Elastic ribbons, slender structures whose length ($L$), width ($W$), and thickness ($b$) satisfy $L \gg W \gg b$, exhibit mechanical behaviors intermediate between one-dimensional rods ($L \gg W, b$) and two-dimensional plates ($L, W \gg b$). In quadratic Kirchhoff-type rod-based frameworks, such as Discrete Elastic Rods (DER), the governing equilibrium equations are independent of width, and therefore these models cannot capture width-dependent mechanical effects. Reduced centerline-based ribbon models attempt to capture width dependence via coupled bending-twisting energies. However, their relative accuracy remain unclear due to the absence of a unified simulation framework. In this work, we formulate a framework grounded in discrete differential geometry where the energy is expressed as functions of coupled bending-twisting strain measures along the centerline, rather than a linear sum of quadratic bending and twisting energies in DER. We derive analytical gradients and Hessians of the energy that enable implicit time integration. Within this unified setting, we compare five ribbon models: Kirchhoff, Sadowsky, Wunderlich, Sano, and Audoly. As a benchmark, a straight ribbon is longitudinally constrained into a pre-buckled arch and subjected to transverse displacement, inducing a supercritical pitchfork bifurcation. Predicted bifurcation thresholds are compared against shell-based finite element simulations, with the Sano model providing the closest agreement in capturing width-dependent shifts. Our high-performance JAX-based implementation achieves $\mathcal{O}(N)$ per-iteration cost and also confirms that Sano model introduces negligible per-iteration overhead relative to standard DER.

翻译：弹性带是一种细长结构，其长度（$L$）、宽度（$W$）和厚度（$b$）满足$L \gg W \gg b$，表现出介于一维杆（$L \gg W, b$）和二维板（$L, W \gg b$）之间的力学行为。在基于 Kirchhoff 型二次杆的框架中（如离散弹性杆（DER）），控制平衡方程与宽度无关，因此这类模型无法捕捉依赖于宽度的力学效应。基于中心线的简化带模型试图通过耦合的弯曲-扭转能量来捕捉宽度依赖性，但由于缺乏统一模拟框架，其相对精度仍不明确。在本工作中，我们构建了一个基于离散微分几何的框架，其中能量表示为沿中心线的弯曲-扭转应变测度的函数，而非 DER 中二次弯曲和扭转能量的线性叠加。我们推导了能量的解析梯度和 Hessian 矩阵，从而能够实现隐式时间积分。在此统一框架下，我们比较了五种带模型：Kirchhoff、Sadowsky、Wunderlich、Sano 和 Audoly。作为基准测试，将一条直带纵向约束为预屈曲拱形，并施加横向位移，从而引发超临界叉形分岔。预测的分岔阈值与基于壳的有限元模拟进行了对比，其中 Sano 模型在捕捉宽度依赖性偏移方面具有最佳一致性。我们基于 JAX 的高性能实现达到了每次迭代 $\mathcal{O}(N)$ 的成本，并确认了 Sano 模型相对于标准 DER 几乎不引入额外迭代开销。