On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps

Here, the dynamic thermal buckling behavior of functionally graded spherical caps is studied considering geometric nonlinearity based on von Karman's assumptions. The formulation is based on first-order shear deformation theory and it includes the in-plane and rotary inertia effects. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the material constituents. The effective material properties are evaluated using homogenization method. The governing equations obtained using finite element approach are solved employing the Newmark's integration technique coupled with a modified Newton–Raphson iteration scheme. The pressure load corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure is taken as the dynamic buckling load. The present model is validated against the available isotropic case. A detailed numerical study is carried out to highlight the influences of shell geometries, power law index of functional graded material and boundary conditions on the dynamic buckling load of shallow spherical shells.     

Acoustic pulse interaction with a submerged functionally graded material hollow cylinder

A detailed study is undertaken to analyze the two-dimensional transient fluid-structure interaction of a plane acoustic pressure pulse with an arbitrarily thick, isotropic, functionally graded, hollow cylinder of infinite length, submerged in and filled with non-viscous compressible fluids. A laminate approximate model is adopted to deal with the assumed power-law variation of the constituents’ volume fractions across the thickness of the inhomogeneous cylinder. The problem solution is obtained by employing the classical method of modal expansion in conjunction with the powerful Transfer matrix solution technique and Durbin’s numerical Laplace inversion algorithm. Detailed numerical examples for the transient responses of water-filled and submerged thick-walled TiC-Al FGM cylinders with ceramic or metal rich material compositional gradient profiles under wideband and narrowband Gaussian incident shock loadings are presented and discussed. Many of the interesting dynamic features in the transient shell-shock interaction are addressed through appropriate plots of the internal/external pressure field as well as the induced dynamic stress concentrations within the shell material. Also, the response curves for the FGM cylinders are compared with those of equivalent bi-laminate shells containing comparable total volume fractions of constituent materials. A limiting case is considered and the validity of the work is established by comparison with the data obtained with the aid of a commercial finite element package.     

Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition

Linear thermal buckling and free vibration analysis are presented for functionally graded cylindrical shells with clamped-clamped boundary condition based on temperature-dependent material properties. The material properties of functionally graded materials (FGM) shell are assumed to vary smoothly and continuously across the thickness. With high-temperature specified on the inner surface of the FGM shell and outer surface at ambient temperature, 1D heat conduction equation along the thickness of the shell is applied to determine the temperature distribution; thereby, the material properties based on temperature distribution are made available for thermal buckling and free vibration analysis. First-order shear deformation theory along with Fourier series expansion of the displacement variables in the circumferential direction are used to model the FGM shell. Numerical studies involved the understanding of the influence of the power-law index, r/h and l/r ratios on the critical buckling temperature. Free vibration studies of FGM shells under elevated temperature show that the fall in natural frequency is very drastic for the mode corresponding to the lowest natural frequency when compared to the lowest buckling temperature mode.     

Fracture analysis of a functionally graded interfacial zone between two dissimilar homogeneous materials

In this paper the plane elasticity problem for a functionally graded interfacial zone containing a crack between two dissimilar homogeneous materials has been considered. It is assumed that in the interfacial zone the reciprocal of the shear modulus is a linear function of the coordinate, while Possion’s ratio keeps constant. By utilizing the Fourier transformation technique and the transfer matrix method, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters and the graded parameter on the stress intensity factors are investigated. The numerical results show that the graded parameters, the thickness of interfacial zone, the crack size and location have significant effects on the stress intensity factors.     

Nonlinear free vibration behavior of functionally graded plates

In this paper, an analytical solution is provided for the nonlinear free vibration behavior of plates made of functionally graded materials. The material properties of the functionally graded plates are assumed to vary continuously through the thickness, according to a power-law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular plates of functionally graded materials are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermal loading on the dynamic behavior of the plates is determined and discussed. The results reveal that nonlinear coupling effects play a major role in dictating the fundamental frequency of functionally graded plates.     

A New Higher Order Shear Deformation Model for Static Behavior of Functionally Graded Plates

In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concerned flexural behavior of FG plates with Metal-Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. It can be concluded that the proposed theory is accurate and simple in solving the static behavior of functionally graded plates.     

