Exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders

Exact solutions are obtained for transient torsio- nal responses of a finitely long, functionally graded hollow cylinder under three different end conditions, i.e. free-free, free-fixed and fixed-fixed. The cylinder with its external surface fixed is subjected to a dynamic shearing stress at the internal surface. The material properties are assumed to vary in the radial direction in a power law form, while keep invariant in the axial direction. With expansion in the axial direction in terms of trigonometric series, the governing equations for the unknown functions about the radial coordinate r and time t are deduced. By applying the variable substitution technique, the superposition method and the separation of variables consecutively, series-form solutions of the equations are obtained. Natural frequencies and the transient torsional responses are finally discussed for a functionally graded finite hollow cylinder.     

Elastic and viscoelastic solutions to rotating functionally graded hollow and solid cylinders

Analytical solutions to rotating functionally graded hollow and solid long cylinders are developed. Young＇s modulus and material density of the cylinder are assumed to vary exponentially in the radial direction, and Poisson＇s ratio is assumed to be constant. A unified governing equation is derived from the equilibrium equations, compatibility equation, deformation theory of elasticity and the stress-strain relationship. The governing second-order differential equation is solved in terms of a hypergeometric function for the elastic deformation of rotating functionally graded cylinders. Dependence of stresses in the cylinder on the inhomogeneous parameters, geometry and boundary conditions is examined and discussed. The proposed solution is validated by comparing the results for rotating functionally graded hollow and solid cylinders with the results for rotating homogeneous isotropic cylinders. In addition, a viscoelastic solution to the rotating viscoelastic cylinder is presented, and dependence of stresses in hollow and solid cylinders on the time parameter is examined.     

Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft

In the present paper, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented. The variation of material properties is assumed to follow a power law along the radial direction of the shaft. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different shaft geometries with different profiles of inhomogeneity are also graphically displayed.     

Elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders by analytical method

An elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders subjected to uniformly-distributed dynamic pressures at boundary surfaces is presented. The material properties, except Poisson’s ratio, are assumed to vary through the thickness according to a power law function. To achieve an exact solution, the dynamic radial displacement is divided into two quasi-static and dynamic parts, and for each part, an analytical solution is derived. The quasi-static solution is obtained by means of Euler’s equation, and the dynamic solution is derived using the method of the separation of variables and the orthogonal expansion technique. The radial displacement and stress distributions are plotted for various functionally graded material (FGM) hollow cylinders under different dynamic loads, and the advantages of the presented method are discussed. The proposed analytical solution is suitable for analyzing various arrangements of hollow FGM cylinders with arbitrary thickness and arbitrary initial conditions, which are subjected to arbitrary forms of dynamic pressures distributed uniformly on their boundary surfaces.     

Elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders by analytical method

An elastodynamic solution for plane-strain response of functionally graded thick hollow cylinders subjected to uniformly-distributed dynamic pressures at boundary surfaces is presented. The material properties, except Poisson’s ratio, are assumed to vary through the thickness according to a power law function. To achieve an exact solution, the dynamic radial displacement is divided into two quasi-static and dynamic parts, and for each part, an analytical solution is derived. The quasi-static solution is obtained by means of Euler’s equation, and the dynamic solution is derived using the method of the separation of variables and the orthogonal expansion technique. The radial displacement and stress distributions are plotted for various functionally graded material (FGM) hollow cylinders under different dynamic loads, and the advantages of the presented method are discussed. The proposed analytical solution is suitable for analyzing various arrangements of hollow FGM cylinders with arbitrary thickness and arbitrary initial conditions, which are subjected to arbitrary forms of dynamic pressures distributed uniformly on their boundary surfaces.     

Wave characteristics in functionally graded piezoelectric hollow cylinders

Based on linear three-dimensional piezoelasticity, the Legendre orthogonal polynomial series expansion approach is used for determining the wave characteristics in hollow cylinders composed of the functionally graded piezoelectric materials (FGPM) with open circuit. The displacement and electric potential components, expanded in a series of Legendre polynomials, are introduced into the governing equations along with position-dependent material constants so that the solution of the wave equation is reduced to an eigenvalue problem. Dispersion curves for FGPM and the corresponding non-piezoelectric hollow cylinders are calculated to show the piezoelectric effect. The influence of the ratio of radius to thickness is discussed. Electric potential and displacement distributions are used to show the piezoelectric effect on the flexural torsional mode. The influence of the polarizing direction on the piezoelectric effect is illustrated. For the radial and axial polarization, the piezoelectric effect reacts mostly on the longitudinal mode. For circumferential polarization, the piezoelectric effect reacts mostly on the torsional mode. In the FGPM hollow cylinder, piezoelectricity can weaken the guided wave dispersion.     

