4.1. Meso-Scale: Effect of Crack Presence
In the meso-scale, the electrical response of models with 39 different crack lengths and 9 different depths (from −0.1 to −0.9) has been simulated. The crack length and depth are normalized with regards to the dimension of the Representative Volume Element (RVE) [
15],
. The input parameters and material properties of the meso-scale model are listed in
Table 1. Note that the resistance of the circuit elements does not affect the results.
The predicted
drop and
span values for the different
and
ratios have been listed in tables.
Table 2 lists the results for the case of
. The other tables are omitted for the sake of briefness. To visualize the data, 3D plots have been created from tabular data in the MATLAB software using fitting functions.
Figure 5 shows the 3D plot of
drop vs.
and
for the case of maximum
drop while
Figure 6 shows the 3D plot of span vs.
and
for the case of maximum
span. We observe a decrease in the sensitivity of the response with increasing the depth ratio
, which means that the deeper the crack is the less sensitive the method becomes.
The opposite procedure, i.e., the characterization of a cracked model, lies in the determination of crack length and position using the tabular data or the graphs. To demonstrate the process, we select random values of
drop and
span, for instance,
drop = 6.2% and
span = 1.18%, and from the graphs of
Figure 7 and
Figure 8 using intersection lines we gather all sets of the
and
ratios which have given the selected values. The outcome of this first step are projection-curves (see
Figure 7 and
Figure 8). Then, we plot the two lines in a
vs.
system (
Figure 9) and we find their intersections. The intersection points give sets of
and
ratios which lead to the selected
drop and
span values. Since the intersections are more than one, this means we have more than one sets of
drop and
span values that lead to the same electrical result. The selection of a unique solution requires more data. Nevertheless if we use the solution with the maximum
ratio, i.e.,
and
, we get a crack length of
and a depth of
.
4.2. Macro-Scale: Crack Detection
The input parameters and material properties of the macro-scale model are listed in
Table 3. In the macro-scale, we examine the model’s capability of detecting a crack in the
ZY plane introduced through the reduction of the element’s electrical conductivity derived from the micro-scale analysis [
14]. Two cracks (crack1 and crack2) of different length have been modeled at the same location; their characteristics are listed in
Table 4.
By comparing the current values taken from the circuit elements for the reference and the cracked model, we plot the results at the
XZ plane in
Figure 10. The obtained inhomogeneity of the current distribution is due to the inhomogeneous electrical properties of the model’s elements.
The computed contours of
drop at the
Y direction due to the presence of crack1 and crack2 are plotted on the
XZ plane in
Figure 10 and
Figure 11, respectively. The results have been normalized by the current of the reference model. As shown, in both cases the location of the current drop at the
Y direction matches the location of the crack. For crack1, given the crack spans through the width of the modified element, the ratio of the crack surface to the
XZ surface of the model is
. Hence, the response is quite large compared to the modeled crack length. The reason for choosing the width of 0.2 mm (20% of the model’s width) is because the meso-scale analyses have shown that smaller crack widths give very weak responses. In the case of crack2, the magnitude of electrical response increases by 643.3% for a 100% increase in crack length from 50 μm to 100 μm. This finding is an indication of the high sensitivity of the model to the crack length.
To further exploit the above findings, we note that the same technique does not have the same sensitivity in metallic materials since in that case the electric current from the cracked area is redistributed to the remaining material volume and the drop is not detectable. On the contrary, due to the very small volume of the CNT network compared to the volume of the composite material, any interruption of the network, which leads to the redistribution of electric current, is detectable.
Aiming to characterize crack1, we use the technique described in the previous section. Based on the computed values of
drop and
span, we delimit the area around the crack until the nodes where the electrical response is fully recovered (zero value). The dimension of the perturbation area is defined as the dimension in the direction of the denser mesh (
Z direction). For crack1, we define an area of dimension of 250 μm and for this area we read
drop = 3% and
span = 1.1 (see
Figure 3). The intersection of the
drop and
span intersection-curves shown in
Figure 12 gives
= 2.1, a crack length of 52.5 μm and
= −1.88. The characterized values of crack length and crack depth are close to the actual values, something which validates the proposed crack sensing methodology.