Three-dimensional analytical solution for functionally graded magneto–electro-elastic circular plates subjected to uniform load

The problem of a functionally graded, transversely isotropic, magneto–electro-elastic circular plate acted on by a uniform load is considered. The displacements and electric potential are represented by appropriate polynomials in the radial coordinate, of which the coefficients depends on the thickness coordinate, and are called the generalized displacement functions. The governing equations as well as the boundary conditions for these generalized displacement functions are derived from the original equations of equilibrium for axisymmetric problems and the boundary conditions on the upper and lower surfaces of the plate. Explicit expressions are then obtained through a step-by-step integration scheme, with five integral constants determinable from the boundary conditions at the cylindrical surface in the Saint Venant’s sense. The analytical solution is suited to arbitrary variations of material properties along the thickness direction, and can be readily degenerated into those for homogeneous plates. A particular circular plate, with some material constants being the exponential functions of the thickness coordinate, is finally considered for illustration.     

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.     

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.     

A periodic array of cracks in functionally graded materials subjected to transient loading

A periodic array of cracks in an infinite functionally graded material under transient mechanical loading is investigated. In-plane normal (mode I) and shear (mode II) loading conditions are considered. For each individual loading mode, a singular integral equation is derived, in which the crack surface displacements are unknown functions. Numerical results are obtained to illustrate the variation of the stress intensity factors as a function of the crack periodicity for different values of material inhomogeneity, either at the transient state or steady state. The material inhomogeneity can increase or decrease the mode I and mode II stress intensity factors. Compared with the single crack solution, it is also shown that multiple cracking may decrease the mode I stress intensity factors, but enhance the mode II stress intensity factors significantly.     

Interaction integrals for thermal fracture of functionally graded materials

This paper addresses finite element evaluation of the non-singular T-stress and mixed-mode stress intensity factors in functionally graded materials (FGMs) under steady-state thermal loads by means of interaction integral. Interaction integral provides an accurate and efficient numerical framework in evaluating these fracture parameters in FGMs under thermal as well as mechanical loads. We use a non-equilibrium formulation and the corresponding auxiliary (secondary) fields tailored for FGMs. Graded finite elements have been developed to account for the spatial gradation of thermomechanical properties. This paper presents various numerical examples in which the accuracy of the present method is verified.     

均匀热载荷作用下功能梯度圆板的非线性振动

基于经典板理论,研究了热载荷作用下功能梯度圆板的大幅振动问题.在经典板理论下利用物理中面概念,导出了功能梯度圆板的非线性运动方程.利用Ritz?Kantorovich方法消去时间变量,将非线性运动方程转换成了一组关于空间变量的非线性常微分方程.采用打靶法数值求解所得方程,并利用数值结果研究了热载荷作用下功能梯度圆板静态...     

An interaction integral method for analysis of cracks in orthotropic functionally graded materials

This paper presents two new interaction integrals for calculating stress-intensity factors (SIFs) for a stationary crack in two-dimensional orthotropic functionally graded materials of arbitrary geometry. The method involves the finite element discretization, where the material properties are smooth functions of spatial co-ordinates and two newly developed interaction integrals for mixed-mode fracture analysis. These integrals can also be implemented in conjunction with other numerical methods, such as meshless method, boundary element method, and others. Three numerical examples including both mode-I and mixed-mode problems are presented to evaluate the accuracy of SIFs calculated by the proposed interaction integrals. Comparisons have been made between the SIFs predicted by the proposed interaction integrals and available reference solutions in the literature, generated either analytically or by finite element method using various other fracture integrals or analyses. An excellent agreement is obtained between the results of the proposed interaction integrals and the reference solutions. The authors would like to acknowledge the financial support of the U.S. National Science Foundation (NSF) under Award No. CMS-9900196. The NSF program director was Dr. Ken Chong.     

