192
7 Integration of Functionals, PCM and Stochastic IntegralEquations
Appendix 1: The Numerical Model
In its principal-axis coordinate system, the electrical conductivity tensor for hexagonal crystals, such as pure titanium and its most common alloys, is given by
σ =
⎡
⎣
σ 11 0 0
0 σ 11 0
0 0 σ 33
⎤
⎦ ,
(7.60)
where σ 11 is the conductivity in the basal plane (plane of isotropy), and σ 33 is the
conductivity normal to the basal plane. Such a system is said to possess transverse
isotropy, and the crystal class is labeled 6 mm. This notation means that the crystal
contains a sixfold axis of rotational symmetry, as well as six mirror planes that
contain that axis.
For pure titanium, σ 11 = 2.205 × 10 6 S/m, and σ 33 = 2.083 × 10 6 S/m. If
φ denotes the angle between the electric field vector and the normal to the basal
plane, then the conductivity in the φ−direction can be represented by the ellipsoid
of revolution
σ (φ) = σ 33 cos
2 φ + σ 11 sin
2 φ .
(7.61)
We will use (7.61) in establishing our numerical model. See the next section for a
further discussion and generalization of (7.61).
Consider a half-space host of pure titanium, whose crystal axis is oriented in the
z−direction, normal to the surface of the half-space. Lying at the surface of this host
is a rough patch of randomly oriented crystallites of titanium, as shown in Fig. 7.22.
The use of a single conductivity value for the host would be rigorously correct if the
coil were the only current source in the problem. The electric field induced into the
host by this source lies within the basal plane, so only the basal-plane conductivity,
2.205 × 10 6 S/m, enters the picture. The anomalous currents within the random
crystallites, however, produce a scattered field within the host that is not confined to
the basal plane, so the use of a single conductivity is an approximation.
We can use any number of cells in the model in order to get a statistically
reasonable answer. The value of the conductivity to be assigned each cell (or
crystallite or grain) is determined by randomly choosing cos φ in (7.61). We use
a uniform distribution function for this purpose. The volume-fractions for the
cells, from which the conductivities are determined, are computed off-line, and
then imported into VIC-3D® in a routine manner. After simulating the impedance
response of the bad patch, we can run several small slot responses to gain insight
into how deleterious the grain noise is to detecting a crack in its presence.
The relationship between volume-fractions, V F , and conductivity of a cell is
given by
σ = σ max + V F (σ min − σ max ) ,
(7.62)
7 Integration of Functionals, PCM and Stochastic IntegralEquations
Appendix 1: The Numerical Model
In its principal-axis coordinate system, the electrical conductivity tensor for hexagonal crystals, such as pure titanium and its most common alloys, is given by
σ =
⎡
⎣
σ 11 0 0
0 σ 11 0
0 0 σ 33
⎤
⎦ ,
(7.60)
where σ 11 is the conductivity in the basal plane (plane of isotropy), and σ 33 is the
conductivity normal to the basal plane. Such a system is said to possess transverse
isotropy, and the crystal class is labeled 6 mm. This notation means that the crystal
contains a sixfold axis of rotational symmetry, as well as six mirror planes that
contain that axis.
For pure titanium, σ 11 = 2.205 × 10 6 S/m, and σ 33 = 2.083 × 10 6 S/m. If
φ denotes the angle between the electric field vector and the normal to the basal
plane, then the conductivity in the φ−direction can be represented by the ellipsoid
of revolution
σ (φ) = σ 33 cos
2 φ + σ 11 sin
2 φ .
(7.61)
We will use (7.61) in establishing our numerical model. See the next section for a
further discussion and generalization of (7.61).
Consider a half-space host of pure titanium, whose crystal axis is oriented in the
z−direction, normal to the surface of the half-space. Lying at the surface of this host
is a rough patch of randomly oriented crystallites of titanium, as shown in Fig. 7.22.
The use of a single conductivity value for the host would be rigorously correct if the
coil were the only current source in the problem. The electric field induced into the
host by this source lies within the basal plane, so only the basal-plane conductivity,
2.205 × 10 6 S/m, enters the picture. The anomalous currents within the random
crystallites, however, produce a scattered field within the host that is not confined to
the basal plane, so the use of a single conductivity is an approximation.
We can use any number of cells in the model in order to get a statistically
reasonable answer. The value of the conductivity to be assigned each cell (or
crystallite or grain) is determined by randomly choosing cos φ in (7.61). We use
a uniform distribution function for this purpose. The volume-fractions for the
cells, from which the conductivities are determined, are computed off-line, and
then imported into VIC-3D® in a routine manner. After simulating the impedance
response of the bad patch, we can run several small slot responses to gain insight
into how deleterious the grain noise is to detecting a crack in its presence.
The relationship between volume-fractions, V F , and conductivity of a cell is
given by
σ = σ max + V F (σ min − σ max ) ,
(7.62)