# Reliability modeling of tensile properties in recycled carbon fibers: A comparative fit of statistical distributions pre- and post-textile processing

Jean Ivars, Kim Phuc Tran, Ahmad Rashed Labanieh, Damien Soulat

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## To cite this version:

Jean Ivars, Kim Phuc Tran, Ahmad Rashed Labanieh, Damien Soulat. Reliability modeling of tensile properties in recycled carbon fibers: A comparative fit of statistical distributions pre- and post-textile processing. Materials Today Communications, 2025, Materials Today Communications, 43, pp.111818. 10.1016/j.mtcomm.2025.111818. hal-05007418

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Submitted on 26 Mar 2025

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# Reliability modeling of tensile properties in recycled carbon fibers: A comparative fit of statistical distributions pre- and post-textile processing

Jean Ivars* , Kim Phuc Tran , Ahmad Rashed Labanieh , Damien Soulat

Univ. Lille, ENSAIT, ULR 2461 - GEMTEX - Génie et Matériaux Textiles, Lille F-59000, France

## ARTICLE INFO

### Keywords:

- Statistical fitting

- Recycled carbon fiber

- Tensile properties

- Statistical distributions

- Reliability modeling

## ABSTRACT

Due to their extensive use in the aerospace, automotive, wind energy, and sports industries, the recycling of carbon fiber composites has become increasingly important. Characterization of the tensile properties of these recycled fibers is essential, using single fiber tensile tests that provide data on tensile strength, elongation at break, and tensile modulus. This study compares four statistical distributions—2-parameter Weibull, 3-parameter Weibull, Normal, and Lognormal—to fit tensile data from two batches of recycled carbon fibers: one after recycling, and the same type of fiber after a further carding process. Maximum Likelihood Estimation (MLE) was used for the fitting process, while the Sum of Squared Errors (SSE) and Kolmogorov-Smirnov (KS) tests were employed to evaluate the fitting performance. The 2-parameter Weibull distribution provided the best fit overall for tensile strength, elongation at break, and tensile modulus. However, the Normal distribution performed slightly better for tensile strength and tensile modulus in the post-carding batch, indicating that different models might be more appropriate at various processing stages. This study illustrates the challenge of selecting a single model and balancing computational accuracy with mechanical consistency across different properties and conditions, providing insights into the complex behavior of discontinuous fiber composites and contributing to a better understanding of the mechanical challenges associated with recycled materials and their processing.

## 1. Introduction

The recycling of carbon fiber composites is becoming increasingly important as their applications expand in high-performance industries such as aerospace, automotive, wind energy, and sports [35]. These materials are valued for their high strength-to-weight ratio, stiffness, and durability, making them essential for advanced engineering applications. However, the environmental impact and high production costs of carbon fibers have driven the development of effective recycling technologies to promote sustainability and circular economy principles. Recycling typically involves separating the resin matrix from the fibers, using methods that cause slight degradation to the mechanical properties of the recovered fibers [21,23,28]. Nevertheless, uncertainties regarding the mechanical behavior of recycled carbon fibers (rCFs), arising from recycling processes, including length degradation, reduced adhesion to some resins, and residual chars on the surface... remain barriers to their reuse in structural applications.

To evaluate the mechanical performance of rCFs, single-fiber tensile testing is the most widely used method. This approach provides data on

tensile strength, elongation at break, and tensile modulus, which are essential for understanding fiber behavior. However, these tests are time-consuming and require meticulous sample preparation to minimize variability [2,14,16]. The tensile failure of carbon fibers is mainly influenced by surface defects that act as stress concentration points, initiating rupture under load [15]. Understanding how these defects translate into variability in mechanical properties requires probabilistic modeling.

The Weibull distribution is the most frequently applied model in the literature for brittle fibers, including carbon fibers, due to its capacity to account for variability caused by defect distributions. Two-parameter and three-parameter Weibull models are commonly used to characterize the statistical variation of fiber strength, with the choice of model depending on experimental conditions [11–13,25,30,9]. While widely accepted for virgin fibers, the applicability of these models to rCFs has not been systematically evaluated, particularly when additional transformation processes such as carding are introduced during recycling.

Carding, a preliminary step in textile transformation, is used to create either a nonwoven fabric or a yarn by reforming a continuous

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: jean.ivars@ensait.fr (J. Ivars), kim-phuc.tran@ensait.fr (K.P. Tran), ahmad.labanieh@ensait.fr (A.R. Labanieh), damien.soulat@ensait.fr (D. Soulat).</footnote>
structure from discontinuous fibers. This process may also introduce
changes to defect distributions and surface morphology, potentially
affecting mechanical variability.

In addition to Weibull models, other statistical distributions such as normal and lognormal distributions have been employed to model tensile properties. Studies on the tensile behavior of natural fibers have highlighted notable differences between two Weibull models (2-parameter and 3-parameter) for describing strength, elongation at break, and tensile modulus as a function of varying gauge lengths. Amroune et al. [3] analyzing a large dataset of flax fiber tensile behavior, demonstrated that despite the wide dispersion observed in tensile tests, statistical analysis combining multiple models (2- and 3-pa- rameters Weibull) and fitting estimators (Maximum Likelihood Esti- mation (MLE) and Least Squares (LS)) effectively captured the distribution of these mechanical properties across different gauge lengths. They concluded that the 2-parameter Weibull model consis- tently provided better results compared to its 3-parameter counterpart. Similarly, Belaadi et al. [5], focusing on jute fiber tensile behavior, identified the 2-parameter Weibull model as the most effective tool for predicting failure probabilities based on their tensile data. For hemp fibers, Pepi et al. [22] recently compared 3-parameter Weibull and Gaussian distributions, concluding that the Weibull distributions offered the best fit, with the highest correlation coefficients compared to the Gaussian models.

