# Estimation of compressive strength in thick continuous fibre composites: Quantifying the role of initial fibre misalignment

P.-Y. Mechin, M. Grabow, A. Launay, Jean-Claude Grandidier, Z. Zhang, V. Keryvin

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## To cite this version:

P.-Y. Mechin, M. Grabow, A. Launay, Jean-Claude Grandidier, Z. Zhang, et al., Estimation of compressive strength in thick continuous fibre composites: Quantifying the role of initial fibre misalignment. Composites Part A: Applied Science and Manufacturing, 2025, 196, pp.108950. 10.1016/j.compositesa.2025.108950. hal-05333782

HAL Id: hal-05333782

https://hal.science/hal-05333782v1

Submitted on 27 Oct 2025

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# Estimation of compressive strength in thick continuous fibre composites:
## Quantifying the role of initial fibre misalignment

P.-Y. Mechin<sup>a</sup>, M. Grabow<sup>b</sup>, A. Launay<sup>b,c</sup>, J.C. Grandidier<sup>d</sup>, Z. Zhang<sup>e</sup>, V. Keryvin<sup>bID,*</sup>

## GRAPHICAL ABSTRACT

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## ARTICLE INFO

**Keywords:**

Polymer-matrix composites

Fibre misalignment

Compressive strength

Fibre waviness

## ABSTRACT

The present study investigates the compressive strength of continuous fibre composites. These materials comprise almost ten different fibre/matrix pairs with four epoxy-based thermoset resin systems, three different carbon fibres and three manufacturers. The experimental values obtained from flexural testing on thick laminates (with 10 samples for each pair) are compared to an analytical compressive failure criterion. This latter model incorporates ease of use in engineering design and mechanistic contributions, including material (non-linear shear behaviour of the ply), manufacturing (initial fibre misalignment) and structural effect (loading, neighbourhood of ply). A comparison between the experimental and estimated values, using the standard deviation of the frequency distribution of fibre misalignment, demonstrates a satisfactory or excellent outcome. The comparison for a pair produced less favourable results, with greater discrepancies observed

<footnote><sup>a</sup> Dassault-Systèmes, CATIA Composites, F-78946 Vélizy-Villacoublay, France</footnote>

<footnote><sup>b</sup> Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56321 Lorient, France</footnote>

<footnote><sup>c</sup> Gsea Design, F-56100 Lorient, France</footnote>

<footnote><sup>d</sup> ISAE-ENSMA, UPR CNRS 3346, PPrime, F-86360 Chasseneuil-du-Poitou, France</footnote>

<footnote><sup>e</sup> Dassault-Systèmes, CATIA AI, F-78946 Vélizy-Villacoublay, France</footnote>

<footnote>* Corresponding author.
E-mail address: vincent.keryvin@univ-ubs.fr (V. Keryvin).</footnote>
between two different manufacturers. The spatial distribution of fibre misalignment provides evidence that, in certain instances, a single value may be insufficient for accurately characterising the role of defects in compressive failure.

# 1. Introduction

Carbon fibres reinforced polymers (CFRs) are employed extensively due to their elevated specific stiffness and strength, which are the result of the unique properties of the constituents. The mechanical properties of CFRP are the result of an optimal arrangement between the constituents, namely fibres and thermosets (or thermoplastics) resins. With regard to mechanical design, the principal weakness of CFRP is the somewhat low compressive strength of unidirectional plies (UDs) in the fibre direction.

It is widely accepted that the failure mechanism due to compressive loading is fibre micro-buckling. The maximum compressive stress level that can be tolerated before the onset of instability is retained as the compressive strength. The underlying failure mechanisms encompass both material and process characteristics. In the case of a single fibre or a collection of fibres, the surrounding environment acts as a lateral stiffener, preventing or delaying buckling. This environment comprises the surrounding polymer matrix and carbon fibres, whose behaviour may be approximated to that of the UD itself. With regard to the latter, it should be noted that manufacturing defects, particularly during the curing process, result in the continuous fibres not being straight but rather exhibiting an initial distribution of fibre misalignment. For relatively minor degrees of fibre misalignment, the UD shear behaviour becomes a significant factor in the overall stability assessment. However, the polymer matrices display non-linear behaviour due to their viscoelasticity, thereby indicating that the material aspect of the failure mechanism is the UD non-linear shear behaviour.

