强Eshelby猜想与弹性矩张量极值结构的研究进展

An extension of the Eshelby conjecture to domains of general shape in anti-plane elasticity

According to the Eshelby conjecture, an ellipse or ellipsoid is the only shape that induces an interior uniform strain under a uniform far-field loading. We extend the Eshelby conjecture to domains of general shape for anti-plane elasticity. Specifically, we show that for each positive integer $N$, an inclusion induces an interior uniform strain under a polynomial loading of degree $N$ if and only if the exterior conformal map of the inclusion is a Laurent series of degree $N$. Furthermore, for the isotropic case, we characterize the shape of an inclusion by only using the first-degree polynom…

基于Eshelby与Lippmann-Schwinger积分公式的各向同性介质弹性反问题研究

On the inverse elastic problem for isotropic media using Eshelby and Lippmann-Schwinger integral formulations

本研究针对各向同性介质弹性反问题，分别基于Eshelby方程和Lippmann-Schwinger积分方程，结合数值近似与迭代方法，实现静态与动态场景下非均匀夹杂弹性参数的重构，并分析了对应平面应变问题。

Counter-examples to the high-order version and strong version of the generalized Eshelby conjecture for anisotropic media

In this work, we prove that in anisotropic media possessing cubic, transversely isotropic, orthotropic, and monoclinic symmetries, there exist non-ellipsoidal inclusions that can transform particular quadratic eigenstrains into quadratic elastic strain fields in them. Further, we prove that in these anisotropic media, there exist non-ellipsoidal inclusions that can transform particular polynomial eigenstrains of even degrees into polynomial elastic strain fields of the same even degrees in them. A sufficient condition for the existence of those counter-examples is provided. These results const…

三维各向同性介质 Eshelby 猜想强版本的一个受约束证明

A constrained proof of the strong version of the Eshelby conjecture for the three-dimensional isotropic medium

针对三维各向同性介质中本征应力两特征值相等时的 Eshelby 强猜想，本文引入材料参数约束并结合必要条件，证明仅椭球可同时满足均匀性。

Polya-Szego和Eshelby的解

Solutions to the conjectures of Polya-Szego and Eshelby

Eshelby表明，如果夹杂物是椭圆形或椭球形，那么对于任何均匀的弹性载荷，夹杂物内部的场是均匀的。他接着又说，反之亦然，即，如果夹杂内部的场对所有均匀载荷都是均匀的，则夹杂是椭圆形或椭球形的。我们称之为弱Eshelby猜想。在本文中，我们证明了这个猜想在三维空间。在二维空间中，证明了一个更强的猜想，我们称之为强Eshelby猜想：如果包含体内部的场在单一均匀载荷下是均匀的，则该包含体是椭圆形的。我们给出了另一种证明Eshelby猜想在二维使用速端变换。作为弱Eshelby猜想的结果，我们在二维和三维空间中证明了Polya和Szego关于极化张量的等周不等式的一个猜想. Polya-Szego猜想认为，其电极化张量具有最小迹线的包裹体具有圆盘或球的形状。

关于正交各向异性极值弹性张量与极值多项式的关系

On the relation between extremal elasticity tensors with orthotropic symmetry and extremal polynomials

证明了具有正交各向异性对称性的弹性张量是极值的，如果它的声学张量的行列式是一个极值多项式，且不是完满平方。