78
3 Modeling Composite Structures
3.8 Comments and Conclusions
We should make clear at this point that the ‘permittivity’ that we are computing is
not that of the host polymer matrix. That can only be inferred by measurements on
a pristine sample before embedding carbon fibers within it. The VIC-3D® model
that we are using assumes a homogeneous sample of cfrp that is characterized by
a conductivity tensor and a scalar permittivity. The conductivity tensor allows an
anisotropy to be present due to the fibers, but we are assuming a scalar permittivity
because we have no justification not to. In any case, it is a trivial matter to add a
tensor permittivity.
Thus, we are talking about composite features, conductivity and permittivity, of
the composite material. In this sense we are treating the material as an ‘artificial
dielectric’, in which a number of identical conducting obstacles are arranged in a
regular pattern in a dielectric filler material. The net effect of such an arrangement
under the action of an applied electromagnetic field is to produce a net average
dipole polarization per unit volume, which increases the effective permittivity of
the system [26, Chapter 12]. With a regular arrangement of conductors, the system
will produce an anisotropic permittivity, which we have not considered in this paper.
The anisotropy will disappear with a random arrangement of conductors, which is
probably a more reasonable model of the cfrp composite in the first place. We will
attack the random model of cfrps in coming work.
We have demonstrated that rigorous electromagnetic models, supplemented with
equivalent electrical circuits, can be used to analyze carbon-fiber reinforced polymer
(cfrp) composites. The use of equivalent circuits facilitates the interpretation of
the field solution, especially when the ‘observable’ of each model is an electrical
impedance. The volume-integral code, VIC-3D®, is well suited for these analyses,
and we have further demonstrated that it is capable of providing useful results in the
gigahertz range, well out of the normal range for eddy-current models.
An important further result of this modeling effort is the demonstration that
simple eddy-current NDE models, even with a standard circular probe, can produce
useful results for characterizing certain properties, such as fiber-areal weight
(FAWT) of prepreg cfrp. VIC-3D® also has the capability of modeling more
complex probes, such as those that are designed to operate in the gigaherts or
terahertz range, which could also be useful in modeling and characterizing cfrp.
3.9 Eigenmodes of Anisotropic Media
The previous calculations in this chapter were based on a model in which the
anisotropy was a bounded anomaly within an isotropic host. This model allowed
us to use the usual isotropic Green’s function, which has been the basis of
our work to this point. There may be occasions in which one might wish to
generalize the model, so that the host, itself, is anisotropic, and any departures
3 Modeling Composite Structures
3.8 Comments and Conclusions
We should make clear at this point that the ‘permittivity’ that we are computing is
not that of the host polymer matrix. That can only be inferred by measurements on
a pristine sample before embedding carbon fibers within it. The VIC-3D® model
that we are using assumes a homogeneous sample of cfrp that is characterized by
a conductivity tensor and a scalar permittivity. The conductivity tensor allows an
anisotropy to be present due to the fibers, but we are assuming a scalar permittivity
because we have no justification not to. In any case, it is a trivial matter to add a
tensor permittivity.
Thus, we are talking about composite features, conductivity and permittivity, of
the composite material. In this sense we are treating the material as an ‘artificial
dielectric’, in which a number of identical conducting obstacles are arranged in a
regular pattern in a dielectric filler material. The net effect of such an arrangement
under the action of an applied electromagnetic field is to produce a net average
dipole polarization per unit volume, which increases the effective permittivity of
the system [26, Chapter 12]. With a regular arrangement of conductors, the system
will produce an anisotropic permittivity, which we have not considered in this paper.
The anisotropy will disappear with a random arrangement of conductors, which is
probably a more reasonable model of the cfrp composite in the first place. We will
attack the random model of cfrps in coming work.
We have demonstrated that rigorous electromagnetic models, supplemented with
equivalent electrical circuits, can be used to analyze carbon-fiber reinforced polymer
(cfrp) composites. The use of equivalent circuits facilitates the interpretation of
the field solution, especially when the ‘observable’ of each model is an electrical
impedance. The volume-integral code, VIC-3D®, is well suited for these analyses,
and we have further demonstrated that it is capable of providing useful results in the
gigahertz range, well out of the normal range for eddy-current models.
An important further result of this modeling effort is the demonstration that
simple eddy-current NDE models, even with a standard circular probe, can produce
useful results for characterizing certain properties, such as fiber-areal weight
(FAWT) of prepreg cfrp. VIC-3D® also has the capability of modeling more
complex probes, such as those that are designed to operate in the gigaherts or
terahertz range, which could also be useful in modeling and characterizing cfrp.
3.9 Eigenmodes of Anisotropic Media
The previous calculations in this chapter were based on a model in which the
anisotropy was a bounded anomaly within an isotropic host. This model allowed
us to use the usual isotropic Green’s function, which has been the basis of
our work to this point. There may be occasions in which one might wish to
generalize the model, so that the host, itself, is anisotropic, and any departures