Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems
Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems
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DOI：
10.1098/rspa.2011.0698
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发表时间：
2012-06-08
影响因子：
3.5
通讯作者：
Kutsenko, A. A.
中科院分区：
文献类型：
--
作者：
Norris, A. N.;Shuvalov, A. L.;Kutsenko, A. A.
研究了三维周期弹性系统运动方程的均匀化问题。利用平面波展开法，得到了控制空间平均场的全动态有效材料参数表达式。有效方程为威利斯型，动量、应力和张量惯性耦合。该公式表明，弹性动力学的Willis方程在均匀化条件下是封闭的。获得了任意频率和波数组合的有效材料参数，包括但不限于波在周期性介质中传播的布洛赫波分支。一维系统的数值例子说明了参数对布洛赫波分支的频率依赖性，并与另一种动态有效介质理论进行了比较，该理论也简化为威利斯形式，但具有不同的有效模量。
Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.