The problem of a rigid punch on a non-homogeneous elastic half space

The elastostatic problem of a rigid punch on an elastic half space is considered. The medium is assumed to exhibit a non-homogeneity varying with depth. Using the Fourier Transform Technique, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically. The effect of non-homogeneity on the stress distribution under the punch and on the stress singularity is studied. The influence of Poisson's ratio on the results is also considered.     

Elastic-plastic stress analysis in a long functionally graded solid cylinder with fixed ends subjected to uniform heat generation

Elastic-plastic deformation of a solid cylinder with fixed ends, made of functionally graded material (FGM) with uniform internal heat generation is investigated, based on Tresca’s yield criterion and its associated flow rule, considering four of the material properties to vary radially according to a parabolic form. These four material properties are yield strength, modulus of elasticity, coefficients of thermal conduction and thermal expansion, assumed to be independent of temperature as Poisson’s ratio which is taken as constant. The materials which compose the functionally graded cylinder are supposed to be elastic-perfectly plastic materials. Expressions for the distributions of stress, strain and radial displacement are found analytically in terms of unknown interface radii. After determining these radii numerically by means of Mathematica 5.2, the distributions are plotted versus dimensionless radius, increasing heat generation, to compare the FGM cylinder with the homogeneous one. The numerical values used in this work for material parameters are arbitrarily chosen to point out the effect of the non-homogeneity on the stress distribution. The results obtained show that the stress distribution, as well as the development of plastic region radii, is influenced substantially by the material non-homogeneity.     

Flow and heat transfer of a micropolar fluid in an axisymmetric stagnation flow on a cylinder with variable properties and suction (numerical study)

Summary. In this paper, an analysis is presented to study the effects of variable properties, density, viscosity and thermal conductivity of a micropolar fluid flow and heat transfer in an axisymmetric stagnation flow on a horizontal cylinder with suction, numerically. The fluid density and the thermal conductivity are assumed to vary linearly with temperature. However, the fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The similarity solution is used to transform the problem under consideration into a boundary value problem of nonlinear coupled ordinary differential equations which are solved numerically by using the Chebyshev finite difference method (ChFD). Numerical results are carried out for various values of the dimensionless parameters of the problem. The numerical results show variable density, variable viscosity, variable thermal conductivity and micropolar parameters, which have significant influences on the azimuthal and the angular velocities and temperature profiles, shear stress, couple stress and the Nusselt number. The numerical results have demonstrated that with increasing temperature ratio parameter the azimuthal velocity decreases. With increasing variable viscosity parameter the temperature increases, whereas the azimuthal and the angular velocities decrease. Also, the azimuthal and the angular velocities increase and the temperature decreases as the variable conductivity parameter increases. Finally, the pressure increases as the suction parameter increases.     

Eddy currents in a transverse MRI gradient coil

A transverse gradient coil (x- or y-coil) of an MRI-scanner is modeled as a network of curved circular strips placed at the surface of a cylinder. The current in this network is driven by a time-harmonic source current. The low frequency applied allows for an electro-quasi-static approach. The strips are thin and the current is assumed to be uniformly distributed in the thickness direction. For the current distribution in the width direction of the strips, an integral equation is derived. Its logarithmically singular kernel represents inductive effects related to the occurrence of eddy currents. For curved circular strips of width much smaller than the radius of the cylinder one may locally replace the curved circular strip by a tangent plane circular strip. This plane geometry preserves the main characteristics of the transverse current distribution through the strips. The current distribution depends strongly on the in-plane curvature of the strips. The Petrov–Galerkin method, using Legendre polynomials, is applied to solve the integral equation and shows fast convergence. Explicit results are presented for two examples: a set of 1 strip and one of 10 strips. The results show that the current distributions are concentrated near the inner edges and that resulting edge-effects, both local and global, are non-symmetric. This behavior is more apparent for higher frequencies and larger in-plane curvatures. Results have been verified by comparison with finite-element results.     

Dynamic analysis of composite coil springs of arbitrary shape

The dynamic behavior of composite coil springs of arbitrary shape is investigated. The Timoshenko beam theory is adopted in the derivation of the governing equation. The material of the rod is assumed to be homogeneous, linear elastic and anisotropic. The effects of the ratio maximum diameter of the cylinder/thickness (D_max/d), the number of active turns (n), the helix pitch angle (α) and the ratio of the minimum to maximum cylinder radii (R_min/R_max) on the dynamic behavior of the composite barrel and hyperboloidal springs are investigated. The free and forced vibrations of composite coil springs of arbitrary shape such as barrel and hyperboloidal springs are analyzed through various examples.     