Free vibration analysis of functionally graded curved panels using a higher-order finite element formulation

Free vibration analysis of functionally graded curved panels is carried out using a higher-order formulation. A C⁰ finite element formulation is used to carry out the analysis. The element consists of nine degrees of freedom per node with higher-order terms in the Taylor's-series expansion, which represents the higher-order transverse cross-sectional deformation modes. The formulation includes Sanders’ approximation for doubly curved shells considering the effects of rotary inertia and transverse shear. A realistic parabolic distribution of transverse shear strains through the shell thickness is assumed and the use of shear correction factor is avoided. Material properties are assumed to be temperature independent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. Heat conduction between ceramic and metal constituents is neglected. The accuracy of the formulation is validated by comparing the results with those available in the literature. Effects of panel geometry parameters and boundary conditions are studied.     

Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method

The main aim of this paper is to provide a simple yet efficient solution for the free vibration analysis of functionally graded (FG) conical shells and annular plates. A solution approach based on Haar wavelet is introduced and the first-order shear deformation shell theory is adopted to formulate the theoretical model. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. The separation of variables is first performed; then Haar wavelet discretization is applied with respect to the axial direction and Fourier series is assumed with respect to the circumferential direction. The constants appearing from the integrating process are determined by boundary conditions, and thus the partial differential equations are transformed into algebraic equations. Then natural frequencies of the FG shells are obtained by solving algebraic equations. Accuracy and reliability of the current method are validated by comparing the present results with the existing solutions. Effects of some geometrical and material parameters on the natural frequencies of shells are discussed and some selected mode shapes are given for illustrative purposes. It’s found that accurate frequencies can be obtained by using a small number of collocation points and boundary conditions can be easily achieved. The advantages of this current solution method consist in its simplicity, fast convergence and excellent accuracy.     

A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems

A solution of a non-homogeneous orthotropic elastic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions as well as the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to the homogeneous ones. Then by virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally presented, which can degenerate in a rather straightforward way to the solution for a homogeneous orthotropic cylindrical shell and isotropic solid cylinder as well as that for a non-homogeneous isotropic cylindrical shell. Using the present method, integral transform can be avoided. It is fit for a cylindrical shell with arbitrary thickness subjected to arbitrary thermal loads. It is also very convenient to deal with dynamic thermoelastic problems for different boundary conditions. Besides, the numerical calculation involved is very easy to be performed. Several examples are presented.     

Elastodynamic wave propagation in graded materials: simulations, experiments, phenomena, and applications

A two-dimensional numerical simulation model for the elastodynamic wave propagation in two linear elastic, isotropic, joint half-spaces is presented. The border between the two half-spaced is graded in a way, that the values of the elastic properties and the densities vary smoothly (sinusoidally) from the values of one continuum to the values of the other continuum within a transition zone of a defined thickness. It is demonstrated, that a graded layer leads to a frequency and wavelength dependent refraction and reflection behavior of elastodynamic waves. Numerical results show that wavelengths which are long compared with the transition layer thickness are dominantly reflected whereas short waves are dominantly transmitted, a phenomena which does not occur in the case of an infinitely thin transition layer. Furthermore the frequency dependent reflection and transmission behavior of elastodynamic waves is verified experimentally. There the interface between two vapor deposited films is graded due to intermetallic diffusion effects. These graded microstructures are analyzed with a short-pulse-laser-acoustic set-up. The corresponding frequencies of the elastodynamic waves which are filtered with these functionally graded microstructures are in the range of 0.5 THz.     

Modeling of the electric-field distribution of plasma photonic crystals having inhomogeneity in the materials

We study the electric-field distributions in binary 1D-PPCs in the presence of inhomogeneity in the materials of unit cells. Here we consider four 1D-PPC structures: the first two structures have linear and exponential graded dielectric materials with a homogeneous plasma, and the other two structures have homogeneous dielectric material with linear and exponential density profiles of the plasma. Matching the electric fields and their derivatives on the dielectric-plasma interface, we obtain the required field-distribution relations. We observe that the amplitude of electric field changes in the presence of inhomogeneity in the materials, and the angle of incidence of electromagnetic waves highly affects the electric-field distributions.     

Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations

In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge–Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency–amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.     

Axially symmetric vibrations of fluid-filled poroelastic circular cylindrical shells

Employing Biot's theory of wave propagation in liquid saturated porous media, axially symmetric vibrations of fluid-filled and empty poroelastic circular cylindrical shells of infinite extent are investigated for different wall-thicknesses. Let the poroelastic cylindrical shells are homogeneous and isotropic. The frequency equation of axially symmetric vibrations each for a pervious and an impervious surface is derived. Particular cases when the fluid is absent are considered both for pervious and impervious surfaces. The frequency equation of axially symmetric vibrations propagating in a fluid-filled and an empty poroelastic bore, each for a pervious and an impervious surface is derived as a limiting case when ratio of thickness to inner radius tends to infinity as the outer radius tends to infinity. Cut-off frequencies when the wavenumber is zero are obtained for fluid-filled and empty poroelastic cylindrical shells both for pervious and impervious surfaces. When the wavenumber is zero, the frequency equation of axially symmetric shear vibrations is independent of nature of surface, i.e., pervious or impervious and also it is independent of presence of fluid in the poroelastic cylindrical shell. Non-dimensional phase velocity for propagating modes is computed as a function of ratio of thickness to wavelength in absence of dissipation. These results are presented graphically for two types of poroelastic materials and then discussed. In general, the phase velocity of an empty poroelastic cylindrical shell is higher than that of a fluid-filled poroelastic cylindrical shell.The phase velocity of a fluid-filled bore is higher than that of an empty poroelastic bore. Previous results are shown as a special case of present investigation. Results of purely elastic solid are obtained.     

Dynamic stability of functionally graded cantilever cylindrical shells under distributed axial follower forces

In this paper, flutter of functionally graded material (FGM) cylindrical shells under distributed axial follower forces is addressed. The first-order shear deformation theory is used to model the shell, and the material properties are assumed to be graded in the thickness direction according to a power law distribution using the properties of two base material phases. The solution is obtained by using the extended Galerkin's method, which accounts for the natural boundary conditions that are not satisfied by the assumed displacement functions. The effect of changing the concentrated (Beck's) follower force into the uniform (Leipholz's) and linear (Hauger's) distributed follower loads on the critical circumferential mode number and the minimum flutter load is investigated. As expected, the flutter load increases as the follower force changes from the so-called Beck's load into the so-called Leipholz's and Hauger's loadings. The increased flutter load was calculated for homogeneous shell with different mechanical properties, and it was found that the difference in elasticity moduli bears the most significant effect on the flutter load increase in short, thick shells. Also, for an FGM shell, the increase in the flutter load was calculated directly, and it was found that it can be derived from the simple power law when the corresponding increase for the two base phases are known.     

Interaction of a plane progressive sound wave with a functionally graded spherical shell

An exact analysis is carried out to study interaction of a time-harmonic plane progressive sound field with a radially inhomogeneous thick-walled elastic isotropic spherical shell suspended in and filled with compressible ideal fluid mediums. Using the laminated approximation method, a modal state equation with variable coefficients is set up in terms of appropriate displacement and stress functions and their spherical harmonics. Taylor’s expansion theorem is then employed to obtain the solution to the modal state equation ultimately leading to calculation of a global transfer matrix. Numerical example is given for a water-submerged/air-filled Aluminum/Zirconia elastic spherical sandwich shell containing a functionally graded interlayer and subjected to an incident progressive plane sound wave. The mechanical properties of the interlayer are assumed to vary smoothly and continuously across the thickness with the change of volume concentration of its constituents. The effect of incident wave frequency, thickness and compositional gradient of the interlayer on the form function amplitude and the average radiation force acting on the composite shell are examined. Limiting cases are considered and fair agreements with well-known solutions are established.     