In the context of virgin carbon fibers, Wang and Bian [31] applied the 2-parameter Weibull distribution to analyze the dispersion in tensile strength, characterizing it through the Weibull modulus concerning to the micropore radius. Their study established a significant link between the micropore radius and the tensile strength variability of carbon fibers, underscoring the utility of Weibull models in describing defect-driven mechanical behavior. El Azm et al. [1] investigated the reliability of the Weibull distribution using various methods, including MLE, on datasets of different sizes associated with tensile strength in jute and carbon fibers. They employed Kolmogorov-Smirnov (KS) statistics with p-values to evaluate model fit and standard errors (SE) to assess accu- racy. Their results highlighted the efficiency of different estimation methods and the role of sample size in determining Weibull parameter accuracy. Moreover, Mesquita et al. [18] analyzed the influence of sample size on the variability of Weibull parameters using tensile behavior data for carbon and glass fibers. Employing a 2-parameter Weibull model with MLE, they concluded that datasets containing 50 samples, a common size in the literature, could result in a deviation of the Weibull modulus by ±1 compared to estimations from datasets of approximately 200 samples. Collectively, these studies illustrate the complexity of selecting an appropriate statistical distribution model for fiber mechanical characterization. The choice of model depends on the fiber type, dataset size, and testing conditions, with no consensus on a universally applicable statistical distribution. However, to the authors' knowledge, most of these investigations focus on virgin or natural fibers, leaving uncertainty regarding the applicability of these models to recycled carbon fibers. While studies on virgin and natural fibers pro- vide valuable insights, the statistical modeling of rCFs, particularly after post-recycling transformation steps such as carding, remains largely unexplored. Identifying whether the most suitable statistical distribu- tions can vary across mechanical properties and processing stages is essential for advancing the reliable reuse and mechanical modeling of rCFs.

This work aims to address this challenge by comparing four statis-
tical distributions, 2-parameter Weibull, 3-parameter Weibull, normal,
and lognormal, to model the tensile properties of recycled carbon fibers.
Two distinct batches of rCFs are analyzed: one after recycling process

(Batch 1) and another of the same batch but with a further carding
process (Batch 2). The selection of these distributions is motivated by
both their widespread use in the literature for Weibull, and their rele-
vance to the mechanical behavior of fibers. Indeed, the Normal distri-
bution is included because it is a fundamental statistical model that can
approximate the Weibull distribution when the Weibull modulus is in
the range of 3–4, as shown in previous studies [27]. This resemblance
makes the Normal distribution a benchmark for understanding fiber
behavior under specific conditions. Finally, the Lognormal distribution
is chosen due to its suitability for positively skewed data, such as tensile
strength, elongation at break, and tensile modulus, where all values are
positive. This characteristic is relevant for describing mechanical
properties that cannot be negative.

MLE, a method used for fitting probabilistic models to experimental
data, is employed in this study. It provides parameter estimates by
maximizing the likelihood function, ensuring the model aligns with the
observed data. In addition to MLE, the LS method, which minimizes the
sum of squared differences between observed and predicted values, is
also a well-established technique for parameter estimation. Nketiah [19]
conducted a comparison between MLE and LS. Building on this work, an
evaluation of the performance of these two methods will be conducted to
confirm the choice of MLE for this study.

To evaluate the statistical performance of each distribution, three
metrics are used: the Sum of Squared Errors (SSE), the KS-test value and
the KS p-value. The KS-test compares an empirical distribution with a
theoretical one using the KS statistic and the KS p-value to assess sta-
tistical similarity [17]. The SSE quantifies the difference between
observed values and model predictions by summing the squared dif-
ferences. Together, these metrics allow for a comparison of the distri-
butions' fitting performance. This study aims not only to identify the
most suitable statistical distribution for each batch but also to analyze
the performance of these models across three mechanical properties:
tensile strength, elongation at break, and tensile modulus. Beyond
developing a fitting methodology for recycled carbon fibers before and
after textile processing, the study stresses out the importance of
balancing statistical performance with mechanical consistency, enabling
meaningful comparisons between batches.

The remainder of this paper is organized as follows: Section 2 pre-
sents the experimental framework, including data preparation for both
batches along with an overview of the mechanical characteristics of the
fibers, descriptions of the four statistical models, and an explanation of
the fitting methods. Section 3 focuses on the results for tensile strength,
elongation at break, and tensile modulus, detailing the fitting parame-
ters for each statistical distribution and their performance metrics.
Section 4 discusses the findings, and the conclusion summarizes insights
and highlights perspectives for future work.

## 2. Statistical analysis methodology

Following the presentation and the objectives of this study, this
section provides an overview of the proposed approach for the statistical
analysis of the tensile properties of recycled carbon fibers.

2.1. *Experimental framework*

This study examines the statistical fitting of tensile properties for
recycled carbon fibers using tensile data. The carbon fibers used are
Standard Modulus (SM) fibers reclaimed through steam thermolysis
[20]. This process preserves the diameter of the fibers, causes only a
minor reduction in tensile properties, approximately 10%, and results in
a clean surface [4,6]. The fibers are unsized and cut to a length of
<figure><img src="image_3.png" /><figcaption>Fig. 1. Carding machine used to process discontinuous rCF into continuous web.</figcaption></figure>

60 mm.

