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

The meshless local Petrov–Galerkin (MLPG) method is used to solve the inverse fracture problems in two-dimensional (2D) piezoelectric body. Electrical boundary conditions on the crack surfaces are not specified due to unknown dielectric permittivity of the medium inside the crack. Both stationary and transient dynamic boundary conditions are considered here. The analyzed domain is covered by small circular subdomains surrounding nodes spread randomly over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIE) on the boundaries of the chosen subdomains. The Laplace-transform technique is applied to eliminate the time variation in the governing equation. The local integral equations are nonsingular and take a very simple form. The spatial variation of the Laplace transforms of displacements and electrical potential are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. The singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion to retrieve the time-dependent solutions.

Inverse fracture problems in piezoelectric solids by local integral equation method

A gradient theory is applied to the mechanical constitutive equations for piezoelectric semiconductor nanostructures. This is achieved by considering the strain gradients in the constitutive equati...

Size effect in piezoelectric semiconductor nanostructures

A meshless method based on the local Petrov–Galerkin approach is proposed to analyze bending of a porous piezoelectric cylinder under thermal loading. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors for solid and fluid phases. The influence of thermal expansion coefficients in solid and fluid phases on the plate deflection and on the induced electric potential is investigated via the local integral equation method developed in this paper. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares (MLS) scheme. The heat conduction equation is considered as uncoupled with respect to the mechanical and electrical fields.

Bending of a porous piezoelectric cylinder under a thermal load

This paper presents an application of a direct Trefftz method with domain decomposition to the twodimensional elasticity problem. Trefftz functions are substituted into Betti’s reciprocity theorem to derive the boundary integral equations for each subdomain. The values of displacements and tractions on subdomain interfaces are tailored by continuity and equilibrium conditions, respectively. Since Trefftz functions are regular, much less requirements are put on numerical integration than in the traditional boundary integral method. Then, the method can be utilized to analyse also very narrow domains. Linear elements are used for modelling of the boundary geometry and approximation of boundary quantities. Numerical results for a rectangular plate with varying aspect ratio and cantilever beam are presented. keyword: plynomial Trefftz functions, direct formulation, linear approximation, boundary integral equation

Vladimir Sladek

Papers

Inverse fracture problems in piezoelectric solids by local integral equation method

Size effect in piezoelectric semiconductor nanostructures

Bending of a porous piezoelectric cylinder under a thermal load

Application of Multi-Region Trefftz Method to Elasticity

Analysis of coupling effects in FGM piezoelectric plates by a meshless method