Macroscopic stress and strain in a doubly periodic array of dislocation dipoles.
It is known that in two-dimensional periodic arrays of dislocations the summation of the periodic image fields is conditionally convergent. This is due to the long-range character of the elastic fields of dislocations. As a result, the stress field obtained for a doubly periodic array of dislocation dipoles may contain a spurious constant stress that depends on the adopted summation scheme. In the present work, we provide, based on micromechanical considerations, a simple physical explanation of the origin of the conditional convergence of lattice sums of image interactions. In this context, the spurious stresses are found in a closed form for an arbitrary elastic anisotropy, and this is achieved without using the stress field of an individual dislocation. An alternative procedure is also developed where the macroscopic spurious stresses are determined using the solution of the Eshelby's inclusion problem.
Department of Civil, Environmental and Mechanical Engineering , University of Trento, via Mesiano 77 , 38123 Trento, Italy.
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Gourgiotis
Department of Civil, Environmental and Mechanical Engineering , University of Trento, via Mesiano 77 , 38123 Trento, Italy ; Department of Mechanics of Materials , Institute of Fundamental Technological Research (IPPT), Pawińskiego 5b , 02-106 Warsaw, Poland.
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Stupkiewicz
Eshelby's inclusion problem; conditional convergence; dislocation dynamics; micromechanics