We establish a dynamic homogenization framework catering for the linear elastic wave motion in periodic origami structures. The latter are modeled via “bar-and-hinge” paradigm where: (i) the folding of the structure and the bending of individual panels are modeled via elastic hinges, and (ii) the in-plane deformation of each panel is modeled with elastic bars. Using the so-formulated discrete model of an origami structure, we pursue finite wavenumber-finite frequency (FW-FF) homogenization of the wave motion in a spectral neighborhood of simple, repeated, and nearby eigenfrequencies at an arbitrary wavenumber within the first Brillouin zone. The lynchpin of the proposed approach is the “projection” of the nodal displacements over each unit cell onto a suitable Bloch eigenvector, evaluated at the “center” of the spectral region of interest. For completeness, we make an account for: (i) the source term acting at the nodes of a discrete structure, and (ii) periodic Dirichlet boundary conditions. We obtain the leading-order (system of) effective equation(s) synthesizing the wave motion in a selected spectral neighborhood, and we describe asymptotically the corresponding dispersion relationship. We illustrate the proposed framework by comparing numerically the Bloch dispersion relationship to its asymptotic approximation for (a) a 2D-periodic Miura-ori structure, and (b) a 1D-periodic Miura tube. The dispersion analysis is complemented by evaluating the effective wave motion (in terms of both “macroscopic” and “microscopic” essentials) in a 2D-periodic Miura-ori structure due to spatially-localized source term acting either inside a band gap or within a passband.

我们建立了一个动态均匀化框架，以适应周期性折纸结构中的线性弹性波运动。后者通过“杆和铰链”范式建模，其中：（i）结构的折叠和单个面板的弯曲通过弹性铰链建模，以及（ii）每个面板的平面内变形建模弹性棒。使用如此制定的折纸结构的离散模型，我们在第一布里渊区内任意波数的简单、重复和附近特征频率的光谱邻域中追求波动的有限波数 - 有限频率 (FW-FF) 均匀化. 所提出方法的关键是节点位移的“投影”在每个晶胞上到合适的布洛赫特征向量上，在感兴趣的光谱区域的“中心”进行评估。为了完整起见，我们解释了：（i）作用于离散结构节点的源项，和（ii）周期性狄利克雷边界条件。我们获得了在选定光谱邻域中合成波动的有效方程的前阶（系统），并且我们渐近地描述了相应的色散关系. 我们通过数值比较 Bloch 色散关系与其渐近逼近来说明所提出的框架（a）2D 周期 Miura-ori 结构和（b）1D 周期 Miura 管。由于空间局部源项在带隙内或在通带。