The accurate estimation of compressive strength is of paramount importance in guaranteeing the functionality of composite structures. However, numerous challenges impede the process of engineers designing composite structures. A primary challenge arises from the fact that estimations can be derived from either semi-analytical models [1–3] or computational models [4,5]. In the former case, the requisite material data is often challenging to assess, while the latter category typically necessitates in-house numerical capabilities that are not fully available in commercial codes. As a result, there is considerable uncertainty in the estimates, necessitating the use of substantial safety margins in engineering design. A second issue arises from the fact that compressive strength is not a material property at the UD scale. It should be noted that, in addition to the key mechanisms outlined previously, the compressive strength of a UD is also influenced by external factors, including the stiffness of neighbouring plies in the laminate, the thickness of consecutive UD 0° plies and the type of loading. This phenomenon is referred to as the structural effect [6,7] and it is notable that the majority of analytical models do not account for this contribution. This contribution can be illustrated through a comparison of the compressive strength of a 0° ply in pure compression between two laminates: one consisting solely of 0° plies and the other with off-axis plies [7]. The latter will be higher than the former. A second comparison involves a laminate made solely of 0° plies: the compressive strength of the most compressed 0° ply in a bending test is higher than that of the same ply under pure compression [6]. These observations highlight that axial compressive strength is not an intrinsic material property, but rather depends on mesoscopic mechanisms.

A third issue pertains to the lack of proven validity in the models. Three principal difficulties emerge. Firstly, any model must be validated on a robust experimental database. In the majority of cases, only partial experimental data are employed, with additional ad hoc parameters being used [8]. As a result, comprehensive validation campaigns are uncommon, with only a few instances of such campaigns being

conducted on a single material [9,10] or limited to a few [1,11–13]. Secondly, it has long been demonstrated that the experimental measurement of compressive strength itself presents a significant challenge. The nearly 35-year-old round-robin test programme [14] demonstrated that the pure compression test results in questionable data, which is either mainly unreliable or at best very scattered in many situations. These shortcomings require great expertise to prevent [3]. Thirdly, the measurement of initial fibre waviness, its data analysis and its use in estimations is again a challenge by itself. A number of alternative approaches may be employed to enhance the image analysis, including the use of structural tensors, direct tracking, Fourier transformation (FTMA) and Hough transformation. Another key technique is Multiple Field Image Analysis (MFIA) [15]. (For a comprehensive overview of the advantages and limitations of these diverse methodologies, please refer to the survey presented in [16].) There is a paucity of quantitative comparisons between these methods in the literature. In such a study by Creighton and Clyne [17], STE and MFIA methods were compared on a pultruded composite. It was emphasised that the misalignment distributions were similar, but that the STE method would produce slightly higher values. However, only 180 fibres were considered in their image analyses...

Table 1 provides an overview of various studies that have addressed this issue. One of the earliest quantitative measurement methodologies was proposed by Yurgartis [18]. This stereological methodology (STE) is based on the analysis of image data, specifically the orientation of ellipses obtained by cutting test samples. X-ray computed tomography (XCT) is the most promising method for accurately measuring the fibre misalignment [19]. This method is both time-consuming and limited in terms of the size of the coupons used, which raises questions about the representativeness of the results.

Once the distribution of initial fibre misalignment has been obtained, certain parameters are then extracted. As reported by Stewart and Poursartip [29], the majority of research utilising the methodology proposed by Yurgartis [18] interprets the standard deviation of the fibre misalignment distribution as analogous to the initial fibre misalignment defined in models. In contrast, using an advanced version of MFIA, Wilhelmsson and Asp [30] proposed considering 99% of the fibres to be misaligned (equivalent to three times the standard deviation). Daum et al. [5] emphasised that solely relying on the standard deviation to quantify initial misalignment would be an insufficient approach. Instead, it is essential to consider additional information and extract supplementary data to gain a more comprehensive understanding. A further crucial factor is the impact of composite part thickness on the extent of initial fibre misalignment. In their study, Lee and Soutis [9] employed the stereological method to determine the initial fibre misalignment values for composite samples with varying thicknesses. They reported a value of 0.90° for a 2 mm thick sample and 1.90° for an 8 mm thick sample. This further supports the assertion that compressive strength is not a material property. This latter point indicates the necessity for careful consideration of the representativeness of the coupons employed, given that numerous industries utilise composite structures with thicknesses exceeding 10 mm. Nevertheless, the majority of tests are conducted using composite materials with a thickness of approximately 3 mm, which is classical for the axial pure compression standard test. The stereological methodology proposed by Yurgartis [18] quantifies the initial misalignment with respect to both in-plane and out-of-plane misalignments. This is not always the case with the other methods or studies presented in Table 1. JACK is a method that inherently measures the three-dimensional path of fibre misalignment. Even if they used the stereological method, Creighton and Clyne [17] studied only the out-of-plane misalignment on pultruded rods
<caption>**Table 1**</caption>

Overview of the literature related to compressive strength and initial fibre misalignment measurement for in-plane (IN) and/or out-of-plane angles (OP). The acronyms STE, MFIA, FTMA, HRMA and XCT are used to denote, respectively, the stereological method, multiple field image analysis, Fourier transform analysis, high resolution misalignment analysis and X-ray computed tomography.

