Higher order vibrations of thin periodic plates

A problem of higher order vibrations of thin periodic plates related to a plate periodicity is discussed. The applied model, based on the Kirchhoff plate theory assumptions and additional hypothesis of tolerance averaging [Wo?niak Cz, Wierzbicki E. Averaging techniques in thermomechanics of composite solids. Tolerance averaging versus homogenization. Poland, Cz?stochowa: Cz?stochowa University of Technology; 2000], makes it possible to take into account the effect of the periodicity cell size on the overall plate behaviour. Assuming more functions describing oscillations of the periodicity cell, we can investigate within the model not only fundamental vibrations but also higher order vibrations related to a periodic plate structure. These functions have to be properly chosen approximations of solutions to eigenvalue problem for natural vibrations of separated cell with periodic boundary conditions, [J?drysiak J. On vibrations of thin plates with one-dimensional periodic structure. Int J Eng Sci 2000;38:2023-2043]. Using the known two-dimensional plate models—the orthotropic plate model and finite element method as well as the discrete model, certain justifications of the proposed model are made.     

薄板的高阶振动

对薄板的高阶振动进行了讨论。应用模型基于Kirchhoff平板理论的假定和公差平均值的假定[Wo■ niak Cz,Wierzbicki E.Averaging techniques in thermomechanics of composite solids.Tolerance averaging versus homogenization.Poland,Czestochowa:Czestochowa University of Technology;2000],这使得模型可以考虑到周期单元对于整块板性能的影响。假设对周期单元振动的影响参数过多,可以对模型进行比基本振型更高阶的振动分析。这些函数是通过从边界条件中分离周期单元,对其自振时的特征值方程分析选取的近似值[JedrysiakJ.On vibrations of thin plates with one-dimensional periodic structure.Int J Eng Sci 2000;38:2023-2043]。通过应用平面板模型-正交各向异性模型、有限元模型和离散模型,对提议模型进行了论证。     

Buckling of imperfect functionally graded plates under in-plane compressive loading

Buckling behavior of rectangular functionally graded plates with geometrical imperfections is studied in this paper. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the classical plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power function of the thickness variable. The plate is assumed to be under in-plane compressive loading. Simultaneous solving of the stability and compatibility equations in conjunction with the equilibrium equations leads to the buckling relation of the plate. The critical buckling load of a sample plate is obtained and compared for different geometrical ratios. The results are reduced and compared with the results of perfect functionally graded and imperfect isotropic plates.     

Coupled thermoelasticity of functionally graded plates based on the third-order shear deformation theory

In this paper, the analytical solution is presented for a plate made of functionally graded materials based on the third-order shear deformation theory and subjected to lateral thermal shock. The material properties of the plate, except Poisson's ratio, are assumed to be graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The solution is obtained under the coupled thermoelasticity assumptions. The temperature profile across the plate thickness is approximated by a third-order polynomial in terms of the variable z with four unknown multiplier functions of (x,y,t) to be calculated. The equations of motion and the conventional coupled energy equation are simultaneously solved to obtain the displacement components and the temperature distribution in the plate. The governing partial differential equations are solved using the double Fourier series expansion. Using the Laplace transform, the unknown variables are obtained in the Laplace domain. Applying the analytical Laplace inverse method, the solution in the time domain is derived. Results are presented for different power law indices and the coupling coefficients for a plate with simply supported boundary conditions. The results are validated based on the known data for thermomechanical responses of a functionally graded plate reported in the literature.     

Nonlinear vibration of initially stressed functionally graded plates

In this paper, nonlinear partial differential equations for the vibration motion of an initially stressed functionally graded plate (FGP) are derived. The formulations are based on the classical plate theory and derived for the nonlinear vibration motion of an FGP in a general state of arbitrary initial stresses. The material properties of the FGP are assumed to vary continuously from one surface to another, according to a simple power law distribution in terms of the constituent volume fractions. The motion of functionally graded ceramic/metal plate is obtained by employing the Galerkin method and then solved by Runge–Kutta method. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the ceramic/metal plate. It is found that the frequency responses of nonlinear vibration are sensitive of the state of initial stress, the amplitude of vibration and the volume fraction of FGP.     

