Material tailoring and analysis of functionally graded isotropic and incompressible linear elastic hollow cylinders

We use the Airy stress function to derive exact solutions for plane strain deformations of a functionally graded (FG) hollow cylinder with the inner and the outer surfaces subjected to different boundary conditions, and the cylinder composed of an isotropic and incompressible linear elastic material. For the shear modulus given by either a power law or an exponential function of the radius r, we derive explicit expressions for stresses, the hydrostatic pressure and displacements. Conversely, we find the variation with r of the shear modulus for a linear combination of the radial and the hoop stresses to have a pre-assigned variation in the cylinder; this inverse problem is usually called material tailoring. The shear modulus found while solving the inverse problem must be positive everywhere. Results for a few problems are computed and presented graphically. It seems that the Airy stress function approach is used here for the first time to analyze two-dimensional problems for incompressible materials. When studying axisymmetric deformations of an FG cylinder, it is found that for the hoop stress to be uniform through the cylinder thickness the shear modulus must be proportional to the radial coordinate r as found earlier by Batra [Batra RC. Optimal design of functionally graded incompressible linear elastic cylinders and spheres. AIAAJ 2008;46(8):2005–7.] and for the maximum in-plane shear stress to be constant the shear modulus must vary as r². The expression for the maximum in-plane shear stress in terms of pressures and the radii of the inner and the outer surfaces of the cylinder is a universal result valid for all materials for which the shear modulus is proportional to r². For a hollow cylinder fixed on the inner surface and subjected to tangential tractions on the outer surface (or vice versa) the through-the-thickness in-plane shear stress distribution is also universal and is determined by surface tractions and the outer radius of the cylinder; it is independent of the spatial variation of the shear modulus.     

Static deformations of functionally graded polar-orthotropic cylinders with elliptical inner and circular outer surfaces

Functionally graded materials (FGMs) enable one to tailor the spatial variation of material properties so as to fully use the material everywhere. For example, in a hollow circular cylinder one can vary, in the radial direction, the material moduli to make the hoop stress constant. Whereas the problem for a hollow cylinder with the inner and the outer surfaces circular has been studied, that of a cylinder with a circular outer surface and a non-circular inner surface or vice versa has not been investigated. We study here such a plane-strain problem when the cylinder material is polar-orthotropic, material properties vary exponentially in the radial direction, and deformations are independent of the axial coordinate. The problem is challenging since the cylinder thickness varies with the angular position of a point, and the cylinder material is inhomogeneous. Equilibrium equations are solved by expanding the radial and the circumferential displacements in Fourier series in the angular coordinate. The method of Frobenius series is used to solve ordinary differential equations for coefficients of the Fourier series, and boundary conditions are satisfied in the sense of Fourier series. A parametric study has been conducted that delineates effects on stresses of the eccentricity of the ellipse, the material property gradation index and loads applied on boundaries of the cylinder. The analytical solutions presented here will serve as benchmarks for comparing solutions derived by numerical methods.     

Analytical elastic solutions for pressurized hollow cylinders with internal functionally graded coatings

The main objective of this work is to obtain analytical solutions for thick-walled cylinders subjected to internal and external pressure in which the entire wall is made of functionally graded material or of only a thin functionally graded coating present on the internal homogeneous wall. We assume that the materials are isotropic with constant Poisson’s ratio; as far as the Young modulus is concerned, we consider a power and an exponential. The proposed analytical solutions show the effects of the different profiles describing the graded properties of the materials on the stress and displacement fields; in addition, comparisons between graded coating and conventional homogeneous coating highlight the advantage of the graded material on the interface stress reduction. Furthermore, we show how even a thin graded coating can be useful to satisfy the requirements of a specific application without having to make an entire wall with graded properties. This investigation permits us to optimize the elastic response of cylinders under pressure by tailoring the thickness variation of the elastic properties and to reduce manufacturing costs given by the technological limitations that occur to produce entire functionally graded walls.     

