60
3 Modeling Composite Structures
The heart of the problem, as we have emphasized throughout this book, is to
determine a Green’s function for the composite material. Much work has been done
in recent years on the subject of electromagnetic interactions with composite materials, mostly in the context of electromagnetic shielding of avionics equipment from
electromagnetic pulses [4, 5, 45, 120, 132]. Some of this work is directly applicable
to the problem of computing eddy-current flow within composites, but the Green’s
function problem must be attacked by applying rigorous electromagnetic theory to
anisotropic media.
3.2 Constitutive Relations for Advanced Composites
Advanced composite materials are laminates made up of a number of individual
layers bonded together. Each layer consists of a unidirectional array of long fibers
embedded in, and firmly bonded to, a matrix. The basic building blocks of any
specific composite are defined by the types of fibers and matrix involved. Some
fiber-matrix systems are: boron-epoxy, graphite-epoxy, Kevlar-epoxy, graphitepolymide and graphite-thermoplastic [5]. The matrix for each of these materials
is normally a good dielectric, whereas the fibers vary in electrical conductivity from
modest (graphite) to a poor dielectric (boron) to a good dielectric (Kevlar). These
materials are nonmagnetic, so that the magnetic permeability is μ 0 .
Composites have anisotropic conductivities because of the unidirectional arrays
of fibers within. For example, for graphite-epoxy the average macroscopic conductivity along the fiber direction is 20,000 S/m, whereas in the direction transverse
to the fibers, the conductivity is 100 S/m. It may be surprising to find a nonzero
transverse conductivity in graphite-epoxy, in view of the earlier statement that the
matrix is a good dielectric. The fact is that there is enough local fiber-to-fiber
contact that the average macroscopic conductivity is not zero, as illustrated in
Fig. 3.1 [83]. Other materials, of course, have different longitudinal and transverse
conductivities, as shown in Table 3.1 [5]. The reason that R for graphite-epoxy is
indeterminate is because the fiber-to-fiber contact effectively shunts the capacitors
between fibers with a fairly low resistance path, making it impossible to measure
dielectric permittivities at frequencies less than 100 MHz, or so. Thus, in Fig. 3.1b,
which shows a possible AC equivalent circuit for eddy-current flow, the capacitors
are effectively short-circuited by the fiber-to-fiber resistors at the lower frequencies.
The anisotropy of the composite manifests itself in a complex-permittivity tensor,
the tensor being diagonal in a coordinate system (ξ 1 , ξ 2 , ξ 3 ), where ξ 1 is parallel to
the average fiber direction, ξ 2 is perpendicular to the average fiber direction, but lies
in the plane of the composite layer, and ξ 3 is perpendicular to both fibers and the
plane of the layer:
=
⎡
⎣
ˆ
11 0 0
0 ˆ
22 0
0 0 ˆ
33
⎤
⎦ ,
(3.1)
3 Modeling Composite Structures
The heart of the problem, as we have emphasized throughout this book, is to
determine a Green’s function for the composite material. Much work has been done
in recent years on the subject of electromagnetic interactions with composite materials, mostly in the context of electromagnetic shielding of avionics equipment from
electromagnetic pulses [4, 5, 45, 120, 132]. Some of this work is directly applicable
to the problem of computing eddy-current flow within composites, but the Green’s
function problem must be attacked by applying rigorous electromagnetic theory to
anisotropic media.
3.2 Constitutive Relations for Advanced Composites
Advanced composite materials are laminates made up of a number of individual
layers bonded together. Each layer consists of a unidirectional array of long fibers
embedded in, and firmly bonded to, a matrix. The basic building blocks of any
specific composite are defined by the types of fibers and matrix involved. Some
fiber-matrix systems are: boron-epoxy, graphite-epoxy, Kevlar-epoxy, graphitepolymide and graphite-thermoplastic [5]. The matrix for each of these materials
is normally a good dielectric, whereas the fibers vary in electrical conductivity from
modest (graphite) to a poor dielectric (boron) to a good dielectric (Kevlar). These
materials are nonmagnetic, so that the magnetic permeability is μ 0 .
Composites have anisotropic conductivities because of the unidirectional arrays
of fibers within. For example, for graphite-epoxy the average macroscopic conductivity along the fiber direction is 20,000 S/m, whereas in the direction transverse
to the fibers, the conductivity is 100 S/m. It may be surprising to find a nonzero
transverse conductivity in graphite-epoxy, in view of the earlier statement that the
matrix is a good dielectric. The fact is that there is enough local fiber-to-fiber
contact that the average macroscopic conductivity is not zero, as illustrated in
Fig. 3.1 [83]. Other materials, of course, have different longitudinal and transverse
conductivities, as shown in Table 3.1 [5]. The reason that R for graphite-epoxy is
indeterminate is because the fiber-to-fiber contact effectively shunts the capacitors
between fibers with a fairly low resistance path, making it impossible to measure
dielectric permittivities at frequencies less than 100 MHz, or so. Thus, in Fig. 3.1b,
which shows a possible AC equivalent circuit for eddy-current flow, the capacitors
are effectively short-circuited by the fiber-to-fiber resistors at the lower frequencies.
The anisotropy of the composite manifests itself in a complex-permittivity tensor,
the tensor being diagonal in a coordinate system (ξ 1 , ξ 2 , ξ 3 ), where ξ 1 is parallel to
the average fiber direction, ξ 2 is perpendicular to the average fiber direction, but lies
in the plane of the composite layer, and ξ 3 is perpendicular to both fibers and the
plane of the layer:
=
⎡
⎣
ˆ
11 0 0
0 ˆ
22 0
0 0 ˆ
33
⎤
⎦ ,
(3.1)