Vibration analysis of piezoelectric FGM sensors using an accurate method

In this study, free vibration analysis of moderately thick smart FG annular/circular plates with different boundary conditions is presented on the basis of the Mindlin plate theory. This structure comprised a host FG plate and two bonded piezoelectric layers. Piezoelectric layers are open circuit therefore this plate can be used as a sensor. According to power-law distribution of the volume fraction of the constituents, material properties vary continuously through the thickness of host plate while Poisson's ratio is set to be constant. Using Hamilton's principle and Maxwell electrostatic equation yields six complex coupled equations which are solved via an exact closed-form method. The accuracy of the frequencies is verified by the available literature, finite element method (FEM) and the Kirchhoff theory. The effects of plate parameters like boundary condition and gradient index are investigated and significance of coupling between in-plane and transverse displacements on the resonant frequency is proved.     

Stochastic hybrid numerical method for transient analysis of stress field in functionally graded thick hollow cylinders subjected to shock loading

This investigation aims to study the random stresses in a functionally graded (FG) thick hollow cylinder with uncertain material properties subjected to mechanical shock loading using a hybrid numerical method. The mechanical properties are considered to vary across thickness of FG cylinder as a nonlinear power function of radius. The stresses are obtained by solving Navier equation and using Galerkin finite element and Newmark finite difference methods. The Monte Carlo simulation is used to generate the random mechanical properties for the problem. The failure probabilities and time history analysis of stresses are determined for various coefficient of variation considering various grading patterns of mechanical properties. The presented hybrid numerical method is effective, with high capability for stochastic analysis of dynamic and transient analysis of FG structures with various boundary conditions.     

Semi-analytical solution of magneto-thermo-elastic stresses for functionally graded variable thickness rotating disks

In this paper, a semi-analytical solution for magneto-thermo-elastic problem in functionally graded (FG) hollow rotating disks with variable thickness placed in uniform magnetic and thermal fields is presented. Stresses and perturbation of magnetic field vector in FG rotating disks are determined using infinitesimal theory of magneto-thermo-elasticity under plane stress conditions. The material properties except Poisson’s ratio are modeled as power-law distribution of volume fraction. The profile of disk thickness is assumed to be a parabolic function of radius. The non-dimensional distribution of temperature, displacement, stresses and perturbation of magnetic field vector throughout radius are shown. Effects of material grading index, geometry of the disk and magnetic field on the stress and displacement fields are investigated. The results of stresses and radial displacements for two different boundary conditions with and without the effect of magnetic field are compared for a FG rotating disk with concave thickness profile. It has been found that imposing a magnetic field significantly decreases tensile circumferential stresses. Therefore the fatigue life of the disk will be significantly improved by applying the magnetic field. Results of this investigation could be applied for optimum design of FG hollow rotating disks with variable thickness.     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

Dynamic relaxation method for nonlinear buckling analysis of moderately thick FG cylindrical panels with various boundary conditions

Using new approach proposed by Dynamic relaxation (DR) method, buckling analysis of moderately thick Functionally graded (FG) cylindrical panels subjected to axial compression is investigated for various boundary conditions. The mechanical properties of FG panel are assumed to vary continuously along the thickness direction by the simple rule of mixture and Mori-Tanaka model. The incremental form of nonlinear formulations are derived based on First-order shear deformation theory (FSDT) and large deflection von Karman equations. The DR method combined with the finite difference discretization technique is employed to solve the incremental form of equilibrium equations. The critical mechanical buckling load is determined based on compressive load-displacement curve by adding the incremental displacements in each load step to the displacements obtained from the previous ones. A detailed parametric study is carried out to investigate the influences of the boundary conditions, rule of mixture, grading index, radius-to-thickness ratio, length-to-radius ratio and panel angle on the mechanical buckling load. The results reveal that with increase of grading index the effect of radius-to-thickness ratio on the buckling load decreases. It is also observed that effect of distribution rules on the buckling load is dependent to the type of boundary conditions.

     

Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers

In this paper, a free vibration analysis of moderately thick circular functionally graded (FG) plate integrated with two thin piezoelectric (PZT4) layers is presented based on Mindlin plate theory. The material properties of the FG core plate are assumed to be graded in the thickness direction, while the distribution of electric potential field along the thickness of piezoelectric layers is simulated by sinusoidal function. The differential equations of motion are solved analytically for two boundary conditions of the plate: clamped edge and simply supported edge. The analytical solution is validated by comparing the obtained resonant frequencies with those of an isotropic host plate. The emphasis is placed on investigating the effect of varying the gradient index of FG plate on the free vibration characteristics of the structure. Good agreement between the results of this paper and those of the finite element analyses validated the presented approach.     

