Functionally Graded Cylinder Research Articles

Exact solution of magneto-elastoplastic problem of functionally graded cylinder subjected to internal pressure

With the development of composite material technology, the multi-field coupling analysis of functionally graded (FG) structures has attracted extensive attention. For FG cylindrical pressure vessels, due to the non-uniform distribution of stresses filed in the structure, plastic deformation occurs irreversibly under external load, and Lorentz's force under static magnetic field is also one of the key factors affecting the elastoplastic state. Assuming that the elastic modulus of FG cylinder changes as a power function along the thickness and Poisson's ratio is constant, the hollow cylinder under the combined action of applied magnetic field and internal pressure may have four deformation states containing pure elastic deformation, partial plastic deformation yielding from the inner surface, partial plastic deformation yielding from the outer surface, and full plasticity. In this paper, analytical solutions of stress and displacement fields of the FG cylinder under these four different elastoplastic states are given, and the critical load conditions corresponding to the transition between these four different states are proposed. The correctness of the present analytical solution in this paper is verified by comparing it with the existing results of related simplified problems. In addition, the analytical solution of residual stress distribution caused by unloading internal pressure is discussed for self-reinforcing processing. Based on the analytical solution in this paper, the influence of FG parameter, applied magnetic field and internal pressure on the elastoplastic state, plastic zone size, stresses and displacements distribution of FG cylinder is analyzed in detail. This study shows that for this magneto-elastoplastic problem of FG cylinder, plastic deformation may appear from the inner surface or outer surfaces. With the increase of FG parameter, the first yielding position of FG cylinder changes from the inner surface to the outer surface. In addition, the existence of an applied magnetic field reduces the critical internal pressure required for FG cylinders to yield, making the FG hollow cylinder more prone to plastic deformation.

Secondary Creep Analysis of FG Rotating Cylinder with Exponential, Linear and Quadratic Volume Reinforcement.

Creep is an irreversible time-dependent deformation in which a material under constant mechanical stress and elevated temperature for a considerably prolonged period of time, starts to undergo permanent deformation. Creep deformation occurs in three stages namely, primary, secondary and tertiary. Out of these three stages, secondary or steady state creep is particularly an area of engineering interest as it has almost a constant creep rate. Creep deformation plays a significant role in understanding effective service life of an engineering component working under high temperature conditions as such components such as super-heater and re-heater tubes and headers in a boiler, jet engines operating at temperature as high as 1200 C, usually experience a failure or rupture due to creep phenomenon. Design engineers keep a close attention on working stress conditions and elevated temperature under which an engineering component is expected to work as these conditions determine the onset of creep behavior in an engineering component. By recognizing the parameters of material response to creep behavior, engineers can analyse the useful service life and hazardous working conditions for an engineering components. Recognizing the creep phenomenon as high temperature design limitation, ASME Boiler and Pressure Vessel Code have provided guidelines on maximum allowable stresses for materials to be used in creep range. One of the criteria for determination of allowable stresses is 1% creep deformation of material in 100,000 h of service. Thus, the study of creep behavior in engineering components pertaining to high stress and temperature working conditions is very important as it affects the reliability and performance of the engineering components. The aim of our study is to understand the behavior of secondary creep deformation so that an advanced reinforced functionally graded material with better creep resistance, can be designed. In this paper, a secondary creep analysis of functionally graded (FG) thick-walled rotating cylinder under internal and external pressure is conducted. The novelty of the model intends to specify secondary creep stresses and strains by employing exponential, linear and quadratic volume reinforcement for ceramic in metal matrix in radial direction. This will help us to understand the effect of volume reinforcement in FG cylinder under internal/external pressure and rotating centrifugal body force by obtaining secondary creep stresses and strains. The response of the FG cylinder with isotropic material is analyzed and the solution for stress–strain rates in radial and tangential directions are obtained in closed form. Comparison of steady state creep stresses and strains under exponential, linear and quadratic volume reinforcement profiles are discussed and presented graphically.

