多相材料有效性质的理论研究
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摘要
多相介质有极为广泛的应用，但是关于它的有效性质的已有各种方法都存
在不同程度的问题，有的甚至在概念上含糊不清。本文着重对多相材料的有效
性质做了理论研究。并与文献中和郑泉水老师的研究小组所做的大量数值结果
做了对比。具体完成的工作可划分为两个方面：
1．提出和发展了如下新方法：
IDD法和等效自洽法：提出并发展了具有简单表示、计及夹杂形状和空间
分布、适合于多相和各向异性材料的具有体积分数二阶精度的新的细观力
学方法－相互作用直推法（IDD）。大量数值模拟和试验结果表明，IDD确
实具有很好的精度，可望用之替换普遍采用的Mori－Tanaka方法。同时IDD
可以普遍适用于各种有效线性物理特征的估计。本文具体针对弹性、电导
和热弹性给出了IDD估计。
等效加载法：提出了等效加载法，结合IDD估计给出的等效载荷，使得可
以用简单的半解析的方法，达到分析缺陷间的空间分布相互作用并获得较
精确的局部场的目的。该方法是宏细观关联的范例，做为一个应用，该方
法简便地给出了等效强度。
复合相互作用模型：针对具有多重拓扑结构的细观准周期复合材料，在
IDD估计的基础上，提出了复合相互作用的概念，并具体应用于分析核工
业材料Zr－2．5Nb。复合相互作用概念对于分析一大类高等材料提供了新的
处理思路。克服了传统细观力学方法只能处理简单空间分布的局限。
2．指出并修正了现有方法中普遍忽视的几个关键缺陷问题：
证明了Eshelby张量的若干性质，在此基础上分析了以伪面力法为代表的
一类叠加法应用于分析有限大和无限大复合材料时所存在的问题，并分别
提出了简单修正方法；
证明了广义自洽法与无关性定理的一致性。提出了利用无关性的内禀概
念，不仅可以保证数值模拟结果的无关性，还可以用来化简形式；
讨论了Cauchy－Voigt争论在平面问题中的结论，证明了对于特定的平面压
电问题，对应的Cauchy－Voigt争论中Cauchy的观点是正确的，由此也发现
了CLM不变性平移的内在本质。
There have been a lot of studies on the effective properties of multiphase materials widely used in industry. But various existing methods have their own drawbacks, some even ambiguous in concepts. This dissertation focuses on the theoretical study of some problems associated with the effective properties of multiphase materials, and compared the results with the enormous numerical works accomplished by the research group of Prof. Zheng. The specific works finished are listed below:
1.Developed the following new methods
IDD and Effective self-consistent method: Starting from the topological structure and spatial distribution, a new micromechanical method-interaction direct derivation (IDD) was developed, which accounts for inclusion shapes and spatial distribution, suitable for multiphase and anisotropic materials, while having the precision up to the second order of volume fraction. Comparison with various numerical and experimental results showed that IDD indeed has excellent precision and it can expected to replace the widely used Mori-Tanaka method. At the same time, IDD is applicable for the estimation of various effective linear physical properties. This dissertation gives the IDD estimation for elasticity, electric conductivity and thermoelasticity specifically.
Equivalent loading method: In combination with IDD, this method can use simple semi-analytic method to analyze the spatial distribution induced interactions among defects and to give more precise local fields. This method is an paradigm of micro-macro correlation, as an application, it gives effective strength quite easily.
Double interaction model: Based on IDD estimate, the concept of double interaction was proposed for micro-quasi-periodic composites with multi-level topological structure. As an example, this concept is used to analyze Zr-2.5Nb used in nuclear industry. This concept also provides a new approach for analyzing a wide class of advanced materials, in contrast with traditional micro -mechanical models which can only deals with simple spatial distributions.
2.Pointed out and revised the several essential drawbacks of existing method:
A series of properties of Eshelby tensor were proved. Based on these properties, the problems associated with the kind of method similar to pseudo traction
method for both finite and infinite composites were analyzed, simple revisions were proposed correspondingly.
The consistency between the generalized self-consistent method and the independence theorem was confirmed. The concept of intrinsic was proposed, which utilizes the independence properties, this concept can both guarantee the independence of the numerical simulation results and simplify the results.
The plane version of Cauchy-Voigt dispute was discussed. It is proved that for certain plane piezoelectricity problem, Cauchy's view is correct. As a byproduct, the essence of the CLM invariant transformation was discovered.
