Investigation of the Discrepancy between Optically and Gravimetrically Calculated Fiber Volume Fraction in Flax Fiber-Reinforced Polymer

1. Introduction

The fiber volume fraction is critical for accurately estimating mechanical properties, including tensile strength, stiffness [1], and fatigue performance [2], as well as electrical properties such as conductivity [3]. When designing load-bearing components from composite materials like carbon, glass, or natural fiber-reinforced polymers, it is essential to accurately determine the fiber volume fraction to obtain reliable material properties. An incorrectly assessed fiber volume fraction can lead to premature failure of components.

Several factors influence the fiber volume fraction, including the manufacturing process, as well as the types of fiber and matrix used (thermoset and thermoplastic materials) [4]. Typically, it is measured using indirect methods based on gravimetric and volumetric measurements [5]. These methods offer significant advantages due to their quick execution, ease of use, and non-destructive nature. They only require the measurement of fiber and total composite weight. The gravimetric approach (hereafter referred to as indirect measurement) can determine fiber and matrix volume fractions by utilizing the densities of the fiber and matrix, which the manufacturer must provide. Moreover, thermo-gravimetric methods can be employed for composites with glass or carbon fiber-reinforcement, where the composite is burned to measure the fiber weight content accurately, but the composite is destroyed in the process [6]. ASTM D2734 [7] can be applied to materials that can be measured using thermo-gravimetric analysis to determine the void fraction. This standard also mandates measuring composite density following ASTM D792 [8], which specifies that the composite must not absorb moisture. In general, direct methods have limitations, as they do not provide information about the morphology of the fiber contents and depend on precise weight and volume measurements. However, thermo-gravimetrical analysis is favorable over the gravimetric approach.

Optical (hereafter referred to as direct) methods, based on micrographs or micro-CT images, involve examining the cross-section of the composite material and using algorithms like Otsu thresholding [9] or the Fiber Counting Method [10] to determine the fiber volume fraction. These methods are particularly effective for carbon and glass fiber-reinforced polymers since synthetic fibers have a homogeneous structure, consistent shape, and strong contrast with the matrix [11], simplifying the analysis of these cross-sections.

The analysis becomes much more complex when dealing with natural fibers [12], which have emerged as an eco-friendly alternative [13,14] to glass fiber-reinforced polymers, with comparable material properties [15]. In our case, flax fibers replace traditional fibers in the composite. However, analyzing these natural fibers poses challenges due to their non-uniformity, fine structure, inability to be thermo-gravimetrically measured, and low contrast with the epoxy resin (cf. Figure 1). Therefore, gravimetric methods continue to be utilized for natural fibers [16,17,18].

To compare indirect with direct measurements, we use a convolutional neural network architecture known as U-Net [19], which is commonly employed in the medical field [20], to analyze micro-CT data. Existing research has explored determining fiber volume fraction using convolutional neural networks. For example, Blarr et al. developed a convolutional neural network to extract threshold for fiber volume fractions from 3D micro-CT images of carbon fiber-reinforced polymers [21]. Galvez-Hernandez et al. found that U-Net works well with small training data in determining fiber volume fraction. In addition, the architecture can process spatial information well [22]. They also utilized a U-Net architecture to segment dry areas and voids in carbon fiber-reinforced polymers [23] and compared Thresholding methods, as well as the U-Net structure, concluding that the latter produced superior results. To our knowledge, no study compares indirect and direct methods for natural fiber-reinforced polymer.

We intend to analyze and compare various pressure ranges using direct and indirect methods to evaluate the influence of pressure—the main parameter in compression molding—on different volume fractions. We will additionally compare the discrepancy between indirect and direct methods and analyze the 2D void distribution within the composite.

2. Materials and Methods

2.1. Manufacturing of Flax Fiber-Reinforced Polymer Samples Using Compression Molding

The samples were produced using a press equipped with heatable plates. Their dimensions, limited by the size of the heating plate, were 150×200 $m{m}^2$. Each laminate consisted of 8 layers of unidirectional flax [24], and the epoxy resin used was Entropy Resin 305 [25] with the hardener CPF [26]. Before pouring, the weight of the fiber and the epoxy resin was measured using a scale [27]. During the insertion of the fiber layers into the mold, a layer of epoxy resin was applied to the flax every two layers to ensure proper epoxy distribution [28]. Curing occurred at 82°C [25,26] under four different pressures: 2.33 bar, 5.7 bar, 9.4 bar, and 13.5 bar. These pressures were selected for comparability reasons [29]. The weight of each laminate is equal to 99.05 g for 2.33 bar, 93.22 g for 5.7 bar, 89.46 g for 9.4 bar, and 87 g for 13.5 bar. Due to the system, pressures lower than 2.33 or higher than 13.5 bar were not feasible. The chosen pressures were verified directly on the pressure plates using the KM115-200kN load cell from ME-Messsysteme GmbH [30]. Preliminary tests indicated that the best mechanical properties (tensile test, stiffness) were achieved at pressures around 9.4 bar. The pressing process caused excess resin to escape from the mold. The original fiber weight was measured against the weight of the finished composite plate to determine the fiber weight fraction [5]

