二维扩张元材料中的共形弹性动力学
Conformal Elastodynamics in 2D Dilational Metamaterials
AI总结：
本文研究了一种二维扩张元材料，通过刚性正方形与柔性铰链连接实现均匀扩张，表现出泊松比-1的共形对称性，揭示了低频响应由旋转与边界扩张主导，高频下动量守恒特性为非线性波控制提供新途径。
摘要：
灵活的机械结构在小负载下可承受大变形，从而在有限频率驱动下产生大、复杂和非线性波响应。本文研究了一种由刚性正方形通过柔性铰链连接而成的动态驱动柔性机械超材料，其支持均匀扩张机制，并在理想铰链极限下表现出泊松比为-1。该扩张变形模式导致共形对称性，动态响应近似在广泛物理变换下保持不变。发现系统低频响应由空间变化的旋转和边界集中扩张组成。即使在高频下，每个共形映射也意味着空间复杂动量的守恒。探讨了实验参数如材料刚度、单元格几何形状和数量如何使实验共形动量接近这种守恒，其变化速度远慢于同类非共形动量。这些结果构成了一个新的框架，为研究扩张超材料中的共形波现象以及非线性波和振动的控制策略提供了新的途径。
英文摘要：
Flexible mechanical structures can undergo large deformations under small loads, enabling large, complex, and nonlinear wave responses under finite-frequency driving. Here, we study a dynamically driven canonical flexible mechanical metamaterial composed of rigid squares connected at their corners by flexible hinges. This metamaterial supports a uniform dilational mechanism and, in the limit of ideal joints, exhibits a Poisson ratio of -1. The presence of this dilational mode of deformation gives rise to a conformal symmetry, in which the dynamics are approximately invariant under a wide class of physical transformations -- conformal maps. We find that the low-frequency response of the system is dominated by conformal deformations consisting of spatially varying rotations and dilations concentrated at the boundary. Even at high frequencies, each conformal map implies a conserved spatially complex momentum. We explore how experimental parameters such as material stiffnesses and the geometry and number of unit cells allow experimental conformal momenta to approach this conservation, varying slowly compared to the non-conformal momenta of same order. These results constitute a new framework opening fundamental avenues for the study of conformal wave phenomena in dilational metamaterials as well as potential strategies for controlling nonlinear waves and vibrations.
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