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A compatible‐incompatible decomposition of symmetric tensors in L p with application to elasticity
Author(s) -
Maggiani Giovanni Battista,
Scala Riccardo,
Goethem Nicolas Van
Publication year - 2015
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3450
Subject(s) - mathematics , compatibility (geochemistry) , pure mathematics , mathematical analysis , symmetric tensor , elasticity (physics) , physics , exact solutions in general relativity , geochemistry , thermodynamics , geology
In this paper, we prove the Saint‐Venant compatibility conditions in L p for p ∈(1,+ ∞ ), in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in L p are provided. We also use the Helmholtz decomposition in L p to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence‐free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami‐type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where 1 < p < 2. This justifies the need to generalize and prove these rather classical results in the Hilbertian case ( p = 2), to the full range p ∈(1,+ ∞ ). Copyright © 2015 John Wiley & Sons, Ltd.