Donati compatibility conditions on a surface - Application to shell theory

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Original languageEnglish
Pages (from-to)173-202
Journal / PublicationJournal des Mathematiques Pures et Appliquees
Issue number1
Online published25 Nov 2013
Publication statusPublished - Jul 2014


The Donati compatibility conditions constitute a characterization of symmetric 3×3 matrix fields defined over a domain Ω⊂R3 as linearized strain tensor fields. They take various forms, according to which boundary conditions are to be satisfied by the corresponding displacement vector field. A typical result in this direction is the following one, due to G. Geymonat and F. Krasucki and C. Amrouche, the first author, L. Gratie, and S. Kesavan: Given a symmetric matrix field (eij) with components eij in the space L2(Ω), there exists a displacement vector field (vi) with components vi in the space H01(Ω) such that eij=12(∂jvi+∂ivj) in Ω if, and only if, ∫Ωeijsijdx=0 for all symmetric matrix fields (sij) with sij∈L2(Ω) that satisfy ∂jsij=0 in Ω.The main objective of this paper is to identify and justify various Donati-like compatibility conditions on a surface, guaranteeing that the components of two symmetric matrix fields (cαβ) and (rαβ) with cαβ and rαβ in the space L2(ω), where ω is now a domain in R2, are the covariant components of the linearized change of metric and linearized change of curvature tensors associated with a displacement vector field of a surface θ(ω-), where θ:ω-→R3 is a smooth immersion.Such compatibility conditions, the expressions of which again depend on the boundary conditions satisfied by the displacement vector field, in turn allow us to reformulate the quadratic minimization problem for a linearly elastic shell in terms of the fields (cαβ) and (rαβ) as the sole unknowns, thus providing an example of an intrinsic approach to shell theory. © 2013 Elsevier Masson SAS.

Research Area(s)

  • Donati compatibility conditions, Intrinsic elasticity, Linear shell theory, Linearized elasticity