We combine discrete differential geometry (DDG)-based models and desktop experiments to study supercritical pitchfork bifurcation of a pre-compressed elastic plate under lateral end translation, with a focus on its width effect. Based on the ratio among length, width, and thickness, the elastic structures in our study fall into three different structural categories: rods, ribbons, and plates. In order to numerically simulate the mechanical response of these structures, we employ two DDG-based numerical frameworks — Discrete Anisotropic Rods method and Discrete Elastic Plates method. Even though the multi-stability and bifurcation of a narrow strip can be precisely captured by a naive one dimensional rod model, it fails to match with experiments as the ribbon increases in width. A two dimensional approach using a plate model, on the other hand, accurately predicts the geometrically nonlinear deformations and the supercritical pitchfork points for plate even when the width is as large as half of the length. Exploiting the efficiency and robustness of the simulator, we perform a systematic parameter sweep on plate size and lateral displacement to build a phase diagram of different configurations of the elastic plates. We find that the deformed configuration of the nearly developable strips can be described, up to a very good approximation, using the bending and twisting of the centerline. This indicates that a one dimensional energy model for the simulation of nearly developable strips can potentially be developed in the future. The results can serve as a benchmark for future numerical investigations into modeling of ribbons. Our study can also provide guidelines on the choice of the appropriate structural model – rod vs. ribbon vs. plate – in simulation of thin elastic structures.

我们结合基于离散微分几何（DDG）的模型和桌面实验，研究了预压缩弹性板在侧向端平移下的超临界干草叉分叉，重点是其宽度效应。根据长度，宽度和厚度之间的比率，我们研究中的弹性结构分为三种不同的结构类别：棒，带和板。为了对这些结构的机械响应进行数值模拟，我们采用了两个基于DDG的数值框架-离散各向异性杆法和离散弹性板法。即使天真的一维杆模型可以精确地捕获窄带的多稳定性和分叉性，但随着带的宽度增加，它无法与实验匹配。另一方面，使用平板模型的二维方法 即使宽度为长度的一半，也可以准确地预测板的几何非线性变形和超临界干草叉点。利用模拟器的效率和鲁棒性，我们对板的尺寸和横向位移进行了系统的参数扫描，以构建弹性板不同配置的相图。我们发现，利用中心线的弯曲和扭曲，可以描述几乎可显影的带材的变形构造，直至非常好地近似。这表明将来可能会开发出一维能量模型来模拟几乎可开发的带钢。该结果可作为将来对色带建模进行数值研究的基准。