Buckling of cracked functionally graded plates under tension

Buckling of functionally graded cracked plates under tension has not been investigated so far. In this paper critical buckling load of functionally graded plates containing a crack has been obtained using classical plate theory through the finite element method. Displacement in vicinity of crack tips has been approximated using previous solutions related to bending of cracked plates. Effect on buckling of plate under uni-axial and bi-axial tension of different parameters, such as plate dimensions and material properties, are studied. Results show that the critical load decreases as material gradient index increases, while bi-axial loading leads to higher critical loads compared to uni-axial case.     

Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection

Nonlinear thermomechanical post-buckling of an imperfect functionally graded material (FGM) circular plate, subjected to both mechanical load and transversely non-uniform temperature rise, is presented. The material properties of FGM plates 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. Based on von Kármán's plate theory, equilibrium equations governing a large axi-symmetric deformation of the FGM circular plate under thermomechanical loads are derived. In the analysis, the geometric imperfections of the plate are taken into account. By using a shooting method the nonlinear ordinary differential equations with immovably clamped boundary conditions are solved numerically. Responses for the nonlinear thermomechanical post-buckling responses of the FGM plate are obtained. Numerical examples are presented that relate to the performances of perfect and imperfect, homogenous and graded plates. Characteristic curves of the post-buckling deformation of the imperfect FGM circular plate varying with thermal loads, imperfection parameters and volume fraction index are plotted. And then effects of the load parameters, materials constitution, and the geometric imperfection of the plate on the deformation are discussed in detail.     

对带几何缺陷的FGM圆板的非线性热力学后屈曲现象的分析

带有几何缺陷FGM圆板的非线性热力学后屈曲现象是由力学荷载及横向不均匀温度升高导致的。基于组成材料的体积分数导致的能量分布法则,FGM板的材料特性假定在板厚度方向呈梯度变化。根据VonK偄rm偄n平板理论,推导出FGM板在温度荷载作用下的轴对称大变形的计算公式。在此分析中同时考虑了板的几何缺陷。通过运用目标法,解决了那些带有固定边界条件的非线性方程。数值算例中考虑了是否带有几何缺陷、均质或非均质的钢板。带缺陷FGM圆板的后屈曲变形特征曲线根据温度荷载、缺陷参数和体积分数的变化绘制而成。文中同时讨论了荷载参数、材料构成及板的几何缺陷对变形的影响。     

Manufacturing technique and performance of functionally graded concrete segment in shield tunnel

The quality of segment is very important to the service life of shield tunnel. Concerning the complex engineering environment of the Wuhan Yangtze River Shield Tunnel, the principle of functionally graded materials was introduced to design and produce the functionally graded concrete segment (FGCS). Its key manufacturing technique was proposed and its performance was tested. __________ Translated from Construction Technology, 2008, 37(1): 106–108 [译自:施工技术]     

基于传统板理论的智能圆形FGM薄板自由振动研究

基于传统板理论（CPT）对圆形功能梯度（FG）薄板的自由振动性能进行分析。这种板具有2个由压电（PZT4）材料构成的均布驱动器。对FG底层板的材料特性进行假定,考虑了材料体积分数的能量分布,将材料沿厚度方向进行分级,并采用二次方程模拟沿压电覆层厚度方向分布的电势场。采用运动微分方程来模拟板的固定端边界条件。介绍了详细的数学推导过程,并采用数值分析来研究FG板的梯度变化对结构自由振动的影响。其分析结果得到了三维有限元分析结果的证实。     

A deep feed-forward neural network for damage detection in functionally graded carbon nanotube-reinforced composite plates using modal kinetic energy

