Eshelby's problem of non-elliptical inclusions
Eshelby's problem of non-elliptical inclusions
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DOI：
10.1016/j.jmps.2009.11.008
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发表时间：
2010-03
影响因子：
5.3
通讯作者：
W. Zou;Q. He;Mojia Huang;Q. Zheng
中科院分区：
文献类型：
--
作者：
W. Zou;Q. He;Mojia Huang;Q. Zheng
埃谢尔比问题在于确定无限线弹性均匀介质的应变场，该应变场归因于介质的子域（称为包含体）上规定的均匀特征应变。椭圆体夹杂物的埃谢尔比解的显着特征是后者内部的应变张量场是均匀的。这种均匀性具有重要的意义，即在包含嵌入式椭球不均匀性并受到远程均匀载荷的无限线弹性均匀介质中确定应变场的基本问题的解决方案可以很容易地从施加适当的均匀本征应变的椭圆体包含物的埃谢尔比解决方案中推导出来。基于这一结果，大多数致力于估计非均匀材料有效特性的现有微观力学方案已应用于许多实际感兴趣的材料，其中非均匀性实际上是非椭圆体的。为了检验各种微观力学方案中非均匀性的椭球近似的有效性，我们首先推导了一个新的边界积分表达式，用于在二维各向同性弹性的背景下计算埃谢尔比张量场（ETF）。新边界积分表达式的简单紧凑的结构使我们能够获得各种非椭圆包裹体（包括任意多边形包裹体和有限洛朗级数特征的包裹体）的 ETF 及其平均值的显式表达式。根据这些新的分析结果，我们表明： (i) ETF 平均值的椭圆近似对于凸非椭圆包含体是有效的，但对于非凸非椭圆包含体则变得不可接受； (ii) 一般来说，非椭圆包裹体内部的埃谢尔比张量场非常不均匀，不能用其平均值代替； (iii) 用平均埃谢尔比张量代替各种微观力学方案中涉及的广义埃谢尔比张量来代替非椭圆不均匀性通常是不可接受的。
The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.