These fibers are used as input material for the carding step. Carding is a textile transformation process that individualizes the fibers, forming a continuous structure, such as a web, from the initially discontinuous recycled fibers. This step facilitates further processing into nonwoven materials or yarn. Fig. 1 illustrates the carding machine.

The fibers are processed using the carding machine with defined machine settings. In this study, the tensile data for post-carding fibers were obtained from fibers directly recovered from a web composed entirely of 100 % recycled rCF, then prepared for mechanical testing to evaluate their tensile properties alongside those of fibers before carding.

Tensile tests were then performed following the NF ISO 11566 standard [2] for single carbon fibers. Individual fibers were mounted on cardboard frames using cyanoacrylate glue. The frames were placed on an INSTRON MTS tensile testing machine with a 10 N load cell, using a gauge length of 25 mm and a speed of 2 mm/min. Compliance effects were not included in this analysis [16]. While the NF ISO 11566 standard recommends a minimum of 20 tests per fiber type, the inherent variability of carbon fibers and the demands of statistical modeling require a larger sample size as mentioned before.

In this study, 51 reliable tensile data points were gathered for batch 1 and 45 for batch 2. Analyzing the differences in the standard deviation of the Weibull modulus and its convergence with increasing sample size confirmed the validity of comparisons between the two datasets, even with a slight difference in the number of data points.

Diameter measurements were taken at five points along each fiber length through longitudinal observation using the OLYMPUS DSX1000 microscope. A circular cross-section was assumed based on the circularity coefficient [7] that was calculated at 0.956 ± 0.014 across 11 samples, close to 1, confirming that the fibers are nearly circular before carding. These diameter measurements were used as input data for the tensile property calculations.

Two batches of recycled fibers were tested: Batch 1 represents fibers directly after the recycling process, and Batch 2 includes fibers from the same source that underwent carding. Variables recorded for each batch include tensile strength (MPa), tensile modulus (GPa), elongation at break (%), and fiber diameter (µm). These data serve as the basis for fitting, and for comparing the fitting performance of different statistical distributions across mechanical properties and processing stages.

The mechanical properties of the two batches are summarized in Table 1.

Table 1 indicates that Batch 2, after carding, shows a slightly higher tensile strength and tensile modulus compared to Batch 1. The standard

errors for tensile strength and elongation at break are similar across both batches, while the modulus shows a lower standard error in Batch 2, suggesting a higher consistency. The fiber diameter remains consistent for the two batches.

Taking the framework and mechanical context elements into account, and while few studies analyze statistical data on elongation and tensile modulus compared to tensile strength, this work aims to enhance understanding of these properties from a statistical perspective. Moreover, statistical analyses combining all three mechanical properties have not, to our knowledge, been conducted for recycled carbon fibers, leading to the detailed exploration of distribution models in the next part of this study.

## 2.2. Distribution models

The elastic-brittle behavior of carbon fiber is based on the assumption that it is an assembly of links, with tensile failure occurring when the weakest link in the assembly breaks [8]. This modelling principle for the brittle fibers enables the implementation of the Weibull model. It is employed with 2 and 3-parameter types in addition to Normal and Lognormal distribution models.

Although the Weibull distribution, used widely in carbon fiber mechanical analysis, is not an intrinsic characteristic of carbon fiber, it is a statistical model applied to assess the strength variability in carbon fiber composites which can facilitate the prediction of failure probabilities under varying stress conditions. The Weibull Cumulative Distribution Function (CDF) is used to represent the probability of failure *P* at a given value of *σ* the applied stress on the fiber. CDF provides the probability that a randomly selected value from the distribution is less than or equal to the studied value. The 3-parameters Weibull model is given in Eq. 1 with the key parameters: *σ*₀ – scale parameter (indicating the average lifetime under stress at which 63.2 % of fiber are broken), *m* – shape of the distribution, or Weibull modulus (measuring the variability, the spread or scatter of fiber strength), *σ*<sub>u</sub> – location parameter (reflecting the threshold for fiber failure initiation, below this value the failure

<figure><img src="image_4.png" /><figcaption>Fig. 2. Strength probability distribution of a carbon fiber by Weibull model with σ₀ = 4000MPa (scale parameter), remade from [31].</figcaption></figure>

<table><caption>**Table 1**
Summary of fiber properties depending on the batch.</caption><thead><tr><th rowspan="2"></th><th colspan="1">Data points</th><th colspan="2">Diameter (µm)</th><th colspan="2">Tensile strength (MPa)</th><th colspan="2">Tensile modulus (GPa)</th><th colspan="2">Elong. at break (%)</th></tr><tr><th>Numb.</th><th>Mean</th><th>S-E</th><th>Mean</th><th>S-E</th><th>Mean</th><th>S-E</th><th>Mean</th><th>S-E</th></tr></thead><tbody><tr><td>Batch 1</td><td>51</td><td>7.41</td><td>0.25</td><td>3143</td><td>773</td><td>186.2</td><td>20.8</td><td>1.57</td><td>0.37</td></tr><tr><td>Batch 2</td><td>45</td><td>7.36</td><td>0.45</td><td>3221</td><td>769</td><td>192.8</td><td>13.7</td><td>1.57</td><td>0.36</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Fig. 3. 2-parameters and 3-parameters Weibull CDF.</figcaption></figure>

probability is null) and $\sigma$ – applied stress.

$$P(\sigma) = 1 - e^{-\left(\frac{\sigma - \sigma_u}{\sigma_0}\right)^m} \qquad (1)$$

The interpretation of the scale and modulus parameters on the strength probability distribution of a carbon fiber with Weibull model is illustrated on Fig. 2. The 3-parameters Weibull CDF can also be applied to estimate the probability of two other tensile parameter values for carbon fiber which are tensile modulus and elongation at break.

