Modelling the effective conductivity function of an arbitrary two–dimensional polycrystal using sequential laminates
Modelling the effective conductivity function of an arbitrary two–dimensional polycrystal using sequential laminates
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使用顺序层压板对任意二维多晶的有效电导率函数进行建模
DOI：
10.1017/s030821050002864x
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
G. Milton
中科院分区：
文献类型：
--
作者：
K. E. Clark;G. Milton
二维多晶材料的有效电导率张量σ* 取决于构成多晶的纯晶体的电导率张量σ0和多晶中晶粒的几何构型，由旋转场R（x）表示，给出每个点x处晶体的取向。在此，我们确定，在R（x）保持不变的情况下，任何多晶中σ* 对σ0的依赖关系，都可以用一个由连续叠层构成的多晶来精确地模拟。本文首先指出，如果将多晶中场的希尔伯特空间截断为有限维空间，则有效电导率函数只受到轻微的扰动。然后，这个有限维空间的字段的结构被证明是同构的顺序层压板相关联的有限维空间的字段的结构。特别地，存在对应于剥离顺序层压件中的层并连续减小场空间的尺寸的操作。
The effective conductivity tensor σ* of a two-dimensional polycrystalline material depends on the conductivity tensor σ0 of the pure crystal from which the polycrystal is constructed and on the geometrical configuration of grains in the polycrystal, represented by a rotation field R(x) giving the orientation of the crystal at each point x. Here it is established that the dependence of σ* on σ0 in any polycrystal, with R (x) held fixed, can be mimicked exactly by a polycrystal constructed by sequential lamination. It is first shown that the effective conductivity function is perturbed only slightly if we truncate the Hilbert space of fields in the polycrystal to a finitedimensional space. Then the structure of this finite-dimensional space of fields is shown to be isomorphic to the structure of the finite-dimensional space of fields associated with the sequential laminate. In particular, there is an operation which corresponds to peeling away the layers in the sequential laminate and successively reducing the dimension of the space of fields.