Abstract Asymptotic homogenisation of an elastic metamaterial consisting of a series of plates interspaced by a fluid is considered. It is shown that the usual method must be extended by the inclusion of a second macroscale, the “emergent scale”, in order to correctly capture the behaviour of this metamaterial at low frequency. The leading order solutions for plane wave propagation are found and the effective constitutive equations derived. At leading order, the dispersion curves for plane wave propagation depend on the direction of propagation and the branch associated with the emergent scale is dispersive. The effective constitutive equations contain some terms, associated with the shear normal to the plates, for which stress is not proportional to strain but rather depends on the third derivative of the normal displacement with respect to transverse position. These terms are not present in the usual form of the Willis equations and the effective fields are non-analytic functions of frequency as the frequency tends to zero.

中文翻译：

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弹性超材料有效方程中的紧急尺度行为

摘要 考虑了由一系列由流体隔开的板组成的弹性超材料的渐近均匀化。结果表明，必须通过包含第二个宏观尺度（“紧急尺度”）来扩展通常的方法，以便正确捕获这种超材料在低频下的行为。找到了平面波传播的主阶解并导出了有效的本构方程。在超前阶，平面波传播的色散曲线取决于传播方向，并且与涌现尺度相关的分支是色散的。有效本构方程包含一些与垂直于板的剪切相关的项，其中应力与应变不成正比，而是取决于法向位移相对于横向位置的三阶导数。这些项在 Willis 方程的通常形式中不存在，并且有效场是频率的非解析函数，因为频率趋于零。