The Pressurized Hollow Cylinder or Disk Problem for Functionally Graded Isotropic Linearly Elastic Materials

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic hollow circular cylinders or disks under uniform internal or external pressure. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic Lamé problem for a pressurized homogeneous isotropic hollow circular cylinder or disk is considered. The special case of a body with Young"s modulus depending on the radial coordinate only, and with constant Poisson"s ratio, is examined. It is shown that the stress response of the inhomogeneous cylinder (or disk) is significantly different from that of the homogeneous body. For example, the maximum hoop stress does not, in general, occur on the inner surface in contrast with the situation for the homogeneous material. The results are illustrated using a specific radially inhomogeneous material model for which explicit exact solutions are obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Effects of curvilinear anisotropy on radially symmetric stresses in anisotropic linearly elastic solids

It has been known for some time that certain radial anisotropies in some linear elasticity problems can give rise to stress singularities which are absent in the corresponding isotropic problems. Recently related issues were examined by other authors in the context of plane strain axisymmetric deformations of a hollow circular cylindrically anisotropic linearly elastic cylinder under uniform external pressure, an anisotropic analog of the classic isotropic Lamé problem. In the isotropic case, as the external radius increases, the stresses rapidly approach those for a traction-free cavity in an infinite medium under remotely applied uniform compression. However, it has been shown that this does not occur when the cylinder is even slightly anisotropic. In this paper, we provide further elaboration on these issues. For the externally pressurized hollow cylinder (or disk), it is shown that for radially orthotropic materials, the maximum hoop stress occurs always on the inner boundary (as in the isotropic case) but that the stress concentration factor is infinite. For circumferentially orthotropic materials, if the tube is sufficiently thin, the maximum hoop stress always occurs on the inner boundary whereas for sufficiently thick tubes, the maximum hoop stress occurs at the outer boundary. For the case of an internally pressurized tube, the anisotropic problem does not give rise to such radical differences in stress behavior from the isotropic problem. Such differences do, however, arise in the problem of an anisotropic disk, in plane stress, rotating at a constant angular velocity about its center, as well as in the three-dimensional problem governing radially symmetric deformations of anisotropic externally pressurized hollow spheres. The anisotropies of concern here do arise in technological applications such as the processing of fiber composites as well as the casting of metals.     

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.     

Stress analysis of functionally graded rotating discs: analytical and numerical solutions

This study deals with stress analysis of annular rotating discs made of functionally graded materials(FGMs).Elasticity modulus and density of the discs are assumed to vary radially according to a power law function,but the material is of constant Poisson’s ratio.A gradient parameter n is chosen between 0 and 1.0.When n = 0,the disc becomes a homogeneous isotropic material.Tangential and radial stress distributions and displacements on the disc are investigated for various gradient parameters n by means of the diverse elasticity modulus and density by using analytical and numerical solutions.Finally,a homogenous tangential stress distribution and the lowest radial stresses along the radius of a rotating disc are approximately obtained for the gradient parameter n = 1.0 compared with the homogeneous,isotropic case n = 0.This means that a disc made of FGMs has the capability of higher angular rotations compared with the homogeneous isotropic disc.     

Inflation and eversion of functionally graded non-linear elastic incompressible circular cylinders

We study axisymmetric radial deformations of a circular cylinder composed of an inhomogeneous Mooney-Rivlin material with the two material parameters varying continuously through the cylinder thickness either by a power law or an affine relation. It is found that for the exponent of the power law function equal to 1, the hoop stress for an internally pressurized cylinder is uniform in the cylinder. One can tailor the gradation of these two material parameters to make the maximum tensile hoop stress occur either on the inner surface or on the outer surface. Also, the stress concentration in a pressurized thick cylinder strongly depends upon the value of the exponent of the power law variation of the two material parameters. For an affine through-the-thickness variation of the two elastic moduli the hoop stress at the point is nearly the same as that in a cylinder composed of a homogeneous material. Here R_in and R_ou equal, respectively, the inner and the outer radii of the cylinder in the unstressed reference configuration, and R is the radial coordinate of a point in the reference configuration. The stress distribution in an everted cylinder strongly depends upon its thickness in the reference configuration.     

