A meshless procedure for transient heat conduction in functionally graded materials

A meshless numerical procedure is developed for analyzing the transient heat conduction problem in non-homogeneous functionally graded materials. In the proposed method the time derivative of temperature is approximate by the finite difference. At each time step the original nonlinear boundary value problem is transform into a hierarchy of non-homogeneous linear problem by used the homotopy analysis method. In this method a sought solution is presented by using a finite expansion in Taylor series, which consecutive elements are solutions of series linear value problems defining differential deformations. Each of linear boundary value problems with the corresponding boundary conditions is solved by using the method of fundamental solutions and radial basis functions which are used for interpolation of the inhomogeneous term. The accuracy of the obtained approximate solution is controlled by the number of components of the Taylor series, while the convergence of the process is monitored by an additional parameter of the method. Numerical experiments demonstrate the efficiency and accuracy of the present scheme in the solution of the heat conduction problem in nonlinear functionally graded materials. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical modelling of heat conduction in certain functionally graded composites

The aim of contribution is to formulate a certain extended version of the tolerance modelling technique for functionally graded composites. For the sake of simplicity, the considerations are restricted to the bidirectionally graded heat conductors. It is shown that the proposed approach enables to determine an entire class of mathematical models for which applications can be found in various specific problems. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.     

Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients

The mean and variance of the temperature are analytically obtained in a functionally graded annular disc with spatially random heat transfer coefficients (HTCs) on the upper and lower surfaces. This annular disc has arbitrary variations in the HTCs (i.e., arbitrary thermal interaction with the surroundings) and gradient material composition only along the radial direction and is subjected to deterministic axisymmetrical heating at the lateral surfaces. The stochastic temperature field is analysed by considering the annular disc to be multilayered with spatially constant material properties and spatially constant but random HTCs in each layer. A type of integral transform method and a perturbation method are employed in order to obtain the analytical solutions for the statistics. The correlation coefficients of the random HTCs are expressed in the form of a linear function with respect to the radial distance as a non-homogeneous random field of discrete space. Numerical calculations are performed for functionally graded annular discs composed of stainless steel and ceramic, which comprise two types of material composition distributions. The effects of the magnitude of the means of HTCs, volume fraction distributions of the constitutive materials and correlation strengths of the HTCs on the standard deviation of the temperature are discussed.     

The radial basis reproducing kernel particle method for geometrically nonlinear problem of functionally graded materials

In this paper, the radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is proposed for solving geometrically nonlinear problem of functionally graded materials (FGM). Compared with the RKPM, the advantages of the proposed method are that it can eliminate the negative effect of different kernel functions on the computational accuracy, and has higher computational accuracy and stability. Using the Total Lagrange (T.L.) formulation and the weak form of Galerkin integration, the corresponding formulae for geometrically nonlinear problem of FGM are derived. The penalty factor, shaped parameter of the RBF, the control parameter of influence domain radius, loading step number and node distribution are discussed. Furthermore, the effects of different gradient functions and exponents on displacement and stress are analyzed. Newton-Raphson (N-R) iterative method is utilized for numerical solution. The proposed method is correct and effective for solving geometrically nonlinear problem of FGM, which can be demonstrated by several numerical examples.     

Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories

This study investigates the nonlinear free vibration of functionally graded material (FGM) beams by different shear deformation theories. The volume fractions of the material constituents and effective material properties are assumed to be changing in the thickness direction according to the power-law form. The von Kármán geometric nonlinearity has been considered in the formulation. The Ritz method and Lagrange equation are adopted to yield the discrete formulations. A direct numerical integration method for the motion equation in matrix form is developed to solve the nonlinear frequencies of FGM beams. Comparing with the global concordant deformation assumption (GCDA), a new deformation assumption named as local concordant deformation assumption (LCDA) is proposed in this study. The LCDA fits with the real deformation of the vibrating beam better, thus more accurate results of the nonlinear frequency can be expected. In numerical results, the comparison study of the GCDA and LCDA is carried out. In addition, the effects of power-law index, slenderness ratio and maximum deflection for different shear deformation theories and boundary conditions on the nonlinear frequency of the beam are discussed.     