A simpler form of the Weibull CDF is also considered to simulate the brittle fracture of fiber by setting the location parameter $\sigma_u$ in Eq. (1) to zero which is known as 2-parameters Weibull CDF. The impact of this parameter on the CDF diagram can be visualized in Fig. 3 at the same scale and shape of the distribution parameter value. It represents a threshold below which failures do not occur. This threshold might improve the accuracy of the adjustment, especially if the data show an

initial delay in observations.

The normal distribution is a widely used model in natural sciences to describe real-valued random variables. It has been applied in the literature to characterize the failure probability of brittle fibers [25]. As previously noted, the Weibull distribution can approximate a normal distribution when its modulus ranges between 3 and 4. The normal distribution is a continuous probability distribution characterized by the symmetry around the mean value. The CDF of a normal distribution is defined by two parameters, the mean $\mu$ and the standard deviation $\delta$ given in Eq. 2.

$$F(x) = \frac{1}{2} \left[ 1 + \operatorname{erf}\left(\frac{x - \mu}{\delta\sqrt{2}}\right) \right] \qquad (2)$$

where erf is the error function, defined as: $erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt$.

The CDF lognormal distribution, Eq. 3, on the other hand, is useful for

<figure><img src="image_6.png" /><figcaption>Fig. 4. Normal and Lognormal CDF.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 5. LS/MLE performance comparison on tensile strength.</figcaption></figure>

<table><caption>**Table 2**
Fitting results for distributions' parameters on elongation at break.</caption><thead><tr><th>Distribution</th><th>Weibull Modulus</th><th>Location</th><th>Scale</th><th>Mean</th><th>Std</th></tr></thead><tbody><tr><td><strong>BATCH 1</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>weibull_2p</td><td>4.794050</td><td>0.000000</td><td>1.722959</td><td></td><td></td></tr><tr><td>weibull_3p</td><td>2.806089</td><td>0.639831</td><td>1.054514</td><td></td><td></td></tr><tr><td>norm</td><td></td><td></td><td></td><td>1.577843</td><td>0.3673931</td></tr><tr><td>lognorm</td><td></td><td></td><td></td><td>1.577844</td><td>0.3674833</td></tr><tr><td><strong>BATCH 2</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>weibull_2p</td><td>5.236785</td><td>0.000000</td><td>1.703487</td><td></td><td></td></tr><tr><td>weibull_3p</td><td>6.322325</td><td>-0.319789</td><td>2.029051</td><td></td><td></td></tr><tr><td>norm</td><td></td><td></td><td></td><td>1.566222</td><td>0.3522328</td></tr><tr><td>lognorm</td><td></td><td></td><td></td><td>1.567434</td><td>0.3520257</td></tr></tbody></table>

modeling data that are positively skewed, as is the case with tensile strength, elongation or tensile modulus measurements.

$$F(x) = \frac{1}{2} \left[ 1 + \text{erf} \left( \frac{\ln(x) - \mu}{\delta\sqrt{2}} \right) \right] \quad (3)$$

Fig. 4 shows the representations of two CDFs, normal and lognormal, with the same mean and standard deviation values.

## 2.3. Model fitting on experimental data and algorithm

Fitting models are applied to estimate the parameters of the probability distributions with respect to the collected tensile data. MLE and LS methods are performed on tensile tests on carbon and sisal fiber [5,9,10, 19]. Nketiah [19] reported the reliability of MLE in comparison with LS method. The application of MLE method can be demonstrated for the 3-parameter Weibull model as an example to identify its three key parameters ($\sigma_0$, $\sigma_u$ and $m$) [34]. The likelihood function is expressed in Eq. 4 with a set of points $x_1, x_2, ..., x_n$

$$L(m, \sigma_0, \sigma_u; x_i) = \prod_{i=1}^{n} \left( \frac{m}{\sigma_0} \left( \frac{x_i - \sigma_u}{\sigma_0} \right) \right)^{m-1} e^{-\left( \frac{x_i - \sigma_u}{\sigma_0} \right)^m} \quad (4)$$

To make the calculation easier, the logarithm is applied to the likelihood function in Eq. 5:

$$\ln L(m, \sigma_0, \sigma_u; x_i) = \sum_{i=1}^{n} \left[ \ln \frac{m}{\sigma_0} + (m-1) \ln \frac{x_i - \sigma_u}{\sigma_0} - \left( \frac{x_i - \sigma_u}{\sigma_0} \right)^m \right] \quad (5)$$

