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 Reissner–Mindlin theory is applied to shallow shells with material properties of orthorhombic quasicrystals (QC). A quasi-periodic arrangement of atoms through the thickness of the shell is considered. Due to the intrinsic characteristics of QCs two excitations need to be described under dynamic loads, namely, phonon (elastic waves) and phason (long wavelength fluctuations) fields. The Bak and elasto-hydrodynamic models are applied for phason governing equation in the elastodynamic case. The phason displacement for the orthorhombic QC in the first-order shear deformation shallow shell theory depends only on the in-plane coordinates on the mid-surface of the shell. A weak formulation for the set of governing equations in the Reissner–Mindlin theory is transformed into local integral equations on local subdomains on the mid-surface of the shell by using a unit-step test function. Nodal points are randomly spread on the mid-surface of the shell and each node is surrounded by a circular subdomain to which local integral equations are applied. The meshless approximation based on the Moving Least-Squares (MLS) method is employed for the implementation. The influences of the shell curvature and the coupling parameter of quasicrystals on the shell deflection are investigated.

Modelling of orthorhombic quasicrystal shallow shells

The flexoelectric effect on elastic waves is investigated in nano-sized cracked structures. The strain gradients are considered in the constitutive equations of a piezoelectric solid for electric displacements and the higher-order stress tensor. The governing equations with the corresponding boundary conditions are derived from the variational principle. The finite element method (FEM) is developed from the principle of virtual work. It is equivalent to the weak-form of derived governing equations in gradient elasticity. The computational method can be applied to analyze general 2D boundary value problems in size-dependent piezoelectric elastic solids with cracks under a dynamic load. The FEM formulation is implemented for strain-gradient piezoelectricity under a dynamic load.

Gradient piezoelectricity for cracks under an impact load

The paper addresses the three-dimensional problem on steady-state vibrations of an elastic body consisting of two perfectly joined dissimilar half-spaces with an elliptic mode I crack located in one of the half-spaces normally to the interface. The problem is reduced to a boundary integral equation for the crack opening function. The integration domain of the equation is bounded by the crack domain, and the interaction between the crack and the interface is described by a regular kernel. The equation is solved using the mapping method. Numerical results are obtained for the case where the surfaces of the elliptic crack are subjected to harmonic loading with constant amplitude. The dependences of the stress intensity factors on the wave number are presented for various relationships among the mechanical constants that ensure the absence of near-surface waves

Stress Concentration Near an Elliptic Crack in the Interface Between Elastic Bodies under Steady-State Vibrations

The nonsingular traction BIEs are derived for the Laplace transforms in elastodynamic crack problems. Two different forms of the final nonsingular traction BIEs are received with respect to the leading singularity of the integral kernels involved. In the first one, the traction BIE is derived from the integral representation of stresses which involves hypersingular kernels. In the second way, the partially regularized integral representation of stresses with strongly singular kernels is used as a starting point. The singular integrals are transformed to nonsingular ones by making use of the Stokes theorem.

Nonsingular traction BIEs for crack problems in elastodynamics

This work proposes a modified procedure, based on analytical integrations, to analyse poroelastic models discretized by time-domain Meshless Local Petrov-Galerkin formulations. In this context, Taylor series expansions of the incognita fields are considered, and the related integrals of the meshless formulations are solved analytically, rendering a so called modified methodology. The work is based on the u−p formulation and the incognita fields of the coupled analysis in focus are the solid skeleton displacements and the interstitial fluid pore pressures. Independent spatial discretization is considered for each phase of the model, rendering a more flexible and efficient methodology. The Moving Least Squares approximation is employed for the spatial variation of the displacement and porepressure fields and two variants of the meshless local Petrov-Galerkin formulation are discussed here, which are based on the use of Heaviside or Gaussian weight test functions. Modified expressions to properly compute the shape function derivatives are also considered. At the end of the paper, numerical examples illustrate the performance and potentialities of the proposed techniques.

Vladimir Sladek

Papers

Modelling of orthorhombic quasicrystal shallow shells

Gradient piezoelectricity for cracks under an impact load

Stress Concentration Near an Elliptic Crack in the Interface Between Elastic Bodies under Steady-State Vibrations

Nonsingular traction BIEs for crack problems in elastodynamics

Porous Media Analysis by Modified MLPG Formulations