This paper proposes a new Deep Feed-forward Neural Network (DFNN) approach for damage detection in functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates. In the proposed approach, the DFNN model is developed based on a data set containing 20 000 samples of damage scenarios, obtained via finite element (FE) simulation, of the FG-CNTRC plates. The elemental modal kinetic energy (MKE) values, calculated from natural frequencies and translational nodal displacements of the structures, are utilized as input of the DFNN model while the damage locations and corresponding severities are considered as output. The state-of-the art Exponential Linear Units (ELU) activation function and the Adamax algorithm are employed to train the DFNN model. Additionally, in order to enhance the performance of the DFNN model, the mini-batch and early-stopping techniques are applied to the training process. A trial-and-error procedure is implemented to determine suitable parameters of the network such as the number of hidden layers and the number of neurons in each layer. The accuracy and capability of the proposed DFNN model are illustrated through two distinct configurations of the CNT-fibers constituting the FG-CNTRC plates including uniform distribution (UD) and functionally graded-V distribution (FG-VD). Furthermore, the performance and stability of the DFNN model with the consideration of noise effects on the input data are also investigated. Obtained results indicate that the proposed DFNN model is able to give sufficiently accurate damage detection outcomes for the FG-CNTRC plates for both cases of noise-free and noise-influenced data.     

Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method

In the present study, the free vibration of laminated functionally graded carbon nanotube reinforced composite beams is analyzed. The laminated beam is made of perfectly bonded carbon nanotubes reinforced composite (CNTRC) layers. In each layer, single-walled carbon nanotubes are assumed to be uniformly distributed (UD) or functionally graded (FG) distributed along the thickness direction. Effective material properties of the two-phase composites, a mixture of carbon nanotubes (CNTs) and an isotropic polymer, are calculated using the extended rule of mixture. The first-order shear deformation theory is used to formulate a governing equation for predicting free vibration of laminated functionally graded carbon nanotubes reinforced composite (FG-CNTRC) beams. The governing equation is solved by the finite element method with various boundary conditions. Several numerical tests are performed to investigate the influence of the CNTs volume fractions, CNTs distributions, CNTs orientation angles, boundary conditions, length-to-thickness ratios and the numbers of layers on the frequencies of the laminated FG-CNTRC beams. Moreover, a laminated composite beam combined by various distribution types of CNTs is also studied.     

三维轴对称功能梯度材料瞬态热传导问题的自然单元法

为了更有效地求解三维轴对称功能梯度材料瞬态热传导问题,对无网格自然单元法应用于此类问题进行了研究,并发展了相应的计算方法。基于几何形状和边界条件的轴对称性,三维的轴对称问题可降为二维平面问题。为了简化本质边界条件的施加,轴对称面上的温度场采用自然邻近插值进行离散。功能梯度材料特性的变化由高斯点的材料参数进行模拟。时间域上,采用传统的两点差分法进行离散求解,进而得到瞬态温度场的响应。数值算例结果表明,提出的方法是行之有效的,理论及方法不仅拓展了自然单元法的应用范围,而且对三维轴对称瞬态热传导分析具有普遍意义。     

基于高阶剪、正变形板理论和径向基函数的MLPG方法进行功能分级厚板分析

采用MLPG和HOSNDPT方法对分级厚弹性板的极小变形进行了分析。两类径向基函数,即二次多项式型和薄板条带型,用于构建试解。在局部域,四阶样条函数被用作重力/测试函数。采用Mori-Tanaka均匀化技术计算了由仅在厚度方向变化的各向同性材料的要素所构成的板的有效模量。对于简单支撑的铝/陶瓷板,计算结果与解析解十分吻合。两对边自由、另两对边简单支撑板的计算结果,与采用有限元方法分析板的三维变形所获得结果具有一致性。对于不同边界条件下,板挠度和应力的分配也进行了分析。研究发现,两种类型的径向基函数均给出了板挠度的精确值,而二次多项式函数给出了比薄板条带函数更为精确的应力值。     

Thermoelastic dynamic stability of thin-walled beams with graded material properties

The dynamic stability of functionally graded thin-walled beams allowing for shear deformability is investigated in this article. The analysis is based on a model that has small strains and moderate rotations which are formulated through the adoption of a second-order non-linear displacement field. The beam is subjected to axial external dynamic loading. The model takes into account thermoelastic effects. The heat conduction equation is solved in order to characterize the temperature in the cross-sectional domain. Galerkin's and Bolotin's methods are employed with the scope to discretize the governing equations and to determine the regions of dynamic stability, respectively. Regions of stability are evaluated and expressed in non-dimensional form. The influence of the longitudinal vibration on the unstable regions is investigated. The numerical results show the importance of this effect when the forcing frequency approaches to the natural longitudinal frequency, obtaining substantially wider parametric stability regions. The effects of temperature gradients, shear flexibility and axial inertia, in beams with different cross-sections and different types of graded material are analyzed as well.     