Tensile fracture characteristics of double convex-faced cylindrical powder compacts

Doubly-convex cylindrical tablets compacted uniaxially from different particle-size fractions of Di-Pac-Sugar powder, have been fractured under diametral loading conditions employing the standard diametral compression test. The ratio of cylinder length to diameter (W/D) ranged from 0.476–1.06; the ratio of cylinder diameter to radius of curvature of the tablet faces (D/R) was varied from 0.0–1.184. An equation based on geometrical volume equivalence conditions, of a doubly-convex cylinder to that of a plane-faced cylinder, relating the tensile strength of the material to the fracture load and dimensions of doubly-convex cylindrical specimens, has been developed. This equation is valid for any compacted cylindrical tablet. Using this equation, it was possible to assess the tensile strength of doubly-convex tablets, and a shape factor has been defined. Also, the predicted fracture loads obtained by this equation are found to compare well with those values determined by utilizing a stress factor based on the photoelastic stress analysis. For higher ratios of W/D and smaller diameter values, the variation in the results calculated is found to be less than 5%. Meanwhile, for higher values of W/D, the geometrical end effect is reduced and the tensile fracture stress tends to become similar to that of a plane-faced specimen.     

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.     

Stress intensity factors in a reinforced thick-walled cylinder

In this paper an elastic thick-walled cylinder containing a radial crack is considered. It is assumed that the cylinder is reinforced by an elastic membrane on its inner surface. The model is intended to simulate pressure vessels with cladding. The formulation of the problem is reduced to a singular integral equation. Various special cases including that of a crack terminating at the cylinder-reinforcement interface are investigated and numerical examples are given. Among the interesting results found one may mention the following: In the case of the crack touching the interface the crack surface displacement derivative is finite and consequently the stress state around the corresponding crack tip is bounded, and generally, for realistic values of the stiffness parameter, the effect of the reinforcement is not very significant.     

Effects of an airfoil and shock waves on vortex shedding process behind a square cylinder

Summary Effects of an airfoil and shock waves on vortex shedding process behind a square cylinder have been examined experimentally at a Mach number of about 0.91 and at a Reynolds number (based on the side lengthD of the square cylinder) of about 4.2×10⁵. The main experimental parameter is the spacing ratioL/D, and is varied from 1.125 to 5.5, whereL is the spacing between the square cylinder and the airfoil.It is found that similarly to the case at subcritical Mach numbers at the supercritical Mach number there exist three patterns of the flow around the square cylinder and airfoil arranged in tandem depending upon the spacing ratioL/D: In the first flow pattern with small spacing ratio, the downstream airfoil is enclosed completely in the vortex formation region of the square cylinder. In the second flow pattern, the shear layers separating from the square cylinder reattach to the airfoil. In the third flow pattern with large spacing the shear layers roll up upstream of the airfoil. The Strouhal number at the supercritical Mach number is higher than that at the subcritical Mach numbers. Shock waves hasten the vortex shedding behind the square cylinder by decreasing the area of asymmetrical part of the vortex formation region with respect to the wake axis, and let the streamwise length of the separating shear layers longer than otherwise.With 8 Figures     

One-Dimensional Transient Dynamic Response in Functionally Graded Layered Media

In this study, one-dimensional transient wave propagation in multilayered functionally graded media is investigated. The multilayered medium consists of N different layers of functionally graded materials (FGMs), i.e., it is assumed that the stiffness and the density of each layer are varying continuously in the direction perpendicular to the layering, but isotropic and homogeneous in the other two directions. The top surface of the layered medium is subjected to a uniform dynamic in-plane time-dependent normal stress; whereas, the lower surface of the layered medium is assumed free of surface tractions or fixed. Moreover, the multilayered medium is assumed to be initially at rest and its layers are assumed to be perfectly bonded to each other. The method of characteristics is employed to obtain the solutions of this initial-boundary-value problem. The numerical results are obtained and displayed in curves showing the variation of the normal stress component with time. These curves reveal clearly the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers. The curves also display the effects of non-homogeneity in the wave profiles. The curves further show that the numerical technique applied in this study is capable of predicting the sharp variations in the field variables in the neighborhood of the wave fronts. By suitably adjusting the material constants, solutions for the case of isotropic, homogeneous and linearly elastic multilayered media and for some special cases including two different functionally graded layers are also obtained. Furthermore, solutions for some special cases are compared with the existing solutions in the literature; very good agreement is found.     

