constituent volume fraction and the influence of feedback control gain values on the dynamic responses of the FGM shell system are examined in detail. Received 13 November 2000     

Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers

This paper presents analytical solution for functionally graded material (FGM) beams integrated with piezoelectric actuator and sensor under an applied electric field and thermo-mechanical load. In FGM host properties is assumed to vary exponentially in thickness direction and the Poisson’s ratio is held constant. The hybrid beam is in a state of plane stress and the piezoelectric is composed of orthotropic materials. The beam is simply supported with the bottom surface traction free and zero temperature. By using of state-space method in thickness direction and Fourier series in longitudinal direction, the solution can be made. To verify the accuracy of the present formulation, numerical results for the simple case is compared with results obtained in the published literature. Finally, effects of FGM index, electromechanical coupling, thickness ratio and thermo-mechanical surface boundary condition on the bending behaviour of beam are investigated.     

A comparison of buckling and postbuckling behavior of FGM plates with piezoelectric fiber reinforced composite actuators

Compressive postbuckling under thermal environments and thermal postbuckling due to a uniform temperature rise are presented for a simply supported, shear deformable functionally graded plate with piezoelectric fiber reinforced composite (PFRC) actuators. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and PFRC layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. A two step perturbation technique is employed to determine buckling loads (temperature) and postbuckling equilibrium paths. The numerical illustrations concern the compressive and thermal postbuckling behaviors of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded PFRC actuators under different sets of thermal and electric loading conditions. The results for monolithic piezoelectric actuator, which is a special case in the present study, are compared with those of PFRC actuators. The results reveal that, in the compressive buckling case, the applied voltage usually has a small effect on the postbuckling load–deflection curves of the plate with PFRC actuators, whereas in the thermal buckling case, the effect of applied voltage is more pronounced for the plate with PFRC actuators, compared to the results of the same plate with monolithic piezoelectric actuators.     

Fracture problems,vibration, buckling,and bending analyses of functionally graded materials: A state-of-the-art review including smart FGMS

Composite materials fail under extreme working conditions, particularly at high temperature, due to delamination (separation of fibers from matrix). And therefore it is needed to switch over functionally graded materials (FGMs) which can sustain at high temperature conditions (250–2000°C). There is a need to analyze the fracture and fatigue characteristics of FGM structures and so through this review the emphasis is given on fracture analysis of FGM materials. It has been reported that a combination of extended finite element method and isogeometric analysis methodologies has been used for general mixed-mode crack propagation problems after the introduction of extended isogeometric analysis. Furthermore, recent computational advances have been in the form of multiscale simulations where the part of model is simulated by a finer modeling scale, which can represent details of the material behavior and the interacting effects of material constituents in the finest way. The review is also focused on new advances in analytical and numerical methods for the stress, vibration, and buckling analyses of FGMs. Emphasis has been primarily on to restrict 2D analysis with sorts of compromise in the accuracy of results. First shear deformation theory (FSDT) and third-order shear deformation theory have been extensively used among the various 2D plate theories. FSDT can help us in terms of getting reasonably accurate results with less computational afford. This paper also outlines review on carbon nanotubes (CNT) reinforced FGMs, functionally graded nanocomposites, functionally graded single-walled CNT, FG nanobeam as well as functionally graded piezoelectric materials. Future applications would be based on these smart materials which are supposed to serve us in adverse conditions. Of course, with rise and advent of promising nanotechnology and its potential impact on aerospace industry as well as on other areas, it becomes important to us to compile this review article.     

Nonlinear thermal bending response of FGM plates due to heat conduction

Nonlinear thermal bending analysis is presented for a simply supported, shear deformable functionally graded plate without or with piezoelectric actuators subjected to the combined action of thermal and electrical loads. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations of an FGM plate are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. A two step perturbation technique is employed to determine the thermal load–deflection and thermal load–bending moment curves. The numerical illustrations concern nonlinear bending response of FGM plates without or with surface bonded piezoelectric actuators due to heat conduction and under different sets of electric loading conditions. The results reveal that for the case of heat conduction the nonlinear thermal bending responses are quite different to those of FGM plates subjected to transverse mechanical loads, and the temperature-dependency of FGMs could not be neglected in the thermal bending analysis.     