Stress intensity factors for functionally graded solid cylinders

This paper presents the mode I stress intensity factors for functionally graded solid cylinders with an embedded penny-shaped crack or an external circumferential crack. The solid cylinders are assumed under remote uniform tension. The multiple isoparametric finite element method is used. Various types of functionally graded materials and different gradient compositions for each type are investigated. The results show that the material property distribution has a quite considerable influence on the stress intensity factors. The influence for embedded cracks is quite different from that for external cracks.     

Transient piezothermoelastic analysis for a functionally graded thermopiezoelectric hollow sphere

This paper is concerned with the theoretical treatment of transient piezothermoelastic problem involving a functionally graded thermopiezoelectric hollow sphere due to uniform heat supply. The transient one-dimensional temperature is analyzed by the method of Laplace transformation. The thermal, thermoelastic and piezoelectric constants of the hollow sphere are expressed as power functions of the radial coordinate. The one-dimensional solution for the temperature change in a transient state, and piezothermoelastic response of a functionally graded thermopiezoelectric hollow sphere is obtained herein. Some numerical results for the temperature change, displacement, stress and electric potential distributions are shown. Furthermore, the influence of the nonhomogeneity of the material upon the temperature change, displacement, stresses and electric potential is investigated.     

Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness

We analyze axisymmetric deformations of a rotating disk with its thickness, mass density, thermal expansion coefficient and shear modulus varying in the radial direction. The disk is made of a rubberlike material that is modeled as isotropic, linear thermoelastic and incompressible. We note that the hydrostatic pressure in the constitutive relation of the material is to be determined as a part of the solution of the problem since it cannot be determined from the strain field. The problem is analyzed by using an Airy stress function φ. The non-homogeneous ordinary differential equation with variable coefficients for φ is solved either analytically or numerically by the differential quadrature method. We have also analyzed the challenging problem of tailoring the variation of either the shear modulus or the thermal expansion coefficient in the radial direction so that a linear combination of the hoop stress and the radial stress is constant in the disk. For a rotating annular disk we present the explicit expression of the thermal expansion coefficient for the hoop stress to be uniform within the disk. For a rotating solid disk we give the exact expressions for the shear modulus and the thermal expansion coefficient as functions of the radial coordinate so as to achieve constant hoop stress. Numerical results for a few typical problems are presented to illuminate effects of material inhomogeneities on deformations of a hollow and a solid rotating disk.     

Stochastic dynamic analysis of a functionally graded thick hollow cylinder with uncertain material properties subjected to shock loading

This paper presents an investigation of the stochastic dynamic response of a functionally graded (FG) thick hollow cylinder with uncertain material properties subjected to mechanical shock loading. The mechanical properties are considered to vary across thickness of FG cylinder as a non-linear power function of radius. To obtain the radial displacement in each point, the Navier equation in displacement form is derived using linear functionally graded elements. The Galerkin finite element and Newmark finite difference methods along with the Monte Carlo simulation are employed to deal with the statistical response of the FG cylinder. The mean and variance of radial displacements are calculated in various points across thickness for different values of volume fraction exponents. The results are used to quantify the effects of variations in the mechanical properties on the dynamic response and safety within the FG cylinder.     

Optimisation of sandwich panels with functionally graded core and faces

The distributions of properties across the thickness (core) and in the plane (face sheets) that minimise the interlaminar stresses at the interface with the core are determined solving the Euler–Lagrange equations of an optimisation problem in which the membrane and transverse shear energy contributions are made stationary. The bending stiffness is maximised, while the energy due to interlaminar stresses is minimised. As structural model, a refined zig-zag model with a high-order variation of displacements is employed. Simplified, sub-optimal distributions obtainable with current manufacturing processes appear effective for reducing the critical interfacial stress concentration, as shown by the numerical applications.     

