## Abstract

Electrical resistance (ER), also known as direct current potential drop, has been demonstrated as an enabling means to monitor damage evolution in SiC-based ceramic matrix composites (CMCs). For laminate composites, it has become apparent that the location and orientation of SiC fibers, free Si, and in some cases insertion of C rods can greatly affect the measured resistance. In addition, the nature of crack growth through the different plies which consist of different constituents will have different effects on the change in resistance. Therefore, both experimental and modeling approaches as to the resistance and change in resistance for different laminate architectures based on the nature of constituent content and orientation are needed to utilize and optimize ER as a health-monitoring technique. In this work, unidirectional and cross-ply laminate composites have been analyzed using a ply-based electrical model. Based on a ply-level circuit model, the change in resistance was modeled for damage development. It is believed that this can serve as a basis for tailoring the architecture/constituent content to create a “smarter” composite.

## 1 Introduction

The demand for more efficient and powerful aircraft engines is increasing. It is essential to implement materials that can withstand high stresses at higher temperatures under hostile environments. Ceramic matrix composites (CMCs) are one class of material currently in use and being developed for hot-section aero applications. For example, CMCs have been implemented into hot segment parts of aero engines and hypersonic hot structures to maintain the high thrust to weight ratio of the overall performance of the engine [14]. However, to fully implement these composites into industrial applications, a thorough understanding of damage evolution is therefore necessary. In a composite system, damage can occur in the form of matrix microfracture, concomitant debonding, and/or fiber breakage [510]. Typically, these composites are made from nonoxide materials which comprise at least one conductive constituent. The semiconductive properties of some of the constituents of SiC-based composites enable the material to possess “smart” in situ sensing capability to quantify damage progression and location. Proper understanding of the electrical network in the composite system via the conductive constituents, such as carbon (C) or silicon (Si), could be helpful to strategically tailor the material to demonstrate and manipulate the understanding of where, when, and how much local damage has occurred, and can also enhance the ability to monitor the change of resistance during operation.

The direct current potential drop technique has been used as in situ monitoring technique [11,12] to detect damages in metals and more recently in composite materials at ambient and elevated temperatures [1316]. This approach has been demonstrated in metallic systems [17], polymer composite systems [18,19], and nonoxide CMC systems [20,21]. The typical constituents of SiC-reinforced SiC matrix composites are silicon carbide (SiC), Si, Cm and BN (boron nitride). Where C is the most conducting phase followed by Si then SiC then BN, the latter being an electrical insulator. The electrical resistivity values of these constituents can be found elsewhere [22]. Provided the constituent is a continuous phase in the composite and there is enough fractional content of the constituent, the constituent with lowest resistivity typically dominates the overall electrical resistance (ER) of composite system. In carbon fiber and/or carbon interphase composites, the change in resistance when subjected to loading depends on how the carbon constituent is affected as the material dramatically rupture [23]. Thus, matrix microfracture may not significantly change the measured resistance because majority of the current is being carried by the carbon fiber. However, if fibers or interphases are exposed to an oxidizing environment due to matrix crack opening with fiber bridging or fiber breakage, changes in resistance will be obvious. In a melt infiltrated composite, the change in resistance is dominated by the present of Si in the system because it is the most conducting constituent in the material. For clear understanding of the complex electrical network in a composite system, it is imperative to fully investigate the current path at a constituent and ply level.

The aim of this study is to understand the nature of electrical current flow in prepreg laminate melt infiltrated composite based on constituent resistivities, ply resistivities, and volume fractions. The study is based on two different laminate architectures from earlier works [24,25]. The architectures are defined by the orientation of the fiber-reinforced plies and include unidirectional $[0 deg]8$ and a balanced cross ply $[0 deg/90 deg]2s$. The model will highlight the resistivity of different plies, and a combination of parallel and series resistors are constructed at a ply level to monitor damage.

## 2 Background Work and Materials

This study first utilized specimens and results from two earlier studies [24,25]. The composite specimens all contained Hi-Nicalon Type S fibers (Nippon Carbon, Tokyo, Japan) coated with a chemical vapor deposited BN interphase. The material was fabricated via prepreg melt infiltrated technique; therefore, the matrix constituted SiC matrix and free silicon. Initially, specimens from the work of Morscher and Gordon [24], unidirectional $[0 deg]8$ and cross ply $[0 deg/90 deg]2s$, were used in this work to examine and better understand the ply-level contribution to resistivity by measuring the resistivity in the [0] and [90] directions of a unidirectional composite. Figure 1 shows polished cross sections of the two architectures. Note that in between the fiber-containing layers there is a “matrix-only” layer, i.e., the composite is made up of fiber-containing plies and matrix-only plies. The results from Morscher and Maxwell's work [25] were also incorporated in this work. They measured the change in ER with crack growth emanating from a notch and fully characterized the growth of the crack(s) in the different ply layers via acoustic emission (AE) and digital image correlation (DIC). The results from Morscher and Maxwell were used to apply the ply-level resistivity electrical model extracted from the measured resistances of the different architecture specimens measured in different directions.

