We address the numerical simulation of periodic solids (phononic crystals) within the framework of couple stress elasticity. The additional terms in the elastic potential energy lead to dispersive behavior in shear waves, even in the absence of material periodicity. To study the bulk waves in these materials, we establish an action principle in the frequency domain and present a finite element formulation for the wave propagation problem related to couple stress theory subject to an extended set of Bloch-periodic boundary conditions. A major difference from the traditional finite element formulation for phononic crystals is the appearance of higher-order derivatives. We solve this problem with the use of a Lagrange-multiplier approach. After presenting the variational principle and general finite element treatment, we particularize it to the problem of finding dispersion relations in elastic bodies with periodic material properties. The resulting implementation is used to determine the dispersion curves for homogeneous and porous couple stress solids, in which the latter is found to exhibit an interesting bandgap structure.

中文翻译：

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耦合应力弹性动力学中的变分原理和有限元Bloch分析

我们在耦合应力弹性的框架内解决了周期性固体（声子晶体）的数值模拟。即使在没有材料周期性的情况下，弹性势能中的附加项也会导致剪切波的色散行为。为了研究这些材料中的体波，我们在频域中建立了一个作用原理，并提出了与耦合应力理论相关的波传播问题的有限元公式，该问题受一组扩展的 Bloch 周期边界条件约束。与传统的声子晶体有限元公式的主要区别在于高阶导数的出现。我们使用拉格朗日乘数方法解决了这个问题。在介绍了变分原理和一般有限元处理之后，我们将其具体化为在具有周期性材料特性的弹性体中寻找色散关系的问题。结果实现用于确定均质和多孔偶应力固体的色散曲线，其中发现后者表现出有趣的带隙结构。