Dispersion and characteristic surfaces of waves in hybrid multilayered piezoelectric circular cylinders

The frequency and group velocity dispersion behaviors, and characteristic surfaces of waves in a hybrid multilayered piezoelectric circular cylinder are investigated. The associated frequency dispersion equation is developed using an analytical-numerical method. In this method, the cylinder is modeled using the three-nodal-line layer element; the coupling between the elastic field and the electric field is considered in each element. A system of governing differential equations of each layer element is obtained following the Hamilton Principle. The phase velocity and slowness as well as the group velocity and slowness are established in terms of the Rayleigh quotient. Six characteristic wave surfaces, e.g. the phase velocity, slowness and wave surfaces as well as the group velocity, slowness and wave surfaces, are introduced to visualize the effects of anisotropy and piezoelectricity on wave propagation. A corresponding program code is developed and numerical examples are presented for hybrid multilayered piezoelectric circular cylinders with two ratios of radius to thickness.     

Material tailoring for functionally graded hollow cylinders and spheres

We present a technique to tailor materials for functionally graded (FG) linear elastic hollow cylinders and spheres to attain through-the-thickness either a constant hoop (or circumferential) stress or a constant in-plane shear stress. The volume fractions of two phases of a FG material (FGM) are assumed to vary only with the radius and the effective material properties are estimated by using either the rule of mixtures or the Mori-Tanaka scheme; the analysis is applicable to other homogenization methods. For a FG cylinder we find the required radial variation of the volume fractions of constituents to make a linear combination of the radial and the hoop stresses uniform throughout the thickness. The through-the-thickness uniformity of the hoop stress automatically eliminates the stress concentration near the inner surface of a very thick cylinder. The through-the-thickness variations of Young’s moduli obtained with and without considering the variation of Poisson’s ratio are very close to each other for a moderately thick hollow cylinder but are quite different in a very thick hollow cylinder. For an FG sphere the required radial variation of the volume fractions of the two phases to get a constant circumferential stress is similar to that in an FG cylinder. The material tailoring results presented here should help structural engineers and material scientists optimally design hollow cylinders and spheres comprised of inhomogeneous materials.     

Inverse problem of material distribution for desired fracture characteristics in a thick-walled functionally graded material cylinder with two diametrically-opposed edge cracks

This study concerns the inverse problem of evaluating the optimum material distribution for desired fracture characteristics in a thick-walled functionally graded material (FGM) cylinder containing two diametrically-opposed edge cracks emanating from the inner surface of the cylinder. The thermal eigenstrain developed in the cylinder material due to nonuniform coefficient of thermal expansion as a result of cooling from sintering temperature is taken into account. Based on a generalized method of evaluating stress intensity factors developed in a previous study, an inverse method is developed to optimize material distribution intending to realize prescribed apparent fracture toughness in the FGM cylinder. To present some numerical results, a TiC/Al₂O₃ FGM cylinder is considered and the inverse problems are solved to evaluate material distributions for two examples of prescribed apparent fracture toughness. The effect of cylinder wall thickness on the material distribution and comparison of material distributions corresponding to a single and two cracks are also discussed. The numerical results reveal that the apparent fracture toughness of FGM cylinders can be controlled by choosing the material distributions properly.     

Vibro-acoustic behavior of a hollow FGM cylinder excited by on-surface mechanical drives

The linear three-dimensional elasticity theory in conjunction with the powerful transfer matrix solution technique is employed to investigate the steady-state nonaxisymmetric sound radiation characteristics of an arbitrarily thick functionally graded hollow cylinder of infinite length subjected to arbitrary time-harmonic on-surface concentrated mechanical drives. A formal integral expression for the radiated pressure field in the frequency domain is obtained by utilizing the spatial Fourier transform along the shell axis and Fourier series expansion in the circumferential direction. The method of stationary phase is subsequently employed to evaluate the integral for an observation point in the far-field. The analytical results are illustrated with numerical examples in which water-submerged metal-ceramic FGM cylinders are driven by harmonic concentrated radial/transverse surface forces and circumferential moment. The far-field radiated pressure amplitudes and directivities are calculated and compared with those of equivalent bi-laminate hollow cylinders with comparable volume fractions of constituent materials. The effects of FGM material profile, cylinder thickness, excitation frequency and type on the radiated far-field are examined. Limiting cases are considered and the validity of results is established by comparison with the data in the existing literature as well as with the aid of a commercial finite element package.     

