Finite Element Analysis of Rotating Truncated Functionally Graded Conical Shell

Finite element analysis of a generalized rotating FGM vessel subjected to thermo-mechanical loadings: Effect of Poisson ratio and inhomogeneity parameters

The present study deals with the analysis of rotating spherical pressure vessels made of functionally graded materials (FGMs). Material properties are graded along the radial direction, which is achieved by varying volume fractions of the constituting materials according to Mori-Tanaka scheme. Shells are subjected to hydrostatic internal pressure and have free boundary condition at the outer surface. Governing equations are derived using principle of stationary total potential. The effects of grading parameter and thickness parameter on stress and deformation behavior of the shells are investigated. Results obtained show that there is a significant reduction in stress and deformation of the FGM shells as compared to homogeneous shell. Cite this Article Amit Kumar Thawait, Lakshman Sondhi, Rupesh Kumar Verma. An Investigation of Stress and Deformation States of Rotating Functionally Graded Spherical Pressure Vessels . Journal of Experimental & Applied Mechanics . 2017; 8(3): 22–3 1 p. Keywords: Functionally graded material (FGM), elastic analysis, rotating spherical pressure vessel, Mori-Tanaka scheme

A time-dependent third-order shear deformation theory (TSDT) approach on the displacements of thick functionally graded material (FGM) conical shells under dynamic thermal vibration is studied. Dynamic equations of motion with TSDT for thick FGM conical shells are applied directly with the partial derivative of variable R*θ in the curve coordinates (x, θ, z) instead of y in the Cartesian coordinates (x, y, z) for thick FGM plates, where R* is the middle-surface radius at any point on conical shells. The generalized differential quadrature (GDQ) numerical method is used to solve the dynamic differential equations in equilibrium matrix forms under thermal loads. It is the novelty of the current study to identify the parametric effects of shear correction coefficient, environment temperature, TSDT model, and FGM power law index on the displacements and stresses in the thick conical shells only subjected to sinusoidal heating loads. The physical parts with values on the length-to-thickness ratio equals 5, and 10 FGMs can be used in an area of an airplane engine that usually operates near more than 1000 K of temperatures when the thermal stress is considered and affected. The important findings of the presented study are listed as follows. The values of normal stress are in decreasing tendencies with time in cases when the coefficient c1 equals 0.925925/mm2 in TSDT and length-to-thickness ratio equals 5. The shear stress values in x plane z direction on the minor middle-surface radius (r) equals the major middle-surface radius (R) over 8 and length-to-thickness ratio equals to 5 can withstand T = 1000 K of pressure.

In this study, static bending of an edge cracked cantilever nanobeam composed of functionally graded material (FGM) subjected to transversal point load at the free end of the beam is investigated based on modified couple stress theory. Material properties of the beam change in the height direction according to exponential distributions. The cracked nanobeam is modelled using a proper modification of the classical cracked-beam theory consisting of two sub-nanobeams connected through a massless elastic rotational spring. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new non-classical beam model reduces to the classical beam model when the length scale parameter is set to zero. The considered problem is investigated within the Euler-Bernoulli beam theory by using finite element method. In order to establish the accuracy of the present formulation and results, the deflections are obtained, and compared with the published results available in the literature. Good agreement is observed. In the numerical study, the static deflections of the edge cracked FGM nanobeams are calculated and discussed for different crack positions, different lengths of the beam, different length scale parameter, different crack depths, and different material distributions. Also, the difference between the classical beam theory and modified couple stress theory is investigated for static bending of edge cracked FGM nanobeams. It is believed that the tabulated results will be a reference with which other researchers can compare their results.

In the present work, a new axisymmetric weak form meshless method is presented for analysis of free vibration of functionally graded material (FGM) cylinders. This method is based on weak form of equilibrium equation and moving least squares (MLS) approximation. Essential boundary conditions are imposed by transformation method. In this method, shape functions that do not satisfy the Kronecker delta condition are corrected, then essential boundary conditions are imposed easily as in the finite element method (FEM). In the present work, the material is assumed to be functionally graded in the radial direction. Variations in the material properties such as Young’s modulus and Poisson’s ratio may be arbitrary functions of the radial coordinate. The FGM cylinder material varies continuously from silicon carbide (SiC) on the inner surface to stainless steel (SUS304) on the outer surface. Free vibration analysis of FGM cylinders with any arbitrary combination of boundary conditions is possible by the proposed model. Natural frequencies obtained from the presented model are in good agreement with results of finite element simulation. Effects of various types of boundary conditions, geometrical parameters, and mechanical properties on the natural frequencies are studied.