Free vibration analysis of orthotropic plates with variable thickness resting on non-uniform elastic foundation by element free Galerkin method

This study intends to investigate the vibration behavior of a thin square orthotropic plate resting on non-uniform elastic foundation and its thickness varying in one or two directions. By using the classical plate theory and employing element free Galerkin method, it is shown that the fundamental frequency coefficients obtained are in good agreement with available results in the literature. The effects of thickness variation, foundation parameter and boundary conditions on frequency are investigated. The results show that the method converges very fast regardless of parameters involved.     

Free and forced vibration analysis of FG beam considering temperature dependency of material properties

This paper presents a finite element method (FEM) free and forced lateral vibration analysis of beams made of functionally graded materials (FGMs). The temperature dependency of material properties along with damping had not previously been taken into account in vibration analysis. In the present study, the material properties were assumed to be temperature-dependent, and were graded in the thickness direction according to a simple power law distribution of the volume fractions of the constituents. The natural frequencies were obtained for functionally graded (FG) beams with various boundary conditions. First, an FG beam was assumed to be isotropic (metal rich) and the results were compared with the analytical solution and the results for ANSYS and NASTRAN software. Finally, dynamic responses were obtained for damped and un-damped systems. Numerical results were obtained to show the influences of the temperature dependency of the materials properties, the boundary conditions, the volume fraction distribution (the index of power law, N) and the geometrical parameters.     

功能梯度压电材料壳体热残余应力

功能梯度压电材料结构成型冷却后会出现热残余现象,影响结构强度.借鉴复合材料层合结构的研究方法,将功能梯度压电材料球壳和圆柱壳沿厚度分为若干层,各层视为均匀材料,根据层间连续条件导出递推关系,得到显式的力—电—热多场耦合热残余解.统一了多层功能梯度压电材料壳体和连续功能梯度压电材料壳体热残余解.对于前者,其解为精确解;对于后者,其解为渐近解,随层数增加而收敛于精确解.其解也适用于功能梯度压电材料涂层.该方法对材料性能的变化方式(函数)没有要求,适应性强.并讨论影响热残余应力和界面强度的因素,球壳因双曲率的影响,热残余应力显著大于柱壳.     

Free vibration analysis of functionally graded plates with multiple circular and non-circular cutouts

Cutouts are inevitable in structures due to practical consideration.In order to investigate the free vibration of functionally graded plates with multiple circular and non-circular cutouts,finite eleme...     

Effects of different restraints on the weld-induced residual deformations and stresses in a steel plate

Effects of three different plate boundary constraints on the residual stress field and deformation are investigated numerically during butt-joint welding. For the numerical solution of the heat transfer equations, the finite element method is used to predict the temperature profile as well as residual stress field due to three different plate boundary conditions. The distortion of the welded plate is modeled as a nonlinear problem in geometry and material, adopting a finite element solution based upon the thermo–elastic–plastic large deflection theory. High-strength shipbuilding steel AH36 with temperature-depending material properties and nonlinear stress–strain material properties (bilinear isotropic hardening option uses the von Mises yield criteria) are assumed for the numerical analysis. For verifying the results, the temperature profile is compared with the result obtained in a previous research. In the mechanical analysis, three different boundary conditions are applied. Effects of plate thickness, plate width, and mesh model on the residual stress with boundary constraint are studied.     

Eshelby-Mori-Tanaka approach for vibrational behavior of functionally graded carbon nanotube-reinforced plate resting on elastic foundation

In this study, based on the three-dimensional theory of elasticity, free vibration characteristics of functionally graded (FG) nanocomposite plates reinforced by randomly-oriented straight single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation are considered. Material properties are graded in the thickness direction of the plate according to the volume fraction power law distribution. An embedded carbon nanotube (CNT) in a polymer matrix and its surrounding inter-phase which is perfectly bonded to surrounding resin is replaced with an equivalent fiber to predict the mechanical properties of the carbon nanotube/polymer composite. The Mori-Tanaka approach is employed to calculate the effective elastic moduli of the plate. The natural frequencies of the plate are obtained by means of the generalized differential quadrature (GDQ) method. Detailed parametric studies have been carried out to investigate the influences of the CNT volume fraction, Winkler foundation modulus, shear elastic foundation modulus and various geometrical parameters on the vibration behavior of the functionally graded carbon nanotube-reinforced (FG-CNTR) plates.     