Fig. 1
Fig. 1
Close modal

## 3 Electrical Resistance Measurements for Different Architectures

Electrical resistance was measured using the four-probe method to minimize contact resistance. A conductive silver epoxy (EPO-TEK, Billerica, MA) was pasted around the specimen to attach the leads and to ensure bulk measurements. A constant direct current was applied using a current source digital multimeter (Keithley 2450 Source Meter, Colorado Springs, CO), and the potential drop was monitored using a multifunction switch/measure unit (Agilent, model 34980A, Santa Clara, CA). A sketch of the ER setup is presented in Fig. 2.

Fig. 2
Fig. 2
Close modal

The unidirectional system was first investigated using specimens left over from Ref. [24] to determine the resistivity of the 0-ply, the 90-ply, and the matrix-only ply which could then be used to develop a ply-level model for the $[0 deg/90 deg]2s$ architecture. The unidirectional material consists of eight fiber plies alternating with seven matrix plies (Fig. 3(a)). The “0” direction corresponds to the axial loading direction. The 0/90 composite has four fiber plies in the 0 direction, four fiber plies in the 90 direction, and seven matrix plies (Fig. 3(b)). On the other hand, for both architectures ($[0 deg]8$ and $[0 deg/90 deg]2s$), the fiber plies consist of fiber, interphase, matrix, and free silicon, while the matrix plies have only matrix and some bulk free silicon as constituents.

Fig. 3
Fig. 3
Close modal
For unidirectional specimens, the material resistivity (ρ) was measured in both axial and transverse directions according to the following equation:
$ρ=RAL$
(1)

where R is the measured resistance, A is the cross-sectional area, and L is the length between the potential leads. Resistivity is measured in Ω mm. For axial measurements in the unidirectional material (30 mm in length, 12.5 mm in width, and 2 mm in depth), the current was sent in a parallel direction to the fiber orientation in the fiber plies, and the resistance $(ρ0)$ was measured between the current leads (Fig. 4(a)). However, for the transverse measurements, three specimens were cut into smaller strips in the 90 direction (12.5 mm long with a cross-sectional area of 1–2 $mm2$). The current was sent in a direction perpendicular to the fiber orientation in the fiber plies (Fig. 4(b)), and the resistivity $(ρ90)$ was measured. In this scenario, the fibers are oriented at a 90 deg compared to the flowing direction of the current. For resistivity of 0/90 architecture specimens, data for pristine tensile bars from Morscher and Gordon [24] were used. The volume fractions of “matrix” and “fiber” ply were determined using imagej, an open-source java-written program [26]. All measurements are shown in Table 1 [24].

Fig. 4
Fig. 4
Close modal
Table 1

Measured physical properties of unidirectional and cross-ply composite systems

Specimen architectureFiber-ply volume fraction-0 directionFiber-ply volume fraction-90 directionMatrix-ply volume fraction-0 directionMatrix-ply volume fraction-90 directionResistivity (Ω mm)Standard deviation (no. of measurements)
Unidirectional [0]80.620.380.160.06 [4]
Unidirectional [90]80.620.381.160.01 [3]
Cross ply [0/90]2 s0.3150.3150.370.260.06 [11]
Specimen architectureFiber-ply volume fraction-0 directionFiber-ply volume fraction-90 directionMatrix-ply volume fraction-0 directionMatrix-ply volume fraction-90 directionResistivity (Ω mm)Standard deviation (no. of measurements)
Unidirectional [0]80.620.380.160.06 [4]
Unidirectional [90]80.620.381.160.01 [3]
Cross ply [0/90]2 s0.3150.3150.370.260.06 [11]

## 4 Modeling Ply-Level Electrical Resistance

The unidirectional composite system presented in Sec. 3 consists of two main plies (fiber-containing ply and matrix-only ply). Therefore, they can be treated as parallel circuit of these continuous plies
$1RCMC=1Rm−ply+ 1R0−ply$
(2)
where $RCMC$ is the resistance of the composite, $Rm−ply$ is the resistance of the matrix-ply, and $R0−ply$ is the resistance of the fiber-ply. From Eq. (1), Eq. (2) can be expanded and can be written as the following for both measurements in the 0 direction and 90 direction:
$LRA=1ρ0=fmρm+f0−plyρ0−ply$
(3)
$LRA=1ρ90=fmρm+f90−plyρ90−ply$
(4)
where $ρ0$ is the measured resistivity of the unidirectional composite in the 0 direction, and $ρ90$ is the measured resistivity of the unidirectional composite in the 90 direction (Table 1). $ρ0−ply$ is the resistivity of the 0 fiber-ply, $ρ90−ply$ is the resistivity of “90” fiber-ply,$ρm$ is the resistivity of the matrix-ply, and the subscript $f$ refers to the volume fraction of a given ply in the composite. It is assumed that the matrix-ply resistivity, $ρm$, is the same in both directions. Then, $ρm$ in Eqs. (3) and (4) would be equal, and the equations can be rearranged as follows:
$ρm= fm1ρ0 − f0−plyρ0−ply=fm1ρ90 − f90−plyρ90−ply$
(5)