Nonlinear dynamic buckling of functionally graded cylindrical shells subjected to time-dependent axial load

This paper is presented to solve the nonlinear dynamic buckling problem of a new type of composite cylindrical shells, made of ceram/metal functionally graded materials. The material properties vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. The dynamic axial load is set in a linear increase form with regard to time. By taking the temperature-dependent material properties into account, the effect of environmental temperature rise is included. The nonlinear dynamic equilibrium equation of the shell was obtained by applying an energy method, and was then solved using the four-order Runge–Kutta method. The critical condition was eventually determined using B-R dynamic buckling criterion. Numerical results show the dynamic buckling load is higher than its static counterpart. Meanwhile, various effects of the inhomogeneous parameter, loading speed, dimension parameter, environmental temperature rise and initial geometrical imperfection on nonlinear dynamic buckling are discussed.     

Elastica solution for thermal bending of a functionally graded beam

The problem of nonlinear thermal bending of a pinned slender beam fabricated of functionally graded material is considered. Based on the concept of physically neutral surface, the problem is reduced to a system of two coupled transcendental equations in terms of Legendre’s elliptic integrals. Solutions of these equations are presented in graphical and tabular form. Specific features of the nonlinear response of the functionally graded beam under thermal loading are discussed.     

Dynamic response of functionally graded circular cylindrical shells subjected to radial impulse load

Dynamic response of simply supported circular cylindrical shell made of functionally graded material (FGM) subjected to lateral impulse load is investigated. The effective material properties are assumed to vary continuously along the thickness direction according to a volume fraction power law distribution. First order shear deformation theory and Love’s first approximation theory are utilized in the equilibrium equations. Finally time response of displacement components is derived using mode superposition method. The influence of material composition, FGM configuration and geometrical parameters (length-to-radius and thickness-to-radius ratios) on the dynamic response is investigated. The results show that even though the shell is thin, the value of power law exponent has a considerable effect on the natural frequencies as well as the dynamic response of the functionally graded shell. Verification of the results of natural frequencies and time response of the FGM shell is achieved by making comparison with those available in the literature and those obtained using commercial software.     

Interaction integrals for thermal fracture of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials subjected to steady-state thermal loading. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium.     

A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials under thermal shock

In the present work, a study of thermoelastic analysis of functionally graded cylindrical shells subjected to transient thermal shock loading is carried out. A semi-analytical axisymmetric finite element model using the three-dimensional linear elasticity theory is developed. The three-dimensional equations of motion are reduced to two-dimensional ones by expanding the displacement field in Fourier series in the circumferential direction involving circumferential harmonics. The material properties are graded in the thickness direction according to a power law. The model has been verified with the results of simple analytical isotropic cylindrical shells subjected to a transient thermal loading. Additional FGM results for stresses and displacements are presented.     

Thermo-elasto-plastic stresses in functionally graded materials subjected to thermal loading taking residual stresses of the fabrication process into consideration

Functionally graded materials (FGMs) have recently been received with considerable interest, primarily as high temperature resistant materials for space vehicles subjected to high temperature environment. FGMs are one of the composite materials and consist of continuous change of composition of different material components from one surface to the other. FGMs usually fabricated at high temperature at which the FGMs have stress free condition. After the FGMs cooled from the fabrication temperature to the room temperature residual thermal stresses produced. In this paper, elasto-plastic thermal stresses in a rectangular plate (FGP) of a particle reinforced composite FGM are treated by finite element method due to the microscopic combination law when the FGP is subjected to three kinds of temperature conditions, first is cooling from the fabricated temperature to the room temperature, second is heating and last is heating after cooling from the fabricated temperature. In the analysis, the thermal stress constitutive equation of a particle-reinforced composite taking temperature change and damage process into consideration is used. The effects of the particle volume fraction and the three kinds of temperature conditions on the stresses in the matrix, stresses in the particle and macroscopic stress are discussed.     

