Fourier series based FE analysis of a disc under prescribed displacements–elastic stress study

This paper examines the numerical displacements and stresses developed around a disc under horizontal prescribed displacements and at the interface separating it from the surrounding elastic soil. Since the geometry of the problem exhibits axial symmetry and the loading is non-axisymmetric, the semi-analytical FE approach is used as it proves to be efficient and economical. First, both analytical and numerical expressions for soil reaction are established and compared. Results of comparison show a very good agreement. Then, for different values of the soil Poisson’s ratio, normal radial stresses, orthoradial stresses and shear stresses distributions along radial distance reaching 20r _d (r _d is the disc radius) are presented for a disc that has either perfectly smooth or perfectly rough interfaces with the elastic medium. The paper finishes by showing the effect of the soil Poisson’s ratio as well as the relative soil/interface stiffness on the stresses developed at the interface locations.     

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.      

Stress analysis of a functional graded material plate with a circular hole

This paper is to study the two-dimensional stress distribution of a functional graded material plate (FGMP) with a circular hole under arbitrary constant loads. With using the method of piece-wise homogeneous layers, the stress distribution of the functional graded material plate having radial arbitrary elastic properties is derived based on the theory of the complex variable functions. As examples, numerical results are presented for the FGMPs having given radial Young’s modulus or Poisson’s ratio. It is shown that the stress is greatly reduced as the radial Young’s modulus increased, but it is less influenced by the variation of the Poisson’s ratio. Moreover, it is also found that the stress level varies when the radial Young’s modulus increased in different ways. Thus, it can be concluded that the stress around the circular hole in the FGMP can be effectively reduced by choosing the proper change ways of the radial elastic properties.     

Analytical Solution for Radial Deformations of Functionally Graded Isotropic and Incompressible Second-Order Elastic Hollow Spheres

We analytically analyze radial expansion/contraction of a hollow sphere composed of a second-order elastic, isotropic, incompressible and inhomogeneous material to delineate differences and similarities between solutions of the first- and the second-order problems. The two elastic moduli are assumed to be either affine or power-law functions of the radial coordinate R in the undeformed reference configuration. For the affine variation of the shear modulus μ, the hoop stress for the linear elastic (or the first-order) problem at the point R=(R _ou R _in (R _ou +R _in )/2)^1/3 is independent of the slope of the μ vs. R line. Here R _in and R _ou equal, respectively, the inner and the outer radius of the sphere in the reference configuration. For μ(R)∝R ⁿ , for the linear problem, the hoop stress is constant in the sphere for n=1. However, no such results are found for the second-order (i.e., materially nonlinear) problem. Whereas for the first-order problem the shear modulus influences only the radial displacement and not the stresses, for the second-order problem the two elastic constants affect both the radial displacement and the stresses. In a very thick homogeneous hollow sphere subjected only to pressure on the outer surface, the hoop stress at a point on the inner surface depends upon values of the two elastic moduli. Thus conclusions drawn from the analysis of the first-order problem do not hold for the second-order problem. Closed form solutions for the displacement and stresses for the first-order and the second-order problems provided herein can be used to verify solutions of the problem obtained by using numerical methods.     

Resonant-Based Identification of the Poisson’s Ratio of Orthotropic Materials

The resonant-based identification of the in-plane elastic properties of orthotropic materials implies the estimation of four principal elastic parameters: E ₁ , E ₂ , G ₁₂ , and ν ₁₂ . The two elastic moduli and the shear modulus can easily be derived from the resonant frequencies of the flexural and torsional vibration modes, respectively. The identification of the Poisson’s ratio, however, is much more challenging, since most frequencies are not sufficiently sensitive to it. The present work addresses this problem by determining the test specimen specifications that create the optimal conditions for the identification of the Poisson’s ratio. Two methods are suggested for the determination of the Poisson’s ratio of orthotropic materials: the first employs the resonant frequencies of a plate-shaped specimen, while the second uses the resonant frequencies of a set of beam-shaped specimens. Both methods are experimentally validated using a stainless steel sheet.     