Since ρm is same for both directions, ρ0-ply and ρ90-ply values can be varied until the same ρm value is achieved, and the volume fractions used in this equation are from Table 1. Figure 5 shows the relationship between the fiber-ply resistivity for 0 and 90 and the matrix-ply resistivity. The same value for rm (∼0.62 Ω mm) occurs at r0-ply = 0.11 Ω mm and r90-ply = 125 Ω mm. It should be noted that the value of rm varies little above a r90-ply = 25 Ω mm which has little consequence on the results of this study, since the 90 fiber-ply would be over 2 orders of magnitude greater in resistivity than the 0-ply and at least five times that of the matrix-ply. It is also evident from this analysis that the 0 fiber-ply is more conductive compared to the matrix-ply. The 0 fiber-ply consists of fibers, interphase, Si/SiC, and bulk free Si which are the brightest regions in Fig. 1 and appear to be aligned with the fibers as observed in the polished 90 fiber-ply of the [0/90]2 s architecture (Fig. 1(a)). Evidently, it is these bulk Si areas that probably contribute the greatest to conductivity, since the fiber conductivity is ∼1 Ω mm [26] and the matrix (predominantly Si/SiC) is presumed to be about 0.62 Ω mm.

Fig. 5
Fig. 5
Close modal
Moving forward from the unidirectional system to the 0/90 composite system, Eqs. (2) and (5) can be formulated as
$1RCMC−0/90=1Rm−ply+1R0−ply+1R90−ply$
(6)
$ρm= fm−uni1ρ0 − f0−plyρ0−ply=fm−uni1ρ90 − f90−plyρ90−ply=fm−0/901ρ0/90 − f0−1/90ρ90−ply − f90−0/90ρ90−ply$
(7)

based on the values established from the unidirectional material, for same vintage 0/90 composites, resistivity predicted based on the ply-level electrical model for 0/90 (ρ(0/90)) composite is 0.287 Ω mm, which is in the range of what was measured in Ref. [24] (Table 1, ρ_(0/90) = 0.26±0.06 Ω mm). Therefore, the results established from Eqs. (6) and (7) align with the experimental results obtained in Ref. [24].

## 5 Results and Discussion

### 5.1 Model Validation.

To establish our ply-level hypothesis, the single notch crack growth results of Morscher and Maxwell [25] were examined using a parallel resistor circuit that was constructed using 0 plies, 90 plies, and matrix plies. Morscher and Maxwell performed a tension–tension fatigue test at room temperature on $[0 deg/90 deg]2s$ SiC-based laminate composite single notched specimens with AE and DIC (Fig. 6(a)). DIC monitored the surface strain field, whereas AE monitored the axial and transverse location of matrix crack formation or propagation. Damage (matrix cracks emanating from the notch) occurred within a narrow axial length (∼±1 mm from notch radius) as depicted in Fig. 6(b). The nature of transverse crack growth was determined to be comprised of internal crack propagation or “tunnel” crack propagation along the inner 90 plies located by AE and lateral extension of the matrix crack through 0 plies to the surface measured by DIC (Fig. 6(c)). The change in ER is plotted in Fig. 6(d) as a function of time. The length of the damage zone was based on DIC strain field imaging in the axial direction (∼2 mm).

Fig. 6
Fig. 6
Close modal
To validate our model approach, a simple parallel resistor circuit was constructed on a ply level using the surface crack length and presumed inner tunnel crack length from Morscher and Maxwell's study [25]. The function of the circuit is to model the change in ER (ΔR) for 26 mm lead distance (gray shaded region). The model circuit setup is presented in Fig. 7. In this study, it is assumed that the damage occurred near the notch plane, the notch length (n), surface crack length (s), and center crack length (c) of Fig. 7 represents the pertinent transverse dimensions. From the circuit presented in Fig. 7, the damage region can be set as parallel resistors where each resistor represents a ply; the parallel circuit can be represented as
$1Rdamage=∑(1Rm)+∑(1R0−ply)+∑(1R90−ply)$
(8)
Fig. 7
Fig. 7
Close modal
And the total axial resistance ($RTotal)$ between the potential leads can be written as a series of resistors
$Rtotal=Rpristine+Rdamage+Rpristine$
(9)