The maximum likelihood for each of the parameters can be found using the partial derivatives and setting them to zero [33] in Eqs. 6–8:

$$\frac{\partial \ln L(m, \sigma_0, \sigma_u; x_i)}{\partial m} = \sum_{i=1}^{n} \left[ \frac{1}{m} + \ln \frac{x_i - \sigma_u}{\sigma_0} - \left( \frac{x_i - \sigma_u}{\sigma_0} \right)^m \ln \frac{x_i - \sigma_u}{\sigma_0} \right] = 0 \quad (6)$$

$$\frac{\partial \ln L(m, \sigma_0, \sigma_u; x_i)}{\partial \sigma_0} = \sum_{i=1}^{n} \left[ \frac{-m}{\sigma_0} + \frac{m(x_i - \sigma_u)^m}{\sigma_0^{m+1}} \right] = 0 \quad (7)$$

$$\frac{\partial \ln L(m, \sigma_0, \sigma_u; x_i)}{\partial \sigma_u} = \sum_{i=1}^{n} \left[ \frac{m-1}{x_i - \sigma_u} + \frac{m(x_i - \sigma_u)^{m-1}}{\sigma_0^m} \right] = 0 \quad (8)$$

Once the Weibull CDF parameters are determined by MLE method, Hessian Matrix [19] is computed to assess the confidence intervals for the estimated parameters. Hessian Matrix is a square matrix of the second derivatives of the log-likelihood function. By inverting this matrix, the covariance matrix is obtained, as the inverse of the Hessian approximates the covariance of the parameter estimates in the MLE framework [29].

For the 2-parameter Weibull model, the covariance matrix is expressed as follows in Eq. 9:

$$\text{Covariance Matrix} = \begin{bmatrix} \text{Var}(m) & \text{Cov}(m, \sigma_0) \\ \text{Cov}(m, \sigma_0) & \text{Var}(\sigma_0) \end{bmatrix} \quad (9)$$

Here:

* $Var(m)$ represents the variance of the Weibull modulus

* $Var(\sigma_0)$ represents the variance of the scale parameter

* $Cov(m, \sigma_0)$ captures the covariance, or linear relationship between $m$ and $\sigma_0$

By taking the square root of the variances $\sqrt{Var}$ the standard errors of $m$ and $\sigma_0$ can be derived. These standard errors provide a measure of the reliability of the estimated parameters and allow for comparisons between batches. For example, the standard error of $m$ reflects the confidence in the Weibull modulus.

The computation route, involving the application of MLE methods and the computation of the Hessian Matrix, is applied to determine the normal and lognormal distribution parameters and obtain the necessary information about the precision of these estimations.

To further understand the fitting procedure, a detailed algorithm in pseudocode is written:

**Algorithm 1.** : Fitting method of statistical distributions
**Input:**

* Experimental data for two batches of carbon fibers, tensile strength (MPa), tensile modulus (GPa), and elongation at break (%):

* Batch 1: Recycled carbon fibers after the recycling process.

* Batch 2: Recycled carbon fibers after both recycling and the carding process.

**Output:**

* Estimated parameters for each distribution (shape, scale, location if applicable for Weibull distribution, mean and standard deviation for Normal and Lognormal distributions).

* Goodness-of-fit metrics: SSE, KS-test, KS p-value.

**Algorithm steps:**

1. **Data preprocessing**

* Collect the tensile test data: from two batches after fiber individualization

* Prepare the data for tensile strength, elongation at break, and tensile modulus, by identifying the strength at fiber breakage for tensile strength and the slope of the stress-strain curve for tensile modulus.

* Selection of the parameter estimation method, weighing the performance of LS and MLE.

2. **MLE algorithm (Weibull, Normal and Lognormal models):**

* For each batch and property (tensile strength, elongation at break, and tensile modulus), the log-likelihood function is computed based on the observed data.

* Parameters for the distributions are adjusted to maximize the log-likelihood function. This includes estimating the shape, scale, and location parameters for the Weibull models, and the mean and standard deviation for the Normal and Lognormal distributions.

* Once the log-likelihood is maximized, the Hessian matrix is computed to estimate confidence intervals for the parameters.

3. **Goodness-of-Fit assessment**

* For each distribution:

* **KS Test:** Calculate the KS-test statistic for each distribution.

* **p-value:** Compute the p-value based on the KS-test statistic to evaluate the fit.

* **SSE:** Calculate SSE between observed data and model-predicted values.

4. **Compare model performance**

* 1: For each property and batch, compare the goodness-of-fit metrics (SSE, KS-test, p-value).

* 2: Identify the best-fitting model for each property (elongation, strength, modulus) based on the highest p-value and lowest SSE.

5. **Post-fit analysis**

* Comparison of variance made possible between the two batches using the Weibull modulus.

* Evaluation of the parameters value significance with regard to the mechanical justification

* Optionally adjust the location parameter manually and observe the impact on the KS p-value (Fig. 9).
<figure><img src="image_8.png" /><figcaption>Fig. 6. Metrics for fitting assessment – Elongation at break.</figcaption></figure>

To implement this algorithm, we used the Python programming language along with the NumPy, SciPy and Reliability libraries. With the output of Algorithm 1, the best model can be identified using goodness-of-fit metrics. The statistical significance of differences between batches using Weibull modulus and other metrics can be used to conclude the fibers' mechanical properties.

## 2.4. Performance assessment of the fitted models

After fitting the tensile test data to the distribution models, the performance of the fitting (goodness-of-fit) needs to be assessed with respect to the experimental data. Three metrics are applied in this study: KS-test, p-value and SSE

KS-test, which is a non-parametric test, measures the largest distance

between the cumulative distribution functions and the reference distribution for a sample data [5,9,17,32]. This test is performed with 2-parameter Weibull distribution model on static strength and fatigue life of T300 carbon fiber by Song et al. [26]. Rao et al. carried out the KS-test for the strength probability distribution of a virgin carbon fibers modelled with 2-parameters Weibull CDF associated with the assessment of the confidence interval using the bootstrapping method [24].

