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Proceedings Paper

Basic solutions and regular expansions in the mechanics of periodic composites
Author(s): Alexei I. Borovkov; Vladimir A. Palmov
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Paper Abstract

We consider an inhomogeneous elastic body. We suppose that it represents so called periodic composite. It means that the body is composed of many similar cells with dimensions a, b, c in the direction of coordinate ox, oy and oz respectively. We suppose that all cells have similar complex structure. At last we suppose that all cells have full contact with their neighbors, so that displacements and stresses are continuous on the bounds of cells. Bodies of this type, periodic composites, are the subject of investigations in many papers and books. We confine ourselves by noting only some of them. [1-5]. The transactions [6] are devoted to the advanced problems of periodic and nonperiodic composites, and contain great number of literature sources on the mechanics of composites. This paper is devoted to the new technology of analysis of periodic composites. This technology is based on using so called "basic solutions" for one cell and on application of linear combinations of these basic solutions for presentation of all variables of problem. We call these combinations by the name 'regular expansions.' For simplification of motivations and in order to get simple results, we confine ourselves by considering the case when any cell has three planes of symmetry, which are parallel to the coordinate planes. Now we enumerate new results which can be obtained using new approach and corresponding way of motivation. Firstly, new method for getting effective moduli of elasticity for composite. For example, we present simple formulae for these moduli. We convince a reader that we get exact values of the moduli for uniform averaged stresses and strains. Secondly, we present new motivation, which leads to the homogenized problem. We prove that "regular expansion" represents effective solution of all equations of elasticity theory for an initial nonhomogenized composite if averaged stresses and strains satisfy to the homogeneous equation. Thirdly, the boundary layer concept is applied for correct formulation of the boundary conditions for homogenized problem which permits to satisfy boundary condition for initial nonhomonized problem.

Paper Details

Date Published: 9 June 2006
PDF: 18 pages
Proc. SPIE 6253, Ninth International Workshop on Nondestructive Testing and Computer Simulations, 62530F (9 June 2006); doi: 10.1117/12.676395
Show Author Affiliations
Alexei I. Borovkov, St. Petersburg State Polytechnical Univ. (Russia)
Vladimir A. Palmov, St. Petersburg State Polytechnical Univ. (Russia)


Published in SPIE Proceedings Vol. 6253:
Ninth International Workshop on Nondestructive Testing and Computer Simulations
Alexander I. Melker, Editor(s)

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