<table><caption>**Table 2**</caption><thead><tr><th>Authors</th><th>Method</th><th>Fibre</th><th>Misalignment</th><th>Orientation</th><th>Process</th><th>Year</th></tr></thead><tbody><tr><td>Yurgartis [18]</td><td>STE</td><td>AS4</td><td>0.75-0.82° (std)</td><td>OP/IN</td><td>Tape</td><td>1987</td></tr><tr><td>Clarke et al. [20]</td><td>STE</td><td>T800</td><td>2.3-3.0° (std)</td><td>OP/IN</td><td>Wet process</td><td>1995</td></tr><tr><td rowspan="2">Paluch [21]</td><td>STE</td><td>AS4/GY70</td><td>0.89-2.14° (std)</td><td>3D</td><td>Tape</td><td>1996</td></tr><tr><td>STE</td><td>T300/T800</td><td>0.54-1.36° (std)</td><td>3D</td><td>Tape</td><td>1996</td></tr><tr><td>De Morais and Marques [1]</td><td>Unknown</td><td>T300</td><td>0.75-1.47° (std)</td><td>n.d.</td><td>Tape</td><td>1997</td></tr><tr><td>Creighton and Clyne [17]</td><td>STE, MFIA</td><td>IM7</td><td>0.7-1.0° (max)</td><td>OP</td><td>Pultrusion</td><td>2000</td></tr><tr><td>Creighton et al. [15]</td><td>MFIA</td><td>n.d.</td><td>5.0° (max)</td><td>OP</td><td>Tape</td><td>2001</td></tr><tr><td>Lee and Soutis [9]</td><td>STE</td><td>T800</td><td>0.90-1.90° (std)</td><td>IN</td><td>Tape</td><td>2006</td></tr><tr><td>Thomsen and Kratmann [19]</td><td>FTMA</td><td>–</td><td>0.90-1.30° (std)</td><td>IN</td><td>Pultrusion</td><td>2010</td></tr><tr><td>Jumahat [22]</td><td>STE</td><td>HTS40</td><td>0.62-1.19° (std)</td><td>IN</td><td>Tape</td><td>2011</td></tr><tr><td rowspan="2">Sutcliffe et al. [23]</td><td>MFIA</td><td>–</td><td>0.7-1.12°/1.37-1.63° (std)</td><td>OP/IN</td><td>RTM/Tape</td><td>2012</td></tr><tr><td>XCT</td><td>–</td><td>0.75-1.21° (std)</td><td>3D</td><td>Tape</td><td>2012</td></tr><tr><td>Nguyen et al. [24]</td><td>XCT</td><td>AS7/AS4</td><td>0.79-1.8°/1.2-3.8° (std)</td><td>3D</td><td>ATL/VARTM</td><td>2018</td></tr><tr><td>Wilhelmsson et al. [25]</td><td>HRMA</td><td>HTS45</td><td>1.7-3.8° (mean)</td><td>OP</td><td>NCF</td><td>2018</td></tr><tr><td>Sebaey et al. [26]</td><td>XCT</td><td>IM7</td><td>0.23-0.28 rad (std)</td><td>3D</td><td>Tape</td><td>2019</td></tr><tr><td>Mechin et al. [10]</td><td>STE</td><td>HR40</td><td>0.6-0.83° (std)</td><td>OP/IN</td><td>Tape</td><td>2019</td></tr><tr><td rowspan="3">Jeppesen et al. [27]</td><td>XCT</td><td>Carbon</td><td>1.18-1.75° (std)</td><td>3D</td><td>Pultrusion</td><td>2021</td></tr><tr><td>XCT</td><td>Carbon</td><td>2.75-3.42° (std)</td><td>3D</td><td>Tape</td><td>2021</td></tr><tr><td>XCT</td><td>Glass</td><td>4.52-5.42° (std)</td><td>3D</td><td>NCF</td><td>2021</td></tr><tr><td>Keryvin et al. [28]</td><td>STE</td><td>IM2C</td><td>0.93-1.13° (std)</td><td>OP/IN</td><td>Tape</td><td>2024</td></tr></tbody></table>

Overview of the fibre/resin pairs used in the study.