A model of the heat conduction for functionally graded laminate with uniformly distributed micro inclusions

The aim of the contribution is to formulate an asymptotic model of heat conduction for functionally graded two component laminate reinforced by periodically spaced micro inclusions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Natural frequencies of a functionally graded cracked beam using the differential quadrature method

The present work is concerned with the free vibration analysis of an elastically supported cracked beam. The beam is made of a functionally graded material and rested on a Winkler–Pasternak foundation. The line spring model is employed to formulate the problem. The method of differential quadrature is applied to solve it. The obtained results agreed with the previous similar ones. Further, a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the problem on the values of natural frequencies and mode shape functions.     

A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional hyperbolic equation that combines classical and integral boundary conditions using collocation points and approximating the solution using radial basis functions (RBFs). The results of numerical experiments are presented, and are compared with analytical solution and finite difference method to confirm the validity and applicability of the presented scheme.     

Wave propagation and transient response of a functionally graded material plate under a point impact load in thermal environments

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load in thermal environments are studied. The thermal effects and temperature-dependent material properties are taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varies in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived from Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Using the dispersion relation and integral transforms, exact integral solutions of the functionally graded plate under a point impact load in thermal environments are obtained. The influences of the volume fraction distributions and temperature field on the wave propagation and transient response of functionally graded plates are discussed in detail. The results carried out can be used in the ultrasonic inspection techniques and provide a theoretical basis for engineering applications.     

Exploration of the encased nanocomposites functionally graded porous arches: Nonlinear analysis and stability behavior

This paper explores the nonlinear stability mechanism of the functionally graded porous (FGP) arch reinforced by graphene nanocomposites. Both the pores and the nanocomposites are distributed symmetrically to the mid-surface of the arch but not uniformly in the cross-section so that the bending stiffness can be best improved. The arch is confined in an elastic medium with a radially-pointed concentrated load at the crown position. The confinement of the medium results in a symmetrical deformed shape of the arch, which can be described by an admissible displacement function. Associated with the thin-walled arch theory and the principle of minimum potential energy, analytical predictions are obtained to express the critical buckling load, as well as the hoop force and bending moment. Subsequently, a numerical model is developed to simulate the medium and the arch in ABAQUS software. By introducing the modified arc-length method, the equilibrium paths of the encased arch are traced. After comparison in terms of the critical buckling load and the equilibrium paths, it is found the numerical results are in good accordance with the analytical solutions. Finally, particular attention is paid to the parameters that may impact the buckling load, such as the porosity coefficient, the weight fraction, the central angle, the geometry of the Graphene platelets (GPLs) et al.     

Free vibration analysis of discrete-continuous functionally graded circular plate via the Neumann series method

The Neumann series method has been used for the first time to solve the boundary value problem of free axisymmetric and nonaxisymmetric vibrations of continuous and discrete-continuous functionally graded circular plate on the basis of the classical plate theory. The equation of motion and the general solution for a functionally graded circular plate with a very complex system of a discrete elements attached, such as concentric ring masses, elastic supports, rotational springs, and damping elements are presented for the first time. The particular continuous solutions to the defined differential equations are obtained as the Neumann power series rapidly, absolutely, and uniformly convergent to the exact eigenfrequencies for any physically justified values of the plate's parameters on the basis of the properties of the obtained closed-form kernels of the Volterra integral equations. The multiparametric nonlinear characteristic equations for plate with classical and nonclassical boundary conditions are defined and numerically solved to obtain the full spectrum of eigenfrequencies in a simple way. The effects of the position and stiffness of ring supports and of singularities as the radii of supports shrink to the center of the plate on the dimensionless eigenfrequencies of homogeneous and functionally graded circular plate with sliding support and elastic constraints are comprehensively studied and presented for the first time. The accuracy of the proposed low-computational-cost method is demonstrated by comparison of the numerical results with those available in the literature.     