$$W_f={\displaystyle \frac{w_f}{w_c}}$$

(1)

with $W_f$ being the fiber weight fraction, $w_f$ being the fibers’ weight and $w_c$ being the weight of the composite. Depending on the pressure, the fiber weight fractions were 56.08%, 62.19%, 64.7%, and 66.34% (from lowest to highest pressure). As the pressure increased, the thickness of the samples decreased, measuring 2.55 mm, 2.35 mm, 2.25 mm, and 2.15 mm, while the density remained approximately constant (4% between the lowest and highest pressure).

2.2. Indirect Measurement of the Fiber Volume Fraction

There are several indirect methods for determining the fiber volume fraction. These methods require additional knowledge about the composite and its components, such as density, weight, and volume. The usual approach is to divide the fibers’ volume by the composite material’s volume. Methods are categorized based on whether they utilize two phases (fiber and matrix) or three phases (fiber, matrix, and voids). A standard method used by Roe and Ansell [31], as well as Oksman [32], requires knowledge of the weight of the composite $w_c$, the weight of the fiber $w_f$, the volume of the composite $V_c$, and the density of the matrix ${\rho }_m$ to calculate the fiber volume fraction $FVF$.

$$FV{F}_{Roe}={\displaystyle \frac{V_c-\frac{w_c-w_f}{{\rho }_m}}{V_c}}$$

(2)

Before processing, weighing the fiber blankets is necessary to determine the optimum amount of resin. The weight of the fiber blankets is, therefore, known. The composite can also be easily weighed, requiring only knowledge of the resin’s density, which the manufacturer has provided in this instance. According to Madsen, this method is only suitable for fiber volume fractions not approaching the maximum fiber volume fraction because $V_f$ is overestimated (the voids are included in the fiber volume) ([33] p. 113). Another method was developed by Bisanda and Ansell [34]. Here, the fiber volume fraction is determined directly via the fiber’s weight $w_f$, measured using a scale and density of the fiber ${\rho }_f$, which the manufacturer also provides, as well as the volume of the composite $V_c$.

$$FV{F}_{Bisanda}={\displaystyle \frac{\frac{w_f}{{\rho }_f}}{V_c}}$$

(3)

According to Madsen [33], this is the more precise of the two methods. Madsen additionally developed a three-phase model that considers the voids in the composite, a factor that had not been addressed in previously mentioned models [35]. To apply this, we need to know the portions of voids in the composite. To the authors’ knowledge, this information is not available in the literature for the specific material combination and manufacturing process. Therefore, this method could not be applied.

2.3. Direct Measurement of the Fiber Volume Fraction

Direct methods require images of the composite cross-sections. Madsen suggests micrographs for this purpose [33]. However, preliminary tests revealed that grinding smears the voids, and micrographs can be highly user-dependent and time-consuming. Instead of micrographs, we opted for micro-CT imaging. The composite samples were imaged using a micro-CT equipped with a 225 kV Microfocus X-ray tube and a 2048×2048 pixel flat panel detector. One of the most commonly used direct approaches for determining the fiber volume fraction is Otsu’s thresholding [9]. This method is non-user-dependent and can find an appropriate image segmentation threshold based on finding a threshold that maximizes the intra-class variance. This method works best for data with high contrast between the different phases, i.e., distinct peaks [36]. This is the case for identifying fiber and matrix for carbon [21] and glass fiber-reinforced polymers [37]. In our case, there is a distinct peak for the void at a grayscale value around zero but only a single peak for fiber and matrix (cf. Figure 2). It can also be noted that using a filter (e.g., wiener filter) on the image removes fine structures. This leaves the large fibers but not the surrounding fine structures, making Otsu’s thresholding unviable in this case.

Nevertheless, we want to compare Otsu’s thresholding performance later on with the test data and the results of our neural network. Further information can be found in Appendix C. Other studies have already shown that the performance of convolutional neural networks is better, such as U-Net, than Otsu thresholding for the fiber volume fraction of carbon fiber-reinforced polyamide 6 [21]. This neural network architecture is often used in medical applications [38] and is well-suited for image segmentation tasks, especially when dealing with limited data [19].

