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On the Cavities and Rigid Inclusions Correspondence and the Cosserat Spectrum
Author(s) -
Markenscoff X.,
Paukshto M.
Publication year - 1996
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19961770111
Subject(s) - mathematics , uniqueness , traction (geology) , mathematical analysis , boundary value problem , uniqueness theorem for poisson's equation , elasticity (physics) , eigenvalues and eigenvectors , linear elasticity , physics , quantum mechanics , geomorphology , finite element method , thermodynamics , geology
The correspondence in two‐dimensional elasticity between the stress fields of cavities and rigid inclusions has been obtained by Dundurs [1] and Markenscoff [3]. It was shown that if the limit of the stress of the inclusion boundary‐value problem, which depends on the elastic constants, exists when the Poisson's ratio v tends to 1, then this solves the traction boundary‐value problem for the cavity problem since it satisfies equilibrium and boundary conditions, and, by the uniqueness theorem, exists and is unique. In three dimensions the solution of the traction boundary‐value problem of elasticity does depend on Poisson's ratio since the Beltrami‐Mitchell compatability conditions for the stress depend on Poisson's ratio. So the similar argument for the correspondence between cavities and rigid inclusions cannot in principle be made. However, the Beltrami‐Mitchell compatability conditions are independent of v if the dilatation is a constant or a linear function of the position. In this case we can show that the same result goes through for the correspondence. In order to investigate the behavior of the solutions in the vicinity of v = 1, we use some results obtained for the Cosserat spectrum by Mikhlin [4], Maz'ya and Mikhlin [3], see also [6]. The existence of the limit for 2D and 3D when v tends to 1 is proved on the basis of the fact that the eigenvalue ω = — 1 of the Cosserat spectrum is isolated.