Dynamic response of interfacial finite cracks in orthotropic materials subjected to concentrated loads

A cohesive zone model has been proposed to model crack growth with a part-through process zone in a thin solid. With the solid being modeled in Kirchhoff’s plate theory, the crack with a relatively long, inclined front is modeled as a line discontinuity with a finite cohesive zone within the plate. A cohesive force law is adopted to capture the effect of residual strength and residual rigidity of a plate cross-section gradually cracking through the thickness. It is derived by a plane-strain elasticity analysis of a cross section normal to the part-through crack. It is then applied in the plate formulation of a line crack to simulate its propagation within the plate plane. This model essentially resolves the originally three-dimensional crack problem in two hierarchical steps, i.e., in the thickness and in the in-plane directions. In the present study, the bending case is considered. A boundary element method is applied to numerically derive the cohesive force law and simulate the crack growth in a thin titanium-alloy plate. The computational efficiency of the model is demonstrated. The plate is shown to fracture in a nominally brittle or ductile manner depending on its thickness.     

Material tailoring and moduli homogenization for finite twisting deformations of functionally graded Mooney-Rivlin hollow cylinders

We analytically analyze finite plane strain twisting deformations of a hollow cylinder made of an isotropic and inhomogeneous Mooney-Rivlin material with material moduli varying in the radial direction. The cylinder is deformed by applying either tangential tractions on the inner surface and tangential displacements on the outer surface or vice versa. The radial variation of the moduli is found that will minimize the tangential displacement of the bounding surface where tangential traction is specified. Furthermore, the modulus of a homogeneous neo-Hookean cylinder is found that is energetically equivalent to the inhomogeneous cylinder.     

Transient solution of temperature field in functionally graded hollow cylinder with finite length using multi layered approach

In this paper by using a multi-layered approach based on the theory of laminated composites, the solution of temperature in functionally graded circular hollow cylinders subjected to transient thermal boundary conditions are presented. The material properties are assumed to be temperature-independent and radially dependent. The cylinder has finite length and is subjected to axisymmetric thermal loads. It is assumed that the functionally graded circular hollow cylinder is composed of N fictitious layers and the properties of each layer are assumed to be homogeneous and isotropic. Employing Laplace transform techniques and series solving method for two-dimensional heat conduction equation in the cylinder, solutions for the variation of temperature with time as well as temperature distribution through the cylinder are obtained.     

Elastic solution of a two-dimensional functionally graded thick truncated cone with finite length under hydrostatic combined loads

In this paper, a thick truncated hollow cone with finite length made of two-dimensional functionally graded materials (2D-FGM) subjected to combined loads as internal, external and axial pressure is considered. The volume fraction distribution of materials and geometry are assumed to be axisymmetric but not uniform along the axial direction. The Finite Element Method based on the Rayliegh-Ritz energy formulation is applied to obtain the elastic behavior of the functionally graded thick truncated cone. By using this method, the effects of semi-vertex angle of the cone and the power law exponents on the distribution of different types of displacements and stresses are considered. The results show that using 2D-FGM leads to a more flexible design so that both the maximum stresses and stress distribution can be controlled by the material distribution.     

Reissner mixed variational theorem-based finite cylindrical layer methods for the static analysis of functionally graded piezoelectric circular hollow cylinders under electro-mechanical loads

A unified formulation of Reissner's mixed variational theorem-based finite cylindrical layer methods is developed for the static analysis of simply-supported, multilayered functionally graded piezoelectric material (FGPM) circular hollow cylinders. The material properties of the cylinder are assumed to obey an exponent-law exponentially varying through the thickness coordinate of this. The trigonometric functions and Lagrange polynomials are used to interpolate the in-surface and thickness variations of the primary variables of each individual layer, respectively. The coupled electro-elastic effects on the static behaviors of multilayered FGPM cylinders are closely examined.     

A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials

In the present work, a study of free vibrations of functionally graded cylindrical shells made up of isotropic properties is carried out. A semi-analytical axisymmetric finite element model using the 3D linear elastic theory is developed. The 3D equations of motion are reduced to 2D by expanding the displacement field in Fourier series in the circumferential direction, involving circumferential harmonics. The material properties are graded in the thickness direction according to a power law. The model has been verified with simple benchmark problems and the results show that the frequency characteristics are found to be close to published results of isotropic cylindrical shells. New results are included for FGM shells.