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)     

Multiple moving cracks in a functionally graded strip

This paper considers several finite moving cracks in a functionally graded material subjected to anti-plane deformation. The distributed dislocation technique is used to carry out stress analysis in a functionally graded strip containing moving cracks under anti-plane loading. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. By utilizing the Fourier sine transformation technique the stress fields are obtained for a functionally graded strip containing a screw dislocation. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by several moving cracks. Numerical examples are provided to show the effects of material properties, the crack length and the speed of the crack propagating upon the stress intensity factor.     

Periodically varying heat source response in a functionally graded microelongated medium

The present paper deals with the response due to periodically varying heat sources in the neighborhood of the origin of a functionally graded isotropic unbounded microelongated medium, in the context of generalized thermoelastic theory. The expressions for displacement, microelongation and temperature fields have been obtained in Laplace-Fourier transformed domain. After computing the inverse Fourier transforms by contour integration technique, the inversion of Laplace transforms has been obtained numerically. The changes of displacement, microelongation, and normal strain have been shown graphically for different types of heat source.     

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.     

End effects for a generalized biharmonic equation with applications to functionally graded materials

In this paper we consider a generalized biharmonic equation modeling two-dimensional inhomogeneous elastic state in the curvilinear rectangle a<r<b, 0<θ<α, where (r,θ) denote plane polar coordinates. Such an arch-like region is maintained in equilibrium under self-equilibrated traction applied on one of the edges, while the other three edges are traction free. Our aim is to derive some explicit spatial estimates describing how some appropriate measures concerning the specific Airy stress function evolve with respect to the distance to the loaded edge. Two types of smoothly varying inhomogeneity are considered: (i) the elastic moduli vary smoothly with the polar distance, (ii) they vary smoothly with the polar angle. Such types of smoothly varying inhomogeneous elastic materials provide a model for technological important functionally graded materials. The results of the present paper prove how the spatial decay rate varies with the constitutive profile.     

Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method

An improved meshless radial point interpolation method, for the analysis of nonlinear transient heat conduction problems is proposed. This method is implemented for the heat conduction analysis of functionally graded materials (FGMs) with non-homogenous and/or temperature dependent heat sources. The conventional meshless RPIM is an appropriate numerical technique for the analysis of engineering problems. One advantage of this method is that it is based on the global weak formulation, and also the associated shape functions possess the Kronecker delta function property. However, in the original form, the evaluation of the global domain integrals requires the use of a background mesh. The proposed method benefits from a meshless integration technique, which has the capability of evaluating domain integrals with a better accuracy and speed in comparison with the conventional integration methods, and therefore a truly meshless technique is attained. This integration technique is especially designed for the fast and accurate evaluation of several domain integrals, with different integrands, over a single domain. Some 2D and 3D examples are provided to assess the efficiency of the proposed method.     

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 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)     

Simulation of brittle crack growth in functionally graded materials

We consider a thermodynamic consistent framework for crack propagation by applying a dissipation inequality to a time dependent migrating control volume. The direction of crack growth is obtained in terms of material forces as a result of the principle of maximum dissipation. In the numerical implementation a staggered algorithm – deformation update for fixed geometry followed by geometry update for fixed deformation – is employed within each time increment. The corresponding mesh is generated by combining Delaunay triangulation with local mesh refinement. A numerical example with inhomogeneous material properties illustrates the capability of the resulting algorithm. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transient response of inhomogeneous thermoelastic media to a dynamic input

A problem involving a half-space or an infinite space with a cylindrical or spherical cavity subjected to a uniform dynamic input on its surface is studied. The material is elastic and has an inhomogeneity in the direction perpendicular to the boundary surface for the half-space problem and an inhomogeneity in the radial direction for the problem with a cylindrical or spherical cavity. The numerical results, obtained when the dynamic input is a step pressure, indicate that the wave profiles are influenced by two kinds of dispersions: one caused by thermal effects and the other by inhomogeneity. The influence of thermal dispersion is observed more pronouncedly along the initial portion of wave profiles. Compared to the homogeneous case the inhomogeneities in which Lamé's constants increase and decrease with the distance measured from the boundary have respectively amplifying and attenuating effects on both the axial stress and velocity.