ASYMPTOTIC ANALYSIS OF MODE Ⅱ STATIONARY GROWTHCRACK ON ELASTIC-ELASTIC POWER LAWCREEPING BIMATERIAL INTERFACE

A mechanical model was established for mode Ⅱ interfacial crack static growingalong an elastic-elastic power law creeping bimaterial interface. For two kinds of boundaryconditions on crack faces, traction free and frictional contact, asymptotic solutions of thestress and strain near tip-crack were given. Results derived indicate that the stress andstrain have the same singularity, there is not the oscillatory singularity in the field; thecreep power-hardening index n and the ratio of Young‘s module notably influence the crack-tip field in region of elastic power law creeping material and n only influences distribution ofstresses and strains in region of elastic material. When n is bigger, the creepingdeformation is dominant and stress fields become steady, which does not change with n.Poisson‘s ratio does not affect the distributing of the crack-tip field.     

An Instrumented Indentation Method for Young’s Modulus Measurement with Accuracy Estimation

The relationships between indentation responses and Young’s modulus of an indented material were investigated by employing dimensional analysis and finite element method. Three representative tip bluntness geometries were introduced to describe the shape of a real Berkovich indenter. It was demonstrated that for each of these bluntness geometries, a set of approximate indentation relationships correlating the ratio of nominal hardness/reduced Young’s modulus H _n /E _r and the ratio of elastic work/total work W _e/W can be derived. Consequently, a method for Young’s modulus measurement combined with its accuracy estimation was established on basis of these relationships. The effectiveness of this approach was verified by performing nanoindentation tests on S45C carbon steel and 6061 aluminum alloy and microindentation tests on aluminum single crystal, GCr15 bearing steel and fused silica.     

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.     

Influence of elastic material compressibility on parameters of an expanding spherical stress wave

We investigated the influence of elastic material compressibility on parameters of an expanding spherical stress wave. The material compressibility is represented by Poisson’s ratio, ν, in this paper. The stress wave is generated by a pressure produced inside a spherical cavity surrounded by the isotropic elastic material. The analytical closed form formulae determining the dynamic state of the mechanical parameters (displacement, particle velocity, strains, stresses, and material density) in the material have been derived. These formulae were obtained for surge pressure p(t) = p ₀ = const inside the cavity. From analysis of these formulae, it is shown that the Poisson’s ratio substantially influences the course of material parameters in space and time. All parameters intensively decrease in space together with an increase of the Lagrangian coordinate, r. On the contrary, these parameters oscillate versus time around their static values. These oscillations decay in the course of time. We can mark out two ranges of parameter ν values in which vibrations of the parameters are “damped” at a different rate. Thus, Poisson’s ratio in the range below about 0.4 causes intense decay of parameter oscillations. On the other hand in the range 0.4 < ν < 0.5, i.e. in quasi-incompressible materials, the “damping” of parameter vibrations is very low. In the limiting case when ν = 0.5, i.e. in the incompressible material, “damping” vanishes, and the parameters harmonically oscillate around their static values. The abnormal behaviour of the material occurs in the range 0.4 < ν < 0.5. In this case, an insignificant increase of Poisson’s ratio causes a considerable increase of the parameter vibration amplitude and decrease of vibration “damping”.      

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.     

Analytical and numerical analysis for the FGM thick sphere under combined pressure and temperature loading

Thermo-mechanical analysis of functionally graded hollow sphere subjected to mechanical loads and one-dimensional steady-state thermal stresses is carried out in this study. The material properties are assumed to vary non-linearly in the radial direction, and the Poisson’s ratio is assumed constant. The temperature distribution is assumed to be a function of radius, with general thermal and mechanical boundary conditions on the inside and outside surfaces of the sphere. In the analysis presented here, the effect of non-homogeneity in FGM thick sphere was implemented by choosing a dimensionless parameter, named β _i (i = 1, . . . , 3), which could be assigned an arbitrary value affecting the stresses in the sphere. It is observed that the solution process for β _i (i = 3) = −1 are different from those obtained for other values of β _i (i = 1, . . . , 3). Cases of pressure, temperature, and combined loadings were considered separately. It is concluded that by changing the value of β _i (i = 1 . . . 3), the properties of FGM can be so modified that the lowest stress levels are reached. The present results agree well with existing results. Using FEM simulations, the analytical findings for FGM spheres under the influence of internal pressure and temperature gradient were compared to finite element results.     

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.     

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 state of an inhomogeneous piezoceramic cylinder subject to bending

The electroelastic state of a bent inhomogeneous piezoceramic cylinder with sliding restraints at the ends is examined. The boundary-value problem is reduced to a system of 12k (k = 1, 2,...) integro-differential equations. Expressions for stresses characterizing the stress state of the cylinder are derived. Relative hoop stresses are calculated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 107–114, August 2006.     