Functionally graded rod with small concentration of inclusions: Homogenization and optimization

This work is devoted to strain analysis and optimal design of a Functionally Graded (FG) rods and beams with small inclusions. The homogenization procedure plays a key role in our investigations. The method is illustrated using an example of the rod longitudinal deformation and bending of a beam. We consider the cases of FG inclusion sizes and FG steps between inclusions separately. Particular problems of optimal design are discussed in some details. The mathematical model of the bending beam, which adapts to the external load action, is proposed and an illustrative example of the adaptation process is given.     

On 3-dimensional non-axisymmetrical deformation analyses for finite hollow circular cylinders

In this paper, 3-dimensional non-axisymmetrical deformation analyses for finite hollow circular cylinders have been carried out by Pickett's double series expansion method^[1]. Through expanding the displacement potentials as the sum of fourier series and Fourier-Bessel series, we could express the coefficients of one series by those of another under certain boundary conditions. Thus, a set of linear algebraic equations were derived. Solving these equations, we could obtain the solutions of the problems. Numerical examples have been given to show that the method presented here is workable for practical applications.     

Anti-plane problem of periodic cracks in a functionally graded coating–substrate structure

The static and dynamic anti-plane problem for a functionally graded coating–substrate structure containing a periodic array of parallel cracks, which are perpendicular to the boundary, is considered. Integral-transform techniques are employed to reduce the problem to the solution of an integral equation with hypersingular kernels. Numerical results are presented to show the influence of geometry, material properties and material gradient parameter on the fracture behavior.     

Thermomagnetic viscoelastic responses in a functionally graded hollow structure

This paper presents an analytical solution for the interaction of electric potentials,electric displacements,elastic deformations,and thermoelasticity,and describes electromagnetoelastic responses and perturbation of the magnetic field vector in hollow structures(cylinder or sphere),subjected to mechanical load and electric potential.The material properties,thermal expansion coefficient and magnetic permeability of the structure are assumed to be graded in the radial direction by a power law distribution.In the present model we consider the solution for the case of a hollow structure made of viscoelastic isotropic material,reinforced by elastic isotropic fibers,this material is considered as structurally anisotropic material.The exact solutions for stresses and perturbations of the magnetic field vector in FGM hollow structures are determined using the infinitesimal theory of magnetothermoelasticity,and then the hollow structure model with viscoelastic material is solved using the correspondence principle and Illyushin’s approximation method.Finally,numerical results are carried out and discussed.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix

This paper is to study the non-axisymmetric two-dimensional problem of thermal stresses in an infinite matrix with a functionally graded coated circular inclusion based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded coating having radial arbitrary elastic properties is derived when the matrix is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded coating can change interfacial thermal stresses, and choosing proper change ways of the radial elastic properties in the coating can obviously reduce the thermal stresses.     

On the deformation of inhomogeneous orthotropic elastic cylinders

This paper is concerned with the deformation of inhomogeneous orthotropic cylinders. The work is motivated by the recent interest on the functionally graded materials. We assume that the elastic coefficients are independent of the axial coordinate. First, we study the Saint-Venant problem. Then we consider the case when the cylinder is subjected to body forces and surfaces tractions on the lateral surface. The results are applied to study the deformation of a circular cylinder with a given inhomogeneity.     