2.3.1. Data Representation Overview

A sample measuring 2 mm x 8 mm was milled out for every pressure from the gravimetrically measured plates. The micro-CT scan for the composite samples was conducted along the fiber (length ≈ 2.7 mm). Due to the milling process, the sample edges are not perfectly smooth, which means parts of the edge area could not be used (see Appendix A). After manually inspecting all the images, no damage was caused by the milling process within the composite. The data is stored as voxel (16-bit grayscale) with a consistent edge length of 1.9603 $\mu m$. The resolution of the micro-CT images are 1462x1633x1632 for 2.33 bar, 1338x1486x1632 for 5.7 bar, 1506x1449x1674 for 9.4 bar and 1486x1511x1632 for 13.5 bar, respectively. The fiber direction of the composite samples aligns with one of the axes of the 3D grid, which is sliced along the aligned axis of the fibers. These extracted images are saved as 16-bit grayscale lossless PNG images for further processing.

2.3.2. Data Impurities

The micro-CT scan data is affected by noise, which varies across the images. The most significant disturbances occur in the middle of the composite samples and diminish towards the outer edges. Additionally, a discernible ghosting effect likely stems from vibrations during the imaging process. We have not conducted any additional signal processing on the provided data.

2.3.3. Summary of Data Processing Pipeline

When developing a supervised learning neural network, it is standard practice to utilize a recognized and verified dataset with clear ground truth for comparison [19,39]. However, to our knowledge, no such dataset currently exists for natural, carbon, or glass fiber-reinforced polymers. Consequently, an expert will perform manual segmentation in this instance, which is the basis for our training and test data, which was done similarly by other research groups [23,40]. The goal is to capture the expert’s expertise in an optimized manner. In this context, "optimized" means that the extrapolated segmentation closely aligns with the unseen validation data the expert has segmented by comparison. To achieve this, we train a neural network on the expert’s segmentation and utilize the trained model to predict or extrapolate the segmentation for the remaining images. Since the neural network’s performance depends on hyperparameters, optimizing them to ensure a close match between the extrapolated and expert segmentation is essential. This optimization is carried out using a Bayesian Optimization algorithm, which iteratively selects and validates the hyperparameters of the neural network. The final step of the pipeline involves testing the iteratively optimized neural network on a dedicated test set of images unknown to the neural network to ensure that it genuinely extrapolates the expert segmentation. This testing step is crucial, as it compares the predicted segmentations with the expert’s segmentation to evaluate the model’s performance across the dataset. By incorporating these steps, the pipeline ensures that the extrapolated segmentation maintains the quality and precision of the expert’s manual work while significantly reducing the time required for segmenting the entire dataset. This approach effectively balances the need for accuracy with the practical limitations of manual segmentation, making it feasible to process large volumes of images efficiently. As a reference, we have based our pipeline on similar approaches using Bayesian Optimization for neural networks, as described in [41] and [42].

2.3.4. Data and Data Augmentation

Data augmentation describes the generation of additional training data by introducing unobserved data through transformations of existing data to improve the volume, quality, and diversity of data [43,44]. Due to the relatively small size of the annotated data set, we employ data augmentation to expand it. The training data set comprises only four images, one from each natural fiber-reinforced polymer sample scan. By utilizing data augmentation, we generate a new data set of any desired size, consisting solely of augmented images. Data augmentation can significantly enhance the neural network’s performance, mainly when dealing with a small data set [45]. The following combination of transformations is applied to the images:

• Rotation (Uniform between 0 and 360 degrees)
• Translation (Uniform over the whole original image)
• Uniform Scaling (Uniform between $80\%$ and $120\%$)
• Mirroring (Horizontal and Vertical)

These similarity transformations maintain important characteristics of the augmented image, such as collinearity and angles ([46] P. 270). Specifically, these transformations enable the generation of new images similar to the original images in terms of fiber, matrix, and void fractions, although not identical in pixel positions. Additionally, noise is introduced to the images, which alters the pixel values. This technique, known as Photometric Data Augmentation, has been proven to enhance the robustness of deep learning models [44]. Furthermore, noise is a common impurity in micro-CT scan data. Due to the scarcity of data, the validation dataset comprises the original images without any data augmentation to ensure that Bayesian Optimization guides the neural network toward a solution that aligns with expert’s segmentation. The test dataset consists of 4 unseen images, one from each pressure, which is not augmented, to assess the neural network’s performance on unseen data.

2.3.5. Expert’s Segmentation Dataset

Due to its time-consuming nature, the expert performed manual segmentation on a small dataset of 8 images. Specifically, two images from each sample pressure were segmented. The dataset was divided into two parts: one for training and validation and one for testing. Each part consisted of 4 images, with one from each composite sample pressure. Only augmented images were used for training, while the validation and test data were not augmented because they were used to compare the model to the expert.