Hybrid finite element method applied to supersonic flutter of an empty or partially liquid-filled truncated conical shell

In this study, aeroelastic analysis of a truncated conical shell subjected to the external supersonic airflow is carried out. The structural model is based on a combination of linear Sanders thin shell theory and the classic finite element method. Linearized first-order potential (piston) theory with the curvature correction term is coupled with the structural model to account for pressure loading. The influence of stress stiffening due to internal or external pressure and axial compression is also taken into account. The fluid-filled effect is considered as a velocity potential variable at each node of the shell elements at the fluid-structure interface in terms of nodal elastic displacements. Aeroelastic equations using the hybrid finite element formulation are derived and solved numerically. The results are validated using numerical and theoretical data available in the literature. The analysis is accomplished for conical shells of different boundary conditions and cone angles. In all cases the conical shell loses its stability through coupled-mode flutter. This proposed hybrid finite element method can be used efficiently for design and analysis of conical shells employed in high speed aircraft structures.     

梯度颗粒复合介质的光学双稳

重点研究了组分的梯度构形对带壳球形颗粒复合介质的光学双稳特性的影响.其中球形颗粒是由非线性核和介电函数具有梯度分布的线性壳组成.对于壳层介电函数具有幂指数分布的情况,通过求解麦克斯韦方程,得到各区域的势能分布函数,从而求得核内电场的数学表达式.数值研究发现,该复合介质的光学双稳阈值和区域与壳层的厚度及壳层的介电幂指数有关,随着壳层厚度增大或幂指数增大,双稳阈值将变宽.此外,还研究了正入射情况下复合材料体系的反射系数随外电场的变化情况, 发现其关系曲线是一条回线. 关键词： 梯度颗粒 复合介质 光学双稳     

Buckling of 2D-FG Cylindrical Shells under Combined External Pressure and Axial Compression

This paper presents the stability of two-dimensional functionally graded (2D-FG) cylindrical shells subjected to combined external pressure and axial compression loads, based on classical shell theory. The material properties of functionally graded cylindrical shell are graded in two directional (radial and axial) and determined by the rule of mixture. The Euler's equation is employed to derive the stability equations, which are solved by GDQ method to obtain the critical mechanical buckling loads of the 2D-FG cylindrical shells. The effects of shell geometry, the mechanical properties distribution in radial and axial direction on the critical buckling load are studied and compared with a cylindrical shell made of 1D-FGM. The numerical results reveal that the 2D-FGM has a significant effect on the critical buckling load.     

Modal theory for the two-frequency mutual coherence function in random media: general theory and plane wave solution: I

In this paper, the modal expansion theory is presented as a new analytical approach together with the resulting new physical parameters. In particular, the features of an arbitrary power-law structure function are investigated. The exact expression for the Gaussian spectrum is rederived. An approximate analytical expression for the two-frequency coherence function evaluated at equal positions for the Kolmogorov spectrum is presented and comparison with the numerical solution in the literature exhibits a remarkable agreement. As a result of the modal decomposition, general properties for a transversally homogeneous and isotropic medium are demonstrated, such as the exponential decay of the amplitude of the solution and the linear phase behaviour at large propagation distances.     

VIBRATION CHARACTERISTICS AND TRANSIENT RESPONSE OF SHEAR-DEFORMABLE FUNCTIONALLY GRADED PLATES IN THERMAL ENVIRONMENTS

Free and forced vibration analyses for initially stressed functionally graded plates in thermal environment are presented. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Theoretical formulations are based on Reddy's higher order shear deformation plate theory and include the thermal effects due to uniform temperature variation. The plate is assumed to be clamped on two opposite edges with the remaining two others either free, simply supported or clamped. One-dimensional differential quadrature technique, Galerkin approach, and the modal superposition method are used to determine the transient response of the plate subjected to lateral dynamic loads. Comprehensive numerical results for silicon nitride/stainless-steel rectangular plates are presented in dimensionless tabular and graphical forms. The roles played by the constituent volume fraction index, temperature rise, shape and duration of dynamic loads, initial membrane stresses as well as the character of boundary conditions are studied. The results reveal that, when thermal effects are considered, functionally graded plates with material properties intermediate to those of isotropic ones do not necessarily have intermediate natural frequencies and dynamic responses.