A FEM model for the active control of curved FGM shells using piezoelectric sensor/actuator layers

In this paper, a generic finite element formulation is developed for the static and dynamic control of FGM (functionally graded material) shells with piezoelectric sensor and actuator layers. The properties of the FGM shell are graded in the thickness direction according to a volume fraction power‐law distribution. The proposed finite element model is based on variational principle and linear piezoelectricity theory. A constant displacement and velocity feedback control algorithm coupling the direct and inverse piezoelectric effects is applied in a closed‐loop system to provide feedback control of the integrated FGM shell structure. Both static and dynamic control of FGM shells are simulated to demonstrate the effectiveness of the proposed active control scheme within a framework of finite element discretization and piezoelectric integration. Copyright © 2002 John Wiley & Sons, Ltd.     

Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environment

In the present study, finite element formulation based on higher order shear deformation plate theory is developed to analyze nonlinear natural frequencies, time and frequency responses of functionally graded plate with surface-bonded piezoelectric layers under thermal, electrical and mechanical loads. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The material properties of functionally graded material (FGM) are assumed temperature-dependent. The temperature field has uniform distribution over the plate surface and varies in the thickness direction. The considered electric field only has non-zero-valued component E_z. Numerical results are presented to study effects of FGM volume fraction exponent, applied voltage in piezoelectric layers, thermal load and vibration amplitude on nonlinear natural frequencies and time response of FGM plate with integrated piezoelectric layers. In addition, nonlinear frequency response diagrams of the plate are presented and effects of different parameters such as FGM volume fraction exponent, temperature gradient, and piezoelectric voltage are investigated.     

Thermal Buckling Analysis of Piezoelectric Functionally Graded Plates with Temperature-Dependent Properties

In this study, the thermal buckling analysis of hybrid laminated plates made of two-layered functionally graded materials (FGMs) that are integrated with surface-bonded piezoelectric actuators referred to as (P/FGM)_s are investigated. Material properties for both substrate FGM layers and piezoelectric layers are temperature-dependent. Uniform temperature rise as a thermal load and constant applied actuator voltage are considered for this analysis. By definition of four new analytic functions, the five coupled governing stability equations, which are derived based on the first-order shear deformation plate theory, are converted into fourth-order and second-order decoupled partial differential equations (PDEs). Considering a Levy-type solution, these two PDEs are reduced to two ordinary differential equations. One of these equations is solved using an accurate analytical solution, which is named as power series Frobenius method. The effects of parameters, such as the plate aspect ratio, ratio of piezoelectric layer thickness to thickness of FGM layer, gradient index, actuator voltage, and the temperature dependency on the critical buckling temperature difference, are illustrated and explained. The critical buckling temperatures of (P/FGM)_s with six various boundary conditions are reported for the first time and can be served as benchmark results for researchers to validate their numerical and analytical methods in the future.     

压电梯度薄壳的高阶理论解

压电功能梯度执行器能产生较大的位移、降低应力峰值并避免了粘结层带来的问题,压电梯度超声换能器能拓展频带宽度。本文作者提出了一个简单而有效的求解压电梯度薄壳力、电行为特性的高阶理论。设定位移分量为壳厚的线性函数,而电势沿厚度方向为二次分布。考虑了压电作动元的驱动信号不同时所具有的不同形式的电荷平衡方程。应用Fourier级数法得到压电系数沿厚度坐标变化的梯度壳的力电耦合的解析解。所得结果可退化至梁、板等多种特殊情况。利用所得方程分析了一非均匀简支压电层合板,并与三维精确结果作了对比,两者吻合得很好,表明该理论的正确性。最后具体求解了压电梯度圆柱壳的力、电特性,给出了位移、应力、电势沿厚度方向的变化规律。     

Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical shell with piezoelectric actuators subjected to lateral or hydrostatic pressure combined with electric loads in thermal environments. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the shell surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation theory with a von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling is extended to the case of FGM hybrid laminated cylindrical shells of finite length. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of pressure-loaded, perfect and imperfect, FGM cylindrical shells with fully covered piezoelectric actuators under different sets of thermal and electric loading conditions. The results reveal that temperature dependency, temperature change and volume fraction distribution have a significant effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells. In contrast, the control voltage only has a very small effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells.     