2.3.6. Neural Network Architecture and Training

The neural network employed is a modified version of the U-Net architecture, as outlined in Figure 3.

Ronneberger describes the functionality of the U-Net as follows: "It consists of a contracting path (left side) and an expansive path (right side)." [19]. The contracting path includes convolutional layers that extract features and a max pooling layer to reduce the spatial size. On the other hand, the expansion path involves restoring the original spatial size (transposed convolution) after each convolutional layer stage using a transposed convolutional layer. Through a skip connection, each stage of the contracting path is linked to the corresponding stage of the expansion path, enabling the network to regain spatial information lost during the contraction path. This design allows the network to learn features at different spatial scales while retaining spatial information for precise spatial predictions, such as segmentation.

As a consequence of the principles of the convolution operation, the spatial information gets subsequently reduced after each layer, which we compensate by padding zeros (extending the spatial size with zero values) around each layer to keep the spatial resolution constant. The neural network’s output is a probability map, calculated by the softmax activation function, representing the likelihood of each pixel belonging to one of the three classes: fiber, matrix, or void.

Formally, the softmax function is defined as:

$${\widehat{y}}_i={\displaystyle \frac{e^{z_i}}{{\sum }_{j=1}^3{e}^{z_j}}}$$

(4)

Where ${\widehat{y}}_i$ is the predicted probability of class i, $z_i$ is the raw score for class i, and ${\sum }_{j=1}^3{e}^{z_j}$ is the sum of all three classes to normalize the probabilities.

There is a class imbalance due to the significant differences in the occurrence of fiber, matrix, and void fractions in the images. We employed a weighted cross-entropy loss function described in [47] to address this. This loss function L is defined as:

$$L=-{\displaystyle \frac{1}{N}}\sum _{i=1}^N{w}_i{y}_{i}log\left({\widehat{y}}_i\right)+(1-y_i)log(1-{\widehat{y}}_i)$$

(5)

where N represents the number of pixels in the image, $y_i$ denotes ground truth of pixel i, which is drawn from the expert’s segmentation, ${\widehat{y}}_i$ signifies the predicted label of pixel i, which is the neural network segmentation, and $w_i$ stands for the class weight of pixel i. The cross-entropy loss is calculated based on regional integrals, measuring the affinity between the regions defined by the probability softmax output of the neural network and the expert’s segmentation [48]. This affinity reflects the similarity between the fiber, matrix, and void volume fractions between the predicted and expert segmentation. It is ultimately determined by the number of pixels in the image, the expert’s ground truth label of each pixel, and its predicted label.

The neural network was trained using Gradient Descent and Gradient Accumulation techniques. The training was stopped after $150.000$ Gradient Accumulation steps, which correspond to $150.000$ images processed by the neural network (an equal number of augmented images were used from each sample). Manual experiments established the effectiveness of combining Gradient Descent with Gradient Accumulation over these images. This procedure led to a promising variety of results with comparable training convergences.

2.3.7. Hyperparameter Optimization

Hyperparameters are parameters not learned during the training process but are set by the user. The selection of the hyperparameters is a critical step in the development of a neural network, as the network’s performance is highly dependent on these values. The process of selecting the hyperparameters is often done through trial and error, which can be time-consuming, computationally expensive, and leads to user-biased results. [49,50] This is especially problematic because the selection of hyperparameters is often nonintuitive, and the possible combinations of hyperparameters are vast (in our case, we have $36.000.000$ possible combinations of hyperparameters). To address this issue, we employ a hyperparameter optimization algorithm to automatically explore the combinations effectively and find the hyperparameters that yield the best performance, according to our objective function 5. The optimization of the hyperparameters for the neural network is accomplished using a bayesian optimization algorithm. This approach is well-suited for the relatively small number of hyperparameters and the computational expense of evaluating the objective function (i.e., training and testing a neural network) [51]. The hyperparameters that are optimized include:

• Learning Rate (0.01 to 1, 20 linearly spaced values)
• Batch Size (1 to 40, 10 non-uniform spaced values)
• Data set Size (1 to 40, 10 non-uniform spaced values)
• Data set Iterations (1 to 40, 10 non-uniform spaced values)
• Class Weights 1-3 (3 Distinctive Hyperparameters, 0 to 1 each, 30 linearly spaced values each)
• Convolutional Kernel Size (3 and 5, 2 values)
• Data Augmentation Noise (0 to 0.05, 10 linearly spaced values)
• Input/Output Size (128 to 256, 3 Uniform spaced Values)

Implementing the Bayesian optimization algorithm involves the utilization of Gaussian process regression, which serves as a model for the objective function [51]. This regression technique enables the algorithm to predict the objective function at points where no evaluation has been made. As an Objective Function or Validation Metric, we use again the Cross-Entropy Loss between the expert segmentation and the predicted segmentation as described in equation 5.

