Understanding the feature-rich buckling-dominated behavior of thin elastic ribbons is ripe with opportunities for fundamental studies exploring the nexus between geometry and mechanics, and for conceiving of engineering applications that exploit geometric nonlinearity as a functioning principle. Predictive mechanical models play an instrumental role to this end. As a direct consequence of their physical appearance, ribbons are usually modeled either as one-dimensional rods having wide cross sections, or as narrow two-dimensional plates/shells. These models employ drastically different kinematic assumptions, which in turn play a decisive role in their predictive capabilities. Here, we critically examine three modeling approaches for elastic ribbons using detailed measurements of their complex three-dimensional deformations realized in quasistatic experiments with annulus-shaped ribbons. We find that simple and practically realizable ribbon deformations contradict assumptions underlying strain-displacement relationships in nonlinear rod and von Kármán plate models. These observations do not point at shortcomings of the theories themselves, but highlight fallacies in their application to modeling ribbon-like structures that are capable of undergoing large displacements and rotations. We identify and validate, seemingly for the first time, the 1-director Cosserat plate theory as a model for elastic ribbons over a useful range of loading conditions. In the process, we demonstrate annular ribbons to be prototypical systems for studying the mechanics of elastic ribbons. Annular ribbons exhibit a tunable degree of geometric nonlinearity in response to simple displacement and rotation boundary conditions— a feature that we exploit here for highlighting the consequences of kinematic assumptions underlying different ribbon models. We additionally provide experimental evidence for the existence of multiple stable equilibria, bifurcation phenomena correlated with the number of zero crossings in the mean curvature, and localization of energy, thus making annular ribbons interesting mechanical systems to study in their own right.

理解薄弹性带的特征丰富的屈曲主导行为已经为基础研究探索几何与力学之间的联系以及构想将几何非线性作为功能原理的工程应用提供了机会。预测性机械模型为此发挥了重要作用。由于其物理外观的直接结果，色带通常被建模为具有宽横截面的一维棒，或建模为狭窄的二维板/壳。这些模型采用截然不同的运动学假设，进而在其预测能力中起决定性作用。这里，我们使用弹性带的准三维实验中实现的复杂三维变形的详细测量，仔细研究了弹性带的三种建模方法。我们发现，简单且可实际实现的带形变形与非线性杆和vonKármán板模型中的应变-位移关系所基于的假设相矛盾。这些观察并没有指出这些理论本身的缺点，而是着重指出了其在建模能够经历大位移和旋转的带状结构时的谬误。我们似乎首次确定并验证了1指向性Cosserat板理论作为在有用载荷条件下的弹性带的模型。进行中，我们证明环形带是研究弹性带力学的原型系统。环形色带在简单的位移和旋转边界条件下表现出可调节的几何非线性程度，这是我们在此处利用的功能，以突出显示不同色带模型所依据的运动学假设的后果。我们还为存在多个稳定平衡，分叉现象与平均曲率过零的数量以及能量的局部化提供了实验证据，因此使环形色带成为有趣的机械系统，可以独立研究。环形色带在简单的位移和旋转边界条件下表现出可调节的几何非线性程度，这是我们在此处利用的功能，以突出显示不同色带模型所依据的运动学假设的后果。我们还为存在多个稳定平衡，分叉现象与平均曲率过零的数量以及能量的局部化提供了实验证据，因此使环形色带成为有趣的机械系统，可以独立研究。环形色带在简单的位移和旋转边界条件下表现出可调节的几何非线性程度，这是我们在此处利用的功能，以突出显示不同色带模型所依据的运动学假设的后果。我们还为存在多个稳定平衡，分叉现象与平均曲率过零的数量以及能量的局部化提供了实验证据，因此使环形色带成为有趣的机械系统，可以独立研究。