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Zusammenfassung Das Problem der gleichmässigen dynamischen Belastung an den Grenzen eines halbunendlichen Körpers oder eines unendlichen Körpers mit einem zylindrischen oder sphärischen Halbraum wurde untersucht.Der Körper ist elastisch und enthält Inhomogenitäten, im halbunendlichen Körper in der Richtung senkrecht zur Grenzebene und im unendlichen Körper mit Hohlräumen in radialer Richtung. Die numerischen Ergebnisse, die für eine Belastung nach einer Schritt-Funktion erzielt wurden, zeigen dass die Profile der Ausbreitungswellen von zwei Streuungen beeinflusst sind: die erste ist eine Folge der thermischen Effekte, die zweite eine solche der Inhomogenitäten. Der Effekt der thermischen Zerstreuung ist im anfänglichen Bereich der Profile stärker. Im Vergleich zum homogenen Fall hat die Zu-beziehungsweise die Abnahme der Lamé'schen Konstanten mit dem Abstand von der Grenze erhöhenden beziehungsweise vermindernden Einfluss sowohl auf die axiale Spannung als auch auf die Geschwindigkeit.

     

Some problems of heat conduction in transversally graded composites

The object of considerations is a two-component layer made of conductors non-periodically distributed in the form of laminas along the layer thickness. It is assumed that the distribution of the macroscopic properties of this laminate is approximated by continuous slowly-varying functions across laminas. Media of this kind can be treated as made of a functionally graded material. The aim of the paper is to apply the tolerance model, [8], to analyse one-directional, non-stationary heat conduction along the axis perpendicular to laminas. Moreover, received results are compared to the solutions obtained in the framework of the higher-order theory, [1]. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems.     

Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems. Received: July 14, 2003     

Contact problems for functionally graded materials of complicated structure

An approximate analytical method allowing one to efficiently solve, to a preassigned accuracy, contact problems for materials with properties arbitrarily varying in depth is developed. Its possibilities are illustrated with the example of torsion of an elastic half-space, having a coating inhomogeneous across its thickness, by a circular stamp. All the results obtained are rigorously substantiated. For the approximate solutions constructed, their error is analyzed. The asymptotic properties of the solutions are investigated. The cases of a nonmonotonic change in the elastic properties are considered. In particular, the analytical solutions are examined in the case where the variation gradient of the elastic properties changes its sign many times. The results derived allow one to solve the inverse problems of elasticity theory of inhomogeneous media (e.g., the problem on controlling the variation in the elastic properties of a covering across its thickness).     

A BEM for transient thermoelastic analysis of a functionally graded layer on a homogeneous substrate under thermal shock

Transient thermoelastic analysis of isotropic and linear thermoelastic bimaterials, which are constituted by a functionally graded (FG) layer attached to a homogeneous substrate, subjected to thermal shock is presented in this paper. For this purpose, a boundary element method for transient linear coupled thermoelasticity is developed. The material properties of the FG layer are assumed to be continuous functions of the spatial coordinates. The boundary-domain integral equations are derived by using the fundamental solutions of linear coupled thermoelasticity for the corresponding isotropic, homogeneous and linear thermoelastic solids in the Laplace-transformed domain. For the numerical solution, a collocation method with piecewise quadratic approximation is implemented. Numerical results for the dynamic stress intensity factors are presented and discussed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two collinear antiplane cracks in functionally graded magnetoelectroelastic composite materials

An analysis of two collinear antiplane cracks in inhomogeneous (functionally graded) anisotropic magnetoelectroelastic materials is presented. In designing components involving functionally graded materials, an important aspect of the problem is the fracture failure. The problem is formulated for transversely isotropic functionally graded magnetoelectroelastic materials. An integral transform is employed to reduce the problem to a singular integral equation that can be solved.     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.Received: January 13, 2004     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.