在不同程度的问题，有的甚至在概念上含糊不清。本文着重对多相材料的有效
性质做了理论研究。并与文献中和郑泉水老师的研究小组所做的大量数值结果
做了对比。具体完成的工作可划分为两个方面：
1．提出和发展了如下新方法：
IDD法和等效自洽法：提出并发展了具有简单表示、计及夹杂形状和空间
分布、适合于多相和各向异性材料的具有体积分数二阶精度的新的细观力
学方法－相互作用直推法（IDD）。大量数值模拟和试验结果表明，IDD确
实具有很好的精度，可望用之替换普遍采用的Mori－Tanaka方法。同时IDD
可以普遍适用于各种有效线性物理特征的估计。本文具体针对弹性、电导
和热弹性给出了IDD估计。
等效加载法：提出了等效加载法，结合IDD估计给出的等效载荷，使得可
以用简单的半解析的方法，达到分析缺陷间的空间分布相互作用并获得较
精确的局部场的目的。该方法是宏细观关联的范例，做为一个应用，该方
法简便地给出了等效强度。
复合相互作用模型：针对具有多重拓扑结构的细观准周期复合材料，在
IDD估计的基础上，提出了复合相互作用的概念，并具体应用于分析核工
业材料Zr－2．5Nb。复合相互作用概念对于分析一大类高等材料提供了新的
处理思路。克服了传统细观力学方法只能处理简单空间分布的局限。
2．指出并修正了现有方法中普遍忽视的几个关键缺陷问题：
证明了Eshelby张量的若干性质，在此基础上分析了以伪面力法为代表的
一类叠加法应用于分析有限大和无限大复合材料时所存在的问题，并分别
提出了简单修正方法；
证明了广义自洽法与无关性定理的一致性。提出了利用无关性的内禀概
念，不仅可以保证数值模拟结果的无关性，还可以用来化简形式；
讨论了Cauchy－Voigt争论在平面问题中的结论，证明了对于特定的平面压
电问题，对应的Cauchy－Voigt争论中Cauchy的观点是正确的，由此也发现
了CLM不变性平移的内在本质。
There have been a lot of studies on the effective properties of multiphase materials widely used in industry. But various existing methods have their own drawbacks, some even ambiguous in concepts. This dissertation focuses on the theoretical study of some problems associated with the effective properties of multiphase materials, and compared the results with the enormous numerical works accomplished by the research group of Prof. Zheng. The specific works finished are listed below:
1.Developed the following new methods
IDD and Effective self-consistent method: Starting from the topological structure and spatial distribution, a new micromechanical method-interaction direct derivation (IDD) was developed, which accounts for inclusion shapes and spatial distribution, suitable for multiphase and anisotropic materials, while having the precision up to the second order of volume fraction. Comparison with various numerical and experimental results showed that IDD indeed has excellent precision and it can expected to replace the widely used Mori-Tanaka method. At the same time, IDD is applicable for the estimation of various effective linear physical properties. This dissertation gives the IDD estimation for elasticity, electric conductivity and thermoelasticity specifically.
Equivalent loading method: In combination with IDD, this method can use simple semi-analytic method to analyze the spatial distribution induced interactions among defects and to give more precise local fields. This method is an paradigm of micro-macro correlation, as an application, it gives effective strength quite easily.
Double interaction model: Based on IDD estimate, the concept of double interaction was proposed for micro-quasi-periodic composites with multi-level topological structure. As an example, this concept is used to analyze Zr-2.5Nb used in nuclear industry. This concept also provides a new approach for analyzing a wide class of advanced materials, in contrast with traditional micro -mechanical models which can only deals with simple spatial distributions.
2.Pointed out and revised the several essential drawbacks of existing method:
A series of properties of Eshelby tensor were proved. Based on these properties, the problems associated with the kind of method similar to pseudo traction
method for both finite and infinite composites were analyzed, simple revisions were proposed correspondingly.
The consistency between the generalized self-consistent method and the independence theorem was confirmed. The concept of intrinsic was proposed, which utilizes the independence properties, this concept can both guarantee the independence of the numerical simulation results and simplify the results.
The plane version of Cauchy-Voigt dispute was discussed. It is proved that for certain plane piezoelectricity problem, Cauchy's view is correct. As a byproduct, the essence of the CLM invariant transformation was discovered.
引文
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2． Woo C H. in: Proceeding on the international conference on materials for nuclear reactor core applications. London: British Nuclear Energy Society, 1987． 65．
3． Tome C N, So C B, Woo C H. Philosophical Magazine A.1994, 67(4) : 917-930
4． Oleinik O.A, Shamaev A S, Yosifian, G A. Mathematical Problems in Elasticity and Homogenization. North-Holland, New York. 1992．
5． Huang Y, Hwang K C, Hu K X, Chandra, A. A unified energy approach to a class of micromechanics models for composite materials Acta Mechanica Sinica, 1995, 11(1) : 59-75．
6． Nemat-Nasser S, Hori M. Micromechanics: Overall properties of heterogeneous elastic solids, Lecture Notes. initiated at UCSD, 1993．
7． Budiansky Y. On the elastic moduli of some heterogeneous material J. Mech. Phys. Solids, 1965, 13: 223-7．
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