Dynamic response of an FGM cylindrical shell under moving loads

This paper presents an analytical study on the dynamic behavior of the infinitely-long, FGM cylindrical shell subjected to combined action of the axial tension, internal compressive load and ring-shaped compressive pressure with constant velocity. It is assumed that the cylindrical shell is a mixture of metal and ceramic that its properties changes as a function of the shell thickness. The problem is studied on the basis of the theory of vibrations of cylindrical shells. Derived formulas for the maximum static and dynamic displacements, dynamic factors and critical velocity for the FGM cylindrical shell subjected to moving loads. Numerical calculations have been made for fully metal, fully ceramic and FGM (Si₃N₄/SUS304) cylindrical shells. A parametric study is conducted to demonstrate the effects of the material property gradient, the radius to thickness ratio and the velocity of the moving load on the dynamic displacements and dynamic factors of the inner and ring-shaped pressures for FGM cylindrical shells.     

Exact Solutions for the Stability and Free Vibration of Multilayered Functionally Graded Material Hollow Cylinders under Axial Compression

Exact three-dimensional stability and free vibration analyses of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders and laminated composite ones under axial compression are presented. The material properties of each FGM layer are assumed to obey a power-law distribution of the volume fractions of constituents through the thickness coordinate. The Pagano method, which is based on the principle of virtual displacement and is conventionally used for the analysis of laminated composite structures, is modified to be feasible for the study of multilayered FGM cylinders, in which Reissner's mixed variational theorem, the successive approximation and transfer matrix methods, and the transformed real-valued solutions of the system equations are used. The present modified Pagano solutions for laminated composite cylinders are in excellent agreement with the exact 3D ones available in the literature, and those for sandwich FGM cylinders may be used as the benchmark solutions to assess the ones obtained using various two-dimensional theories and numerical models. The influence of some effects on the lowest critical load parameters of multilayered FGM cylinders and laminated composite ones is investigated, such as the derivation between using von Karman nonlinearity and full kinematic one, and the difference between using the uniform stress assumption and the uniform strain one. In addition, a parametric study with regard to some effects on the lowest frequency parameters of axially loaded, multilayered FGM cylinders is carried out, such as the magnitude of the applied compressive loads, the radius-to-thickness, length-to-radius and orthotropic ratios, and the material-property gradient index.     

Thermal stability analysis of circular functionally graded sandwich plates of variable thickness using pseudo-spectral method

In the present study, the thermal stability of laminated functionally graded (FGM) circular plates of variable thickness subjected to uniform temperature rise based on the first-order shear deformation plate theory is presented. Furthermore, two models for FGM plates with variable thickness, corresponding with two manufacturing methods, are proposed. The laminated FGM plate with variable thickness is considered as a sandwich plate constituted of a homogeneous core of variable thickness and two constant thickness FGM face sheets whose material properties are assumed to be graded in the thickness direction according to a simple power law. In order to determine the distribution of the prebuckling thermal load along the radius, the membrane equation is solved using the shooting method. Subsequently, employing the pseudo-spectral method that makes use of Chebyshev polynomials, the stability equations are solved numerically to evaluate the critical temperature rise. The results demonstrate that the thermal stability is significantly influenced by the thickness variation profile, aspect ratio, the volume fraction index, and the core-to-face sheet thickness ratio.     