Analytical Solution for Strain Gradient Plasticity of Rotating Functionally Graded Thick Cylinders

The elastic-plastic deformation of rotating functionally graded (FG) cylinders is investigated based on strain gradient theory. The governing equations are obtained based on the modified von Mises yield criterion, linear work hardening and plane strain assumptions. An analytical solution for the obtained equations is presented by which the deformation, strain and stress components for any point of the cylinder can be obtained. After verification of the formulation by comparing the obtained results with the reported results in the literature, some studies are presented to investigate the effects of cylinder size on the stress distribution and elastic-plastic interface radius of the rotating FG cylinder under internal and external pressure. The effects of the strain gradient coefficient, angular velocity, and the heterogeneity constant of the material are investigated. The results show that increasing the heterogeneity constant of the material and decreasing the cylinder radius lead to increasing the strength of material and decreasing the elastic-plastic interface radius. Moreover, classical theory is compared with this study and the range of the sizes in which both the theories leading to the same results, are defined.

Fabrication and mechanical properties of aluminum-carbon nanotube functionally-graded cylinders

In spite of strong theoretical interest, limited experimental studies have investigated the production of functionally-graded (FG) Aluminum-Carbon Nanotube (Al-CNT) composites. Such studies focused on axially-layered FG structures because of their ease of manufacturing. In order to take advantage of the full potential of FG Al-CNT, methods to fabricate bulk components of various shapes are needed. In the current work, Al-CNT functionally graded cylinders with varying CNT content along the radial direction were prepared for the first time. A novel and simple method was used to generate the composition gradient. The composite was designed to have 2 wt.% CNT at the cylinder surface where it can provide the highest strength and hardness. Pure as-received Al was placed at the core of the cylinders to provide overall ductility. The FG samples exhibited excellent bonding between the layers, increased strength and hardness at the surface in addition to a unique combination of high strength, hardness, and ductility, which exceeded those of homogeneous Al-CNT composites of the same composition.

On propagation of axisymmetric waves in pressurized functionally graded elastomeric hollow cylinders

Soft materials can be designed with a functionally graded (FG) property for specific applications. Such material inhomogeneity can also be found in many soft biological tissues whose functionality is only partly understood to date. In this paper, we analyze the axisymmetric guided wave propagation in a pressurized FG elastomeric hollow cylinder. The cylinder is subjected to a combined action of axial pre-stretch and pressure difference applied to the inner and outer cylindrical surfaces. We consider both torsional waves and longitudinal waves propagating in the FG cylinder made of incompressible isotropic elastomer, which is characterized by the Mooney-Rivlin strain energy function but with the material parameters varying with the radial coordinate in an affine way. The pressure difference generates an inhomogeneous deformation field in the FG cylinder, which dramatically complicates the superimposed wave problem described by the small-on-large theory. A particularly efficient approach is hence employed which combines the state-space formalism for the incremental wave motion with the approximate laminate or multi-layer technique. Dispersion relations for the two types of axisymmetric guided waves are then derived analytically. The accuracy and convergence of the proposed approach is validated numerically. The effects of the pressure difference, material gradient, and axial pre-stretch on both the torsional and the longitudinal wave propagation characteristics are discussed in detail through numerical examples. It is found that the frequency of axisymmetric waves depends nonlinearly on the pressure difference and the material gradient, and an increase in the material gradient enhances the capability of the pressure difference to adjust the wave behavior in the FG cylinder. This work provides a theoretical guidance for characterizing FG soft materials by in-situ ultrasonic nondestructive evaluation and for designing tunable waveguides via material tailoring along with an adjustment of the pre-stretch and pressure difference.

Overall Heat Transfer Coefficient of Functionally Graded Hollow Cylinder

This paper presents how to calculate the overall heat transfer coefficient of a very long functionally graded hollow circular cylinder subjected to steady state heat transfer. Thermal conductivity coefficient of the functionally graded cylinder (FGC) vary radially and continuously according to an exponential form, which is supposed to be independent of the temperature. Overall heat transfer coefficient is found analytically in terms of the radial coordinate, thermal conductivity, material parameter, inner surface and outer surface temperatures of the cylinder. Once the overall heat transfer coefficient is found, calculation of the heat transfer rate across the cylinder wall is quite straightforward. The equation derived for the overall heat transfer coefficient can be applied to any type of functionally graded hollow circular cylinder playing with the material parameter term.

