Identification and Quantification of 3D Fiber Clusters in Fiber-Reinforced Composite Materials

Abstract

Microscale computed tomography scans of fiber-reinforced composites reveal that fibers are most often not strictly parallel to each other but exhibit varying degrees of misalignment and entanglement. One characteristic of this entanglement is the degree to which fibers stay together as clusters. In this study, a method for identifying and isolating fiber clusters was established, and scans of two different composite microstructures were analyzed. To identify clusters, fiber center points of the first cross-section were triangulated, and the variation of the perimeter and area of triangles along the fiber direction was used to identify fiber triads which stay together. A filtering process eliminated fiber triads not part of a larger cluster. Geometric properties of the clusters such as cluster orientation, radius of gyration, cluster density, and volume fraction were calculated and compared. The metrics revealed fundamental differences between the two samples, suggesting that clusters have origins in manufacturing.

Appendix

Fiber Path Extraction

To confirm the validity of the extracted fiber paths, a second method for fiber detection was used. The second method used a novel algorithm, coded in Mathematica, that mixed tools from mathematical morphology and traditional Gaussian filters. A median filter, with a disk structuring element, was applied to the images, followed by thresholding and Gaussian smoothing. The fiber centers were identified by finding the local maxima of the filtered image. The blending of image foreground and background made traditional gradient-based edge detection methods inapplicable. It was concluded that these optical artifacts were due to surface contamination since they did not persist from one cross-section to the next. The algorithm based on mathematical morphology is computationally more intensive but performs better on noisy images. These difficulties notwithstanding, the two methods were always within 1 pixel on the centers of the fiber cross-sections and agreed with each other on more than 90% of the disks.

Use of Otsu’s Method

Otsu’s method is typically used to threshold greyscale images for the purpose of, for example, separating background from foreground or isolating objects. It was recognized, however, that Otsu’s method provides a reliable way of splitting histograms into specific regions. The implementation of Otsu’s method used in this work is a MATLAB function, multithresh, which can split histograms into multiple regions based on the input number of thresholds.

To find the thresholds for ${V}_{A}$ and ${V}_{P}$, histograms of both values were generated. For all samples, these histograms contained long, discontinuous tails filled with outliers. For each histogram, the average and standard deviation were calculated. The tails were then trimmed by filtering out all values greater than the average plus twice the standard deviation (Fig.

Fig. 11

Histogram plots of (a) untrimmed and (b) trimmed histograms with thresholds for total variation of area created using Otsu’s method

11).

The multithresh function was then used to split the total variation histograms into three bins. The multithresh function splits the distribution of values into $k$ number of bins with the objective of minimizing the variation of values within each bin as expressed in

$$ \arg \mathop {\min }\limits_{S} \mathop \sum \limits_{i = 1}^{k} \mathop \sum \limits_{{x \in S_{i} }} \left\| {x - \mu_{i} } \right\|^{2} $$

(10)

where $S$ is the set of partitioned values of the input data, $x$ is a particular entry of $S$, also known as ${S}_{i}$, and ${\mu }_{i}$ is the mean of the points in $x$. This is also the function used in K-means clustering, a process in which groups of values are clustered together by sets of points in which the variance for those points is minimized.35

All triads with ${V}_{A}$ and ${V}_{P}$ values below the lower threshold were determined to be clustered triads. The triads in the two upper bins were not studied in this work. These bins are hypothesized to contain both intermediate triples, which contain relatively equal proportions of fiber and matrix, and matrix-rich triads. Future works will focus on what these bins truly represent.

Filtering

Filtering algorithms are applied to the clustered triads to remove extraneous fibers which do not appear to be clustered. The subtraction algorithm loops through each clustered triad and removes it from the cluster if two of its neighboring triads do not belong to a cluster. The addition algorithm loops through every nonclustered triad and adds it to a cluster if two of its neighbors belong to one. A generalized version of these two algorithms, where each triad becomes whatever two of its neighbor triads are, is used as a first step in the overall filtering process. A final, subtractive filtering step is used to remove any remaining noise and allow the clusters to converge to their final shapes. The entire filtering process is then repeated until the number of clusters converges (Fig. 5e).