Stability of Boundary Conditions for the Sadowsky Functional

Abstract

AbstractIt has been proved by the authors that the (extended) Sadowsky functional can be deduced as the $$\Gamma $$ Γ -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 $$\Gamma $$ Γ -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.