Stability of boundary conditions for the Sadowsky functional. (English) Zbl 1497.49017
Summary: It has been proved by the authors that the (extended) Sadowsky functional can be deduced as the \(\Gamma\)-limit of the Kirchhoff energy on a rectangular strip, as the width of the strip tends to 0. In this paper, we show that this \(\Gamma\)-convergence result is stable when affine boundary conditions are prescribed on the short sides of the strip. These boundary conditions include those corresponding to a Möbius band. This provides a rigorous justification of the original formal argument by Sadowsky about determining the equilibrium shape of a free-standing Möbius strip. We further write the equilibrium equations for the limit problem and show that, under some regularity assumptions, the centerline of a developable Möbius band at equilibrium cannot be a planar curve.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49S05 | Variational principles of physics |
| 74B20 | Nonlinear elasticity |
| 74K20 | Plates |
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