2.3.8. Neural Network Testing

The neural network is tested using the remaining four images not included in the training or validation process. We independently assess the predicted segmentation against the expert’s segmentation, considering the total fraction error for each class (fiber, matrix, void) and test image. Furthermore, we visually compare the predicted and expert segmentation to evaluate the neural network’s performance.

2.3.9. Software and Hardware Infrastructure

Our software is developed entirely in Julia, a high-level, high-performance programming language for technical computing [52]. Additionally, we extensively utilize the Lux framework, a high-level deep learning framework for Julia [53]. We employ the Surrogates.jl package for Bayesian Optimization, which offers Gaussian Process Surrogates. We use three common desktop PCs with recent Nvidia GPUs configured as a cluster for training. Inference of the neural network is performed on a single desktop PC with an Nvidia GPU. The output data was analyzed using Matlab 2023b.

2.3.10. Comparison Against Commercial Software Segmentation Tools

Commercial tools like Avizo [54] and VGStudio Max [55] provide powerful and user-friendly solutions for image segmentation using neural networks. Avizo includes "a default set of deep learning Python packages and modules based on Keras, a high-level neural networks API running on top of TensorFlow" [54], while VGStudio Max offers a Deep Segmentation module supporting ONNX-format models and the "Paint & Segment" feature for machine learning-assisted segmentation.

While these tools are excellent for general-purpose applications, our approach is tailored to the specific requirements of this study. By combining Bayesian Optimization and a compute cluster with multiple GPUs, we minimize user influence, ensuring consistent and objective segmentation aligned with our objective function (Equation 5). In contrast, commercial tools often prioritize ease of use and rely on user-defined setups, which may introduce variability.

Although a direct comparison is not included in the Results chapter, this is due to differences in scope. Commercial tools and our method address different needs, focusing on maximizing accuracy and reproducibility for this specific application.

3. Results & Discussion

3.1. Neural Network Data Processing

We carried out automated hyperparameter optimization for approximately three days to obtain an acceptable neural network for the segmentation task. Through this process, we effectively explore most hyperparameter configurations that lead to an acceptable final neural network. Following this process, we selected the best-performing neural network, according to our objective function 5, to segment each sample’s micro-CT scan. The best-performing or, respectively, the final neural networks hyperparameters used for segmenting each sample’s micro-CT scan can be taken from Table 1. More information about the neural network’s performance can be found in Appendix B. More Information about the performance of Otsu’s thresholding can be found in Appendix C.

The segmented image is constructed from the combined three classes: fiber, matrix, and void, as shown in Figure 4. The neural network can process significant variations in brightness that occur for each sample. Inference of the Neural Network for one full image of approximately $1500\times 1500pixel$ takes about $50ms$, so a total micro-CT scan with approximately 1000 images takes about $50s$.

3.2. Pressure Dependence of Volume Fractions

Figure 5 illustrates an example of the analysis for each pressure used. The colors red, blue, and green represent the natural fiber, matrix, and voids, respectively. In Figure 5, the purple regions at the fiber’s edge indicate an area of uncertainty, whether it is fiber or matrix. Large matrix regions and voids are visible at a pressure of 2.33 bar, but higher pressures, such as 5.7 bar, reduce these matrix regions. Higher pressures also lead to the fragmentation of large voids into smaller ones located primarily in the middle of the composite. At pressures of 9.4 and 13.5 bar, the uncertainty region is at its peak. The unit cell changes from a square at low pressures to a hexagonal at higher pressures.

We can use images like these in Figure 5 to calculate the volume fraction by counting the sum of red, blue, and green pixels, respectively, and dividing by the number of total pixels in an image. The region of uncertainty needs to be resolved, so each pixel is assigned to the class it is most likely to belong to.

$$FV{F}_{U-Net}={\displaystyle \frac{{\sum }_{i=1}^{N}F\left(i\right)}{N}}$$

(6)

$$MV{F}_{U-Net}={\displaystyle \frac{{\sum }_{i=1}^{N}M\left(i\right)}{N}}$$

(7)

$$VV{F}_{U-Net}={\displaystyle \frac{{\sum }_{i=1}^{N}V\left(i\right)}{N}}$$

(8)

N corresponds to the number of pixels, F to the number of fiber pixels in the image, M to the number of matrix pixels, and V to the number of void pixels in the image. We can repeat this process for each image and pressure used. By plotting these values over the length of the composite, we obtain Figure 6.