Torsin of Functionally Graded Isotropic Linearly Elastic Bars

The purpose of this research is to investigate the effects of material inhomogeneity on the torsional response of linearly elastic isotropic bars. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The classic approach to the torsion problem for a homogenous isotropic bar of arbitrary simply-connected cross-section in terms of the Prandtl stress function is generalized to the inhomogeneous case. The special case of a circular rod with shear modulus depending on the radial coordinate only is examined. It is shown that the maximum shear stress does not, in general, occur on the boundary of the rod, in contrast to the situation for the homogeneous problem. It is shown that the material inhomogeneity may increase or decrease the torsional rigidity compared to that for the homogeneous rod. Optimal upper and lower bounds for the torsional rigidity for nonhomogeneous bars of arbitrary cross-section are established. A new formulation of the basic boundary-value problem is given. The results are illustrated using specific material models used in the literature on functionally graded elastic materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Rotating Variable-thickness Orthotropic Cylinder Containing a Solid Core of Uniform-thickness

This paper presents accurate elastic solutions for the rotating variable-thickness and/or uniform-thickness orthotropic circular cylinders. The present circular cylinder may contain a uniform-thickness solid core of rigid or homogeneously isotropic material. Different cases of rotating cylinders of various cores are investigated. These cylinders include completely isotropic solid cylinder, uniform-thickness orthotropic cylinder containing an isotropic core, variable-thickness orthotropic cylinder containing an isotropic core, uniform-thickness orthotropic cylinder containing a rigid core, and variable-thickness orthotropic cylinder containing a rigid core. For all cases studied, exact elastic solutions are obtained and numerical results are presented. The results include the radial, hoop, and axial stresses and radial displacement of the five cylinder configurations. The distributions of displacement and stresses through the radial direction of the rotating cylinder are obtained and comparisons between different cases are made at the same angular velocity.     

Optimum Young��s Modulus of a Homogeneous Cylinder Energetically Equivalent to a Functionally Graded Cylinder

For a functionally graded (FG) circular cylinder loaded by uniform pressures on the inner and the outer surfaces and Young??s modulus varying in the radial direction, we find lower and upper bounds for Young??s modulus of the energetically equivalent homogeneous cylinder. That is, the strain energies of the FG and the homogeneous cylinders are equal to each other. For a typical power law variation of Young??s modulus in the FG cylinder, it is shown that taking only two series terms, yields good values for bounds of the equivalent modulus. We also study two inverse problems. First, an investigation is made to find the radial variation of Young??s modulus in the FG cylinder, having a constant Poisson??s ratio, that gives the maximum value of the equivalent modulus. Second, the complementary problem of finding the radial variation of Poisson??s ratio in the FG cylinder, having a constant stiffness, that gives the maximum value of the equivalent modulus, is considered. It is found that the spatial variation of the elastic properties, that maximizes the equivalent modulus, depends strongly upon the external loading on the cylinder.     

Theory of gravitational stability of a rotating cylinder

The stability of a rotating dust cylinder against perturbations located in the plane perpendicular to the axis of rotation is investigated. It is shown that a homogeneous rotating cylinder containing a weak inhomogeneity is stable against such perturbations. A weakly inhomogeneous cylinder with opposite streams of equal density is unstable for thel=2 mode in the case of a perturbation of the form_ei(l–t), when the density increases radially. The instability of a system consisting of a homogeneous rotating dust cylinder in a hot homogeneous medium is determined. It is shown that the maximum growth rate corresponds tol = 2 when the density of a cold cylinder is not negligible in comparison with the density of the medium. In the opposite case, the maximum growth rate shifts toward l=3. An attempt is made to associate the existence of the maximum growth rate for l=2 with the presence of two spiral arms in most galaxies. It is shown that, when the longitudinal temperature is high enough, a rotating cylinder which is bounded in the radial direction is stable against arbitrary perturbations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol.10, No. 3, pp. 3–11, May–June, 1969.     

Three-dimensional analytical solution for a rotating disc of functionally graded materials with transverse isotropy

Based on the basic equations for axisymmetric problems of transversely isotropic elastic materials, the displacement components are expressed in terms of polynomials of the radial coordinate with the five involved coefficients, named as displacement functions in this paper, being undetermined functions of the axial (thickness) coordinate. Five equations governing the displacement functions are then derived. It is shown that the displacement functions can be found through progressive integration by incorporating the boundary conditions. Thus a three-dimensional analytical solution is obtained for a transversely isotropic functionally graded disc rotating at a constant angular velocity.The solution can be degenerated into that for an isotropic functionally graded rotating disc. A prominent feature of this solution is that the material properties can be arbitrary functions of the axial coordinate. Thus, the solution for a homogeneous transversely isotropic rotating disc is just a special case that can be easily derived. An example is finally considered for a special functionally graded material, and numerical results shows that the material inhomogeneity has a remarkable effect on the elastic field.     