A closed-form solution for thermal buckling of piezoelectric FGM rectangular plates with temperature-dependent properties

A thermal buckling analysis is presented for functionally graded rectangular plates that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of thermal load and constant applied actuator voltage. The temperature-dependent material properties of the functionally graded plate are assumed to vary as a power form of the thickness coordinate. Derivation of the equations is based on the third-order shear deformation plate theory. Results for the critical buckling temperatures are obtained in closed-form solution, which are convenient to be used in engineering design applications. The effects of the applied actuator voltage, plate geometry, and volume fraction exponent of the functionally graded material on the buckling temperature are investigated.     

A Method of Material Optimization of Cementless Stem Through Functionally Graded Material

Ideally, a bone implant should be such that it exhibits an identical response to loading as real bone and is also biocompatible with existing tissue. A stiff stem, which is usually made of titanium, shields the proximal bone from mechanical loading (stress shielding). On the other hand, decreasing the stem stiffness increases the proximal interface shear stress and the risk of proximal interface failure. Therefore the purpose of this study is to solve these conflicting requirements in order to have more uniform interface shear stress distribution and less stress shielding through the concept of functionally graded material (FGM). FGM is a kind of advanced composite materials, which changes its composition and structure gradually over one or two directions of its volume, resulting in corresponding changes in the properties of the material. This study is divided into two parts; in the first part, the finite element analysis and optimization technique are used to design the stem as one-dimensional FGM, while in the second part, the stem is designed as two-dimensional functionally graded material. The aim of both designs is to overcome the above mentioned problems. In the case of part one (one-dimensional FGM), the gradation of elastic modulus is changed along the vertical direction (model 1) and along the horizontal direction (model 2), in order to find the optimal gradation direction. It is found that the optimal design is to change the elastic modulus gradually from 110 GPa (Hydroxyapatite) at the top of the stem to 1GPa (Collagen) at the bottom (model 1). This optimal gradation decreases stress shielding by 83%, while reduces the maximum interface shear stress by 32% compared to homogenous titanium stem. However, in the second part (two-dimensional FGM, model 3) the materials of optimal design are found to be hydroxyapatite, Bioglass, and collagen. This design leads to the same stress shielding reduction as in model 1, while at the same time, the maximum interface shear stress is reduced by 45% and 63% compared to the optimal one-dimensional FGM design and homogenous titanium stem, respectively.     

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

Summary The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.     

Optimal shape control of functionally graded smart plates using genetic algorithms

This paper deals with optimal shape control of functionally graded smart plate containing patches of piezoelectric sensors and actuators. The genetic algorithm (GA) is designed to search for optimal actuator voltage and displacement control gains for the shape control of the functionally graded material (FGM) plates. The work extends the earlier finite element formulations of the two leading authors, so that it can be readily treated using genetic algorithms. Numerical results have been obtained to study the effect of the shape control of the FGM plates under a temperature gradient by optimising (i) the voltage distribution for the open loop control, and (ii) the displacement control gain values for the closed loop feedback control. The effect of the constituent volume fractions of zirconia, through varying the volume fraction exponent n, on the optimal voltages and gain values has also been examined.     

Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams

In this article, buckling analysis of functionally graded material (FGM) beams with or without surface-bonded piezoelectric layers subjected to both thermal loading and constant voltage is studied. Thermal and mechanical properties of FGM layer is assumed to follow the power law distribution in thickness direction, except Poisson’s ratio which is considered constant. The Timoshenko beam theory and nonlinear strain-displacement relations are used to obtain the governing equations of piezoelectric FGM beam. Beam is assumed under three types of thermal loading and various types of boundary conditions. For each case of boundary conditions, existence of bifurcation-type buckling is examined and for each case of thermal loading and boundary conditions, closed-form solutions are obtained which are easily usable for engineers and designers. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of FGM beam on critical buckling temperature difference are examined.     

