Tensor permeability, conductance

The components of a symmetrical second-rank tensor, referred to its principal axes, transform like the three coefficients of the general equation of a second-degree surface (a quadric) referred to its principal axes (Nye, 1957). Hence, if all three of the quadric s coefficients are positive, an ellipsoid becomes the geometrical representation of a symmetrical second-rank tensor property (e.g., electrical and thermal conductivity, permittivity, permeability, dielectric and magnetic susceptibility). The ellipsoid has inherent symmetry mmm. The relevant features are that (1) it is centrosymmetric, (2) it has three mirror planes perpendicular to the... 

We can introduce now the transformed conductivity and magnetic permeability tensors as... 

In the last formulae we assume that (Txx,( yy,crzz and Myyi Mzz are the principal values of the conductivity and permeability tensors, and the axes of the grid are oriented along the principal axes of these tensors. 

E. However, developing this approach for practically important 3D anisotropic models with arbitrary tensors of electrical conductivity, magnetic and dielectric permeability happens to be very complicated. Yee s algorithm is based on calculation of different electric field components at different space points, but the electrical conductivity tensor relates these components taken at the same point. 

A three-dimensional time-dependent model has been developed by Ma et al. (1998) to solve for the electric field, charge density, and Maxwell current. The last quantity is influenced primarily by the spatially anisotropic tensor conductivity in the ionosphere. Maxwell s equations for the transient fields propagating into the ionosphere are reduced to two equations. If p is the charge density (C/m ), q the dielectric constant (C /N m ), the permeability (N/A ), 0 the conductivity tensor (S/m), and E the electric field (V/m), then the wave equation in SI imits becomes... 

A finite element simulator derived from the Equation (1) and (3) is used. In the equations the parameters of disposal panel such as permeability tensor and thermal conductivity tensor have to be decided. To specify the parameters, the homogenization theory (Ene Polisevski (1987)), Kyoya Terada(2001)) is applied. 

We must not be misled by the examples of MgO, BaTi03 or diamond most ceramics do not crystallize in a cubic group arrd this implies that the many physical properties that are described by a second order terrsor are not isotropic [NYE 87]. Thermal expansion, optical index, electric and thermal conductivities, permittivity and permeability are at first view anisotropic, which can misinform some metallmgists, because the most cormnon metals (irort, alttminum, copper) are cubic. An effect of the anisotropy of thermal expansion is to create residual stress at the grain boundaries of the polycrystals. Beyond the properties described by a second order tensor, low synunetries combine with the properties of iono-covalent bonds to make dislocations rare and relatively immobile, which explains the lack of ductility and the impossibility of plastic deformation. 

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