轴对称任意梯度圆筒的弹性动力学解

基于轴对称平面应变问题的运动方程及弹性梯度材料的应力和位移关系,通过将圆筒分层使材料性质离散为分段常数函数,同时在时域内应用有限差分格式,求得了材料性质沿径向梯度变化的圆筒弹性动力学解。本文解不仅适合任意梯度的弹性圆筒,而且容易满足多种形式的初始条件和边界条件。通过对材料性质沿径向为连续函数分布和分段函数分布的梯度圆筒数值分析,并与已有文献结果比较,得出本文解与已有文献的解吻合较好,验证了本文解的正确性和有效性。对材料性质为分段函数的三层组合圆筒分析发现,中间功能梯度层的指数分布因子对圆筒的径向位移和应力随时间变化都会产生显著影响。     

Double-layered piezo-thermoelastic hollow cylinder under some coupled loadings

In the present paper, a long thick-walled piezo-thermoelastic hollow cylinder with double layers is studied. The effects of temperature on the performance of the cylinder are obtained. Based on the theory of elasticity, the exact solutions of the cylinder under some coupled loadings are found. In the present paper, differences of a piezoelectric parameter between the two layers is taken into account. For comparison, numerical results have been carried out for both double-layered and graded cylinders. At the end of the present paper some discussions are addressed.     

Stochastic thermoelastic problem of a functionally graded plate under random temperature load

This study attempts to derive the statistics of temperature and thermal stress in functionally graded material (FGM) plates exposed to random external temperatures. The thermomechanical properties of the FGM plates are assumed to vary arbitrarily only in the plate thickness direction. The external temperatures are expressed as random functions with respect to time. The transient temperature field in the FGM plate is determined by solving a nonhomogeneous heat conduction problem for a multilayered plate with linear nonhomogeneous thermal conductivity and different homogeneous heat capacity in each layer. The autocorrelations and power spectrum densities (PSDs) of temperature and thermal stress are derived analytically. These statistics for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304) are computed under the condition that the fluctuation in the external temperature can be considered as white noise or a stationary Markov process.     

A novel class of graded cylinders neutral to host shafts of arbitrary cross-sections under torsion

A novel class of graded cylinders is proposed as neutral inclusions inside host shafts of arbitrary cross-section under Saint-Venant's torsion. The graded cylinder is made of cylindrically orthotropic materials with position varying quantities. The profiles of the two distinct shear moduli in the radial and tangential directions follow specific forms based on an arbitrarily selected function along the radial distance. We show that this type of graded cylinders can serve as universal neutral inclusions within host shafts of arbitrary cross-sections. In addition, we find that the associated warping fields can be exactly determined in terms of simple exponents of the selected function. This suggests that, by tuning the gradation parameter, one can manipulate the warping field of the inserted cylinder without disturbing the fields inside the host shaft. This finding is an original contribution to the existing solvable configurations of composite shafts under torsion.     

Stress analysis of circumferentially corrugated hollow orthotropic cylinders

A problem-solving approach based on discrete Fourier series is used to analyze the stress state of corrugated orthotropic and transversely isotropic hollow cylinders __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 87–96, December, 2006.     

Anti-plane shear of an arbitrary oriented crack in a functionally graded strip bonded with two dissimilar half-planes

An internal crack located within a functionally graded material (FGM) strip bonded with two dissimilar half-planes and under an anti-plane load is considered. The crack is oriented in an arbitrary direction. The material properties of strip are assumed to vary exponentially in the thickness direction and two half-planes are assumed to be isotropic. Governing differential equations are derived and to reduce the difficulty of the problem dealing with solution of a system of singular integral equations Fourier integral transform is employed. Semi closed form solution for the stress distribution in the medium is obtained and mode III stress intensity factor (SIF), at the crack tip is calculated and its validity was verified. Finally, the effects of nonhomogeneous material parameter and crack orientation on the stress intensity factor are studied.     

Spectral equivalent inclusion method: Anisotropic cylindrical multi-inhomogeneities

Consider a set of nested infinitely extended elastic cylindrical bodies possessing general cylindrical anisotropy embedded in an unbounded elastic isotropic medium. For general far-field loading, the nature of the elastic fields inside the inhomogeneities is predicted and a number of pertinent attractive properties is noted and proved. Moreover, the associated equivalent inclusion method (EIM) is concisely formulated. The concepts of the homogenization, spectral consistency conditions, and the so-called Eshelby-Fourier tensor are introduced. As a result the tedious and lengthy algebra encountered in the conventional EIM is circumvented and the corresponding large number of unknowns is reduced remarkably. Interestingly, the proposed theory is proved useful in the study of inhomogeneities with coatings made of functionally graded material (FGM). In addition to the relevance of the present work to multiple coated fiber reinforced composites, it is also of great value in the study of multi-shell quantum wire in electronic devices. The robustness and efficacy of the presented theories are demonstrated through consideration of several boundary value problems and various types of materials.