This paper presents a review of the principal developments in functionally graded materials (FGMs) with an emphasis on the recent work published since 2000. Diverse areas relevant to various aspects of theory and applications of FGM are reflected in this paper. They include homogenization of particulate FGM, heat transfer issues, stress, stability and dynamic analyses, testing, manufacturing and design, applications, and fracture. The critical areas where further research is needed for a successful implementation of FGM in design are outlined in the conclusions. DOI: 10.1115/1.2777164

https://web.mst.edu/~vbirman/papers/Modeling%20and%20Analysis%20of%20FGM_Review_2007.pdf

Modeling and Analysis of Functionally Graded Materials and Structures

Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

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Electrodynamics Of Continuous Media

Recent developments of meshfree and particle methods and their applications in applied mechanics are surveyed. Three major methodologies have been reviewed. First, smoothed particle hydrodynamics ~SPH! is discussed as a representative of a non-local kernel, strong form collocation approach. Second, mesh-free Galerkin methods, which have been an active research area in recent years, are reviewed. Third, some applications of molecular dynamics ~MD! in applied mechanics are discussed. The emphases of this survey are placed on simulations of finite deformations, fracture, strain localization of solids; incompressible as well as compressible flows; and applications of multiscale methods and nano-scale mechanics. This review article includes 397 references. @DOI: 10.1115/1.1431547#

Meshfree and particle methods and their applications

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Methods of Numerical Integration

Mixed FEM for quantum nanostructured solar cells

The paper deals with the development of the meshless computational techniques based on the Local Integral Equations and analytical integrations. The development is illustrated on 2-d potential problems in functionally graded media. The MLS-approximation is used for simulation of spatial variations of the potential field. Efficient differentiation is proposed for calculation of derivatives of shape functions. The accuracy and efficiency are studied on simple example providing the exact solution for the benchmark solution.

/pdf/meshless-implementations-of-localintegral-equations-2pjp4r2edi.pdf

Meshless Implementations Of LocalIntegral Equations

In the present work, we propose a consistent derivation of the regularized integral representations for displacement gradients within the boundary element formulation for solution of small strain plasticity problems. The regularization is carried out for both the domain and boundary integrals before any discretization and approximation of integral densities. Two kinds of the integral representations are derived with the leading singularity of the integral kernels being either hypersingularity or strong singularity. Both the two- and three-dimensional problems are considered simultaneously.

Displacement gradients in BEM formulation for small strain plasticity

A numerical analysis based on the meshless local PetrovGalerkin (MLPG) method is proposed for a functionally graded material FGM (FGMfunctionally graded material) beam. The planar bending of the beam is considered with a transversal gradation of Young's modulus and a variable depth of the beam. The collocation formulation is constructed from the equilibrium equations for the mechanical fields. Dirac's delta function is employed as a test function in the derivation of a strong formulation. The Moving Least Squares (MLS) approximation technique is applied for an approximation of the spatial variations of all the physical quantities. An investigation of the accuracy, the convergence of the accuracy, the computational efficiency and the effect of the level of the gradation of Young's modulus on the behaviour of coupled mechanical fields is presented in various boundary value problems for a rectangular beam with a functionally graded Young's modulus.

/pdf/analysis-of-beams-with-transversal-gradations-of-the-young-s-1y74ugjndl.pdf

Analysis of Beams with Transversal Gradations of the Young's Modulus and Variable Depths by the Meshless Method

In this paper we present the advanced form of the boundary integral equations for solution of transient heat conduction problems. The BIE are free of Cauchy principal value integrals in the boundary element formulation with time dependent fundamental solutions. This is made independently of the order of the approximation within boundary elements.

Vladimir Sladek

Papers

Mixed FEM for quantum nanostructured solar cells

Meshless Implementations Of LocalIntegral Equations

Displacement gradients in BEM formulation for small strain plasticity

Analysis of Beams with Transversal Gradations of the Young's Modulus and Variable Depths by the Meshless Method

Nonsingular BIE for transient heat conduction