The resistance values can be calculated from Eq. (1). Resistivity values for each ply (matrix-ply, 0-ply, and 90-ply) are based on the unidirectional measurements presented earlier, and the area of each ply was calculated using imagej software. The axial length considered for this model was 12 mm for pristine region and 2 mm for damage region, leading to a total modeled axial length of the 26 mm (distance between the potential leads in the experimental setup). Measured values are displayed in Table 2. A schematic sketch for the cross section view in the damage region is presented in Fig. 8. Different scenarios for the area of crack progression in the damage region were modeled:

Fig. 8
Fig. 8
Close modal
Table 2

Ply measurements

Average thickness, mmAverage width, mmAverage area, mm2Resistivity, Ω mm
Matrix-ply0.0701100.7010.62
0-ply0.145101.450.11
90-ply0.145101.4555
Matrix-center ply0.085100.850.62
Average thickness, mmAverage width, mmAverage area, mm2Resistivity, Ω mm
Matrix-ply0.0701100.7010.62
0-ply0.145101.450.11
90-ply0.145101.4555
Matrix-center ply0.085100.850.62
• Surface only is defined as the area pertaining to the DIC determined surface crack length.

• Tunnel crack length is defined as the area corresponding to the surface crack length (DIC) + area of additional tunnel crack length (AE) where it is assumed that the tunnel propagates along the inner 90 plies.

• Intermediate width is defined as the area corresponding to the surface crack length (DIC) + the area from the extension of the tunnel crack in the width direction to the outer 0 plies which would halt the crack from further propagation in the transverse direction.

• Through width is defined as the area that would correspond to the case where the crack front length of the tunnel crack was through the width. This would be the maximum crack area consideration.

### 5.2 Discussion.

The results for the four different crack scenarios are presented in Fig. 9 along with the crack length (inner and surface) from Morscher and Maxwell [25] versus time. The two extremes of crack area would be the “surface only” (lowest ΔR scenario) and the “through width” (highest ΔR scenario). The latter overestimates the measured ΔR which is expected, since the observed surface crack length from DIC was considerably shorter than the interior crack length as determined from acoustic emission. Note that there is little difference between the “tunnel crack” and surface only scenario, since the resistivity of the tunnel portion of the cracked area is relatively high, made up of two higher resistant 90 plies and a matrix-only ply—the removal of those regions from the circuit has little effect on overall resistance.

Fig. 9
Fig. 9
Close modal

The initial measure of ΔR is in good agreement with the tunnel crack scenario until about 290,000 s which affirms the ply-based electrical model. After 290,000 s, the ΔR increases dramatically, and the tunnel crack scenario underestimates ΔR significantly. This would imply that the interior crack region ahead of the surface crack has extended beyond the bounds of the inner two 90 plies which is affirmed by the fact that for times greater than 290,000 s the “intermediate scenario” predicts the change in ΔR the best. This further validates the usefulness of a ply-based electrical model. The sharp deviation in electrical response at ∼290,000 s from a 90-ply tunnel cracking assumption to transverse extension of the crack could only be informed by the transverse extension of the tunnel crack, as forecast by the intermediate width scenario.

## 6 Conclusion

The use of electrical resistance was utilized at a ply level to monitor damage progression for a previous study that was conducted on a $[0 deg/90 deg]2s$ SiC-based laminate composite subjected to tension fatigue loading with a single notch. Different damage scenarios were investigated based on the nature of crack growth through different plies. The conductivity of each ply in the system varies based on the volume fraction and directionality of various constituents present in each ply. Ply resistivity was established based on direct measurements conducted on a unidirectional composite system, and a simple parallel circuit was constructed to understand the contribution of the resistivity of each ply to the overall change in resistance as damage progress.

A combination of parallel and series resistors circuit was constructed corresponding to the damage region and flow of current traveling in different plies. The model results were in good agreement with the measured ΔR observed during the actual test. The crack morphology is the combination of two presented models (surface + tunnel crack and an intermediate width tunnel crack). Initially, the crack progression follows the surface + tunnel crack model, but when the measured data start to deviate from the modeled data, it is believed that the tunnel crack portion of the crack grows transversely through the thickness up to the outer zero plies. The model demonstrated that the interior tunnel crack region made up of 90 and matrix plies has a negligible effect on the change in ER due to the high resistivity of those plies.

## Acknowledgment

The authors would like to thank Dr. David Shiffler, Office of Naval Research (ONR) for funding this project. Our profound gratitude also goes to General Electric Aviation for the provision of the material and their supports throughout the project.

## Funding Data

• Office of Naval Research (ONR) (Grant No. N00014-18-1-2646; Funder ID: 10.13039/100000006).

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