Multicomponent composites, electrical networks and new types of continued fraction I
Multicomponent composites, electrical networks and new types of continued fraction I
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多组分复合材料、电网和新型连续分数 I
DOI：
10.1007/bf01217763
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发表时间：
1987
影响因子：
2.4
通讯作者：
G. Milton
中科院分区：
文献类型：
--
作者：
G. Milton
复杂有效电导率张量σ*（将多组分复合材料中的平均电流与平均电场联系起来）的界限的发展由于缺乏合适的σ* 连分数表示而受到阻碍。本文提出了一种新的场方程递推方法，它给出了σ* 作为一种新形式的连分数的表达式，该形式将组分电导率和一组反映复合材料几何结构的基本几何参数作为系数。建立了一个场方程族，使得（j+1）阶方程的解生成j阶方程的解。因此，与j阶场方程相关的有效张量Ω（j）可表示为Ω（j+1）的分数线性矩阵变换。这些变换联合收割机形成了σ*=Ω（0）的连分式展开式，在下面的论文第二部分中利用它来获得界：Ω（j）的粗略界，对于j ∈ 1，给出了σ* 的窄界。连分数是一个推广到多元函数的连分数扩展单变量Stieltjes功能，证明了重要的发展理论的帕德逼近，渐近分析，和理论的正交多项式在上世纪。结果扩展到其他运输问题，包括在多晶介质中的传导，复合材料的粘弹性，和多组分，多端线性电网络的响应。
The development of bounds on the complex effective conductivity tensor σ* (that relates the average current to the average electric field in a multicomponent composite) has been hindered by lack of a suitable continued-fraction representation for σ*. Here a new field equation recursion method is developed which gives an expression for σ* as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry. A hierarchy of field equations is set up such that the solutions of the (j+1)th-order equation generate the solutions of thejth-order equation. Consequently the effective tensor Ω(j) associated with thejth-order field equation is expressible as a fractional linear matrix transformation of Ω(j+1). These transformations combine to form the continued fraction expansion for σ*=Ω(0) which is exploited in the following paper, Part II, to obtain bounds: crude bounds on Ω(j), forj≧1, give narrow bounds on σ*. The continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable Stieltjes functions that proved important in the development of the theory of Páde approximants, asymptotic analysis, and the theory of orthogonal polynomials in the last century. The results extend to other transport problems, including conduction in polycrystalline media, the viscoelasticity of composites, and the response of multicomponent, multiterminal linear electrical networks.