Analysis of elastic wave propagation in a functionally graded thick hollow cylinder using a hybrid mesh-free method

In this paper, a hybrid mesh-free method based on generalized finite difference (GFD) and Newmark finite difference (NFD) methods is presented to calculate the velocity of elastic wave propagation in functionally graded materials (FGMs). The physical domain to be considered is a thick hollow cylinder made of functionally graded material in which mechanical properties are graded in the radial direction only. A power-law variation of the volume fractions of the two constituents is assumed for mechanical property variation. The cylinder is excited by shock loading to obtain the time history of the radial displacement. The velocity of elastic wave propagation in functionally graded cylinder is calculated from periodic behavior of the radial displacement in time domain. The effects of various grading patterns and various constitutive mechanical properties on the velocity of elastic wave propagation in functionally graded cylinders are studied in detail. Numerical results demonstrate the efficiency of the proposed method in simulating the wave propagation in FGMs.     

General analytical solution for elastic radial wave propagation and dynamic analysis of functionally graded thick hollow cylinders subjected to impact loading

An analytical method is proposed for the dynamic response analysis of functionally graded thick hollow cylinders under impact loading. The wave motion equation is solved using an analytical method that is based on the composition of Bessel functions. The mechanical properties are considered as power functions of the radius across the thickness of FG cylinder. The FG cylinder is excited by an impact loading at the inner surface of the cylinder, and the plane strain and axisymmetry conditions are assumed for the problem. The time histories of radial displacement and radial and hoop stresses are presented. Also the dynamic response of the FG cylinder is obtained and discussed for various kinds of power function exponents.     

基于裂纹尖端应力比值的含裂纹功能梯度材料圆筒应力强度因子计算方法

由于功能梯度材料（FGM）性质的特殊性,现有含裂纹FGM结构应力强度因子计算方法难以避免复杂的矩阵运算以及数值积分。该文针对含外表面环向裂纹FGM圆筒,利用FGM圆筒与均匀材料圆筒裂纹尖端应力之间的比例关系,将复杂的FGM圆筒应力强度因子求解问题转化为简单的应力值提取问题以及经验公式计算问题,仅由均匀材料圆筒应力强度因子经验公式、均匀材料圆筒和FGM圆筒裂纹尖端应力比值即可得到任意含裂纹FGM圆筒应力强度因子。该方法仅需建立2D轴对称模型即可满足计算要求,在保证精度的基础上成功回避了传统方法中的复杂矩阵运算以及数值积分,且适用于不同FGM、筒体尺寸、裂纹深度等情况下的应力强度因子计算。通过多组算例对比分析,证明该方法计算精度高、计算过程简便,便于工程应用。     

Stress intensity factors for internal semi-elliptical surface cracks in pressurized thick-walled cylinders using the hybrid boundary element method

The purpose of this paper is to present a comprehensive range of results of mode I SIFs of three-dimensional surface cracks in internally pressurized thick-walled cylinders. The hybrid boundary element method is summarily reviewed and used to calculate the SIFs of surface cracks in pressurized thick-walled cylinders. The analyzed ratio of crack depth to wall thickness ranges from 0.2 to 0.8; the ratio of crack depth to crack length ranges from 0.25 to 1.0; and the ratio of wall thickness to cylinder radius is 0.5, 1.0 and 2.0. The present normalized SIFs are also compared with other solutions from the literature. The recent results of the body force method and the finite element method agree well (ca 3%), and the early ones of the boundary integral equation and the finite element method agree fairly well (ca 10%) with the present results.     

A unified formulation of RMVT-based finite cylindrical layer methods for sandwich circular hollow cylinders with an embedded FGM layer

A unified formulation of finite cylindrical layer methods (FCLMs) based on the Reissner mixed variational theorem (RMVT) is developed for the quasi-three-dimensional (3D) analysis of simply-supported, multilayered composite cylinders and sandwich circular hollow cylinders with an embedded functionally graded material (FGM) cylindrical layer, subject to mechanical loads. The material properties of the FGM layer are assumed to obey an exponent-law varying exponentially with the thickness coordinate. In this formulation, the circular hollow cylinder is divided into a number of finite cylindrical layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the field variables of each individual layer, respectively. Because an h-refinement instead of a p-refinement process is adopted to yield the convergent solutions in this work, the layerwise linear, quadratic or cubic function distribution through the thickness coordinate is assumed for the related field variables. The accuracy of the FCLMs developed in this article is assessed by comparing their solutions with the exact 3D ones available in the literature, and the convergence rate and possibility of numerical instability of these FCLMs are also investigated.     

