A nonlinear Korn inequality on a surface

✍ Scribed by Philippe G. Ciarlet; Liliana Gratie; Cristinel Mardare

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 192 KB
• Volume: 85
• Category: Article
• ISSN: 0021-7824
• DOI: 10.1016/j.matpur.2005.10.010

✦ Synopsis

Let ω be a domain in R 2 and let θ : ω → R 3 be a smooth immersion. The main purpose of this paper is to establish a "nonlinear Korn inequality on the surface θ (ω)", asserting that, under ad hoc assumptions, the H 1 (ω)-distance between the surface θ(ω) and a deformed surface is "controlled" by the L 1 (ω)-distance between their fundamental forms. Naturally, the H 1 (ω)-distance between the two surfaces is only measured up to proper isometries of R 3 .

This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let θ k : ω → R 3 , k 1, be mappings with the following properties: They belong to the space H 1 (ω); the vector fields normal to the surfaces θ k (ω), k 1, are well defined a.e. in ω and they also belong to the space H 1 (ω); the principal radii of curvature of the surfaces θ k (ω), k 1, stay uniformly away from zero; and finally, the fundamental forms of the surfaces θ k (ω) converge in L 1 (ω) toward the fundamental forms of the surface θ (ω) as k → ∞. Then, up to proper isometries of R 3 , the surfaces θ k (ω) converge in H 1 (ω) toward the surface θ (ω) as k → ∞.

Such results have potential applications to nonlinear shell theory, the surface θ(ω) being then the middle surface of the reference configuration of a nonlinearly elastic shell.