Green’s Functions and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction

Gray, L. J.; Kaplan, T. A.; Richardson, J.D.; Paulino, Gláucio H.

doi:10.1115/1.1485753

Cited by 87 publications

(55 citation statements)

References 23 publications

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“…The usefulness of this procedure is limited to two-dimensional problems under particular mappings; nevertheless, an analytical solution of this type has value in understanding the processes involved regarding heat conduction through graded materials. Furthermore, these solutions can be used as kernel functions in BIEM solutions of boundaryvalue problems in order to validate ÿnite di erence or ÿnite element numerical approaches for solving the general equation (1).…”

Section: Resultsmentioning

confidence: 99%

Two dimensional heat conduction in graded materials using conformal mapping

Shaw

Manolis

2002

Commun. Numer. Meth. Engng.

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SUMMARYHeat conduction through non-homogeneous materials has numerous applications, since many of the man-made materials that are used in various disciplines such as aerospace, construction, electronics, etc., are composites with carefully manufactured properties that yield desirable thermal characteristics. To that end, fundamental solutions for the heterogeneous heat conduction problem in two dimensions are derived in this work by employing a conformal mapping technique. These functions, besides being useful in their own right, can also be used within the context of integral equation formulations for the solution of boundary-value problems. Finally, numerical examples for a material described by a family of radially symmetric conductivity are given as an illustration of the proposed method.

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Section: Resultsmentioning

confidence: 99%

Two dimensional heat conduction in graded materials using conformal mapping

Shaw

Manolis

2002

Commun. Numer. Meth. Engng.

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“…Application of the BEM to the heterogeneous media is a formidable task, since fundamental solutions correspond to concentrated loads imposed on such media are difficult to obtain. The fundamental solution of the heat transfer problem in the heterogeneous media has been presented by Gray et al (2003) and Kuo and Chen (2005). The fundamental solutions for 2D and 3D elastostatic problems have recently been developed for layered media and for exponentially graded materials (Chan et al 2004;Criado et al 2007Criado et al , 2008Ashrafi et al 2013).…”

Section: Introductionmentioning

confidence: 99%

A time-domain boundary element method for quasistatic thermoviscoelastic behavior modeling of the functionally graded materials

Ashrafi

Shariyat

Asemi

2013

Int J Mech Mater Des

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In the present paper, a boundary element formulation is presented for two-dimensional thermoviscoelasticity analysis of components fabricated from the functionally graded materials (FGMs). In this regard, a graded viscoelastic element capable of tracing gradual variations of the material properties is developed. Several attempts have been made so far to employ the integral equation approach for the heterogeneous viscoelastic materials. In the present research, Somigliana's displacement identity is considered and implemented numerically for analyzing the two-dimensional exponentially graded viscoelastic components. Employing the common assumptions of the viscoelastic constitutive equations and the weighted residual technique, an efficient boundary element formulation is developed for the heterogeneous Kelvin-Voigt solid viscoelastic models. Finally, three numerical examples are provided to verify the proposed formulation and present practical conclusions.

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“…A principal reason for this is the fact that even for their simplest variants, the FGM with exponential properties, continuous Green's functions are not available in explicit forms. This is true for static Green's functions [6] and more than ever for the dynamic framework such as as for problems of heat conduction [7] in two and three dimensions.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the vibrational mode spectrum of a linear chain with spatially exponential properties

Michelitsch

Maugin

Nowakowski

et al. 2009

International Journal of Engineering Science

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We deduce the dynamic frequency-domain-lattice Green's function of a linear chain with properties (masses and next-neighbor spring constants) of exponential spatial dependence. We analyze the system as discrete chain as well as the continuous limiting case which represents an elastic 1D exponentially graded material. The discrete model yields closed form expressions for the N × N Green's function for an arbitrary number N = 2, .., ∞ of particles of the chain. Utilizing this Green's function yields an explicit expression for the vibrational mode density. Despite of its simplicity the model reflects some characteristics of the dynamics of a 1D exponentially graded elastic material. As a special case the well-known expressions for the Green's function and oscillator density of the homogeneous linear chain are contained in the model. The width of the frequency band is determined by the grading parameter which characterizes the exponential spatial dependence of the properties. In the limiting case of large grading parameter, the frequency band is localized around a single finite frequency where the band width tends to zero inversely with the grading parameter. In the continuum limit the discrete Green's function recovers the Green's function of the continuous equation of motion which takes in the time domain the form of a Klein-Gordon equation.

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Green’s Functions and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction

Cited by 87 publications

References 23 publications

Two dimensional heat conduction in graded materials using conformal mapping

Two dimensional heat conduction in graded materials using conformal mapping

A time-domain boundary element method for quasistatic thermoviscoelastic behavior modeling of the functionally graded materials

Analysis of the vibrational mode spectrum of a linear chain with spatially exponential properties

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