Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure☆
Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure☆
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DOI：
10.1016/j.jmaa.2005.03.093
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发表时间：
2006-02
影响因子：
1.3
通讯作者：
K. Lurie;Suzanne L. Weekes
中科院分区：
文献类型：
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作者：
K. Lurie;Suzanne L. Weekes
我们考虑波在一个时空双周期材料结构中的传播，该结构具有一个空间维度和时间上的矩形微几何形状。假定这种动态材料的空间周期和时间周期具有相同的数量级。当波动方程[公式：见文本]允许分离变量时，在特殊情况下可以得到“双Floquet”解。我们还考虑一个棋盘微几何，其中变量不能分离。假设时空棋盘中的方格由阻抗相等但相速不同的材料填充。在一定的参数范围内，我们在数值上观察到不同的和稳定的极限特征路径（“极限环”）的形成，在几个时间段后吸引邻近的特征。沿极限环的平均传播速度在微观几何参数的一定范围内保持不变（“平台效应”）。我们把棋盘结构在平台上当且仅当它产生稳定的极限环时，作为一个假设表述出来。动态材料是一个热力学开放系统，因为它与环境进行能量和动量的永久交换。产生极限环的材料组合在这方面是特殊的。具体地说，为了使波通过这样的组合，我们解析地发现，可能需要一个外部介质提供无限的能量，而这可能与波的频率无关。然而，对于时空层压板，能量的积累（参数共振）可能只出现在相对于系统的某些特征频率不太低的频率上。
We consider propagation of waves through a spatio-temporal doubly periodic material structure with rectangular microgeometry in one spatial dimension and time. Both spatial and temporal periods in this dynamic material are assumed to be of the same order of magnitude. A “double Floquet” solution is obtained in the special case when the wave equation [Formula: see text] allows for the separation of variables. We also consider a checkerboard microgeometry where variables cannot be separated. The squares in a space–time checkerboard are assumed to be filled with materials having equal impedance but different phase speeds. Within certain parameter ranges, we observe numerically the formation of distinct and stable limiting characteristic paths (“limit cycles”) that attract neighbouring characteristics after a few time periods. The average speed of propagation along the limit cycles remains the same throughout certain ranges of parameters of the microgeometry (the “plateau effect”). We formulate, as a hypothesis, the statement saying that a checkerboard structure is on a plateau if and only if it yields stable limit cycles. A dynamic material is a thermodynamically open system, as it is involved in a permanent exchange of energy and momentum with the environment. Material assemblages that produce the limit cycles are special in this aspect. Specifically, to make a wave travel through such an assemblage, we find analytically that an external agent may need to supply infinite energy and this may be so regardless of the wave frequency. For spatio-temporal laminates, however, an accumulation of energy (parametric resonance) may emerge only for frequencies that are not too low relative to some characteristic frequency of the system.