On the fundamental equations of piezoelasticity of quasicrystal media

The three-dimensional fundamental equations of elasticity of quasicrystals with extension to quasi-static electric effect are expresses in both differential and variational invariant forms for a regular region of quasicrystal material. The principle of conservation of energy is stated for the regular region and the constitutive relations are obtained for the piezoelasticity of material. A theorem is proved for the uniqueness in solutions of the fundamental equations by means of the energy argument. The sufficient boundary and initial conditions are enumerated for the uniqueness. Hamilton’s principle is stated for the regular region and a three-field variational principle is obtained under some constraint conditions. The constraint conditions, which are generally undesirable in computation, are removed by applying an involutory transformation. Then, a unified variational principle is obtained for the regular region, with one or more fixed internal surface of discontinuity. The variational principle operating on all the field variables generates all the fundamental equations of piezoelasticity of quasicrystals under the symmetry conditions of the phonon stress tensor and the initial conditions. The resulting equations, which are expressible in any system of coordinates and may be used through simultaneous approximation upon all the field variables in a direct method of solutions, pave the way to the study of important dislocation, fracture and interface problems of both elasticity and piezoelasticity of quasicrystal materials.     

A generalized variational theorem in elastodynamics,with application to shell theory

Summary With a view toward the consistent derivations and numerical solutions of one- and two-dimensional approximate theories in a class of Cosserat continuum, a variational theorem is, in a straightforward manner, established by means of Hamilton's principle. By the use of this theorem, a linear theory of anisotropic shells for both extensional and flexural motions, including thermal effects, is systematically constructed. A theorem of uniqueness in this theory is then presented.

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Sommario Per mezzo del principio di Hamilton si stabilisce direttamente un teorema variazionale in vista di organiche derivazioni e soluzioni numeriche di teorie approssimate a una e a due dimensioni in una classe di continui di Cosserat. Con questo teorema si costruisce sistematicamente una teoria lineare di membrane anisotropiche per movimenti estensionali e flessionali includendo gli effetti termici. Si presenta poi in questo teorema una teoria di unicità.

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Supported by the U.S. Office of Naval Research.     

Fundamental equations of certain electromagnetic-acoustic discontinuous fields in variational form

We present, in the first part of the paper, the well-known fundamental electromagnetic-acoustic equations, that is, the coupled Maxwells and Newtons equations for an elastic dielectric continuum in differential form, and we also discuss the uniqueness of their linear solutions. In the second part, from a general principle of physics, we deduce a three-field variational principle that operates on the mechanical displacements, the electric potential, and the electromagnetic vector potential of the dielectric continuum. Then, we extend it through an involutory (or Friedrichss) transformation in deriving a nine-field unified variational principle that operates on the mechanical, electrical, and magnetic continuous linear fields under the infinitesimal strains. This variational principle generates Maxwells and Newtons equations, the coupled linear constitutive relations, and the associated natural boundary conditions for the regular region of the dielectric continuum as its Euler-Lagrange equations. In the third part, we further generalise the unified variational principle so as to incorporate the jump conditions across a surface of discontinuity within the dielectric region. We also show that the integral and differential types of variational principles that apply to the linear motions of the elastic dielectric region with a fixed internal surface of discontinuity are in agreement with and recover, as special cases, some of the earlier variational principles. Further, the variational principles may be directly used in linear electromagnetic and/or acoustic field computations and in consistently establishing the lower order one- or two-dimensional equations of the elastic dielectric continuum.Received: 9 January 2002, Accepted: 26 May 2003, Published online: 5 December 2003PACS: 03.40, 41.10, 77.60 Correspondence to: G.A. Altay     

On the variational boundary integral equations in elastodynamics with the use of conjugate functions

This paper proposes a Variational Boundary Integral Equation for time harmonic elasticity, using conjugate functions. A bilinear hermitian form for the variational formulation, as well as an a posteriori error indicator are proposed. The method does not involve hypersingular integrals in the finite part sense and preserves the symmetrical structures of equations.     

A generalized variational principle of composite shallow shells and its application to the folded shell

In this paper, a generalized variational principle of elastodynamics in composite shallow shells with edge beams is presented, and its equivalence to corresponding basic equations, ridge conditions and boundary conditions is proved. Then this variational principle is applied to the folded shell structure. By means of double series, the approximate analytical solutions for statics and dynamics under common boundary conditions are obtained. The comparison of our results with FEM computations and experiments shows the analytical solutions have good convergence and their accuracy is quite satisfactory.     

ON THE GENERAL SOLUTION OF CYLINDRICAL SHELL EQUATIONS

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A bilinear form of the variational principle with vanishing parameter

Summary  A variational principle whose Lagrangian function generates a hyperbolic heat conduction equation is exhibited. The main characteristic of this principle is that it contains two temperature fields that enter bilinearly into the Lagrangian function. These two fields are interpreted as two mutually independent approximate temperature profiles which are potentially competent to describe rationally the real physical temperature distribution. The proposed variational principle is used as a starting point for finding approximate solutions of the classical, i.e. Fourier's, heat conduction theory, by employing the vanishing parameter technique and the direct methods of variational calculus. Received 4 June 1997; accepted for publication 1 July 1997     

Further application of hybrid solution to another form of Boussinesq equations and comparisons

Recently, a new hybrid scheme is introduced for the solution of the Boussinesq equations. In this study, the hybrid scheme is used to solve another form of the Boussinesq equations. The hybrid solution is composed of finite‐volume and finite difference method. The finite‐volume method is applied to conservative part of the governing equations, whereas the higher order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy is provided in both time and space. The solution is then applied to several test cases, which are taken from the previous studies. The results of this study are compared with experimental and theoretical results as well as those of the previous ones. The comparisons indicate that the Boussinesq equations solved here and in the previous study produce quite similar results. Copyright © 2006 John Wiley & Sons, Ltd.     

Elastoplastic equilibrium of a spherical shell in the form of an eccentric ring

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 1, pp. 65–72, January, 1990.     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

The fundamental solutions for the plane problem in piezoelectric media with an elliptic hole or a crack

1.IntroductionItiswell-knownthatthefundame,ltalsolutionsorGreen'sfunctionsplayanimportantroleilllinearelasticity.Forexample,theycanbeusedtoconstructmanyanalyticalsolutionsofpracticalproblems.Itismoreimportantthattheyareusedasthefundamentalsolutionsintheboundaryelementmethod(BEM)tosolvesomecomplicatedproblem.Withthewidely-increasingapplicationofpiezoelectricmaterialsinengineeringproblems,thestudyregardingtheGreen'sfLlnctionsinpiezoelectricsolidshasreceivedmuchinterest.The3DGreen'sfunctionsi…     

Singular perturbation of linear algebraic equations with application to stiff equations

In this paper the singular perturbation problem of linear algebraic equations with a small parameter is presented by an example in practice. The existence and uniqueness theorem of its solution is proved by the perturbation method and the estimation of error for its approximate solution is given. Finally, the example mentioned above explaining how to apply the theory to solve the stiff equations is shown.     

On the stress distribution in a spherical shell with an off-center curvilinear hole

A numerical analysis is made of the joint effect of two factors of asymmetry—ellipticity and eccentricity—on the stress distribution near a free hole in a spherical shell. The nature of deformation is determined by the predominant factor. Whether there are “fixed” points on the graphs of stress distribution around a small hole at which they intersect depends on how rigidly the outer edge is fixed. As the rigidity of fixation is increased, the points smear __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 113–118, January 2006.