Vibration analysis of CNT-reinforced functionally graded rotating cylindrical panels using the element-free kp-Ritz method

In this paper, analysis of free vibration of carbon nanotube (CNT) reinforced functionally graded rotating cylindrical panels is presented. The analysis is performed by using the element-free kernel particle Ritz method or in short the kp-Ritz method. The rotating cylindrical panels are reinforced by single-walled carbon nanotubes (SWCNTs) with different types of distributions along thickness direction of the panels. Extended rule of mixture is selected to estimate the effective material properties of the resulting nanocomposite rotating panels. Two-dimensional displacement fields of the plates are approximated by a set of mesh-free kernel particle functions. The discretized governing eigen-equations are developed via the Ritz procedure. This kp-Ritz method enforces essential boundary conditions through the full transformation method. Detailed parametric studies have been carried out to reveal the influences of volume fraction of carbon nanotubes, edge-to-radius ratio and rotation speed on the frequency characteristics, with mode shape visualization provided. In addition, effects of different boundary conditions and types of distributions of carbon nanotubes are examined in detail.     

Reduction of the stress concentration factor in a homogeneous panel with hole by using a functionally graded layer

This work aims at understanding the effect of a radially heterogeneous layer around the hole in a homogeneous plate on the stress concentration factor. The problem concerns a single hole in a plate under different far-field in-plane loading conditions. By assuming a radial power law variation of Young’s modulus and constant value for Poisson’s ratio, the governing differential equations for plane stress conditions, and general in-plane loading conditions are studied. The elastic solutions are obtained in closed form and, in order to describe localized interface damage between the ring and the plate, two different interface conditions (perfectly bonded and frictionless contact) are studied. The formulae for the stress concentration factors are explicitly given for uniaxial, biaxial and shear in-plane loading conditions and comparisons with interface hoop stress values are performed. The solutions are investigated to understand the role played by the geometric and graded constitutive parameters. The results are validated with numerical finite element simulations in which some simplified hypotheses assumed in the analytical model, are relaxed to explore the range of validity of the elastic solution presented. In this way the results obtained are useful in tailoring the parameters for specific applications.     

An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates

In this paper, an efficient and simple higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3-dimensional and quasi-3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.     

Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.     

Analytical solutions for functionally graded incompressible eccentric and non-axisymmetrically loaded circular cylinders

We study analytically plane strain static deformations of functionally graded eccentric and non-axisymmetrically loaded circular cylinders comprised of isotropic and incompressible linear elastic materials. Normal and tangential surface tractions on the inner and the outer surfaces of a cylinder may vary in the circumferential direction. The shear modulus is taken to vary either as an exponential function or as a power law function of the radius only. The radial and the circumferential displacements, and the hydrostatic pressure are expanded in Fourier series in the angular coordinate, and expressions for their coefficients are derived from equations expressing the balance of mass (or the continuity equation) and the balance of linear momentum. Boundary conditions are satisfied in the sense of Fourier series. For the exponential variation of the shear modulus, the method of Frobenius series is used to solve 4th-order ordinary differential equations for coefficients of the Fourier series. It is shown that the series solutions for displacements and the hydrostatic pressure converge rapidly. Results for eccentric cylinders and non-axisymmetrically loaded circular cylinders are computed and exhibited graphically. Effects on stress distributions of the eccentricity in the cylinders and of the gradation in the shear modulus are illuminated. It is found that in a thin cylinder subjected to cosinusoidally varying pressure on the inner surface, segments of the cylinder between two adjacent cusps in the pressure deform due to bending rather than stretching.     

Material tailoring for orthotropic elastic rotating disks

The tailoring of elastic moduli in the radial direction is studied to design a fiber-reinforced orthotropic linear elastic rotating disk with constant radial or hoop stress or constant in-plane shear stress. For fibers arranged in concentric circles the axes of material symmetry coincide with the radial and the circumferential directions. However, when fibers are aligned along helices, the orientation of material principal axes varies with the radial coordinate of a point. For a solid disk made of an orthotropic material with Young’s moduli proportional to each other, we give explicit expressions for the required variations of the elastic moduli with the radius to attain a given state of stress. For a rotating annular disk composed of a fiber-reinforced composite with fibers placed along concentric circles, the required radial variation of the volume fraction of fibers is calculated numerically and exhibited graphically. For fibers of known volume fraction laid along helices, the radial variation of the fiber orientation angle is determined. We have also analyzed the material tailoring problem for a disk of variable thickness. Results presented herein should help structural engineers and material scientists optimally design rotating disks composed of radially inhomogeneous materials.     