Characterization of the Tungsten-Steel Functionally Graded Cylinder in Elastic Region

This paper presents how to derive Airy stress function to obtain the thermal stresses in a tungsten-steel functionally graded solid cylinder with fixed ends in elastic region. Once Airy stress function is derived, the thermal stresses can be found due to the related equations. There is uniform heat generation inside the tungsten-steel functionally graded solid cylinder. Material properties of the functionally graded cylinder (FGC) are assumed to vary radially according to a parabolic form and assumed to be independent of the temperature. These properties are yield strength, modulus of elasticity, coefficient of thermal conduction and coefficient of thermal expansion (CTE). Poisson’s ratio is assumed to be constant as an average value between tungsten’s and steel’s. Airy stress function is derived in terms of these properties to characterize the FGC entirely.

Time-Dependent Thermomechanical Creep Behavior of FGM Thick Hollow Cylindrical Shells Under Non-Uniform Internal Pressure

In this paper, a theoretical solution for time-dependent thermo-elastic creep analysis of a functionally graded (FG) thick-walled cylinder based on the first-order shear deformation theory is presented. The cylinder is subjected to the non-uniform internal pressure and distributed temperature field due to steady-state heat conduction from inner to outer surface of the cylinder. Mechanical and thermal properties except Poisson’s ratio are assumed to vary along the thickness direction based on a power function. The creep constitutive model is on the basis of the Norton’s law. The effects of the temperature gradient and FG grading index on the creep stresses of the cylinder are investigated. A numerical solution using finite element method is also presented and good agreement was found. Although previous publications presented analytical solutions for creep analysis of thick-walled cylindrical pressure vessels under uniform pressure, to the best of the authors’ knowledge, so far, no analytical solution has been provided for time-dependent creep analysis of FG cylinder under non-uniform internal pressure. The results of this study are applicable for designing optimum FG thick-walled cylinder.

Electro-magneto-thermo-elastic response of infinite functionally graded cylinders without energy dissipation

The electro-magneto-thermo-elastic analysis problem of an infinite functionally graded (FG) hollow cylinder is studied in the context of Green–Naghdi's (G–N) generalized thermoelasticity theory (without energy dissipation). Material properties are assumed to be graded in the radial direction according to a novel power-law distribution in terms of the volume fractions of the metal and ceramic constituents. The inner surface of the FG cylinder is pure metal whereas the outer surface is pure ceramic. The equations of motion and the heat-conduction equation are used to derive the governing second-order differential equations. A finite element scheme is presented for the numerical purpose. The system of differential equations is solved numerically and some plots for displacement, radial and electromagnetic stresses, and temperature are presented. The radial displacement, mechanical stresses and temperature as well as the electromagnetic stress are all investigated along the radial direction of the infinite cylinder.

A meshless local Petrov–Galerkin method for nonlinear dynamic analyses of hyper-elastic FG thick hollow cylinder with Rayleigh damping

This paper is devoted to the geometrically nonlinear analysis of a functionally graded (FG) thick hollow cylinder with Rayleigh damping. The hollow cylinder is subjected to axisymmetric mechanical shock loading on its bounding surfaces. First, the meshless local Petrov–Galerkin (MLPG) method is developed for geometrically nonlinear problems based on total Lagrangian approach. During this process, the local integral equations are obtained using the weak formulation on local sub-domains for the set of governing equations by employing a Heaviside test function. The radial point interpolation method is used to approximate the field variables in terms of nodal displacements. The iterative Newmark/Newton–Raphson method is employed to solve the system of resulting nonlinear equations in suitable time steps. Because of large deformations compared to linear elastic materials, the hyper-elastic neo-Hookean model is considered for the problem. The hollow cylinder is supposed to be in plane strain condition. The properties of the FG cylinder are varied in the thickness direction using the volume fraction that is an exponential function of radius. At the end, to prove the robustness of the proposed method, several numerical tests are performed and effects of relative parameters on the dynamic behavior of the cylinder for various kinds of FGMs are discussed in detail. Findings demonstrate the effectiveness of the presented MLPG method for large deformation problems because of vanishing of the mesh distortion. This paper furnishes a ground to develop the MLPG method for dynamic large deformation problems.

