三维复合材料中强场磁输运的边界
Bounds on strong field magneto-transport in three-dimensional composites
摘要：
本文讨论了强磁场作用下三维复合材料的有效非对称电导率所满足的界。一方面，它表明，对于一般的复合材料的反对称部分的有效电导率不能仅限于反对称部分的局部电导率，相反的柱状的情况下。因此，一个合适的秩二层压板，其导电性具有一个有界反对称部分与一个高对比度的对称部分，可以产生一个任意大的反对称部分的有效导电性。另一方面，提供的界限表明，反对称部分的有效电导率必须为零，如果反对称部分的局部电导率的上界为零，和对称部分的局部电导率保持有界上下。假设局部电导率和有效电导率在与磁场正交的平面内具有横观各向同性，导出了有效模量的初等界。在微结构几何各向同性的条件下，给出了具有非对称电导率的两相三维复合材料的新的Hashin-Shtrikman型边界。边界的推导是基于一个特殊的变分原理对称化的问题，并使用Y-张量涉及在每个阶段的平均值的字段。
英文摘要：
This paper deals with bounds satisfied by the effective non-symmetric conductivity of three-dimensional composites in the presence of a strong magnetic field. On the one hand, it is shown that for general composites the antisymmetric part of the effective conductivity cannot be bounded solely in terms of the antisymmetric part of the local conductivity, contrary to the columnar case. So, a suitable rank-two laminate the conductivity of which has a bounded antisymmetric part together with a high-contrast symmetric part, may generate an arbitrarily large antisymmetric part of the effective conductivity. On the other hand, bounds are provided which show that the antisymmetric part of the effective conductivity must go to zero if the upper bound on the antisymmetric part of the local conductivity goes to zero, and the symmetric part of the local conductivity remains bounded below and above. Elementary bounds on the effective moduli are derived assuming the local conductivity and effective conductivity have transverse isotropy in the plane orthogonal to the magnetic field. New Hashin-Shtrikman type bounds for two-phase three-dimensional composites with a non-symmetric conductivity are provided under geometric isotropy of the microstructure. The derivation of the bounds is based on a particular variational principle symmetrizing the problem, and the use of Y-tensors involving the averages of the fields in each phase.
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