KS p-value (probability value) describes the significance of the model in relation to the data. The significance threshold value ($\alpha$) is considered equal to 0.05. A p-value greater than 0.05 indicates that the hypothesis supposing that the reference data do not follow the distribution model cannot be rejected and the results are not due to random chance.

SSE is the sum of the squared differences between each reference observed data and the predicted probability by the distribution model.
The convergence between the predicted and observed data is achieved with minimum SSE value. While a higher value indicates a disparity between the two values.

The three metrics are used to measure the fit goodness of the model's parameter estimators for the probability distribution models (2 or 3-parameter Weibull, normal and log-normal) and their performance.

## 3. Results and discussion

The tensile properties: elongation at break, strength, and modulus of two types of recycled carbon fibers, both before and after carding, are modeled using a defined computational approach. This approach considers four distribution models: 2-parameter and 3-parameter Weibull, normal, and lognormal distributions. The most suitable distribution determined by MLE for each tensile property is identified based on three fit assessment metrics: KS, KS p-value, and SSE.

Given the comparison of the performance of MLE and LS methods, it was found that MLE provided an overall better fit for both batches, in terms of tensile strength for example as shown in Fig. 5. In batch 1, MLE produced a higher p-value (0.9882) compared to LS (0.9671), and similarly in batch 2, MLE showed a stronger fit with a p-value of 0.8886 versus 0.8194 for LS.

In the following three sections, the results in the plots are sorted by relevance, with the SSE and KS statistics in ascending order to reflect respectively smaller errors and shortest distance, and the KS p-value in descending order to indicate greater statistical confidence. The best value in each category is highlighted with a black outline on the histogram for clarity.

### 3.1. Analysis of elongation at break

Elongation at break is the ratio of the change in length to the initial length of a material at the point of breakage. It describes the material's ability to stretch under tensile stress before breaking. It is often expressed as a percentage. The estimated values of the distribution model parameters by MLE for the probability of attaining an elongation at break are listed in Table 2.

Quasi-identical values are estimated for the mean and standard deviation of the normal and lognormal distribution. These results follow the directly computed values of mean and standard deviation from the dataset presented in Table 1. In comparison between the two batches for these estimated values on the normal and lognormal distribution no significant statistical differences is noted. However, by considering the modulus values of the 2 and 3-parameter Weibull distribution, the two batches do not have the same trend. Batch 1 exhibits slightly greater variability than Batch 2 when modeled with the 2-parameter Weibull distribution. However, when the 3-parameter Weibull model is applied, Batch 1 demonstrates significantly higher disparity, suggesting inconsistencies in the location parameter estimation. A negative value is estimated for the localization parameter with 3-parameter Weibull (reflecting the threshold for the studied variable). This negative value has no clear mechanical justification.

<table><caption>**Table 3**
Fitting results for distributions' parameters on tensile strength.</caption><thead><tr><th>Distribution</th><th>Weibull Modulus</th><th>Location</th><th>Scale</th><th>Mean</th><th>Std</th></tr></thead><tbody><tr><td><strong>BATCH 1</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>weibull_2p</td><td>4.480256</td><td>0.000000</td><td>3442.135100</td><td></td><td></td></tr><tr><td>weibull_3p</td><td>2.681489</td><td>1247.427889</td><td>2132.524194</td><td></td><td></td></tr><tr><td>norm</td><td></td><td></td><td></td><td>3.143118e+03</td><td>7.655978e+02</td></tr><tr><td>lognorm</td><td></td><td></td><td></td><td>4.678303e+09</td><td>2.316004e+19</td></tr><tr><td><strong>BATCH 2</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>weibull_2p</td><td>4.989604</td><td>0.000000</td><td>3512.858500</td><td></td><td></td></tr><tr><td>weibull_3p</td><td>6.828347</td><td>-1163.934186</td><td>4697.453746</td><td></td><td></td></tr><tr><td>norm</td><td></td><td></td><td></td><td>3.221378e+03</td><td>7.600336e+02</td></tr><tr><td>lognorm</td><td></td><td></td><td></td><td>1.808263e+13</td><td>1.003942e+26</td></tr></tbody></table>

**Fig. 6** presents the values of the fitting assessment tests on the probability distribution models for the elongation at the break of the two batches of recycled carbon fibers. A difference is noted between the four distribution models and for the two fiber batches. For batch 1, the 2-parameter Weibull distribution model shows the best fit with an SSE of 0.1304 and a KS-test value of 0.0574, together with a high KS p-value of 0.9925, indicating an excellent fit to the data. While 3-parameter Weibull distribution is the least performing fit model with a relatively high SSE of 0.2249, the highest KS-test value of 0.0787 and a KS p-value of 0.8856 showing an acceptable fit but inferior to the other distributions. The best-fit model remains consistent for Batch 2, although, in this specific case, the lognormal distribution closely competes with the 2-parameter Weibull, while the 3-parameter Weibull is still less effective in fitting the experimental data distribution.

Overall, for elongation at break, the 2-parameter Weibull distribution consistently provided the best fit for both batches, with Batch 1 showing slightly greater variability.

The next step will be to determine whether the 2-parameter Weibull distribution still performs best for other mechanical properties such as tensile modulus and tensile strength.