Finite and transversely isotropic elastic cylinders under compression with end constraint induced by friction

This paper derives an exact solution for the non-uniform stress and displacement fields within a finite, transversely isotropic, and linear elastic cylinder under compression with a kind of radial constraint induced by friction between the end surfaces of the cylinder and the loading platens. The main feature of the present work is the introduction of a general solution form for Lekhnitskii’s stress function such that the governing equation and all end and curved boundary conditions of the cylinder are satisfied exactly. Two different solutions were obtained corresponding to the real or complex characteristic roots of the governing equation, depending on the combination of the elastic material constants. The solution by Watanabe [Watanabe, S., 1996. Elastic analysis of axi-symmetric finite cylinder constrained radial displacement on the loading end. Structural Engineering/Earthquake Engineering JSCE 13, 175s–185s] for isotropic cylinders under compression test can be recovered as a special case. Our numerical results show that both the non-uniform stress distribution and the difference between the apparent and the true Young’s moduli of the cylinder are very sensitive to the anisotropy of Young’s moduli, Poisson’s ratios and shear moduli. A more distinct bulging shape of the cylinder is expected when anisotropy in shear modulus is strong, the cylinder is relatively short, and the end constraint is large. The bulging shape, however, does not depend strongly on anisotropy of either Poisson’s ratio or Young’s modulus.     

Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.     

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.     

A Pressurized Functionally Graded Hollow Cylinder with Arbitrarily Varying Material Properties

The elastic analysis of a pressurized functionally graded material (FGM) annulus or tube is made in this paper. Different from existing studies, this study deals with an axisymmetrical FGM hollow cylinder or disk with arbitrarily varying material properties. A simple and efficient approach is suggested, which reduces the associated problem to solving a Fredholm integral equation. The resulting equation is approximately solved by expanding the solution as series of Legendre polynomials. The stresses and displacements can be represented in terms of the solution to the equation. For radius-dependent Young’s modulus, numerical results of the distribution of the radial and circumferential stresses are presented graphically. Our results indicate that change in the gradient of the FGM tube does not produce a substantial variation of the radial stress, but strongly affects the distribution of the hoop stress. In particular, the hoop stress may reach its maximum at an internal position or at the outer surface when the tube is internally pressurized. The results obtained are helpful in designing FGM cylindrical vessels to prevent failure.      

Stresses in rotating heterogeneous viscoelastic composite cylinders with variable thickness

An analytical solution is presented for the rotation problem of a two-layer composite elastic cylinder under a plane strain assumption. The external cylinder has variable-thickness formulation, and is made of a heterogeneous orthotropic material. It contains a fiber-reinforced viscoelastic homogeneous isotropic solid core of uniform thickness. The thickness and elastic properties of the external cylinder are taken as power functions of the radial direction. By the boundary and continuity conditions, the radial displacement and stresses for the rotating composite cylinder are determined. The effective moduli and Illyushin’s approximation methods are used to obtain the viscoelastic solution to the problem. The effects of heterogeneity, thickness variation, constitutive, time parameters on the radial displacement, and stresses are investigated.     

On Pressurized Curvilinearly Orthotropic Circular Disk, Cylinder and Sphere Made of Radially Nonuniform Material

A unified analysis is presented for the elastic response of a pressurized cylindrically anisotropic hollow disk under assumed conditions of plane stress, or a hollow cylinder under plane strain conditions, and a spherically anisotropic hollow sphere, made of material which is nonuniform in the radial direction according to the power law relationship. The solution for a cylinder under generalized plane strain is also presented. Two parameters play a prominent role in the analysis: the material nonuniformity parameter m, and the parameter ?? which accounts for the combined effects of material anisotropy, represented by the specified parameters (??, ??, ??), and material nonuniformity, represented by the parameter m. The radial and circumferential stresses are the linear combinations of two power functions of the radial coordinate, whose exponents (n ₁ and n ₂) depend on the parameters m and ??. New light is added to the stress amplification and shielding under combined effects of curvilinear anisotropy and radial nonuniformity. Different loading combinations are considered, including the equal pressure at both boundaries, and the uniform pressure at the inner or the outer boundary. While the stress state for the equal pressure loading is uniform in the case of isotropic uniform material (m=0, ??=1), and for one particular radially nonuniform and anisotropic material, it is strongly nonuniform for a general anisotropic or nonuniform material. If the aspect ratio of the inner and outer radii decreases (small hole in a large disk/cylinder or sphere), the magnitude of the circumferential stress at the inner radius increases for n ₁>0 (stress amplification), and decreases for n ₁<0 (stress shielding). Both can be achieved by various combinations of the material parameters m, ??, ??, and ??. While the stress amplification in the case of a pressurized external boundary occurs readily, it occurs only exceptionally in the case of a pressurized internal boundary. The effects of material parameters on the displacement response are also analyzed. The approximate character of the plane stress solution of a pressurized thin disk is discussed and the results are compared with those obtained by numerical solution of the exact three-dimensional disk model.     

