力学中高阶张量结构、张量函数表示及其应用的研究

项目名称： 力学中高阶张量结构、张量函数表示及其应用的研究

项目编号： No.10872086

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 轻工业、手工业

项目作者： 邹文楠

作者单位： 南昌大学

项目金额： 29万元

中文摘要： 本项目的成果包括理论和应用两个方面：（1）理论上，首先通过高阶张量的不可约分解，将高阶张量函数表示问题转化为高阶偏张量函数表示问题，再利用二元形式表示偏张量，我们初步提出一种构造偏张量的偏张量值函数的完备不可约系统的可行方法，给出了单个、两个一般偏张量多项式张量值函数的构造，同时提出了结构张量完备系统的概念修正之前结构张量的不确定性、并明确给出了各种对称性下的结构张量的完备构元系统；（2）在应用方面，利用张量不可约分解的思想，求解了著名的Eshelby问题，给出了弹性、热传导任意非椭圆夹杂问题的解析解，比如对弹性问题可以归结为两个简单且没有奇异性的复变函数边界积分，对平均Eshelby张量也得到类似结果，在此基础上，我们透彻地讨论了Eshelby均匀性和椭圆近似适定性疑难，这些成果被JMPS主编Bhattacharya评价为"该问题的一个主要进展"并"有广泛的应用前景"，此外我们还开展了各向异性材料Eshelby问题的研究，利用Stroh公式，推导出了任意非椭圆夹杂问题的显式积分表达式，得到包括任意多边形、约当曲线和洛朗级数表示的任意形状夹杂的解析解。

中文关键词： 高阶张量函数；结构张量；Eshelby问题；非椭圆夹杂；解析解

英文摘要： The results of this research project include two contents: Theoretically, by decomposing a higher-order tensor (function) into a set of symmetric traceless deviators, we represent every deviator with an algebraic binary form, and so propose a feasible approach to obtain the complete deviator-valued function system of one or more deviators. The constructions of polynomial functions for single and two deviators are given. In the meantime, in order to avoid the uncertainty of previous structure tensors charactering the anisotropy of materials, we advance the concept of complete element system of structure tensors, and explicitly give such systems for all kinds of material symmetries. In application aspects, in virtue of the idea of irreducible decomposition of Eshelby's fourth-order tensor, we have solved the famous Eshelby's inclusion problem for infinite elastic and/or thermal materials with arbitrary non-elliptical inclusion. For instance, the elastic Eshelby's tensor fields can be ascribed to two simple complex functions expressend by boundary integrals without conventional difficulty of singularity, similar to the average Eshelby's tensor. Basing on these analytical and compact results, we thoroughly discuss the uniformity of Eshelby's tensor fields and the applicability of ellipse approximate, and reach some comprehensive conclusions which was reviewed by the JMPS editor, Prof. Bhattacharya as "a major advance in this direction"and having "a wide range of applications". In addition, we also carry out the anisotropic Eshelby's problem. Utilizing the Stroh formulism, we obtain compact integral expressions of eigenfunctions, and derive explicit solutions for arbitrary shaped inclusions, including shapes of polygons, and those characterized by the Jordan curves and Laurent polynomials.

英文关键词： High-order tensor function; Structure tensor; Eshelby's problem; Non-elliptical inclusion; Analytical solution