<table><thead><tr><th>Pair</th><th>Resin</th><th>Fibre</th><th>Manufacturer</th></tr></thead><tbody><tr><td>S1</td><td>Se84 LV (Gurit)</td><td>IM2C (Hexcel)</td><td>Manuf-1</td></tr><tr><td>S2</td><td>Se84 LV (Gurit)</td><td>IM2C (Hexcel)</td><td>Manuf-2</td></tr><tr><td>S3</td><td>Se84 nano (Gurit)</td><td>IM2C (Hexcel)</td><td>Manuf-3</td></tr><tr><td>S4</td><td>Se84 nano (Gurit)</td><td>IM2C (Hexcel)</td><td>Manuf-1</td></tr><tr><td>S5</td><td>XB 3515 (Vitech)</td><td>IMS65 (Teijin)</td><td>Manuf-1</td></tr><tr><td>S6</td><td>MTC9810 (SHD)</td><td>T800SC (Toray)</td><td>Manuf-1</td></tr><tr><td>S7</td><td>BT080 (Delta Preg)</td><td>T800SC (Toray)</td><td>Manuf-1</td></tr><tr><td>S8</td><td>DT120 (Delta Preg)</td><td>T800SC (Toray)</td><td>Manuf-2</td></tr><tr><td>S9</td><td>Se75 (Gurit)</td><td>IM2C (Hexcel)</td><td>Manuf-1</td></tr></tbody></table>

while Jumahat [22] considered only the in-plane misalignment on her prepreg materials. The studies from Wilhelmsson and Asp [30], Shepherd et al. [31] focused on non-crimp fabrics composites (NCF) for which out-of-plane misalignment is the primary cause of instability.

In view of the preceding literature review on initial fibre misalignment measurement methods and the difficulties engineers encounter in utilising reliable tools for compressive strength design, it is evident that a study on compressive strength is necessary and is lacking in the literature. To our opinion, this study should comprise the following elements:

(i) firstly, it would be beneficial for engineers to consider the use of an analytical model, which could assist them in their design process. It is essential to consider the various contributions to compressive failure, including the material (UD non-linear shear behaviour), the process (fibre misalignment) and the structural effect. Furthermore, the ability to perform rapid computations is crucial, as strength design is not the sole aspect of a comprehensive design for structures utilising composite materials. This necessitates the execution of numerous design loops.

(ii) Secondly, an experimental testing campaign should be conducted in accordance with the following characteristics. The number of fibre/matrix pairs should be relatively high. The types of materials and process should be common in order to address comparable composite materials. The aim is to perform reliable measurements of compressive strength with a low coefficient of variation (less than 5%) on thick specimens that are more representative of the fibre misalignment levels found in thick composite structures. All necessary input parameters for the model should be measured and not estimated or taken from literature with possible doubts on the relevance. A straightforward measurement method for initial fibre misalignment and large volumes scanned with rapid and robust image analysis is required.

(iii) Finally, a robust comparison between experimental and estimated values of compressive strength is required to allow for a meaningful evaluation of the results, involving a discussion on the influence of the spatial and frequency distributions of fibre misalignment, as well as their orientation, and a final statement on the adequacy of the model for being used in compressive strength design by engineers.

In this paper, the findings of a study conducted on nine composite materials with four epoxy-based thermoset resin systems and three carbon fibres, produced by three different professional manufacturers, from CFRP tape prepregs cured in autoclaves, are presented. The flexural properties of thick laminates comprising 0° plies will be evaluated, and the compressive strength of these 0° plies will be determined. The stereological method is selected for the purpose of characterising the initial fibre misalignment in both the in-plane and out-of-plane directions. Furthermore, automated image analysis, based on machine learning, will facilitate the analysis of a substantial number of fibres. A comparison between experimental and estimated results will be made by employing the analytical failure criterion proposed by Grandidier et al. [32] based on the plastic kinking model of Budiansky and Fleck [33] and the addition of the structural effect proposed by Gardin et al. [7] in a mathematical approach. The parameters of this criterion will be evaluated through tensile tests on ±45° laminates for the UD non-linear shear behaviour, in addition to the fibre misalignment measurements.