### 3.2. Analysis of tensile strength

Tensile strength is the maximum stress a material can withstand when stretched before breaking. It is commonly expressed in Mega-pascal (MPa). The estimated values of the distribution model parameters using MLE for the probability of reaching a specific strength value are listed in Table 3. The parameter estimates of the lognormal distribution, including mean and standard deviation, significantly differ from those directly computed from the experimental dataset. This contrasts with the results obtained for the elongation at break distribution. Differently, the estimated parameters for the normal distribution match the computed values from the dataset. As for the elongation at break distribution, the 2-parameter Weibull model indicates a slightly higher variability in tensile strength for carbon fiber from batch 1 compared to batch 2, though the normal distribution parameters do not reveal a significant difference. On the other hand, the 3-parameter Weibull model presents an opposite trend, showing a substantial disparity in batch 1, while batch 2 exhibits higher consistency.

Similar to the elongation at break, the 2-parameter Weibull is the best fit model concerning the experiment data among the four distribution applied models for the strength distribution of batch 1, Fig. 7. The computed performance metric tests are 3.900e-08 for SSE, 5.978e-02 for KS and a high KS p-value of 0.9882 indicating a good fit to the data. However, the normal distribution model shows the best fit for batch 2 with SSE of 4.038e-08, KS-test value of 8.176e-2 and KS p-value of 0.9003. While the 2 and 3-parameter Weibull shows very similar performance for batch 2: SSEs of 3.632e-08 and 3.655e-08 respectively and close KS-test values of 8.323e-02 and 8.199e-02 with high KS p-values 0.8886 and 0.8985. Otherwise, the lognormal model doesn't fit the experiment data for both batches and obtains a high SSE value of 6.064e-07 and 6.759e-07, KS-test value of 0.7741 and 7.049e-01 and an
<figure><img src="image_9.png" /><figcaption>Fig. 7. Metrics for fitting assessment – Tensile strength.</figcaption></figure>

<table><caption>**Table 4**

Fitting results for distributions' parameters on tensile modulus.</caption><thead><tr><th>Distribution</th><th>Weibull Modulus</th><th>Location</th><th>Scale</th><th>Mean</th><th>Std</th></tr></thead><tbody><tr><td><strong>BATCH 1</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>weibull_2p</td><td>10.994207</td><td>0.000000</td><td>194.915155</td><td></td><td></td></tr><tr><td>weibull_3p</td><td>11.366356</td><td>-6.412414</td><td>201.356593</td><td></td><td></td></tr><tr><td>norm</td><td></td><td></td><td></td><td>1.862353e+02</td><td>2.059658e+01</td></tr><tr><td>lognorm</td><td></td><td></td><td></td><td>1.878479e+02</td><td>2.516558e+01</td></tr><tr><td><strong>BATCH 2</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>weibull_2p</td><td>16.327605</td><td>0.000000</td><td>198.928269</td><td></td><td></td></tr><tr><td>weibull_3p</td><td>7.177126</td><td>109.377970</td><td>89.022217</td><td></td><td></td></tr><tr><td>norm</td><td></td><td></td><td></td><td>1.928222e+02</td><td>1.351096e+01</td></tr><tr><td>lognorm</td><td></td><td></td><td></td><td>1.989953e+02</td><td>2.741294e+01</td></tr></tbody></table>
<figure><img src="image_10.png" /><figcaption>Fig. 8. Metrics for fitting assessment – Tensile modulus.</figcaption></figure>

extremely low KS p-value of 1.083e-32 and 5.251e-23, which implies the rejection of the hypothesis that the data follow a lognormal distribution.

For tensile strength, the results indicate that the 2-parameter Weibull distribution is the most appropriate model for Batch 1, while the normal distribution provided a better fit for Batch 2.

### 3.3. Analysis of tensile modulus

The tensile modulus is a measure of a material's stiffness or rigidity under tensile force, typically expressed in Gigapascals (GPa) for engineering materials. The estimated parameters of the distribution models, calculated using MLE, for the probability of achieving a given tensile modulus are presented in Table 4. The parameters estimated for the normal distribution align closely with the experimental mean and

standard deviation for both batches. However, slightly higher estimated values are observed for the lognormal distribution model in both batches. Similar to the elongation and strength distributions, no significant differences are noted between the two batches. However, the data shows a noticeable disparity between the two batches when fitted with the 2-parameter and 3-parameter Weibull models.

Similar to strength and elongation at break, the 2-parameter Weibull distribution provides the best fit for the tensile modulus in Batch 1, with the lowest SSE of 4.628e-05, KS-test value of 9.279e-02, and high KS p-value of 7.372e-01 as shown in Fig. 8. But the normal distribution has the best fit for the batch 2 with SSE of 1.128e-04, KS-test value of 6.426e-02, and a high KS p-value of 9.865e-01. However, the 3-parameter Weibull distribution is the worst fitted model with the highest SSE of 2.279e-04, KS-test value of 1.363e-01, and a low KS p-value of 2.739e-04.
01.

Regarding tensile modulus, the 2-parameter Weibull model was
more effective for Batch 1, while the normal distribution performed
better for Batch 2.

3.4. Discussion

To investigate the mechanical properties of rCFs from the two batches, three mechanical parameters were analyzed: elongation at break, tensile strength, and tensile modulus. Four statistical models were employed to fit the experimental data: the 2-parameter Weibull, 3-parameter Weibull, normal, and log-normal distributions. Highlighting differences in fit between batches of fiber depending on the three different mechanical properties.