A unified formulation of RMVT-based finite cylindrical layer methods for sandwich circular hollow cylinders with an embedded FGM layer

A unified formulation of finite cylindrical layer methods (FCLMs) based on the Reissner mixed variational theorem (RMVT) is developed for the quasi-three-dimensional (3D) analysis of simply-supported, multilayered composite cylinders and sandwich circular hollow cylinders with an embedded functionally graded material (FGM) cylindrical layer, subject to mechanical loads. The material properties of the FGM layer are assumed to obey an exponent-law varying exponentially with the thickness coordinate. In this formulation, the circular hollow cylinder is divided into a number of finite cylindrical layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the field variables of each individual layer, respectively. Because an h-refinement instead of a p-refinement process is adopted to yield the convergent solutions in this work, the layerwise linear, quadratic or cubic function distribution through the thickness coordinate is assumed for the related field variables. The accuracy of the FCLMs developed in this article is assessed by comparing their solutions with the exact 3D ones available in the literature, and the convergence rate and possibility of numerical instability of these FCLMs are also investigated.     

Bending,buckling and vibration of size-dependent functionally graded annular microplates

The bending, buckling and free vibration of annular microplates made of functionally graded materials (FGMs) are investigated in this paper based on the modified couple stress theory and Mindlin plate theory. This microplate model incorporates the material length scale parameter that can capture the size effect in FGMs. The material properties of the FGM microplates are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using Hamilton’s principle. The differential quadrature (DQ) method is employed to discretize the governing equations and to determine the deflection, critical buckling load and natural frequencies of FGM microplates. A parametric study is then conducted to investigate the influences of the length scale parameter, gradient index and inner-to-outer radius ratio on the bending, buckling and vibration characteristics of FGM microplates with hinged–hinged and clamped–clamped supports. The results show that the size effect on the bending, buckling and vibration characteristics is significant when the ratio of the microplate thickness to the material length scale parameter is smaller than 10.     

正交各向异性空心圆柱体中的纵向导波

基于三维线弹性理论,采用勒让德正交多项式展开法,推导了正交各向异性材料中纵向导波的耦合波动方程,并对耦合波动方程进行了数值求解。首先,为确定方法的适用性和准确性,利用Disperse软件求解各向同性空心圆柱体中纵向导波的频散曲线,并将其与勒让德正交多项式展开法求解结果相对比,二者结果完全一致。然后,讨论了勒让德正交多项式截止值对轴对称导波频散曲线收敛性的影响,并从数值计算的角度分析了产生影响的原因。最后,针对碳纤维缠绕的复合材料空心圆柱体,分别求解纵向、扭转和弯曲三种不同模态纵向导波的相速度频散曲线。计算了不同径厚比下的弯曲模态相速度频散曲线,分析径厚比的变化对频散曲线的影响。     

Wave propagation in coated cylinders with reference to fretting fatigue

Fretting fatigue is the phenomenon of crack initiation due to dynamic contact loading, a situation which is commonly encountered in mechanical couplings subjected to vibration. The study of fretting fatigue in high frequency regime has gained importance in recent years. However the stress wave effects at high frequency loading is scanty in the literature. The objective of present investigation is to study stress wave propagation in cylinders with reference to high frequency fretting. The case of a coated cylinder is considered since coating is often provided to improve tribological properties of the component. Rule of mixtures is proposed to understand the dispersion phenomenon in coated or layered cylinder knowing the dispersion relation for the cases of homogeneous cylinders made of coating and substrate materials separately. The possibility of stress wave propagation at the interface with a particular phase velocity without dispersion is also discussed. Results are given for two different thicknesses of coating.     