一种基于Fluent与UG的风机叶片反求设计的方法

针对风机叶片进行反求设计.建立叶片坐标系,利用UG获取坐标数据,通过导入Gambit软件,建立空气动力学流场,对其进行网格划分和边界条件的定义等,利用Fluent软件进行CFD计算,得到升力系数和阻力系数的数据;运用Hicks- Henne参数化方法,建立翼型弯度和厚度函数表达式.该研究为叶片的理论研究和优化设计提供了基础.     

Investigating the thermal environment effects on geometrically nonlinear vibration of smart functionally graded plates

An analytical solution for a sandwich circular FGM plate coupled with piezoelectric layers under one-dimensional heat conduction is presented. All materials of the device may be of any functional gradients in the direction of thickness. The solution exactly satisfies all the equilibrium conditions and continuity conditions for the stress, displacement and electric displacement as well as electric potential on the interfaces between adjacency layers. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the FG plate that is subjected to in-plane forces and applied actuator voltage in thermal environment in the case of simply supported boundary conditions. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectric coupled FGM plates are derived. The role of thermal environment as well as control effects on nonlinear static deflections and natural frequencies imposed by the piezoelectric actuators using high input voltages are investigated. Numerical examples are provided and simulation results are discussed. Numerical results for FGM plates with a mixture of metal and ceramic are presented in dimensionless forms. The good agreement between the results of this paper and those of the finite element (FE) analyses validated the presented approach. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage and thermal environment as well as gradient index of FG plate on the dynamics and control characteristics of the structure.     

A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Lévy-type rectangular plates) based on the Reissner-Mindlin plate theory. The material properties change continuously through the thickness of the plate, which can vary according to a power law distribution of the volume fraction of the constituents. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS-SSSC-SCSC-SCSF-SSSF-SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

含裂纹板断裂韧度厚度效应的理论与应用

从理论角度对断裂韧度厚度效应进行了研究。在线弹性断裂力学与含裂纹板的二维位移场的基础上,给出分离变量型的具有待定函数的三维位移场的表达式,进而通过几何方程与物理方程获取三维应变场和三维应力场。进一步,通过虚位移原理,使用变分方法建立待定函数所应满足的支配方程与边界条件,从而确定待定函数。基于上述分析,建立一个断裂韧度与试样厚度关系的理论表达式。最后通过两种试验材料LY12CZ(L-T)与TC4(L-T)进行了验证。     

含裂纹板断裂韧度厚度效应的理论研究

从理论角度对断裂韧度厚度效应进行研究。在线弹性断裂力学与含裂纹板的二维位移场的基础上，给出分离变量型的具有待定函数的三维位移场的表达式，进而通过几何方程与物理方程获取三维应变场和三维应力场。进一步，通过虚位移原理，使用变分方法建立待定函数所应满足的支配方程与边界条件，从而确定待定函数。基于上述分析，建立一个断裂韧度与试样厚度关系的理论表达式。最后通过两种试验材料LY12CZ（L-T）与TC4（L-T）进行验证。     

弹性圆柱壳的稳定性优化设计

研究任意轴对称边界条件下和受均布法向载荷作用圆柱壳的稳定性优化设计问题，即极大化屈曲临界载荷。利用能量原理分析轴对称变厚度圆柱壳的分支点屈曲，将求解屈曲临界载荷变成求解广义特征值方程，使圆柱壳稳定性优化设计成为极大化最小特征值问题。实际算例验证了本方法的有效性。研究结果可用于圆柱壳的加肋优化设计。     

弹性地基上转动功能梯度材料Timoshenko梁自由振动的微分变换法求解

基于Timoshenko梁理论研究弹性地基上转动功能梯度材料（FGM）梁的自由振动。首先确定功能梯度材料Timoshenko梁的物理中面,利用广义Hamilton原理推导出该梁在弹性地基上转动时横向自由振动的两个控制微分方程。其次采用微分变换法（DTM）对控制微分方程及其边界条件进行变换,计算了弹性地基上转动功能梯度材料Timoshenko梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种不同边界条件下横向自由振动的量纲一固有频率,与已有文献的计算结果进行比较,退化后结果一致。最后讨论了不同边界条件、转速、弹性地基模量和梯度指数对功能梯度材料Timoshenko梁自振频率的影响。结果表明：功能梯度材料Timoshenko梁的量纲一固有频率随量纲一转速和量纲一弹性地基模量的增大而增大;在量纲一转速和量纲一弹性地基模量一定的情况下,梁的量纲一固有频率随着功能梯度材料梯度指数的增大而减小。