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3 Modeling Composite Structures
where ˆ
ii = ii − jσ ii /ω, with j =
√ −1, and ω is the angular frequency.
The coordinate system just defined, for which the complex permittivity tensor
is diagonal, is not necessarily the laboratory system, (x, y, z), in which the
electromagnetic field vectors are defined. In any case, the tensor symbol will be
used, and the components in a particular coordinate system may be computed by
applying the usual rules for transforming Cartesian tensors.
From here on, we will consider only graphite-epoxy, for which 11 = 22 =
33 = 0 , σ 11 = 2 × 10 4 S/m, and σ 22 = σ 33 = 100 S/m.
3.3 Example Calculations Using VIC-3D®
Figure 3.2 illustrates the response of a two-layered composite structure with the
bottom layer oriented at 0 and 90 ◦ with respect to the top layer. The principal-axis
conductivities of the two layers are [2 × 10 4 , 100, 100] S/m, which is typical of
cfrp composites. The x-axis is aligned with the fibers, and the y- and z- axes are
transverse to this direction. There is enough fiber-to-fiber contact in the transverse
directions to yield a nonzero transverse conductivity. Excitation of this system is by
means of a circular coil. Each layer has dimensions of 6 × 6 × 0.25 in.
When the same composite structure of Fig. 3.2 is excited by a racetrack coil
at 1 MHz, the resulting response when the coil is rotated through 360 ◦ is shown
in Fig. 3.3. The relative ‘flatness’ of the reactance response between 60 and 120 ◦
for the situtation in which the bottom plate is rotated 90 ◦ relative to the top plate
suggests that this configuration is somewhat more ‘isotropic’ than the configuration
in which both plates are similarly oriented. This seems to be intuitively reasonable.
Another metric is the peak-to-peak change in response divided by the mean value of
the response. Again, this shows that the configuration in which the bottom layer is
rotated 90 ◦ yields a ‘more isotropic’ response than otherwise. Indeed, the purpose
of such a ‘lay-up order’ is to force the stress distribution within the structure to be
more isotropic than with a single plate, or with two plates aligned parallel to each
other. One result of the 0–90 ◦ order is that warping will be reduced.
A Second Example Using VIC-3D® Consider the preceding example with both
plates aligned parallel to each other, but excited by a T/R configuration comprising
a circular transmit and receive coil. The receive coil is raster-scanned in the
(x, y)-plane symmetrically about the transmit coil, which is excited at 1 MHz.
The host is isotropic with a conductivity of 100 S/m, and two types of anomalies
will be presented. The first is an isotropic patch with conductivity 20,000 S/m,
and the second a graphite-epoxy patch with the usual conductivity pattern of
[2 × 10 4 , 100, 100] S/m.
3 Modeling Composite Structures
where ˆ
ii = ii − jσ ii /ω, with j =
√ −1, and ω is the angular frequency.
The coordinate system just defined, for which the complex permittivity tensor
is diagonal, is not necessarily the laboratory system, (x, y, z), in which the
electromagnetic field vectors are defined. In any case, the tensor symbol will be
used, and the components in a particular coordinate system may be computed by
applying the usual rules for transforming Cartesian tensors.
From here on, we will consider only graphite-epoxy, for which 11 = 22 =
33 = 0 , σ 11 = 2 × 10 4 S/m, and σ 22 = σ 33 = 100 S/m.
3.3 Example Calculations Using VIC-3D®
Figure 3.2 illustrates the response of a two-layered composite structure with the
bottom layer oriented at 0 and 90 ◦ with respect to the top layer. The principal-axis
conductivities of the two layers are [2 × 10 4 , 100, 100] S/m, which is typical of
cfrp composites. The x-axis is aligned with the fibers, and the y- and z- axes are
transverse to this direction. There is enough fiber-to-fiber contact in the transverse
directions to yield a nonzero transverse conductivity. Excitation of this system is by
means of a circular coil. Each layer has dimensions of 6 × 6 × 0.25 in.
When the same composite structure of Fig. 3.2 is excited by a racetrack coil
at 1 MHz, the resulting response when the coil is rotated through 360 ◦ is shown
in Fig. 3.3. The relative ‘flatness’ of the reactance response between 60 and 120 ◦
for the situtation in which the bottom plate is rotated 90 ◦ relative to the top plate
suggests that this configuration is somewhat more ‘isotropic’ than the configuration
in which both plates are similarly oriented. This seems to be intuitively reasonable.
Another metric is the peak-to-peak change in response divided by the mean value of
the response. Again, this shows that the configuration in which the bottom layer is
rotated 90 ◦ yields a ‘more isotropic’ response than otherwise. Indeed, the purpose
of such a ‘lay-up order’ is to force the stress distribution within the structure to be
more isotropic than with a single plate, or with two plates aligned parallel to each
other. One result of the 0–90 ◦ order is that warping will be reduced.
A Second Example Using VIC-3D® Consider the preceding example with both
plates aligned parallel to each other, but excited by a T/R configuration comprising
a circular transmit and receive coil. The receive coil is raster-scanned in the
(x, y)-plane symmetrically about the transmit coil, which is excited at 1 MHz.
The host is isotropic with a conductivity of 100 S/m, and two types of anomalies
will be presented. The first is an isotropic patch with conductivity 20,000 S/m,
and the second a graphite-epoxy patch with the usual conductivity pattern of
[2 × 10 4 , 100, 100] S/m.