Fatigue life assessment under a complex multiaxial load history: an approach based on damage mechanics

In order to assess the fatigue behaviour of structural components under a complex (cyclic or random) multiaxial stress history, methods based on damage mechanics concepts can be employed. In this paper, a model for fatigue damage evaluation in the case of an arbitrary multiaxial loading history is proposed by using an endurance function which allows us to determine the damage accumulation up to the final failure of the material. By introducing an evolution equation for the endurance function, the final collapse can be assumed to occur when the damage D is complete, that is when D reaches the unity. The parameters of this model, which adopts the stress invariants and the deviatoric stress invariants to quantify the damage phenomenon, are determined through a Genetic Algorithm once experimental data on the fatigue behaviour of the material being examined are known for some complex stress histories. With respect to traditional approaches to multiaxial fatigue assessment, the proposed model presents the following advantages: (1) the evaluation of a critical plane is not necessary; (2) no cycle counting algorithm to determine the fatigue life is required, because it considers the progressive damage process during the fatigue load history; (3) the model can be applied to any kind of stress history (uniaxial cyclic loading, multiaxial in‐phase or out‐of‐phase cyclic loading, uniaxial or multiaxial random loading).     

Mode-III fracture of bonded non-homogeneous materials

This study deals with the mode-III fracture of bonded non-homogeneous materials. Assuming that the shear moduli of the materials vary exponentially with the spatial coordinates, the problem is reduced to the solution of a singular integral equation with a Cauchy type kernel. It is shown that, as for homogeneous materials, the stresses have the usual square-root singularity. The stress intensity factors and the mode III strain energy release rate are computed for various values of the ratio of shear moduli and the non-homogeneity parameters α, β₁, and β₂. The results indicate that non-homogeneity may have a very significant effect on the fracture of bonded materials.     

Analysis of steady free convection from an axisymmetric heat-generating body embedded in a semi-infinite porous medium

An analysis is presented for the steady state free convective flow and heat transfer from an axisymmetric heat-generating body that is embedded in a fluid-saturated, semi-infinite, porous medium. The porous medium is assumed to be rigid, homogeneous and isotropic, and be in thermal equilibrium with the fluid. The fluid is assumed to be incompressible, with the density changes contributing only towards the buoyancy forces via the Boussinesq approximation. The governing equations for the fluid consist of the equation of continuity, Darcy's law and the equation of energy. After introducing the stream function concept, the equations governing the stream function and pressure are derived. Using the non-dimensional variables, the non-dimensional equations governing the non-dimensional forms of the temperature, stream function and pressure are dervied and the appropriate boundary conditions are stated. The mathematical formulation contains two parameters; D, the non-dimensional depth of the body from the surface of the porous medium, and a product Raθ_s of Rayleigh number (Ra) and the non-dimensional surface temperature of the body (θ_s). The Galerkin finite element method, with linear, isoparametric, quadrilateral elements, is used to reduce the mathematical formulation into a set of algebraic equations. The expressions to calculate the non-dimensional surface temperature and Nusselt number of the body, and the non-dimensional velocity of the fluid, are derived. A computer code has been developed to solve the algebraic equations, using Gauss elimination procedure, in a banded matrix form. The computer code, in addition to the non-dimensional temperature, stream function and pressure, calculates the isothermal lines, non-dimensional surface temperature of the body, Nusselt number of the body, velocity field and isobars. To demonstrate the application of the code, a spherical heat-generating body is considered as an example. Numerical results are obtained for D = 3 and 6, and Raθ_s = 0.001, 0.1, 1 and 5, and presented.     

FORMULATION OF A STOCHASTIC MODEL OF FATIGUE CRACK GROWTH

Based on a global/local energy balance a deterministic model of fatigue crack growth under constant amplitude loading is derived. The energy terms resulting from the continuous plasticity and localized fracture around the crack tip are determined for small scale yielding leading to the fatigue crack growth equation involving the stress intensity factor and its amplitude. Four material parameters which should be identified from experimental data have a physical interpretation; these are eventually assumed to be random variables and model the statistical scatter of the crack growth versus N curves observed in experiments. A Gaussian white noise random field is additionally assumed to describe the stochastic material non-homogeneity within a specimen. Its effect on the crack growth is derived and results in a positive non-stationary random function depending on the crack length. Statistical parameters of the random fields are identified. Verification of the model by comparison with experimental results is undertaken in a subsequent paper.     

Application of fracture mechanics to plastics deformed at high strain-rates

Thin walled cylindrical specimens of poly(methylmethacrylate) containing artificial flaws of different length and machined along a generator of the cylinder, are subjected to internal pressure pulses (shock pulses) of increasing magnitude until fracture occurs. A shock tube is used to generate the shock pulse. The variation of the stress required to induce fracture with crack length is studied. It is found that an equation similar to the Griffith relationship applies over the range of crack lengths 38 mm>2c>16 mm, the surface energy value required to fit the Griffith type equation to the experimental data in this crack length range being similar to that derived from quasi-static testing of the same material. It is found that the range of applicability of the Griffith equation is determined by the magnitude of the fracture stress of the non-artificially flawed material at the particular strain-rate at which the test is conducted.     