## 2. Materials and methods

### 2.1. Materials

This study considers a range of intermediate modulus (IM) carbon fibres with different epoxy-based thermoset resins, which are labelled S1 to S9 in accordance with the nomenclature presented in Table 2. The selection of IM fibres was driven by their status as the most prevalent in the aerospace and marine industries, thus ensuring the applicability of the findings within an industrial context. Four distinct epoxy-based thermoset resin systems from disparate tape prepreg suppliers and three distinct fibres are employed. In order to facilitate an effective analysis of the fibre cross-section [18], only circular cross-section fibres are employed in this study. The coupons were produced by various commercial manufacturers. A total of nine fibre/resin pairs are under investigation. In some cases, the same pair is considered for different manufacturers. However, as the fibre misalignment may vary between these instances, we will differentiate between them. The UD plies have a fibre weight of 300 g/m² and are laminated individually on a mould, with compaction occurring at three-ply intervals.
The stacking is [±45°/0°/±45°/0°/11/±45°/0°/±45°]. Once all plies have been laminated, the composite is vacuum bagged and cured in an autoclave (at the specified temperature and pressure) in accordance with the data sheet for the prepregs and/or the practices of the manufacturer.

## 2.2. Theory: microbuckling failure criterion

Budiansky and Fleck [33] chose to describe the non-linear shear behaviour of UD $\gamma_{UD}$ (shear strain) vs. $\tau_{UD}$ (shear stress) by a Ramberg-Osgood (RO)-like behaviour (see Eq. (1)).

$$ \frac{\gamma_{UD}}{\gamma_{UD}^{y}} = \frac{\tau_{UD}}{\tau_{UD}^{y}} + \frac{3}{7} \times \left( \frac{\tau_{UD}}{\tau_{UD}^{y}} \right)^{n_{UD}} \quad (1) $$

It consists of four material parameters $G_{UD}$, $n_{UD}$, $\gamma_{UD}^y$ and $\tau_{UD}^y$. $G_{UD}$ is the shear modulus of the composite, $\tau_{UD}^y$ is a nominal shear yield strength at which the secant modulus is reduced to 70% of its initial value $G_{UD}$, and $n_{UD}$ is a hardening-like exponent. $\gamma_{UD}^y$ is defined as the elastic strain at the shear stress $\tau_{UD}^y$ such that $\tau_{UD}^y = G_{UD} \times \gamma_{UD}^y$, reducing the number of material parameters to three. The compressive strength, referred to as the critical stress generating instability ($\sigma_{UD}^{stab}$), is defined as the maximum local stress applied to UD before micro-buckling of the fibre occurs and is given by Eq. (2). The initial fibre misalignment ($\phi_0$) is related to the manufacturing process.

$$ \sigma_{UD}^{\text{stab}} = \frac{G_{UD}}{1 + n_{UD} \left(\frac{3}{7}\right) \frac{1}{n_{UD}} \left(\frac{\phi_0 / \gamma_{UD}^y}{n_{UD} - 1}\right) \left(\frac{n_{UD} - 1}{n_{UD}}\right)} \quad (2) $$

In addition, due to the type of loading (bending) and layup sequence (0° plies thickness and off-axis plies), a structural contribution is calculated to the micro-buckling mechanism [32] as shown in Eq. (3):

$$ \sigma_{UD}^{\text{stru}} = 2 r_{gf} \frac{\pi}{e_b} \sqrt{\frac{E_m E_f}{1 - v_m^2}} V_f (1 - V_f) \quad (3) $$

where $r_{gf}$ is the gyration radius of fibre, $E_m$ and $v_m$ are respectively the Young's modulus and the Poisson's ratio of the matrix, $E_f$ the longitudinal elastic modulus of the fibre, $V_f$ the volume fraction of fibres and $e_b$ the characteristic thickness of UD involved in the instability mode (a fraction of the total thickness of UD). For a laminate stacking under bending, $e_b$ is 0.4 times the thickness of consecutive UD plies undergoing the higher compressive stress [10,32]. The critical stress $\sigma_{UD}^{crit}$, which is taken as the compressive strength of UD, is the sum of both contributions $\sigma_{UD}^{stab}$ and $\sigma_{UD}^{stru}$.

