论文标题
三维各向同性培养基的Eshelby猜想的强大版本的限制性证明
A constrained proof of the strong version of the Eshelby conjecture for the three-dimensional isotropic medium
论文作者
论文摘要
Eshelby在椭圆形包容问题上的开创性工作导致猜想是，椭圆形是具有均匀特性的唯一包含，即将均匀的特征变成均匀的特性。对于三维各向同性培养基，已经证实了Eshelby猜想的弱版本。 Ammari等人的先前工作。在特征者的三个特征值不同或相同的情况下，证明了Eshelby猜想的强烈猜想，而在特征者的两个特征值相同而另一个则是独特的情况下，这是一个困难的问题。在这项工作中，我们研究了后一种情况。为此，首先，我们介绍并证明了凸包含能够将单一均匀特征性的弹性应力转变为均匀的弹性应力场的必要条件。由于必要的条件不足以确定包含的形状，其次，我们引入了与材料参数有关的约束，并证明存在弹性张量和均匀特征性的组合，因此只有椭圆形能够具有同一组合的Eshelby均匀性。最后，我们通过证明，对于由一组指定统一的特征符号所引起的椭圆形限制的均匀应变场，我们提供了一个更具体的限制证明，Eshelby猜想的强大版本对于一组与指定均匀的均匀的Egensortes相关的同型弹性张量是正确的。这项工作为三维各向同性培养基的有趣且长期以来的Eshelby猜想的完整解决方案做出了一些进展。
Eshelby's seminal work on the ellipsoidal inclusion problem leads to the conjecture that the ellipsoid is the only inclusion possessing the uniformity property that a uniform eigenstrain is transformed into a uniform elastic strain. For the three-dimensional isotropic medium, the weak version of the Eshelby conjecture has been substantiated. The previous work of Ammari et al. substantiates the strong version of the Eshelby conjecture for the cases when the three eigenvalues of the eigenstress are distinct or all the same, whereas the case where two of the eigenvalues of the eigenstress are identical and the other one is distinct remains a difficult problem. In this work, we study the latter case. To this end, firstly, we present and prove a necessary condition for a convex inclusion being capable of transforming a single uniform eigenstress into a uniform elastic stress field. Since the necessary condition is not enough to determine the shape of the inclusion, secondly, we introduce a constraint that is concerned with the material parameters, and prove that there exist combinations of the elastic tensors and uniform eigenstresses such that only an ellipsoid can have the Eshelby uniformity property for these combinations simultaneously. Finally, we provide a more specifically constrained proof of the conjecture by proving that for the uniform strain fields constrained to that induced by an ellipsoid from a set of specified uniform eigenstresses, the strong version of the Eshelby conjecture is true for a set of isotropic elastic tensors which are associated with the specified uniform eigenstresses. This work makes some progress towards the complete solution of the intriguing and longstanding Eshelby conjecture for three-dimensional isotropic media.