Effect of magneto-thermo-viscoelasticity in an unbounded body with a spherical cavity subjected to a harmonically varying temperature without energy dissipation

The present work is devoted to study effects of the thermally induced vibration, magnetic field and viscoelasticity in an isotropic homogeneous unbounded body with a spherical cavity. The GN model of thermoelasticity without energy dissipation is applied. The closed form solutions for distributions of displacement, temperature and radial and hoop stresses are illustrated. The boundary conditions for the temperature and mechanical and Maxwell’s stresses are employed. The solutions valid in the case of small frequency are deduced and the results are compared with the corresponding results obtained in other generalized thermoelasticity theories. The results obtained are calculated for a copper material and presented graphically. It’s deduced that the magnetic field, viscosity and thermally induced vibration are very pronounced on displacement, temperature and stresses.     

Exact Solutions and Material Tailoring for Functionally Graded Hollow Circular Cylinders

We employ the Airy stress function to derive analytical solutions for plane strain static deformations of a functionally graded (FG) hollow circular cylinder with Young’s modulus E and Poisson’s ratio v taken to be functions of the radius r. For E ₁ and v ₁ power law functions of r, and for E ₁ an exponential but v ₁ an affine function of r, we derive explicit expressions for stresses and displacements. Here E ₁ and v ₁ are effective Young’s modulus and Poisson’s ratio appearing in the stress-strain relations. It is found that when exponents of the power law variations of E ₁ and v ₁ are equal then stresses in the cylinder are independent of v ₁; however, displacements depend upon v ₁. We have investigated deformations of a FG hollow cylinder with the outer surface loaded by pressure that varies with the angular position of a point, of a thin cylinder with pressure on the inner surface varying with the angular position, and of a cut circular cylinder with equal and opposite tangential tractions applied at the cut surfaces. When v ₁ varies logarithmically through-the-thickness of a hollow cylinder, then the maximum radial stress, the maximum hoop stress and the maximum radial displacements are noticeably affected by values of v ₁. Conversely, we find how E ₁ and v ₁ ought to vary with r in order to achieve desired distributions of a linear combination of the radial and the hoop stresses. It is found that for the hoop stress to be constant in the cylinder, E ₁ and v ₁ must be affine functions of r. For the in-plane shear stress to be uniform through the cylinder thickness, E ₁ and v ₁ must be functions of r ². Exact solutions and optimal design parameters presented herein should serve as benchmarks for comparing approximate solutions derived through numerical algorithms.     

Temperature distribution in a thin rotating disk

The problem of heat conduction in a thin rotating disk with heat input at a fixed point is considered. The disk is cooled by forced convection from its lateral surfaces. By defining a complex temperature, the temperature throughout the disk is presented as a series of Bessel functions of complex argument. Results are given for a range of rotational speeds.Nomenclature R radial coordinate - angular coordinate - a radius of disk - b thickness of disk - T temperature - T ambient temperature - rotational speed of disk - q heat flux into disk - k thermal conductivity of disk - density of disk - c specific heat of disk - h coefficient of convective heat transfer - r dimensionless radial coordinate, R/a - T* characteristic temperature, q ₀ a/ k - t dimensionless temperature, (T–T )/T* - C ₁, C ₂ dimensionless parameters defined in (3)     

The radially nonhomogeneous elastic axisymmentric problem

The plane axisymmetric problem with axisymmetric geometry and loading is analyzed for a radially nonhomogeneous circular cylinder, in linear elasticity. Considering the radial dependence of the stress, the displacements fields and of the stiffness matrix, after a series of admissible functional manipulations, the general differential system solving the problem is developed. The isotopic radially inhomogeneous elastic axisymmetric problem is also analyzed. The exact elasticity solution is developed for a radially nonhomogeneous hollow circular cylinder of exponential Young’s modulus and constant Poisson’s ratio and of power law Young’s modulus and constant Poisson’s ratio. For the isotropic elastic axisymmetric problem, a general expression of the stress function is derived. After the satisfaction of the biharmonic equation and making compatible the stress field’s expressions, the stress function and the stress and displacements fields of the axisymmetric problem are also deduced. Applications have been made for a radially nonhomogeneous hollow cylinder where the stress and displacements fields are determined.