## 2.3. Mechanical testing

Tensile tests on ±45° samples are carried out with a universal testing machine (Instron 5567, 30 kN load cell) to extract the in-plane UD shear behaviour, $\tau_{UD} - \gamma_{UD}$. As Wisnom [34] reported, the ±45° tensile tests are adequate for estimating the shear behaviour of a ply for strains below 7%. In the present study, the range of shear strains used is [0, 2]%, i.e. well below this level. Bi-axial strain gauges (10 mm in length - Kyowa) are glued to one side of the samples. Five samples with dimensions of 250 × 25 × ~2 mm³ and comprising eight plies (symmetric layup) are tested for the sake of reproducibility at a rate of 2 mm/min. A non-linear regression analysis is conducted for each sample in order to extract the Ramberg-Osgood (RO) parameters (see. Eq. (1)) within the 0%–2% range in shear strain. This limited strain range was chosen for two reasons: firstly, it increases the non-linear fit accuracy and, secondly, it is of sufficient length since the failure shear strains at instability are less than 2% for fibre misalignment levels of ~2° [12].

Four-point bending tests are carried out with a universal testing machine (Instron 5567, 30 kN load cell) using a distance of 90 mm

between the upper rollers (25 mm in diameter), where pure bending occurs, and a distance of 460 mm between the lower rollers. Mono-axial strain gauges (10 mm in length - Kyowa) are glued on both the compression and the tension sides. Ten samples are tested for each composite material at 5 mm/min. Five are tested with the mould side in compression, five with the vacuum bagging side in compression.

Sudden failure occurred systematically between the upper rollers. Data of flexural tests are analysed using the method described in Refs. [35–38], taking into account the non-linear elastic behaviour of CFRP. They are recalled briefly here. The force on the assembly and the two strain gauges signals in compression and in tension were recorded during loading and synchronised. The position of the neutral axis, for which the axial strain (beam direction) is null, is calculated following Euler-Bernoulli kinematics as well as the strain gradient. The failure strain ($\epsilon_{UD}^{fail}$) of the most loaded UD in compression, just below the ±45° outer plies, is then determined. The material behaviour of the unidirectional ply is assumed non-linear elastic introducing four elastic parameters: the initial tensile and compressive moduli, as well as, the linear increase of the tensile modulus and the linear decrease of the compressive modulus. These four parameters are evaluated by minimising the relative error between the experimental bending moment and an estimated value of this moment using the Classical Laminate Theory and taking into account the moderate bending rotations and deflections. Knowing eventually the compressive modulus at failure strain, the failure stress ($\sigma_{UD}^{fail}$) of the most loaded UD is calculated. The failure values $\epsilon_{UD}^{fail}$ and $\sigma_{UD}^{fail}$ as such are not to be compared with pure compression tests on 0° laminates as recalled in Section 2.2 where the structural effect is mentioned.

## 2.4. Fibre misalignment measurement method

The measurement of fibre misalignment is conducted in accordance with the methodology proposed by Yurgartis [18]. The procedure is comprised of the following steps:

**Obtaining smooth surfaces of cutting planes.** Samples from four-point bending tests are plane-cut at two specific angles, as illustrated in Fig. 1, with a diamond saw (rotation and cutting speeds: 2000 rpm and 1000 mm/s; water lubrication). The first cut, with $\Phi_{PC} = 10^\circ$, along the height (thickness) of the specimen, is made to assess in-plane fibre misalignment, while the second cut, with $\Phi_{PC} = 15^\circ$, along the width of the specimen, is made for out-of-plane fibre misalignment. These angles have been selected as the optimal compromise for subsequent polishing and imaging issues. Subsequently, the plane-cut specimens are embedded in resin and progressively ground with SiC papers from #500 to #4000. This is followed by polishing with diamond particles of 3 µm and finally 1 µm, in order to obtain a sufficiently smooth surface for optical microscopy.

**Obtaining high quality images.** Subsequently, the specimens are observed through the use of optical microscopy (OM, Olympus BX51), and images are captured via a video camera (Olympus DP25). A sufficient number of images (approximately 20) are captured along the width of the samples within the block of UD plies, either in proximity to the mould side or the bagging side, as illustrated in Fig. 2. The thickness of the outer plies on the bag and mould sides is approximately 300 µm. Consequently, the fibre misalignment measurements were taken between 300 and 600 µm below the free surfaces. The number of fibres observed in each image is greater than 300, resulting in a minimum of 24,000 fibres being observed for each fibre/resin pair. This is a greater number than that employed in previous studies [9,17].