Magneto-thermoelastic response of an infinite functionally graded cylinder using the finite element method

In this article, the magneto-thermo analysis problem of an infinite functionally graded (FG) hollow cylinder is studied. The radial displacement, mechanical stresses and temperature, as well as the electromagnetic stress, are all investigated along the radial direction of the infinite cylinder. Material properties are assumed to be graded in the radial direction according to a novel power-law distribution in terms of the volume fractions of the metal and ceramic constituents. The inner surface of the FG cylinder is pure metal, whereas the outer surface is pure ceramic. The equations of motion and the heat-conduction equation are used to derive the governing second-order differential equations. A finite element scheme is presented for the numerical purpose. The system of differential equations is solved numerically and some plots for displacement, radial and electromagnetic stresses, and temperature are presented.

Two-Dimensional Stress-Wave Propagation in Finite-Length FG Cylinders with Two-Directional Nonlinear Grading Patterns Using the MLPG Method

AbstractThe application of meshless local integral equations based on the meshless local Petrov-Galerkin (MLPG) method for the two-dimensional (2D) dynamic stress analysis of a 2D functionally graded (FG) cylinder with two-directional grading patterns is developed. It is assumed that the inner bounding surface of the cylinder is excited by mechanical shock loading. The problem is studied in the frequency domain using the Laplace transformation, and then the stresses are transferred to the time domain using Talbot techniques. The mechanical properties of the FG materials (FGMs) are simulated using a nonlinear model with radial and axial volume fractions. The 2D propagation of stresses is tracked through the radial and axial directions in a 2D domain for various grading patterns at various time instants. By using the presented meshless technique, the maximum values of the stresses are calculated. The effects of various grading patterns on the maximum values of the stresses are studied in detail. The present...

Stochastic hybrid numerical method for transient analysis of stress field in functionally graded thick hollow cylinders subjected to shock loading

This investigation aims to study the random stresses in a functionally graded (FG) thick hollow cylinder with uncertain material properties subjected to mechanical shock loading using a hybrid numerical method. The mechanical properties are considered to vary across thickness of FG cylinder as a nonlinear power function of radius. The stresses are obtained by solving Navier equation and using Galerkin finite element and Newmark finite difference methods. The Monte Carlo simulation is used to generate the random mechanical properties for the problem. The failure probabilities and time history analysis of stresses are determined for various coefficient of variation considering various grading patterns of mechanical properties. The presented hybrid numerical method is effective, with high capability for stochastic analysis of dynamic and transient analysis of FG structures with various boundary conditions.

Three-dimensional transient analysis of functionally graded cylindrical shells subjected to asymmetric dynamic pressure

AbstractThis paper presents an efficient and accurate numerical method based on the three-dimensional (3-D) elasticity theory for the transient analysis of functionally graded (FG) hollow cylindrical shells subjected to asymmetric dynamic pressure. The Fourier expansion is employed to describe the displacement components and dynamic pressure in the tangential direction. In addition, the layerwise theory is used to accurately account for the displacement components in the radial direction. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. Then, differential quadrature method (DQM) is implemented to discretize the resulting equations in the both spatial and time domains. The convergence, accuracy and performance of the present method are established through the convergence study and comparison with available results in the literature. Also, the effects of different parameters such as thickness-to-inner radius ratio and boundary conditions on the dynamic behavior of hollow FG cylinders are investigated. The present method can accurately predict transient displacement and stress with less computational efforts.

Response of functionally graded cylindrical shells under moving thermo-mechanical loads

The transient thermoelastic analysis of functionally graded (FG) cylindrical shells under moving boundary pressure and heat flux is presented. The material properties are assumed to be temperature-dependent and graded in the radial direction. The hyperbolic heat conduction equations are used to include the influence of finite heat wave speed (i.e., the non-Fourier effect). To benefit from the high accuracy and low computational efforts of the differential quadrature method (DQM) in conjunction with the effectiveness of the finite element method (FEM) in general geometry, loading and systematic boundary treatment, a combination of these methods is employed to discretize the governing equations in the spatial domain. The resulting system of differential equations is solved using Newmark's time integration scheme in the temporal domain. The presented formulation and method of solution are validated by showing their fast rate of convergence and by comparing the results, in the limit cases, with those obtained using the commercial finite element package ANSYS and some other available solutions in the literature. Then, the effects of different geometrical, material and load parameters on the transient thermoelastic behavior of the FG cylinders under moving mechanical and thermal loads are studied.

