The determination of the elastic field of an ellipsoidal inclusion in an anisotropic medium
The determination of the elastic field of an ellipsoidal inclusion in an anisotropic medium
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各向异性介质中椭球体弹性场的测定
DOI：
10.1017/s0305004100053366
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发表时间：
1977
影响因子：
0.8
通讯作者：
L. Walpole
中科院分区：
文献类型：
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作者：
L. Walpole
1. 介绍。在研究非齐次系统的弹性行为时，某些包涵性和非齐次性问题是基本问题。在“变换问题”中，如果没有周围矩阵的约束，无界均匀各向异性弹性介质的一个区域（“包含”）将经历一些规定的无限小的均匀应变（由于其形状的一些自发变化）。当夹杂体为椭球形时，Eshelby（3,4）能够证明约束夹杂体内的应力场和应变场是均匀的，当介质为各向同性时可以完成计算。一般各向异性介质似乎引起了令人生畏的分析，但Eshelby(3)确实指出了对均匀应变的评估方法，几位作者（后来提到）将其发展成适合数值计算的表达式。在这里，我们提供了一个基本的和直接的途径来表示均匀应变，迄今为止，它只能通过傅里叶变换的迂回过程来实现。一旦感知到包体中应变的均匀状态，并在开始另一种评估之前，就可以使用该方法。首先，我们求助于一个定理（似乎不是以前所知道的），该定理（特别是）揭示了位于包涵外的无限小薄椭球同质面中的平均应变的消失。其次，我们只需要反映，在界面的每一点都有一个直接的代数表达式，即包含物外的应变是内部均匀应变的表达式。
1. Introduction. In studying the elastic behaviour of inhomogeneous systems certain inclusion and inhomogeneity problems are fundamental. In the ‘transformation problem’, a region (the ‘inclusion’) of an unbounded homogeneous anisotropic elastic medium would undergo some prescribed infinitesimal uniform strain (because of some spontaneous change in its shape) if it were not for the constraint imposed by the surrounding matrix. When the inclusion has an ellipsoidal shape, Eshelby (3, 4) was able to show that the stress and strain fields within the constrained inclusion are uniform and that calculations could be completed when the medium was isotropic. A generally anisotropic medium seemed to raise forbidding analyses, but Eshelby (3) did point the way to an evaluation of the uniform strain which several authors (referred to later) developed into an expression amenable to numerical computation. Here we offer an elementary and immediate route to this expression of the uniform strain, which has been accessible hitherto only by the circuitous procedures of Fourier transforms. It is available as soon as the uniform state of strain in the inclusion is perceived and before an alternative evaluation is commenced. First, we appeal to a theorem (not it seems previously known) which reveals (in particular) the vanishing of the mean strain in the infinitesimally thin ellipsoidal homoeoid lying just outside the inclusion. Secondly, we need only reflect that at each point of the interface there is an immediate algebraic expression of the strain just outside the inclusion in terms of the uniform strain just inside.