Analysis of stress shielding reduction in bone fracture fixation implant using functionally graded materials

A good understanding of the mechanical behavior of functionally graded material (FGM) cylindrical shells is necessary for designers and researchers. However, the 3D transient response of FGM cylindrical shells under various boundary conditions has not yet been analyzed. In this paper, the problem is addressed by proposing an approach integrating the state space method, differential quadrature method, and Durbina’s numerical inversion method of Laplace transform. The laminate model is used to obtain the transient solution in the radial direction. At the edges, four kinds of boundary conditions are considered: Clamped-Clamped, Clamped-Simply supported, Clamped-Free, and Simply supported-Simply supported. The results of the proposed method and finite element (FE) method agree with each other excellently. Convergence studies show that the proposed method has a fast convergence rate. The natural frequencies obtained by the proposed method, experiment, and other theoretical methods are in close agreement with each other. The effects of the load frequency and duration, length/outer radius ratio, and the (outer radius-inner radius)/outer radius ratio on the transient response of FGM shells are investigated. Two laws of variation of material properties along the radial direction are considered: the first has material properties varying according to an exponential law along the radial direction, while the second has material properties varying according to a power law. The effect of a functionally graded index on the transient response of FGM shells is investigated in both cases. The results obtained in this paper can serve as benchmark data for further research.

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

Recently, the functionally graded material (FGM) concept has been explored inpiezoelectric materials to improve properties and to increase the lifetime of bimorphpiezoelectric actuators. For instance, elastic, piezoelectric, and/or dielectric properties maybe graded along the thickness of a piezoceramic. Thus, the gradation of piezoceramicproperties influences the performance of piezoactuators. The usual FGM modelling usingtraditional finite element formulation and discretization into layers gives a highlydiscontinuous stress distribution, which is undesirable. In this work, we focus onnonhomogeneous piezoelectric materials using a generalized isoparametric formulationbased on the graded finite element concept, in which the properties change smoothly insidethe element. This approach provides a continuum material distribution, whichis appropriate to model FGMs. Both four-node quadrilaterals and eight-nodequadrilaterals for piezoelectric FGMs were implemented using the graded finite elementconcept. A closed form two-dimensional analytical model of piezoelectric FGMs isalso developed to check the accuracy of these finite elements and to assess theinfluence of material property gradation on the behavior of piezoelectric FGMs. Thepaper discusses and compares the behavior of piezoelectric graded elements underfour loading conditions with respect to the analytical solutions (derived in thiswork) considering exponential variation of elastic, piezoelectric, and dielectricproperties separately. The analytical solutions provide benchmark problems to verifynumerical procedures (such as the finite element method and the boundary elementmethod).

Estimation of C(t) and the creep crack tip stress field of functionally graded materials and verification via finite element analysis

In this paper, nonlinear radial and hoop thermoelastic stress analysis of rotating disk made of functionally graded material (FGM) with variable thickness is carried out by using the finite element method. In this method, one-dimensional second order elements with three nodes have been used. The geometrical and boundary conditions are in the shape of nonexistence of the pressure (zero radial stress) in both external and internal layers and zero displacement at the internal layer of rotating disk. Furthermore, it’s assumed that heat distribution is as second order curve while material properties such as elasticity modulus, Poisson’s ratio and thermal expansion coefficient vary by using a power law versus radius of the disk and also vary with the temperature. In a numerical example, the displacements and stresses for various powers (N) and the angular velocities have been calculated in according to the radius. It’s obvious that by increasing the values of the power (N) and the angular velocity, the value of displacements and stresses will be increased consequently. Finally, the effect of varying the thickness and the dependency and in-dependency of the material properties on the temperature has been considered.