**Detecting the ellipses by automated image analysis.** Automated image analysis (AIA) is necessary for the analysis of the large number of ellipses, as well as for the examination of imperfect images, which can present difficulties in analysis. An in-house algorithm based on Mask R-CNN (Mask Region-based Convolutional Neural Network) was
<figure><img src="image_6.png" /><figcaption>Fig. 1. Schematic definition of fibre misalignment (top) and (bottom) specimen sectioning to determine the in-plane ($\Phi_{PC}$) and out-of-plane ($\Theta_{PC}$) fibre misalignments.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 2. Definition of regions of interest for optical microscopy on the cutting planes for in-plane (left) and out-of-plane (right) fibre misalignment measurement.</figcaption></figure>

developed for use in both object detection and instance segmentation tasks in computer vision [39]. A machine learning-based method was employed, whereby training was conducted on both real and virtual images (25% and 75% of the total, respectively). This algorithm aims to recognise certain geometric shapes such as circles or ellipses. The algorithm process is divided into two parts: (i) the first part relies only on basic image processing methods; (ii) the second part is the effective machine learning algorithm based on neural network training. The detection process is comprised of a series of processing stages, including binarisation, the separation of fibres, contour capture and the measurement of contour dimensions. The first step is the binarisation of digital or real images, turning them into pure black and white images. A threshold is carefully chosen to correctly separate the fibres and therefore the algorithm is able to colour the fibres in white and the matrix in black. As illustrated in Fig. 3, instances may occur where fibres are damaged, rendering basic AIA incapable of detecting all contours (or, conversely, may erroneously identify contours). In the case of damaged fibres, the missing parts appear darker than the matrix and therefore a second threshold can be activated to detect only the missing parts of the fibres. The thresholds are then combined to reconstruct the damaged ellipses. Once binarisation is complete, the Machine Learning algorithm detects the contours of ellipses by detecting the white parts of the binarised images. Once the contours (called masks) have been created, the algorithm has been trained to determine the most likely ellipse within a given mask based on probabilistic criteria. Once the fitting ellipse is computed, the algorithm returns the centre of an ellipse for each fibre, denoted by its label i, and the lengths of its minor and major axes, denoted by $b_i$ and $a_i$, respectively. A final image is also drawn with a specific colour for each detected ellipse as seen in Fig. 3.

**Computing the initial fibre misalignments.** Following Yurgartis [18], the angle $\omega_i$ between the sectioning plane and the major axis of the ellipse labelled i is given by Eq. (4):

$$ \sin \omega_i = \frac{b_i}{a_i} \qquad (4) $$

The choice is made here to utilise the minor axis length of each fibre rather than a mean value of the fibre diameter derived from AIA on transverse cross-sections or from data sheets. Grabow [16] evaluated the impact of various potential approaches and observed minimal sensitivity, with a maximum of approximately 0.05° on the forthcoming statistical parameters.

The volume fraction of fibres at an angle $\omega_i$ to the plane-cut surface, $f_v(\omega_i)$, is given by Eq. (5) as cited in Yurgartis [18]:

$$ f_v(\omega_i) = \frac{\frac{N_i}{\tan \omega_i}}{\sum_{i=1}^{n} \frac{N_i}{\tan \omega_i}} \qquad (5) $$

where $N_i$ is the number of fibres counted at $\omega_i$ and n the total number of fibres.

**Analysing the fibre misalignment data.** The distributions obtained from Eq. (5) for each image are finally centred to zero by subtracting their mean values, as previously described by Yurgartis [18]. The aforementioned distributions represent those of the initial fibre misalignments, denoted as $\phi$. It was determined that the distributions adhere to a normal law distribution, from which the standard deviation ($\phi_{std}$) was calculated for each image (approximately 300 fibres). For a set of
<figure><img src="image_8.png" /><figcaption>Fig. 3. Detection of ellipses contours on a picture with damaged fibres (black areas), using a first AIA method (top) and a perfected one (down). The coloured ellipses are detected (and labelled) while the grey ones are not.</figcaption></figure>

images along the width of each sample (approximately 6000 fibres), a standard deviation is also computed.

## 3. Results

The standard deviation of the distribution of the initial fibre misalignment for the different fibre/resin pairs, $\phi_{\text{std}}$, is given in Table 3. The values are predominantly higher for the in-plane (IN) measurement than for the out-of-plane (OP) measurement, although this is not a systematic trend. Additionally, the maximum value of the IN and OP measurements is reported for each side, designated $\phi_{\text{std}}^{\text{BA}}$ for the bag side and $\phi_{\text{std}}^{\text{MO}}$ for the mould side. No discernible trend is evident. In some instances, the maximum value is observed on the mould side, while in others, the opposite is true. The final reporting of the deviations, $\Delta\phi_{\text{std}}^{\text{BA}}$ and $\Delta\phi_{\text{std}}^{\text{MO}}$ (one standard deviation), is finally presented in Table 3. The values can range from very low (pair S9, $\Delta\phi_{\text{std}}^{\text{MO}}$) to quite high (pair S6, $\Delta\phi_{\text{std}}^{\text{MO}}$), reflecting the inherent variability associated with the

manufacturing process, from lay-up to curing. It would appear that there is no discernible impact of the manufacturer on these levels of fibre misalignment (S1 vs S2, S3 vs S4).