基于现场大型承载试验的原级配筑坝堆石料力学参数反演研究

基于水布垭面板坝的现场大型承载试验成果,采用基于IGA的反演方法,研究了栖霞组灰岩、茅口组灰岩两种原级配筑坝堆石料的邓肯E–B模型参数。结果表明:在相同应力加载路径条件下,现场原级配堆石料比经过缩尺后的室内三轴试验试样具有更大的体积变形和更高的切线弹性模量,两者的力学性状存在较大差别。对栖霞组原级配灰岩堆石料的回弹模量参数反演分析得到K ur约为K值的2倍,而nu r远大于邓肯建议的直接引用加载时的弹性模量指数n值。宜加强现场原级配粗粒堆石料的力学性质研究,为超高堆石坝的建设提供更为可靠的理论依据。     

Eccentric low-velocity impact analysis of transversely graded plates with Winkler-type elastic foundations and fully or partially supported edges

Impact analysis of plates with either elastic foundations or partially supported edges has not been accomplished, even for the homogeneous isotropic plates. In the present paper, eccentric low-velocity impact analysis of a functionally graded plate with fully/partially supported edges and a Winkler-type elastic foundation is performed. In contrast to the available researches, stiffness of the underneath layers is also taken into account in determination of the apparent stiffness of the contact region, employing Mori–Tanaka micromechanics-based model instead of the traditional rule of mixtures. The governing equations are derived based on the principle of minimum total potential energy and the first-order shear-deformation plate theory. Some of the included novelties are: modification of the contact law, incorporation of effects of the elastic foundation, considering the material heterogeneity and partially supporting the edges, and proposing various semi-analytical and (Galerkin-type) numerical solutions. Numerical time integration schemes are used to trace time variations of the responses. A sensitivity analysis is performed to investigate influences of the specifications of the plate and the indenter, the eccentricity, and the supporting length ratio on time histories of the indentation and contact force.     

The evolution of spatial patterns of residential segregation in Central European Cities: The Łódź Functional Urban Region from mature socialism to mature post-socialism

Two themes dominated the study of socio-spatial change in Central Eastern Europe (CEE) after 1990. One was that the development of urban regions in CEE after the demise of socialism is characterized by suburbanization. The second topic raises the issue of growing residential segregation. Unfortunately, studies on patterns of residential segregation are still scarce in the CEE context, in particular those that employ traditional measures of segregation. Relying on solid empirical materials from the 1988 and 2002 National Censuses, the purpose of this paper is to map and analyze the patterns of residential segregation in the ?ód? Functional Urban Region, an old industrial urban region in Poland undergoing a rapid deconcentration and shrinkage of the core area. This contribution concludes that the first decade of systemic transition resulted in decreasing residential segregation, as gauged by global measures of segregation.     

The tolerance averaging model of dynamic stability of thin plates with one-directional periodic structure

A dynamic stability of thin plates with one-directional periodic structure is investigated. An averaged non-asymptotic approach, called the tolerance averaging, is applied to derive governing equations of these plates. Obtained tolerance model equations of the Kirchhoff-type plate take into account the effect of the period lengths on the overall plate behaviour [Jędrysiak J. Dispersive models of thin periodic plates. Theory and applications. Sci. Bul. Łódź Tech. Univ., No. 872, series. Sci Trans 289, Łódź, 2001 [in Polish]]. It is shown that this effect plays a crucial role in some special cases of a dynamic stability of such plates.     

A refined FSDT for the static analysis of functionally graded sandwich plates

This paper presents a static analysis of functionally graded plates (FGPs) by using a new first shear deformation theory (FSDT). This theory contains only four unknowns, with is even less than the classical FSDT. In this paper a simply supported FG square sandwich plate is subjected to a bi-sinusoidal load. The governing equations for static bending analysis are derived by employing the principle of virtual works. These equations are then solved via Navier-type, closed form solutions. The accuracy of the present theory is ascertained by comparing it with various available solutions in the literature.