Black markings with Roman I, II, and III show regions of interest (for a full explanation, cf. Appendix A). The neural network cannot evaluate the areas where complete images were unavailable, so only the areas with complete images are considered below, shown within the black dashed lines, which were inserted manually and represented the composite’s first and last complete image. Within these regions, we can calculate the mean and standard deviation of the volume fraction for each pressure variation, respectively, which are depicted in Figure 7a using the same color scheme as in previous figures (red for fiber, blue for matrix, and green for void). Even at the lowest pressure of 2.33 bar, a fiber volume fraction of approximately 69.85% is observed, which increases to 76.53% with a pressure increase to 5.7 bar—further increasing the pressure to 9.4 bar resulting in a fiber volume fraction of around 82.24%. The relationship between fiber volume fraction and pressure is nearly linear at this pressure range (2 ${\displaystyle \frac{\%}{bar}}$ at a pressure increase from 2.33 to 5.7 bar and 1.5 ${\displaystyle \frac{\%}{bar}}$ at a pressure increase from 5.7 to 9.4 bar). However, this linear relationship no longer holds when the pressure is further increased to the maximum of 13.5 bar, with the fiber volume fraction changing by less than 1% compared to the previous pressure, reaching 83.02% (Gradient is equal to 0.19 ${\displaystyle \frac{\%}{bar}}$ at a pressure increase from 9.4 to 13.5 bar). In all cases, the fiber volume fraction’s standard deviation is less than 1%. Madsen also observed a similar stagnation of the fiber volume fraction for flax and jute with various thermoplastics, noting that the fiber volume fraction increases linearly up to a certain fiber weight fraction and then remains constant between 50 and 60%, depending on the material combination [35].

The matrix volume fraction decreases linearly in the first three pressure ranges, from 25.75% at 2.33 bar to 20.34% at 5.7 bar and to 14.85% at 9.4 bar (cf. Figure 7a in blue, gradients equal to -1.6 ${\displaystyle \frac{\%}{bar}}$ and -1.48 ${\displaystyle \frac{\%}{bar}}$). This behavior corresponds to an almost constant drop of about 5.4% for the first three pressure ranges. A behavior change can be observed beyond this point with a further increase in pressure. At a set pressure of 13.5 bar, the matrix volume fraction shows an average value of 15.3%, indicating an increase of about 0.5%. In all cases, the standard deviation is less than 1%. The gradient of the matrix volume fraction at a pressure change from 9.4 to 13.5 bar corresponds to 0.11 ${\displaystyle \frac{\%}{bar}}$.

At the lowest pressure of 2.33 bar, the void volume fraction is around 4.4% and drops to around 3.12% when the pressure is increased to 5.7 bar, resulting in a reduction of around 1.28%. When the pressure is increased to 9.4 bar, there is only a slight reduction in the void volume fraction to a value of 2.9%, a behavior that differs from the fiber and matrix volume fraction. If the pressure is further increased to 13.5 bar, the void volume fraction decreases to 1.65%, corresponding to a reduction of about 1.25%. The gradient corresponds to -0.38 ${\displaystyle \frac{\%}{bar}}$ for a pressure change from 2.33 to 5.7 bar. When the pressure is increased to 9.4 bar, the gradient is close to zero at -0.06 ${\displaystyle \frac{\%}{bar}}$. Finally, when the pressure is further increased to 13.5 bar, the gradient becomes 0.31 ${\displaystyle \frac{\%}{bar}}$. The standard deviation is below 0.6% in all cases.

In the next step, we want to examine the distribution of voids in the composite in more detail. At this point, it should be noted that no 3D voids were examined. The aim is to highlight the general location of the voids (e.g. at the edges or the center of the composite). For example, so-called unfolding algorithms can examine 3D voids from 2D images. These typically lead to errors [56], so 3D voids are not examined directly here but rather void distributions in each discrete layer, which is depicted for every pressure in Figure 8, using the height of the composite. The height is normalized to account for variations in thickness due to different pressures; hence, 0.5 corresponds to the center of the composite. At the lowest pressure range of 2.33 bar, irregularly distributed voids are observed throughout the composite. The highest void volume fraction was found on the left side up to the center of the composite. This uneven distribution could significantly impact the mechanical properties of the composite, particularly under tensile, compressive, and bending loads, as it would cause uneven deformation. This could also explain the additional variance in the behavior of the composite [57,58], which is further exacerbated by the unevenly grown fiber [59,60]. Increasing the pressure to 5.7 bar reduces the overall void volume fraction, which remains irregularly distributed in the composite. A different void volume fraction is observed at the subsequent higher pressure of 9.4 bar compared to the previous pressure levels. Most of the void fraction is concentrated in the center of the composite. As depicted in Figure 5, the voids at these points tend to be smaller, in contrast to the lower pressure variants. With fewer voids at the edges overall, this pressure variant should have less influence on the bending stiffness. The high pressure allows the air to escape from the edge areas of the composite but not from the center. Upon further increasing the pressure to the maximum of 13.5 bar, a lower void volume fraction is observed in the center.