Application of a hybrid meshless technique for natural frequencies analysis in functionally graded thick hollow cylinder subjected to suddenly thermal loading

A hybrid meshless technique based on composition of meshless local Petrov–Galerkin method (for spatial variables) and Newmark finite difference method (for time domain) is developed for natural frequencies analysis of thick cylinder made of functionally graded materials (FGMs). The FG cylinder is assumed to be under suddenly thermal loading, axisymmetric and plane strain conditions. The dynamic behaviors and time history of displacements are obtained in time domain using Green–Naghdi (GN) theory of coupled thermo-elasticity (without energy dissipation). Using fast Fourier transform (FFT) technique, the displacements are transferred to frequency domain and all natural frequencies are illustrated for various grading patterns of FGMs. The variations of mechanical properties in FG thick hollow cylinder are considered to be in nonlinear volume fraction law through radial direction. The presented hybrid meshless technique furnishes a ground to analyze the effects of various grading patterns of FGMs on natural frequencies, which are obtained employing GN coupled thermo-elasticity governing equations. Also, the frequency history and natural frequencies are illustrated for various grading patterns at several points across thickness of cylinder.     

A meshless method using the radial basis functions for numerical solution of the regularized long wave equation

This article discusses on the solution of the regularized long wave (RLW) equation, which is introduced to describe the development of the undular bore, has been used for modeling in many branches of science and engineering. A numerical method is presented to solve the RLW equation. The main idea behind this numerical simulation is to use the collocation and approximating the solution by radial basis functions (RBFs). To avoid solving the nonlinear system, a predictor‐corrector scheme is proposed. Several test problems are given to validate the new technique. The numerical simulation, includes the propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves and the development of an undular bore. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the presented scheme.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Numerical solution of transient heat conduction equation with variable thermophysical properties by the Tau method

In this article, we proposed the operational approach to the Tau method for solving linear and nonlinear one‐dimensional transient heat conduction equations with variable thermophysical properties which can involve heat generation term. To solve heat conduction equation, first we recall the Tau method to obtain a matrix form of the governing differential equation. Then boundary and initial conditions are transformed into a matrix form. Finally the resulting systems of linear or nonlinear algebraic equations are given. Afterwards, efficient error estimation is also introduced for this method. Some numerical examples are given to illustrate the efficiency and high accuracy of the proposed method and also results are compared with solutions obtained by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 964–977, 2014     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

Asymptotic computation for transient heat conduction performance of periodic porous materials in curvilinear coordinates by the second‐order two‐scale method

A novel second‐order two‐scale (SOTS) analysis method is developed for predicting the transient heat conduction performance of porous materials with periodic configurations in curvilinear coordinates. Under proper coordinate transformations, some non‐periodic porous structures in Cartesian coordinates can be transformed into periodic structures in general curvilinear coordinates, which is our particular interest in this study. The SOTS asymptotic expansion formulas for computing the temperature field of transient heat conduction problem in curvilinear coordinates are constructed, some coordinate transformations are discussed, and the related SOTS formulas are given. The feature of this asymptotic model is that each of the cell functions defined in the periodic cell domain is associated with the macroscopic coordinates and the homogenized material coefficients varies continuously in the macroscopic domain behaving like the functional gradient material. Finally, the corresponding SOTS finite element algorithms are brought forward, and some numerical examples are given in detail. The numerical results demonstrate that the SOTS method proposed in this paper is valid to predict transient heat conduction performance of porous materials with periodicity in curvilinear coordinates. By proper coordinate transformations, the SOTS asymptotic analysis method can be extended to more general non‐periodic porous structures in Cartesian coordinates. Copyright © 2017 John Wiley & Sons, Ltd.