The analysis of the mechanical properties of recycled carbon fibers
revealed that the 2-parameter Weibull distribution generally showed the
best fit for elongation at break for both batches, with an SSE of 0.1304
and KS p-value of 0.9925 for batch 1, and an SSE of 0.09797 and KS p-
value of 0.9871 for batch 2. Although the lognormal and normal dis-
tributions also performed well, the 3-parameter Weibull distribution
showed a less satisfactory fit also due to the negative location parameter.
For tensile strength, the 2-parameter Weibull distribution provided the
best fit for batch 1 (SSE of 3.900e-08, KS p-value of 0.9882), while the
normal distribution performed better for batch 2 (SSE of 4.038e-08, KS
p-value of 0.9003). The 3-parameter Weibull distribution, which fitted
well for batch 1, was less reliable for batch 2 due to the negative location
parameter. For tensile modulus, the data from batch 1 fitted the 2-
parameter Weibull distribution better (SSE of 4.628e-05, KS p-value of
0.7372), while the data from batch 2 favored the normal distribution
(SSE of 1.128e-04, KS p-value of 0.9865). The 3-parameter Weibull
distribution showed similar issues concerning the negative location
parameter.

To further investigate the behavior of the 3-parameter Weibull
model, the location parameter was manually adjusted and examined its
effect on the KS p-value and shape parameter for tensile strength of
Batch 2 as represented in Fig. 9. While the MLE algorithm identified the
maximum likelihood at σ₀ = -1163MPa, the manual variation of σ₀
revealed that a higher KS p-value was achieved at σ₀ ≈ 1540MPa,
corresponding approximately to the lowest value of this dataset σ₀ =
1273MPa. This demonstrates that a maximum likelihood estimate does
not always guarantee the highest p-value, and it is essential to integrate
domain expertise with the MLE approach. Depending on the objective,
one might either constrain the location parameter to achieve a higher p-
value or prioritize the parameters that gives the highest likelihood.

Similarly, when manually adding a location parameter to the
lognormal CDF, the mean and standard deviation became close to the
mean and standard deviation calculated from the dataset directly which
differ from the values found at 10e - 13 for the mean and 10e - 26 for
the standard deviation.

In summary, no single statistical model consistently provided the best fit for all three mechanical properties in both batches. The 2-parameter Weibull distribution often performed best for batch 1 and was also appropriate for batch 2 in most cases, while the normal distribution performed best for batch 2, particularly for tensile modulus and tensile strength. Also, the critical analysis of values found with MLE shows that adjusting parameters manually, in conjunction with statistical modeling, allows for more informed decisions regarding the best-fitting models.

4. Conclusions and perspectives

This study aimed to identify the most suitable statistical models for
analyzing the mechanical properties of recycled carbon fibers before and
after carding, focusing on elongation at break, tensile strength and
tensile modulus. 2-parameter and 3-parameter Weibull, normal and log-
normal distributions were used to fit experimental data from 51 tensile

<figure><img src="image_11.png" /><figcaption>Fig. 9. Change in KS p-value with a manual adjustment of location parameter - Tensile strength - Batch 2.</figcaption></figure>

tests for batch 1 and 45 tests for batch 2. The fit was performed using an
MLE technique and was evaluated using SSE, KS-test values and KS p-
values. The analysis showed that the 2-parameter Weibull distribution
generally provided the best fit for elongation at break for both batches,
particularly for batch 1. Regarding tensile strength, this distribution
fitted the batch 1 data well, while the normal distribution was more
appropriate for batch 2. For tensile modulus, the 2-parameter Weibull
distribution was more representative of batch 1, while the normal dis-
tribution was also preferable for batch 2. The values obtained by MLE for
the 3-parameter Weibull distribution estimated that the best fit with
MLE came with negative location values in most cases, making the
mechanical interpretation of this model less reliable. In conclusion,
although the 2-parameter Weibull distribution was not always the
optimal model for each property and each batch, it demonstrated suf-
ficient robustness to be used as a global model for a following study.
However, it is difficult to conclude that a single model is the most
suitable for all situations. No statistically significant difference is noted
between the two rCF batches regarding the mean and standard deviation
values. However, a difference in the two data disparities is concluded by
considering the Weibull modulus parameter which reflects the variance
of the dataset.

Peering into the future, these models could be incorporated into
simulations of composites made from discontinuous fibers processed
after carding. They may help predict the behavior of recycled carbon
fiber composites and optimize their mechanical properties. Further-
more, models of the mechanical properties of recycled carbon fibers
contribute to the understanding of textile processing and its impact on
fiber properties. As larger datasets become available, machine learning
techniques, such as neural networks and ensemble methods, could be
employed to estimate statistical distributions and material behavior
based on processing parameters and material characteristics. Pre-
liminary tests have suggested the feasibility of these approaches but
highlighted the need for further validation with more data on processing
parameters and material characteristics. Expanding data collection and
refining these methods could enhance model accuracy, ultimately
contributing to the design and performance evaluation of composites
from discontinuous rCFs. Future research will focus on integrating these
methods into composite simulations.

Funding

This work was supported by the “Hauts-de-France” region.
CRediT authorship contribution statement

**Ivars Jean:** Writing – original draft, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Soulat Damien:** Writing – review & editing, Supervision, Project administration. **Tran Kim Phuc:** Writing – review & editing, Visualization, Supervision. **Labanieh Ahmad Rashed:** Writing – review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

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