An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems.
An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems.
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DOI：
10.1098/rspa.2017.0080
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发表时间：
2017-05
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影响因子：
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通讯作者：
Abrahams ID
中科院分区：
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作者：
Joyce D;Parnell WJ;Assier RC;Abrahams ID
在帕内尔和亚伯拉罕(2008年)R.Soc.A 464,1461-1482。(DOI：10.1098/rspa.2007.0254)，发展了一种均匀化方案，给出了具有非圆形截面的周期性单向纤维增强介质的有效反平面剪切模数的显式形式。显式表示为体积分数中的有理函数。在该方案中，采用(非稀疏)近似来确定前导阶表达式。除了体积分数很高外，与现有方法的一致性很好。在这里，为了确定展开式中的高阶项，对理论进行了扩展。对于非圆形截面的纤维，可以不借助数值方法得到有效性质的显式表达式。出现在表达式中的术语被识别为与周期性纤维分布的晶格几何、纤维横截面形状和主体/纤维材料属性相关联。结果是在反平面弹性的背景下得到的，但与势问题的类比说明了该方法对热、静电和静磁问题的广泛适用性。通过与已有的渐近均匀化方法的比较，说明了该方法的有效性。对于一般截面的纤维，相应的胞元问题必须用某种计算方案来求解。
In Parnell & Abrahams (2008 Proc. R. Soc. A 464, 1461–1482. (doi:10.1098/rspa.2007.0254)), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.