High frequency homogenization for structural mechanics
High frequency homogenization for structural mechanics
复制标题
DOI：
10.1016/j.jmps.2010.12.004
复制
发表时间：
2011-03-01
影响因子：
5.3
通讯作者：
Kaplunov, J.
中科院分区：
文献类型：
--
作者：
Nolde, E.;Craster, R. V.;Kaplunov, J.
我们以弹性弦形成的网络作为模型结构来研究波在半离散介质中的传播。我们特别感兴趣的是当波长和网络的每个单元具有相似的阶数时，在高频下有效的连续介质模型的发展。Net结构被选为一个一般的二维例子，封装了晶格结构的二维激发所涉及的基本物理，同时保留了处理弹性弦的简单性。这里开发了与细胞尺度相称的波长的均匀化技术。与以前的理论不同，这些技术不限于低频或静态区域，并导致有效的连续方程在宏观尺度上有效，细胞结构的细节仅通过积分量来封装。渐近过程是基于双尺度方法和物理观察，即存在随细胞周期或双周期周期性地产生驻波的频率。构造了一个由弹性弦构成的网的具体例子，该理论是通用的，不依赖于网是无限的，但无限网是Bloch理论的一个有用的特例。这一特殊情况将被详细探讨，以验证理论，并突出退化情况的重要性；正方形网的具体例子将被详细处理。该方法的通用性的另一个例证是对点强迫的响应，它提供了对齐次化方程的严格检验；推导出了网络的精确格林函数，并与渐近性进行了比较。(C)2010爱思唯尔有限公司。保留所有权利。
We consider a net created from elastic strings as a model structure to investigate the propagation of waves through semi-discrete media. We are particularly interested in the development of continuum models, valid at high frequencies, when the wavelength and each cell of the net are of similar order. Net structures are chosen as these form a general two-dimensional example, encapsulating the essential physics involved in the two-dimensional excitation of a lattice structure whilst retaining the simplicity of dealing with elastic strings.Homogenization techniques are developed here for wavelengths commensurate with the cellular scale. Unlike previous theories, these techniques are not limited to low frequency or static regimes, and lead to effective continuum equations valid on a macroscale with the details of the cellular structure encapsulated only through integrated quantities. The asymptotic procedure is based upon a two-scale approach and the physical observation that there are frequencies that give standing waves, periodic with the period or double-period of the cell. A specific example of a net created by a lattice of elastic strings is constructed, the theory is general and not reliant upon the net being infinite, none the less the infinite net is a useful special case for which Bloch theory can be applied. This special case is explored in detail allowing for verification of the theory, and highlights the importance of degenerate cases; the specific example of a square net is treated in detail. An additional illustration of the versatility of the method is the response to point forcing which provides a stringent test of the homogenized equations; an exact Green's function for the net is deduced and compared to the asymptotics. (C) 2010 Elsevier Ltd. All rights reserved.