The behavior we observe differs from Madsen’s 3-phase model. According to his model, the fiber volume fraction remains stagnant, which is consistent with our data [35]. However, the model also predicts an increase in the void fraction and a simultaneous decrease in the matrix volume fraction. In our production process, we observed a linear increase in the fiber volume fraction up to a pressure of 9.4 bar, after which the value remained constant. The matrix volume fraction decreased to this pressure and then slightly increased at higher pressures. The void volume fraction remained constant between 5.7 and 9.4 bar in the pressure range but decreased in other ranges.

Figure 7b illustrates the percentage of fiber weight in the composite material. This value is directly measured after the panel is manufactured, considering the weight of the fibers and the final composite. The figure depicts a degressive trend, starting at 56.08% at 2.33 bar and increasing to 62.18% at 5.7 bar. Subsequent increases in pressure result in only minor changes in the fiber weight fraction, reaching 64.69% at 9.4 bar and 66.34% at the highest pressure of 13.5 bar. When applying higher pressures, more matrix can escape, leaving relatively more fiber in the composite, leading to a higher fiber weight fraction. This also directly impacts the fiber volume fraction, which can be observed as both the weight and volume fraction show a similar degressive trend.

3.3. Discrepancy Between Direct and Indirect Fiber Volume Fraction

This section compares the direct and indirect fiber volume fractions. Initially, these were determined using the methods described by Roe and Ansell [31] (referred to as Roe) and Bisanda and Ansell [34] (referred to as Bisanda). These methods are independent of the matrix weight or volume, making the matrix volume that leaked out irrelevant for the following analysis. The manufacturer provides the density for the matrix and fiber, allowing the determination of both fiber volume fractions. A comparison between the two indirect methods reveals that at low pressures of 2.33 and 5.7 bar, Roe corresponds to approximately 47.83% and 54.12% fiber volume fraction, respectively, while Bisanda corresponds to 47.93% and 54.27%, respectively. The two methods differ by just under 1% at higher pressures. For 9.4 bar, Roe corresponds to 57.07%, while Bisanda corresponds to 56.59%. At the highest pressure of 13.5 bar, Roe corresponds to 58.35% and Bisanda to 59.06%. Both methods show minimal differences in our data. According to Madsen [35], the fiber volume fraction, calculated as proposed by Bisanda, is more accurate when the fiber volume fraction approximates the maximum possible fiber volume fraction. Therefore, the fiber volume fraction, according to Bisanda, is used for the subsequent analysis and calculations. The fiber volume fraction, as per Bisanda, is depicted on the far left of Figure 9. In the figure, red is the fiber, while blue is the matrix. Since the void fraction cannot be determined using this method, it is assumed to be 0%. Figure 9 also presents the results of direct measurements using U-Net, as described in detail in the previous section. A direct comparison reveals that the measured fiber volume fraction is consistently higher than the expected value, according to Bisanda. For instance, at a pressure of 2.33 bar, a matrix volume fraction of 52.07% would be expected, but only about 25.74% is visible. It has to be considered that the matrix is incompressible [61]. Therefore, one possible explanation for this discrepancy is the potential formation of a fourth phase consisting of fiber and matrix (mixed fiber). Consequently, the direct fiber volume fraction is first calculated without accounting for the void fraction. The mixed phase $MxV{F}_{4Phase}$ then corresponds to

$$MxV{F}_{4Phase}=MV{F}_{Bisanda}-MV{F}_{U-Net}$$

(9)

while $MV{F}_{Bisanda}$ corresponds to the matrix volume fraction using Bisanda and $MV{F}_{U-Net}$ correspond to the matrix volume fraction by using U-Net. This results in the pure fiber in the four-phase model $FV{F}_{4Phase}$ being

$$FV{F}_{4Phase}=FV{F}_{U-Net}-MxV{F}_{4Phase}$$

(10)

while $FV{F}_{U-Net}$ is the fiber volume fraction calculated by U-Net. Finally, the pure matrix $MV{F}_{4Phase}$ remains equivalent to the matrix volume fraction estimated by U-Net $MV{F}_{U-Net}$.

$$MV{F}_{4Phase}=MV{F}_{U-Net}$$

(11)

The void fraction should again be considered, as shown on the right-hand side of Figure 9.

