TY - JOUR
T1 - Asymptotic rigidity for shells in non-Euclidean elasticity
AU - Alpern, Itai
AU - Kupferman, Raz
AU - Maor, Cy
N1 - Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/9/15
Y1 - 2022/9/15
N2 - We consider a prototypical “stretching plus bending” functional of an elastic shell. The shell is modeled as a d-dimensional Riemannian manifold endowed, in addition to the metric, with a reference second fundamental form. The shell is immersed into a (d+1)-dimensional ambient space, and the elastic energy accounts for deviations of the induced metric and second fundamental forms from their reference values. Under the assumption that the ambient space is of constant sectional curvature, we prove that any sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell into ambient space, having the reference second fundamental form. In particular, if the ambient space is Euclidean space, then the reference metric and second fundamental form satisfy the Gauss-Codazzi-Mainardi compatibility conditions. This theorem can be viewed as a (manifold-valued) co-dimension 1 analog of Reshetnyak's asymptotic rigidity theorem. It also relates to recent results on the continuity of surfaces with respect to their fundamental forms.
AB - We consider a prototypical “stretching plus bending” functional of an elastic shell. The shell is modeled as a d-dimensional Riemannian manifold endowed, in addition to the metric, with a reference second fundamental form. The shell is immersed into a (d+1)-dimensional ambient space, and the elastic energy accounts for deviations of the induced metric and second fundamental forms from their reference values. Under the assumption that the ambient space is of constant sectional curvature, we prove that any sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell into ambient space, having the reference second fundamental form. In particular, if the ambient space is Euclidean space, then the reference metric and second fundamental form satisfy the Gauss-Codazzi-Mainardi compatibility conditions. This theorem can be viewed as a (manifold-valued) co-dimension 1 analog of Reshetnyak's asymptotic rigidity theorem. It also relates to recent results on the continuity of surfaces with respect to their fundamental forms.
KW - Non-Euclidean elasticity
KW - Riemannian manifolds
KW - Rigidity
KW - Shell theory
UR - http://www.scopus.com/inward/record.url?scp=85131397441&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2022.109575
DO - 10.1016/j.jfa.2022.109575
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AN - SCOPUS:85131397441
SN - 0022-1236
VL - 283
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 6
M1 - 109575
ER -