High-order free vibration analysis of sandwich plates with both functionally graded face sheets and functionally graded flexible core

Free vibration analysis of functionally graded material sandwich plates is studied using a refined higher order sandwich panel theory. A new type of FGM sandwich plates, namely, both functionally graded face sheets and functionally graded flexible core are considered. The functionally graded material properties follow a power-law function. The first order shear deformation theory is used for the face sheets and a 3D-elasticity solution of weak core is employed for the core. On the basis of continuities of the displacements and transverse stresses at the interfaces of the face sheets and the core, equations of motion are obtained by using Hamilton’s principle. The accuracy of the present approach is validated by comparing the analytical results obtained for a degradation model (functionally graded face sheets and homogeneous flexible core) with ones published in the literatures, as well as the numerical results obtained by finite element method and good agreements are reached. Then, parametric study is conducted to investigate the effect of distribution of functionally graded material properties, thickness to side ratio on the vibration frequencies.     

Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions

The objective of this work is to present a Haar Wavelet Discretization (HWD) method-based solution approach for the free vibration analysis of functionally graded (FG) spherical and parabolic shells of revolution with arbitrary boundary conditions. The first-order shear deformation theory is adopted to account for the transverse shear effect and rotary inertia of the shell structures. Haar wavelet and their integral and Fourier series are selected as the basis functions for the variables and their derivatives in the meridional and circumferential directions, respectively. The constants appearing in the integrating process are determined by boundary conditions, and thus the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations. The proposed approach directly deals with nodal values and does not require special formula for evaluating system matrices. Also, the convenience of the approach is shown in handling general boundary conditions. Numerical examples are given for the free vibrations of FG shells with different combinations of classical and elastic boundary conditions. Effects of spring stiffness values and the material power-law distributions on the natural frequencies of shells are also discussed. Some new results for the considered shell structures are presented, which may serve as benchmark solutions.     

Stability and natural frequencies of functionally graded stringer-reinforced panels

The paper presents an analysis of stability and free vibrations of rectangular functionally graded panels reinforced by a system of parallel stringers. The exact solution of the problem is illustrated for large aspect ratio panels with simply supported long edges and arbitrary boundary conditions along the short edges (hereafter the reference to an “exact solution” implies a closed-form solution in the content of the theory of plates). The spacing between the stringers and the cross sections of individual stringers can be arbitrary. In the particular case where identical stringers are equally spaced, the solution is simplified using the smeared stiffeners technique. The optimization problem concerned with the choice of stringers and their spacing in the situations where the buckling loads or fundamental frequencies are prescribed is also considered. The closed-form solution of the optimization problem is shown in the case of blade stringers.     

Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates

This article introduces new methods for static and free vibration analyses of functionally graded annular and circular micro-plates, which can take into account spatial variation of the length scale parameter. The underlying higher order continuum theory behind the proposed approaches is the modified couple stress theory. A unified way of expressing the displacement field is adopted so as to produce numerical results for three different plate theories, which are Kirchhoff plate theory (KPT), Mindlin plate theory (MPT), and third-order shear deformation theory (TSDT). Governing partial differential equations and corresponding boundary conditions are obtained following the variational approach and the Hamilton's principle. Derived systems of differential equations are solved numerically by utilizing the differential quadrature method (DQM). Comparisons to the results available in the literature demonstrate the high level of accuracy of the numerical results generated through the developed methods. Extensive analyses are presented in order to illustrate the influences of various geometric and material parameters upon static deformation profiles, stresses, and natural vibration frequencies. In particular, the length scale parameter ratio -which defines the length scale parameter variation profile-is shown to possess a profound impact on both static and dynamic behaviors of functionally graded annular and circular micro-plates.