A thin circular rotating disk having a concentric hole and a functionally graded material (FGM) coating at the outer surface is considered with a view to analyzing the thermoelastic characteristics due to a thermal load and rotation of the disk. The FGM coating is assumed to have exponentially varying Young's modulus, coefficient of thermal expansion (CTE), and density in the radial direction of the disk. The Poisson's ratio is assumed to be constant throughout the disk. The incompatible eigenstrain developed in the disk owing to the nonuniform CTE and variation of temperature is taken into consideration. Using the two-dimensional thermoelastic theories, the two-dimensional plane stress axisymmetric problem is formulated as a second order differential equation. A finite element model is developed using the variational approach and Ritz method to obtain the numerical solution of the differential equation. The validity of the finite element model is justified for a rotating circular disk of homogeneous material by comparing the finite element results with analytical solution obtained by Timoshenko. Then the finite element model is applied to the problem of an Al disk with an Al2O3/Al FGM coating at its outer surface. The numerical results of the thermoelastic field demonstrate that the temperature distribution profile, angular speed of the disk, and FGM coating thickness are the crucial factors to be considered in controlling the thermoelastic characteristics of a rotating disk with an FGM coating.

In this paper, a study is carried out to investigate the stress concentrations at junctions of cylindrical-conical shell structures using the finite element method. A finite element solution is developed and used to model the structure. The solution utilizes curved finite elements that are capable to model the cylindrical and conical shell surfaces with high degree of accuracy. Elevated storage cylindrical-conical tanks and silos supported by "unyielding" stiff ring girders, which sit on a cylindrical skirt or on multi equally spaced columns, are analyzed. The opportunity is also taken to investigate the effects of various geometrical parameters, such as the conical angle, the height/radius ratio of the cylindrical shell, the location of the ring girder relative to the junction and the influence of adding ring stiffeners on the behavior of the structure. It is concluded that the geometrical discontinuity at the junction causes concentrations of compressive inplane as well as bending stresses. These stresses can be significantly reduced by either increasing the inclination angle of the conical bottom or by placing the ring girder closer to the junction.

This work presents a finite element formulation for the vibration analysis of functionally graded material (FGM) plates containing microstructural defects operating in thermal environments. The material properties are assumed to be temperature-dependent and are modeled as higher-order functions of temperature. The volume fractions of the constituent materials are varied through the plate thickness according to a power-law distribution. The effective material properties of the FGM are evaluated using both the Mori–Tanaka homogenization scheme and the rule of mixtures. The kinematics of the FGM plates are described using an efficient layerwise theory. The governing equations of motion, constitutive relations, and boundary conditions are derived using on Hamilton’s principle. The stiffness coefficients of the plates are computed using a sub-laminate mechanics approach, wherein the plate thickness is discretized into several thin sublayers, each assumed to possess uniform material properties. This approach is particularly effective for micromechanical models where exact analytical integration of stiffness coefficients is difficult due to the complex polynomial variation of properties across the thickness. A four-noded quadrilateral finite element, previously developed for laminated and FGM shells, is extended in this study to analyze the dynamic behavior of FGM plates with microstructural defects and temperature dependent material properties. The natural frequencies are obtained using a subspace iteration technique, while the transient responses under step and impulse excitations are computed using the state-space form of the governing dynamic equations. The proposed finite element model is validated against three-dimensional elasticity and Ritz-based semi-analytical solutions, demonstrating excellent accuracy and reliability across varying thermal environments. A detailed parametric study is also carried out to investigate the effects of temperature, inhomogeneity index, porosity concentration, boundary conditions, and geometric parameters on the natural frequencies and transient responses of porous FGM plates.

In this research, the free vibration analysis of lap joint laminated conical shell connected to laminated conical shell with adhesive layer for various boundary conditions are investigated. Based on the first-order shear deformation theory (FSDT), the main field equations of the analytical model are derived and the governing differential equations of laminated conical shell to laminated conical shell adherends lap joint are obtained for homogeneous and elasticity adhesive layers by using Hamilton principle method. Then, based on the generalized differential quadrature method (GDQM), the equilibrium equations of the structure adherends are investigated. The influence of the boundary conditions, the type of the composite materials and adhesive layer, the circumferential wave number, the length to radius ratio of the shell, and the length of overlap to length of shell ratio are studied. Furthermore, the thickness to radius ratio and the thickness of adhesive to thickness of shell ratio, as well as the various cone angles of conical shell on the natural frequency of laminated conical adherends shells connected to the conical shells with adhesive layer are investigated. For validation of the numerical results, the results are compared with the previous research and the comparisons show very good agreement. The numerical results show that with the increase of the internal shell and external shell of semi-angle, the non-dimensional frequency of lap joint connection of two laminated conical shells with adhesive layer decreased and increased, respectively. Furthermore, with the increase of the adhesive thickness to conical shell thickness ratio and lap joint length to length of conical shell ratio, the dimensionless natural frequency of the structure increased.