Heat Transfer Analysis of Functionally Graded Hollow Cylinders Subjected to an Axisymmetric Moving Boundary Heat Flux

The transient heat transfer analysis of functionally graded (FG) hollow cylinders subjected to a distributed heat flux with a moving front boundary on its inner surface is presented. The heat flux is assumed to be axisymmetric, and its front boundary moves along the axis of the cylinder. A method composed of the finite element and differential quadrature methods is employed to discretize the governing equations in the spatial domain. After demonstrating the convergence and accuracy of the method, the effects of different parameters on the temperature distribution and time history of the temperature at different points of FG cylinder are investigated.

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.

Strain gradient elasticity solution for functionally graded micro-cylinders

In this paper, strain gradient elasticity formulation for analysis of FG (functionally graded) micro-cylinders is presented. The material properties are assumed to obey a power law in radial direction. The governing differential equation is derived as a fourth order ODE. A power series solution for stresses and displacements in FG micro-cylinders subjected to internal and external pressures is obtained. Numerical examples are presented to study the effect of the characteristic length parameter and FG power index on the displacement field and stress distribution in FG cylinders. It is observed that the characteristic length parameter has a considerable effect on the stress distribution of FG micro-cylinders. Also, increasing material length parameter leads to decrease of the maximum radial and tangential stresses in the cylinder. Furthermore, it is shown that the FG power index has a significant effect on the maximum radial and tangential stresses.

Thermal shock analysis and thermo-elastic stress waves in functionally graded thick hollow cylinders using analytical method

Transient stress field and thermo-elastic stress wave propagation are studied in functionally graded thick hollow cylinder under arbitrary thermo-mechanical shock loading, in this article. Thermo-mechanical properties of functionally graded (FG) cylinder are assumed to be temperature independent and vary continuously and smoothly in the radial direction. The governing dynamic equations are analytically solved in temperature and elastic fields. To solve the problem, Laplace transform is used respect to time in all constitutive equations and boundary conditions. At first, temperature field equation analytically solved using Laplace transform and series method. The dynamic behaviors of thermo-elastic stresses are illustrated and discussed for various grading patterns of thermo-mechanical properties in several points across the thickness of FG cylinder. Time history of temperature field and thermal stresses are obtained using the residual theorem and the fast Laplace inverse transform method (FLIT), respectively. Also, the effects of the cylinder thickness and convection heat transfer coefficient on dynamic response of FG cylinder are revealed and discussed. The presented analytical method provides a ground to study the time histories of radial and hoop stresses in FG cylinders with different thickness and various volume fraction exponents. The advantage of this method is its mathematical ability to support simple and complicated mathematical function for the thermo-mechanical boundary conditions. A reasonable agreement can be seen in comparison of obtained results based on the presented analytical method with published data.

Meshless local Petrov–Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder

In this article, coupled thermoelasticity (without energy dissipation) based on Green–Naghdi model is applied to functionally graded (FG) thick hollow cylinder. The meshless local Petrov–Galerkin method is developed to solve the boundary value problem. The Newmark finite difference method is used to treat the time dependence of the variables for transient problems. The FG cylinder is considered to be under axisymmetric and plane strain conditions and bounding surfaces of cylinder to be under thermal shock loading. The mechanical properties of FG cylinder are assumed to vary across thickness of cylinder in terms of volume fraction as nonlinear function. A weak formulation for the set of governing equations is transformed into local integral equations on local subdomains by using a Heaviside test function. Nodal points are regularly distributed along the radius of the cylinder and each node is surrounded by a uni-directional subdomain to which a local integral equation is applied. The Green–Naghdi coupled thermoelasticity equations are valid in each isotropic subdomain. The temperature and radial displacement distributions are obtained for some grading patterns of FGM at various time instants. The propagation of thermal and elastic waves is discussed in details. The presented method shows high capability and efficiency for coupled thermoelasticity problems.