Care was taken in the description of the nanofiller alignment simulations in Chapter4 to distinguish the micro or (˜x, ˜y, ˜z) domain. This distinction is important because different orientations of the microscale domain are next considered in the EITdomain. To clarify, alignment refers to fillers being more disposed to run parallel to each other while orientation refers to the direction in which fillers are aligned. The EITdomain is defined within the (x, y, z) basis. In this basis, the structure of interest is a thin plate measuring 10 cm × 10 cm × 3 mm. The x and y-directions span the plate while the z-direction is through the thickness of the plate. Assuming the thin plate has conductive properties determined by the microscale nanofiller alignment analysis, the conductivity in the EIT domain can be expressed using standard tensor rotation rules, σij = R(α, β, γ)kiσ˜klR(α, β, γ)lj. Here,
Rij is the rotation tensor formed from the Euler angles α, β, and γ. More explicitly, α is
the first rotation of the (˜x, ˜y, ˜z) basis about the z-axis. The second rotation of the (˜x, ˜y, ˜z) basis is by β about the ˜x-axis, and the last rotation of the (˜x, ˜y, ˜z) basis is about the ˜z-axis by γ. For a prescribed degree of alignment, fillers are equally disposed run in either the ˜x
Figure 6.1: Conductivity versus volume fraction in the transverse and alignment directions for three different degrees of alignment..
or ˜y-direcion. Because of this, conductivity in the micro-domain is transversely isotropic and the choice of γ is inconsequential. Allowing for different orientations of the micro- domain in theEITdomain is akin to the fillers to be aligned at an angle with respect to the through-thickness direction of a plate. Three degrees of nanofiller alignment are selected for the sensitivity enhancement analysis as shown in Figure6.1. Because the conductivity is calculated in the microscale domain, ˜σ is a diagonal matrix with ˜σ11= ˜σ22 = ˜σT where
˜
σT is the conductivity transverse to the alignment direction and ˜σ33= ˜σAwhere where ˜σA
is the conductivity in the nanofiller aligment direction. ˜σij = 0 for i 6= j.
Delaminations to be detected by EITare simulated as thin voids in a reference mesh. The voids are formed by the intersection of two slightly overlapping spheres thereby en- suring a zero-thickness edge as is expected from a real delamination. Delaminations are centered through the plate thickness and located in the plane as shown in Figure6.2. Two cases are considered. The first case considers a single delamination with a diameter of 1.7 cm and a maximum opening of 0.12 mm. The second case considers a second delamina-
tion with a diameter of 1.2 cm and a maximum opening of 0.06 mm in the presence of the original delamination. The sensitivity enhancement analysis begins by examining the influ- ence of filler alignment and orientation on boundary voltage measurements. BecauseEIT
depends on in-plane boundary voltages, it is desirable to tailor the nanocomposite conduc- tivity to enhance boundary voltage changes due to delamination. To this end, for a given damage state and conductivity, the norm of the percent change in boundary voltages due to damage is considered as a function of the Euler angle β. If the aligned conductivity is more sensitive, this norm is expected to increase. The Euler angle α is fixed at 45◦ for identical conduction in the x and y-directions. As shown in Figure6.2, both φm and β have a large
influence on the boundary voltage sensitivity. Furthermore, it is observed that the greatest gains are obtained for 1% filler volume fraction. Because the anisotropic ratio is greatest for each value of φmat 1% filler volume fraction, it can be inferred that higher anisotropic
ratios result in greater sensitivities. The symmetry of the curves about β = 90◦is expected because of the symmetry of the ˜σ tensor.
6.2.2
Formulation of the Anisotropic Electrical Impedance Tomogra-
phy Inverse Problem
TheEITforward problem was presented in general for both isotropic and anisotropic con- ductivity in Chapter3. The inverse problem, however, requires more special consideration and will be presented here. In order to proceed, it must first be conceded that a unique solu- tion generally does not exist for anisotropicEIT[131] [132]. Nonetheless, as demonstrated by other researchers [60] [133], a solution can be recovered for mild degrees of anisotropy if it is assumed that the eigenvectors of the conductivity tensor are preserved during the minimization. In light of this assumption, rewrite the conductivity as σij = κσij0 and select
κ such that det(σ0
ij = 1) [133]. Immediately discretize κ such thatEIT now endeavors
to find κ to minimize kVm− F(κσ0ij)k 2
. Similar to the isotropic formulation, linearize F(κσ0
ij) by retaining the linear terms of a Taylor series expansion centered about κ0 as
shown in equation6.1. This approximation is substituted into the minimization expression resulting in equation6.2. F(κσ0ij) ≈ F(κ0σ0ij) + ∂F(κ0σ0ij) ∂κ (κ − κ0) (6.1) Vm− F(κσij0) − ∂F(κ0σij0) ∂κ (κ − κ0) = 0 (6.2)
Figure 6.2: Norm of the percent change in boundary voltage due to single and double delamination cases as a function of β for φm = 45◦, 60◦, 75◦, and isotropic (90◦) conduc-
tivities at 1%-5% filler volume fraction. Thickness dimensions in schematic exaggerated for clarity.
can be rewritten as follows.
J∆κ = Ve (6.3)
The anisotropic sensitivity matrix now takes the form shown in equation6.4.
JM N e = − Z Ωe ∂uM ∂xi σ0ij∂ ¯u N ∂xj dΩe (6.4)
Lastly, ∆κ is recovered again via Tikhonov regularization.
∆κ = JTJ + α2LTL−1JTVe (6.5)
This anisotropic formulation simplifies to the isotropic case for κσ0
ij = κδij. The most
important assumption in the preceding was that the eigenvectors of σ0
ij are preserved dur-
ing minimization. This assumption makes the problem tractable by reducing the possible unknowns per element from six to one. However, anisotropicEIT is not as robust as the isotropic case and prone to diverge for highly anisotropic materials.
6.2.3
Enhanced Delamination Detection through Nanofiller Alignment
The combined effect of anisotropic conductivity induced by filler alignment andEIT for enhanced delamination detection is now presented. Because the greatest gains in Figure
6.2are seen at 1% filler volume fraction, the analysis is restricted to such for β = 0◦, 30◦, 60◦, and 90◦. Figure6.3shows theEITimages for the single delamination case, and Figure
6.4 shows theEIT images for the double delamination case. These figures show that is indeed possible to markedly enhance the sensitivity ofEITto delaminations by microscale nanofiller tailoring. A final example of the enhancement afforded by nanofiller alignment is shown in Figure 6.5 for β = 60◦ and φm = 45◦ on a more refined mesh with a single
delamination.