基于样条无网格法的厚/薄压电功能梯度板动力分析

在将板的挠度和剪切应变作为场变量的基础上增加考虑面内位移,采用样条无网格法建立了热环境下厚/薄压电功能梯度板动力分析新的计算格式。考虑功能梯度材料的物性参数随温度改变以及温度变化引起的面内力,讨论了压电FGM板在不同边界约束下材料梯度指数变化时,面内位移及面内力对热环境下压电功能梯度板的动力特性和动力响应的影响。分析结果表明,新建立的模型对厚/薄压电功能梯度板动力分析具有通用性和有效性,没有剪切闭锁现象,且计算效率和计算精度较高。     

Laser metal deposition of functionally graded Ti6Al4V/TiC

Functionally graded materials (FGMs) are advanced materials with improved properties that enable them to withstand severe working environment which the traditional composite materials cannot withstand. FGM found their applications in several areas which include: military, medicine and aerospace. Various manufacturing processes are used to produce functionally graded materials that include: powder metallurgy, physical vapour deposition, chemical vapour deposition process and laser metal deposition process. Laser metal deposition (LMD) process is an additive manufacturing process that can be used to produce functionally graded material directly from the three dimensional (3D) computer aided design (CAD) model of the part in one single process. LMD process is a fairly new manufacturing process and a highly non-linear process. The process parameters are of great importance in LMD process and they need to be optimized for the required application. In this study, functionally graded titanium alloy composite was produced using optimized process parameters for each material combination as obtained through a model that was developed in an initial study and the FGM was characterized through metallurgical, mechanical and tribological studies. The results show that the produced FGM has improved properties when compared to those produced at constant processing parameters for all material combinations.     

Finite element analysis for geometrically nonlinear deformations of smart functionally graded plates using vertically reinforced 1-3 piezoelectric composite

In this paper, a finite element model has been developed for the geometrically nonlinear static analysis of simply supported functionally graded (FG) plates integrated with a patch of vertically reinforced 1-3 piezoelectric composite material acting as a distributed actuator. The material properties of the functionally graded substrate plate are assumed to be graded only in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The analysis of the electro-elastic coupled problem includes the transverse deformations of the overall plate to utilize the transverse normal actuation by the distributed actuator for counteracting the nonlinear deformations of smart functionally graded plates. The nonlinear governing equations of equilibrium are solved by using direct iteration method with under-relaxation. The numerical illustrations suggest the potential use of the distributed actuator made of vertically reinforced 1-3 piezoelectric composite material for active control of nonlinear deformations of smart functionally graded plates. The effect of variation of piezoelectric fiber orientation in the distributed actuator on its control authority for counteracting the nonlinear deformations of smart functionally graded plates has also been investigated.     

Exact solutions for characterization of electro-elastically graded materials

The present paper deals with a class of functionally graded materials (FGM), called active FGM that has electro-elastically graded material phases. An active FGM system leads to minimization of stress concentration that arises due to mismatch in the electrical and elastic properties of the constituent phases. This work focuses on the characterization of the through thickness stresses of an active FGM subjected to electrical excitation. The structure is comprised of a substrate, an electro-elastically graded layer and an active layer. A formulation for exact solutions of the system based on Euler–Bernoulli theory is presented. Power-law variation of the composition of the two phases in the graded layer is considered. Performance of linearly gradient FGM for a range of stiffness and electrical property ratios of the active and substrate materials have been studied. It is observed that the electrical strain component and the compositional gradation significantly influence the stress characteristics of the active FGM.     

功能梯度材料的断裂与屈曲驱动断裂的简化分析

作为一类先进的复合材料,功能梯度材料(FGM)能综合利用多种材料的物理性能,同时材料性质的连续变化也使其具有许多优越的力学性能。本文对功能梯度材料中平行于界面的裂纹的断裂参数进行了计算,并分析了梯度变化的薄膜在压应力作用下的屈曲驱动扩展。研究结果表明:功能梯度材料能有效地减小界面中的应力集中及它对材料中缺陷的作用,从而不同程度地提高了材料的强度和韧性。