The Mechanics of Ribbons and Möbius Bands by Roger Fosdick, Eliot Fried

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Aside from approximation results regarding convergence of minimizers of (3) to minimizers of (2), important open questions include the possibility of centerlines with inflection points. An anticipated difficulty to be overcome is the lack of a polynomial growth bound, from below, for the integrand of the Sadowsky functional (2). There may be some temptation to view the results gathered here as being relevant to the work of Friesecke, James and Müller [7]. In that work, the elastic energy of a deformed rectangle is defined on the space of surfaces parameterized by functions u ∈ W 2,2 ([0, ] × [−w, w]; R3 ) satisfying ux · uy = |ux |2 − 1 = |uy |2 − 1 = 0.

2(a)), while the normal projection on the yz plane (“elevation”) takes the form of a pure oval (see Fig. 2(b)), which can be approximated by the ellipse y2 + z2 = 1, c2 4 with c = . 5 (10) The power series expansion of m as a function of x in a neighborhood of the vertex C will begin as follows: y = λx 3 + · · · , z=c 1− λ2 6 x + ··· . 2 (11) We now raise the central projection of the midline m from its singular point C(0, 0, c) onto the xy plane—the “stereographic projection” of the elliptical cylinder (10), if you will: ξ= Reprinted from the journal cx , c−z η= 26 cy .

The theorem has the following important corollary. Corollary 1 On the set X +,p = κm >0 X κm ,p , the Sadowsky functional is the Γ -limit of the elastic energy Fε (·, I ) with respect to the sequential weak topology and with respect to the strong topology on W 3,p (I ; R3 ). Proof Let u ∈ X +,p . Then u ∈ X κm ,p for some κm > 0. That the limsup condition (20) is satisfied follows from the Γ convergence of Fε (·, I ) to F (·, I ) in X κm ,p . Let {uj } be a sequence in X +,p such that uj → u either strongly or weakly.