7.11 Consistency of Calculations
193
of cos(φ) in (2) of the
σ = 2.205E6
Ti:
Crystallites (grains)
σ determined by
random distribution
text, using a uniform
distribution function.
a
:
Coil
Fig. 7.22 Illustrating the model setup for VIC-3D®. There is a second layer identical to the one
shown immediately below the one shown. This is due to the fact that VIC-3D® requires a minimum
of two cells in any direction of the grid
where σ max is the maximum conductivity in the region of interest (host plus
anomaly), σ min is the minimum conductivity, and 0 ≤ V F ≤ 1. As Fig. 7.22
indicates, σ max = σ host = 2.205 × 10 6 in this setup.
Returning to (7.61), we can easily determine a relationship for volume-fractions:
σ = σ max sin
2 φ + σ min cos
2 φ
= σ max (1 − cos
2 φ) + σ min cos
2 φ
= σ max + cos
2 φ(σ min − σ max ) ,
(7.63)
from which we conclude that V F = cos 2 φ.
More Physics Let σ be the conductivity tensor of a material in an arbitrary
coordinate system, so that it will have (in general) nonzero off-diagonal entries.
In this coordinate system we still have J = σ · E, where J is the electric current
density, and E is the electric field. The electric power density dissipated within the
material is P = E · J = E · σ · E, which defines an ellipsoid in E-space.
By definition, the conductivity in the direction of the electric field, E, is given by
σ E =
P
E · E
=
P
|E| 2 ,
(7.64)
193
of cos(φ) in (2) of the
σ = 2.205E6
Ti:
Crystallites (grains)
σ determined by
random distribution
text, using a uniform
distribution function.
a
:
Coil
Fig. 7.22 Illustrating the model setup for VIC-3D®. There is a second layer identical to the one
shown immediately below the one shown. This is due to the fact that VIC-3D® requires a minimum
of two cells in any direction of the grid
where σ max is the maximum conductivity in the region of interest (host plus
anomaly), σ min is the minimum conductivity, and 0 ≤ V F ≤ 1. As Fig. 7.22
indicates, σ max = σ host = 2.205 × 10 6 in this setup.
Returning to (7.61), we can easily determine a relationship for volume-fractions:
σ = σ max sin
2 φ + σ min cos
2 φ
= σ max (1 − cos
2 φ) + σ min cos
2 φ
= σ max + cos
2 φ(σ min − σ max ) ,
(7.63)
from which we conclude that V F = cos 2 φ.
More Physics Let σ be the conductivity tensor of a material in an arbitrary
coordinate system, so that it will have (in general) nonzero off-diagonal entries.
In this coordinate system we still have J = σ · E, where J is the electric current
density, and E is the electric field. The electric power density dissipated within the
material is P = E · J = E · σ · E, which defines an ellipsoid in E-space.
By definition, the conductivity in the direction of the electric field, E, is given by
σ E =
P
E · E
=
P
|E| 2 ,
(7.64)