A ruled narrow elastic strip model with corrected energy

TL;DR

This work addresses inflection-point singularities in elastic ribbon models by introducing a 1D ruled narrow-strip model with an auxiliary field $\epsilon(s)$ that relaxes developability, allowing nonzero twist even when curvature vanishes.Equilibrium is derived via a rotation-tensor variational framework, producing balance laws and an energy that smoothly connects Sadowsky-like behavior to Kirchhoff-rod-like regimes, and reproduces Freddi's corrected scalings in appropriate limits without patching.A key result is that for a particular parameter choice the energy becomes convex, yielding unique large-twist, small-curvature solutions, while in general the model exhibits a convex envelope with possible microstructure and a width-dependent embedding jump at inflection points.A perturbation expansion around inflection points shows curvature and moment are continuous, while the auxiliary field jumps, providing practical asymptotics for numerical implementation and deepening the understanding of regularization mechanisms in slender ribbons.

Abstract

We present a new one-dimensional model for elastic strips based on a nondevelopable ruled surface. An auxiliary field regularizes the Sadowsky narrow-strip model to allow nonzero twist with vanishing curvature. The energy exhibits the scalings derived by Freddi and co-workers, and for a certain choice of parameter, convexifies the Sadowsky energy without patching. We present the kinematics and energetics of the model, and employ a variational approach featuring a rotation tensor to derive equilibrium equations. We perform a regular perturbation expansion to study the model behavior close to inflection points. When the energy is convex, curvature and moment are continuous at inflection points, while the auxiliary function suffers a jump, leading to a discontinuity in the ruled embedding for any finite width.

Figures (5)

• Figure 1: Ribbon with $b_\epsilon = b_H/9$ and a width of $w = 2\times 10^{-2}$, showing a straight twisted solution (helicoid) for $\kappa = 0$ and $\tau = 2$. The aspect ratio has been stretched. The left ribbon was generated with the positive $\epsilon_0$ solution and the right ribbon was generated with the negative $\epsilon_0$ solution. The arc-length step size is $\Delta s = 10^{-4}$, and generators are plotted for every 300 steps.
• Figure 2: Energy density $e$, regularization field $\epsilon$, and moment $M_1$ for $b_\epsilon/b_H = 1/9$. In (a) and (b) the energy density is shown in blue alongside that of the Sadowsky model in red.
• Figure 3: Energy density $e$, regularization field $\epsilon$, and moment $M_1$ for $b_\epsilon/b_H = 1/4$. In (a) and (b) the energy density is shown in blue alongside that of the Sadowsky model in red.
• Figure 4: Comparison of the linear approximation \ref{['eq:epasym']} (dashed red) and exact solution \ref{['eq:ep']} (solid blue) of the regularization field $\epsilon$ over part of the constant-$\tau$ slice shown in Figure \ref{['fig:energy1']}(e).
• Figure 5: Inflection point neighborhood in a ribbon of width $w = 2\times10^{-3}$, using $\tau_0 = 20$, $\varepsilon n_1^{(1)}=0$, $\varepsilon n_2^{(1)}= 0.1$, $\varepsilon m_2^{(1)}=0.1$, and an arc-length step size of $\Delta s = 5\times 10^{-4}$. The right image is zoomed in at the inflection point, where $\kappa$ crosses zero. There is a jump in the embedding at this point, as generators change their orientation.