Centrifugal instability and the rib vortices in the cylinder wake

The physical mechanism for generation of streamwise vortices (or rib vortices) in the cylinder wake is numerically investigated with a finite-difference scheme. Rayleigh's theory of centrifugal instability for inviscid axisymmetric flow is extended to analyze the 2-D primary flows. Accordingly, an analytical dimensionless groupRay=−(r/v _θ)∂v _θ/∂r−1 is derived, wherev _θ represents the velocity of a fluid element relative to the oncoming flow,r is the local curvature radius of the element pathline. Centrifugal instability occurs whenRay>0. Stability analyses are carried out with this discriminant for primary flows at different time levels in a half shedding period of the von Kármán (or vK) vortices. Unstable areas are identified and the locations of rib vortices are coincident well with the unstable areas within the first wavelength of vK vortices behind the cylinder. The numerical results also show that rib vortices experience amplification in this region. It is apparent that centrifugal instability plays an important role in the generation of rib vortices in the cylinder wake. The project spported by the National Natural Science Foundation of China     

Potential Flow Past a Slightly Deformed Porous Circular Cylinder

A uniform potential flow past a porous circular cylinder with a core of different permeability is discussed. The porous circular cylinder is slightly deformed whose radius is r=r₁(1+ecosm q){r=r_1(1+\epsilon \cos m \theta)} , where | e | << 1{\mid\epsilon\mid\ll 1} and m is a positive integer. Here r, θ are the polar coordinates and r ₁ is the characteristic radius of the cylinder. The drag force exerted by the exterior flow on the surface of the cylinder is calculated and it depends on the thickness of the porous material and on the permeabilities of the two porous regions. As special cases, porous cylinder with hollow core, rigid core, and deformed cylinder is discussed.     

Some problems of linear elasticity for cylinders in micropolar orthotropic material

Some special problems for axisymmetric solids made of linearly elastic orthotropic micropolar material with central symmetry are dealt with. The first one is a hollow circular cylinder of unlimited length, subjected to internal and external uniform pressure. The second one is a hollow or solid circular cylinder of finite length, subjected to a relative rotation of the bases about its axis. In both cases, one of the axes of elastic symmetry is parallel to the cylinder axis; the other two are arbitrarily oriented in the plane of any cross-section of the solid. The elastic properties are invariant along the cylinder axis. It is shown that the two problems are governed by formally similar sets of ordinary differential equations in the kinematic fields (in-plane displacements and microrotations). In the general case, numerical solutions are derived. The solution for the cylinder subjected to radial pressure does not significantly differ from that obtained in classical elasticity, at least in terms of radial and hoop force stresses. In the case of a cylinder subjected to torsion the difference between the micropolar and the classical solutions is more pronounced. The torque induces twisting couple stresses about the cylinder axis of variable sign. Finally, size effects in terms of torsional inertia are pointed out.     

Effective thermoelastic properties of discrete-fiber reinforced materials with transversally-isotropic components

In the present paper, we will illustrate the application of the method of conditional moments by constructing the algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters of inclusions. A special case of two-component matrix composite with randomly distributed unidirectional spheroidal inclusions is considered. To this end it is assumed that the components of the composite show transversally isotropic symmetry of thermoelastic properties and that the axes of symmetry of the thermoelastic properties of the matrix and inclusions coincide with the coordinate axis x ₃. As a numerical example a composite based on carbon inclusions and epoxide matrix is investigated. The dependencies of Young’s moduli, Poisson’s ratios and shear modulus from the concentration of inclusions and for certain values which characterize the shape of inclusions are analyzed. The results are compared and discussed in context with other theoretical predictions and experimental data.      

Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder

 In this paper the radial deformation and the corresponding stresses in a non-homogeneous hollow elastic cylinder rotating about its axis with a constant angular velocity is investigated. The material of the cylinder is assumed to the non-homogeneous and cylindrically orthotropic. The system of fundamental equations is solved by means of a finite difference method and the numerical calculations are carried out for the temperature, the components of displacement and the components of stress with the time t and through the thickness of the cylinder. The results indicate that the effect of inhomogeneity is very pronounced. Received on 21 December 2000     

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.