In Figure 9 on the right-hand side, the mixed fiber is depicted in purple and corresponds to 24.08% at 2.33 bar, 23.98% at 5.7 bar, 27.31% at 9.4 bar, and 24.94% at 13.5 bar, respectively. This fourth phase may represent an additional phase between the fiber’s interior and the pure matrix, which could be modeled like [62] suggests. No optical differences are observed in the micro-CT images(cf. Figure 5); however, research that used fluorescence epoxy could illustrate a similar complex interphase for the same combination of materials (flax with epoxy) [63]. The fiber is likely to be saturated with matrix at around 25% of the total volume, with no further matrix absorption possible, which forms a mixed region, which is consistent with the general hygroscopic behavior of the fiber [64,65]. Additionally, it could explain the inferior material properties of the natural fiber-reinforced polymer, as higher fiber moisture content brings the fiber closer to its saturation limit, reducing its ability to absorb the matrix, which leads to weaker fiber-matrix adhesion and inferior properties [66]. Considering fiber saturation in future models for natural fiber-reinforced polymers is essential. Otherwise, the indirect measurements of the volume fractions will lead to high errors, resulting in different material properties. The four-phase model introduced here represents the first step towards such a model and must be further refined in future research. A summary of all described volume fractions can be found in the Appendix D.

From our perspective, the production process for natural fiber-reinforced polymer works as follows: Immediately after the matrix is applied, the hygroscopic fiber begins to absorb the matrix. When pressure is applied, the composite is compacted, and excess matrix escapes from the mold. A square unit cell’s maximum fiber volume fraction can be at most ${\displaystyle \frac{\pi }{4}}$, which is approximately 78.54% [67]. To further increase the fiber volume fraction, the fiber can first arrange itself in a hexagonal pattern, whose maximum value is ${\displaystyle \frac{\pi }{2\sqrt{3}}}$, approximately 90.6% [67]. As the fiber volume fraction approaches this value, more pure matrix material must be displaced while the approximate 25% absorbed matrix in the fiber stays constant. This is supported by the fact that the diameter of the fibers, shown in Figure 5, stays the same, no matter which pressure is applied. As more matrix escapes, more voids can also escape from the composite. When the process is slowed down, higher and higher pressure is needed to displace less and less matrix, which causes the shape of the voids to transform into fragmented small voids, which mainly accumulate in the center of the composite and also lead to a degressive behavior in fiber weight fraction, as depicted in Figure 7. If the pressure is further increased beyond our current capabilities, our data suggest that the matrix will escape up to the theoretical maximum of the hexagonally packed unit cell. However, the fiber must deform to reach levels above 90.6% for more matrix to escape. In practical terms, this range should be of minor importance, as the high pressure required for this degree of matrix displacement would also cause damage to the fiber, similar to the damage microcracks shown in [68,69] and result in low matrix content, which could lead to inferior material properties [69].

4. Conclusion

Our study reveals a discernible correlation between applied pressure and fiber volume fraction, indicating that increased pressure leads to higher fiber volume fractions and reduced matrix and void content up to a certain threshold. However, the volume fraction increase plateaued beyond 9.4 bar. Discrepancies between direct and indirect measurement methods were also observed, hinting that indirect methods like Bisanda (cf. Figure 9) underestimate fiber content by around 25% as the absorption of the matrix by the fiber is not considered. The four-phase model can explain the discrepancy between direct and indirect methods. This mixed phase should be the target of further research, as it is essential to explain the fiber-matrix adhesion correctly. To develop a complete model similar to Madsen, it is necessary to investigate significantly more pressures to capture the dependency of the volume fraction with the applied pressure. We also investigate the void distribution, which fragments into smaller voids at higher pressures and concentrate around the middle of the composite. In future mechanical investigations, classical compression molding may need to be replaced with vacuum-assisted compression molding to maximize the composite’s material properties.

Furthermore, U-Net is a dependable tool for fiber volume analysis, surpassing traditional direct methods, like Otsu’s thresholding, especially for evaluating complex materials like natural fiber composites. It is important to note that the current trained neural network is limited explicitly to analyzing our natural fiber-reinforced polymers with flax and would need retraining for other cases. Furthermore, gathering data from various fibers, such as jute and kenaf, as well as matrix combinations, like thermoset and thermoplastic material, will be crucial for developing generalized models in the future. The data set analyzed in the present study can be accessed by the community using the following git repository: https://gitlab.rlp.net/beckmaa/iotdboagcfvfiffrp/. The aim is to continuously extend this data set in the future so that the community can evaluate the performance of their own methods in an even more objective manner.