夹杂内外场奇妙特性研究

This project is concerned with the Eshelby's conjecture associated with the internal field of inclusions and neutral and harmonic inclusions associated with the exterior field of inclusions, with the minimization of stress concentrations as the design goal of composites.. First we discuss the uniformity of stresses inside an inclusion of arbitrary shape with multiple coatings (or interphase layers). For any number of the coatings, we will identify any possible non-elliptical shape of the inclusion permitting an internal uniform stress field. We will rigorously prove that the shape of the inclusin with internal uniform field can indeed be "arbitrary" provided that the number of the intermediate coatings is sufficiently large. We will address the problem first within the framework of anti-plane elasticity (or equivalently two-dimensional conductivity), then we will solve the problem in the more challenging context of plane elasticity.. Second we focus on the design of neutral and harmonic inclusions. The importance of the concept of a neutral inclusion lies not only in the design of composite structures for specific purposes but perhaps more importantly in its relevance to the design of cloaking structures.We will realize the neutrality of the inclusions through the introduction of interphase layers with finite thickness and/or imperfect interface with vanishing thickness. We will consider both the elastic anisotropy and nonlinear elasticity of the fibrous composite. If under some circumstances the neutrality of the inclusions cannot be realized, we will endeavor to accomplish the harmonization of the inclusions. . Last we will also address some new challenging problems in anisotropic elasticity.

以应力集中最小化为复合材料优化设计目的，本项目着重关注于与夹杂内部场相关的Eshelby猜想和与夹杂外部场相关的中性夹杂和调和夹杂的设计。.我们将首先探讨在夹杂和基体间存在多个涂层时任意形状夹杂内部存在均匀应力场的可能性。对于任意层数，我们将辨识可导致夹杂内部均匀场的各种可能的非椭圆夹杂形状。我们将详尽论证只要存在足够多中间界面层，允许内部存在均匀场的夹杂形状确实可以"任意"。首先在较简单的反平面弹性框架内探讨该问题，其次将在更具挑战性的平面弹性框架内探讨该问题。.接着我们将探讨中性夹杂和调和夹杂的设计，而中性夹杂的设计理念可用于隐身结构设计。我们将通过合理设计有限厚度的界面层和/或零厚度非理想界面从而实现夹杂的中性化。在讨论中将考虑以前研究中所未涉及的各向异性弹性以及非线弹性。如果在一些条件下无法实现夹杂的中性化，我们将努力实现夹杂调和化。.最后我们探讨各向异性弹性力学中的挑战性问题。

我们对具有内部均匀场的夹杂、中性夹杂和调和夹杂展开深入研究。发现当存在中间界面层时，非椭圆压电夹杂内部的电弹性场可为有条件或无条件均匀；探讨了三相任意形状夹杂中由于均匀温度变化引起的热应力场；采用保角映射构造了在均匀温度变化下具有内部均匀静水应力的两个相互作用弹性夹杂的非规则形状；研究了在远场作用线性分布的面内应力场时三相椭圆夹杂内部的应力场；在Kirchhoff各向同性及层合板理论框架内研究了与周围基体通过界面层粘结的三相椭圆夹杂中的内部合力场，发现两个条件从而保证椭圆夹杂内部的合力场均匀且静水；严格证明了具有非规则形状的两个弹性夹杂与螺位错相互作用时夹杂内部的应力场仍保持均匀。设计出三维传导中的中性多层球体夹杂；研究了具有内部均匀静水应力场的三相非椭圆调和夹杂；采用保角映射设计出在远场非均匀载荷下两个相互作用调和弹性夹杂；探讨了由可压缩超弹性调和材料所形成的三相圆柱夹杂中调和夹杂设计。获得了Kirchhoff各向异性板界面裂纹裂尖奇异场的简单数学结构。推导出针对Kirchhoff各向异性压电薄板耦合拉伸-弯曲-极化的十二维Stroh公式。推导了各向异性线弹性板在弯曲变形下三个实矩阵H、L和S的显式表达。研究了Kirchhoff各向同性及各向异性层合板中的任意形状Eshelby夹杂问题。探讨了各向同性层合薄板中调和形状设计。发展了一系列多相Kirchhoff各向同性层合板中的Green函数。获得了向同性层合板中界面圆弧裂纹的解析解。对于具有表面弹性的夹杂和裂纹问题展开系统研究。研究了在反平面剪切状态下具有任意变化表面弹性的两个相互作用的圆柱夹杂问题；求解了一螺型位错与具有界面应力的各向异性弹性椭圆夹杂相互作用问题；求解一螺型位错与任意形状纳米弹性夹杂相互作用问题；对具有混合界面的纳米夹杂进行了严格力学分析。研究了表面弹性对反平面剪切变形下各向同性线弹性材料中弧形裂纹的影响；研究了螺型和刃型位错与具有表面弹性的有限裂纹的相互作用；探讨了各向异性表面弹性对各向异性双材料界面裂纹的影响；研究了带具有表面弹性裂纹的圆柱体扭转问题。研究了压电螺型位错与具有表面压电效应的双材料界面相互作用；研究了螺型位错与具有表面应变梯度弹性效应的双材料界面相互作用；推导了具有混合界面各向异性半空间和双材料中的格林函数；研究了各向异性表面弹性对各向异性材料中二维接触问题的贡献。