Chemical Stresses Induced by Boundary Layer Diffusion in a Cylindrical Sandwich Composite

The chemical stresses developed in a cylindrical sandwich composite during radial boundary layer diffusion have been investigated. The system consists of a thin layer A of circular cross section sandwiched between two semi-infinite outer layers B, with the diffusivity of diffusant in A (D_A) being much greater than that in B (D_B). Two boundary conditions, the constant surface concentration and the instantaneous surface concentration, were considered. The concentration distributions were obtained by the Bessel-Laplace transform method. The stress functions were solved analytically based on the linear elasticity. Numerical computations were performed to illustrate the effects of the diffusivity ratio (D_A/D_B) and of the thickness of the central layer A on stress distributions. The results show that the induced stress in layer A increases as the diffusivity ratio, or its thickness, increases, in consistency with the general findings for composites of rectangular geometry.     

Axisymmetric crack problem for a hollow cylinder imbedded in a dissimilar medium

The analytical solution for the linear elastic problem of flat annular crack in a transversely isotropic hollow cylinder imbedded in a transversely isotropic medium is considered. The hollow cylinder is assumed to be perfectly bonded to the surrounding medium. This structure, which can represent a cylindrical coating-substrate system, is subjected to uniform crack surface pressure. Because of the geometry and the loading, the problem is axisymmetric. The z = 0 plane on which the crack lies, is also a plane of symmetry. The composite media consisting of the hollow cylinder and the surrounding medium extends to infinity in z and r directions. The mixed boundary value problem is formulated in terms of the unknown derivative of the crack surface displacement by using Fourier and Hankel transforms. By extending the crack to the inner surface and to the interface, the cases of surface crack and crack terminating at the interface are obtained. Asymptotic analyses are performed to derive the generalized Cauchy kernel and associated stress singularities. The resulting singular integral equation is solved numerically. Stress intensity factors for various crack configurations, crack opening displacements and stresses along the interface and on z = 0 plane are presented for sample material combinations and geometric parameters.     

MHD Flow past a circular cylinder – a numerical study

 The steady, two-dimensional, incompressible MHD flow past a circular cylinder with an applied magnetic field parallel to the main flow is studied using the finite difference method. The magnetic Reynolds number is assumed to be small. Results are presented up to the Reynolds number R=500 and interaction parameter N=1.3. As N increases suppression of the separation is observed. Drag coefficient is decreasing for the small values of N and then increasing as N increases. It is found that a smaller value of far field distance is required as N increases. Received 6 March 2000     

Reissner-Sagoci problem for a non-homogeneous half-space with surface constraint

This paper deals with the problem of twisting of a non-homogeneous, isotropic, half-space by rotating a circular part of its boundary surface (0ra, z = 0) through a given angle. A ring (a<r<b, z = 0) outside this circle is stress-free and the remaining part (r>b, z = 0) is rigidly clamped. The shear modulus is assumed to vary with the cylindrical coordinates r, z by the power law ( = _, r z ). Such a dependence is of practical interest in the context of Soil Mechanics. The problem leads to a Fredholm integral equation of the second kind which is solved numerically, giving an evaluation of the influence of non-homogeneity on the torque at the surface and the stress intensity factor. The homogeneous case studied in [16] is recovered. Expressions for some quantities of physical importance such as the torque applied at the surface and stress intensity factor are obtained. It appears from our investigation that the influence of clamping dies out with increasing and . Quantitative evaluations are given and some curves for the relative increase, due to clamping, in the torque and in the stress intensity factor are presented.     

The cyclically symmetric anti-plane fracture analysis for the arc-shaped interface in a non-homogeneous bimaterial cylindrical structure

A hollow cylinder, which consists of an inner functionally graded elastic substrate and an outer functionally graded elastic layer with cyclically symmetric cracks (a special case of multiple cracks), is considered under anti-plane shear load. The method of variable separation is employed to reduce the mixed boundary value problems to a Cauchy singular integral equation, which is solved numerically by the Lobatto–Chebyshev quadrature technique. Numerical results are presented to show the effects of geometrical and physical quantities on the stress intensity factors (SIFs). Parametric studies are conducted on the SIFs, and practical guidelines are given for the optimization of the fracture performance: (a) the SIFs depend on the ratio between the outer and inner radii, and the ratio should be at least 1.1; (b) the outer elastic layer should be stiffer than the inner elastic substrate; (c) large non-homogeneity parameter of the outer graded layer and small non-homogeneity parameter of the inner graded substrate are beneficial to SIFs reduction; (d) there is a strong interference between the stress fields around multiple cracks when the cyclically symmetric parameter increases.