Effective antiplane elastic properties of an orthotropic solid weakened by a periodic distribution of cracks
Effective antiplane elastic properties of an orthotropic solid weakened by a periodic distribution of cracks
复制标题
正交各向异性固体的有效反平面弹性特性因裂纹的周期性分布而减弱
DOI：
10.1093/qjmam/hbu008
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Williams T
中科院分区：
文献类型：
--
作者：
Williams T
在低频波传播区域中，利用渐近均匀化方法，我们导出了裂纹固体的有效反平面弹性性质，在没有裂纹的情况下，该固体是正交各向异性的。固体是一种周期性介质，它是由一个周期性胞腔所定义的，胞腔中含有N个在z方向上无限大但在y平面上具有任意形状和取向的裂纹。有效的属性定义在一个所谓的细胞问题的解决方案。我们提出了两个方便的计划，通过它可以解决的细胞问题，一个基于近周期绿色的功能，另一个基于双周期多极展开。使用这些方法，我们比较结果与现有的近似表达式的周期性阵列的直裂纹和讨论其制度的有效性，通过评估其偏离本方法获得的确切结果。我们继续考虑更复杂的分布，如椭圆腔，非直裂纹和正交各向异性的主机相的影响。特别是，我们表明，在正交各向异性主介质中的特定类型的规则阵列的裂纹可以诱导一个macroscopallyisotropicresponse反平面剪切波传播thexyplane.
Using the method of asymptotic homogenization in the low-frequency wave propagation regime we derive the effective antiplane elastic properties of a cracked solid which in the absence of cracks would be orthotropic. The solid is a periodic medium defined by a periodic cell containingNcracks of infinite extent in thezdirection but with arbitrary shape and orientation in thexyplane. Effective properties are defined in terms of the solution to a so-calledcell problem. We propose two convenient schemes by which the cell problem can be solved, one based on a nearly periodic Green's function, the other based on doubly periodic multipole expansions. Using these methods we compare results with existing approximate expressions for periodic arrays of straight cracks and discuss their regimes of validity by assessing their departure from the exact results obtained by the present method. We go on to consider more complex distributions such as elliptical cavities, non-straight cracks and effects of an orthotropic host phase. In particular we show that specific types of regular arrays of cracks in an orthotropic host medium can induce a macroscopicallyisotropicresponse to antiplane shear waves propagating in thexyplane.