Elastic-plastic stress analysis in a long functionally graded solid cylinder with fixed ends subjected to uniform heat generation

Elastic-plastic deformation of a solid cylinder with fixed ends, made of functionally graded material (FGM) with uniform internal heat generation is investigated, based on Tresca’s yield criterion and its associated flow rule, considering four of the material properties to vary radially according to a parabolic form. These four material properties are yield strength, modulus of elasticity, coefficients of thermal conduction and thermal expansion, assumed to be independent of temperature as Poisson’s ratio which is taken as constant. The materials which compose the functionally graded cylinder are supposed to be elastic-perfectly plastic materials. Expressions for the distributions of stress, strain and radial displacement are found analytically in terms of unknown interface radii. After determining these radii numerically by means of Mathematica 5.2, the distributions are plotted versus dimensionless radius, increasing heat generation, to compare the FGM cylinder with the homogeneous one. The numerical values used in this work for material parameters are arbitrarily chosen to point out the effect of the non-homogeneity on the stress distribution. The results obtained show that the stress distribution, as well as the development of plastic region radii, is influenced substantially by the material non-homogeneity.     

Coupled thermoelasticity and second sound in finite length functionally graded thick hollow cylinders (without energy dissipation)

In this article, the coupled thermoelasticity behavior of functionally graded thick hollow cylinders is studied. The governing coupled thermoelasticity and the energy equations are solved for a finite length functionally graded cylinder subjected to thermal shock load. The coupled thermoelastic equations are considered based on Green–Naghdi theory. The mechanical properties of cylinder are graded across the thickness as a power law function of radius. The cylinder is assumed to be made of many isotropic sub-cylinders (layers) across the thickness. Functionally graded properties are created by suitable arrangement of layers and governing equations are expanded in longitudinal direction by means of trigonometric function expansion. The Galerkin Finite Element and Newmark Methods are used to analyze the cylinder. The dynamic behavior of temperature distribution, mechanical displacement and thermal stresses is obtained and discussed. The second sound and elastic wave propagation are determined for various kinds of variation in the mechanical properties. The comparison of present results with published data shows the excellent agreement.     

Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations

Nonlinear vibration, nonlinear bending and postbuckling analyses are presented for a sandwich plate with FGM face sheets resting on an elastic foundation in thermal environments. The material properties of FGM face sheets 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. The governing equation of the plate that includes plate-foundation interaction is solved by a two-step perturbation technique. The thermal effects are also included and the material properties of both FGM face sheets and homogeneous core layer are assumed to be temperature-dependent. The numerical results reveal that the foundation stiffness and temperature rise have a significant effect on the natural frequency, buckling load, postbuckling and nonlinear bending behaviors of sandwich plates. The results also reveal that the core-to-face sheet thickness ratio and the volume fraction distribution of FGM face sheets have a significant effect on the natural frequency, buckling load and postbuckling behavior of the sandwich plate, whereas this effect is less pronounced for the nonlinear bending, and is marginal for the nonlinear to linear frequency ratios of the same sandwich plate.     

Elastic behavior of glass-like functionally graded infinite hollow cylinder under hydrostatic loads using finite element method

A glass-like (viscoelastic) functionally graded cylinder is studied by using finite element method to investigate the mechanical responses. A subroutine is developed by using ANSYS parametric design language (APDL) to simulate two nonlinearities, which are the variation of material properties with respect to time and position. The cylinder is made of two different viscoelastic materials, namely, pure material one at inner and pure material two at outer surfaces. The material properties are assumed to be presented by simple power law distribution and moreover, bulk and shear moduli are varying with respect to time using the kernel functions depicted regarding Prony series. It is shown that the hoop stresses take the same values at the mean radius (middle of the thickness) for different values of time and grading index. It is found that the radial stress decreases to certain values for specific grading index and then by increasing the grading index it increases to maximum value that related to pure material cylinder. It is shown that unlike the zero axial stress in pure material cylinders, it varies along the thickness from minimum to maximum at inner and outer surfaces, respectively. It is concluded that the viscoelastic functionally graded (VFG) materials play an important role in steady and transient response of hollow cylinder under hydrostatic load.