The non-linear UD shear behaviour for all fibre/resin pairs is presented in Fig. 4. The extracted RO parameters (see Eq. (1)) are presented in Table 4. The composites comprising Se84-nano resin as the matrix (S3 and S4) exhibit the highest shear modulus. The composites with Se84-LV resin (S1 to S2) are in second place. The composites comprising XB3515 resin (S5), MTC9810 resin (S6) and Se75 resin (S9) are ranked third, exhibiting an average stiffness below 4 GPa. The shear moduli of the composites with BT080 resin (S7) and DT120 resin (S8) are markedly lower. Non-linear behaviours manifest as a hardening-like response, with the exception of Se84 LV resin, which exhibits a transient softening-like response, and Se84-nano, which displays brittle behaviour.

The compressive strength values $\sigma_{\text{UD}}^{\text{fail}}$ as well as the associated compressive failure strains $\epsilon_{\text{UD}}^{\text{fail}}$ are reported for all fibre/resin pairs in
<figure><img src="image_9.png" /><figcaption>Fig. 4. Non-linear in plane shear response of UD for the different fibre/resin pairs (truncated at 5% in shear strain).</figcaption></figure>

<caption>**Table 3**</caption>

Initial fibre misalignment (in degrees) for the different fibre/resin pairs (standard deviation of the associated distribution, $\phi_{std}$) and for the mould and bag sides. IN and OP indicate in-plane and out-of-plane measurements respectively. “Max” values indicate the maximum between IN and OP values, while the ± values represent the standard deviation over 20 images.

<table><caption>**Table 4**</caption><thead><tr><th rowspan="2">Pair</th><th colspan="2">Mould side</th><th colspan="2">Bagging side</th><th rowspan="2">Max. Mould</th><th rowspan="2">Max. Bagging</th></tr><tr><th>IN</th><th>OP</th><th>IN</th><th>OP</th></tr></thead><tbody><tr><td>S1</td><td>0.89</td><td>0.80</td><td>1.09</td><td>0.82</td><td><strong>0.89</strong> ± 0.14</td><td><strong>1.09</strong> ± 0.19</td></tr><tr><td>S2</td><td>1.05</td><td>1.06</td><td>1.10</td><td>0.87</td><td><strong>1.06</strong> ± 0.18</td><td><strong>1.10</strong> ± 0.14</td></tr><tr><td>S3</td><td>0.92</td><td>0.73</td><td>1.18</td><td>0.88</td><td><strong>0.92</strong> ± 0.19</td><td><strong>1.18</strong> ± 0.21</td></tr><tr><td>S4</td><td>0.88</td><td>0.66</td><td>1.05</td><td>1.01</td><td><strong>0.88</strong> ± 0.15</td><td><strong>1.05</strong> ± 0.11</td></tr><tr><td>S5</td><td>1.23</td><td>0.95</td><td>1.10</td><td>1.13</td><td><strong>1.23</strong> ± 0.20</td><td><strong>1.13</strong> ± 0.20</td></tr><tr><td>S6</td><td>1.52</td><td>0.66</td><td>1.41</td><td>0.90</td><td><strong>1.52</strong> ± 0.47</td><td><strong>1.41</strong> ± 0.30</td></tr><tr><td>S7</td><td>1.20</td><td>0.76</td><td>0.99</td><td>0.81</td><td><strong>1.20</strong> ± 0.12</td><td><strong>0.99</strong> ± 0.21</td></tr><tr><td>S8</td><td>1.16</td><td>0.93</td><td>0.97</td><td>0.90</td><td><strong>1.16</strong> ± 0.12</td><td><strong>0.97</strong> ± 0.10</td></tr><tr><td>S9</td><td>0.75</td><td>1.03</td><td>0.99</td><td>1.21</td><td><strong>1.03</strong> ± 0.07</td><td><strong>1.21</strong> ± 0.12</td></tr></tbody></table>

Ramberg-Osgood (Eq. (1)) parameters for the non-linear UD shear response for the different fibre/resin pairs.

<table><thead><tr><th>Pair</th><th>G<sub>UD</sub> [GPa]</th><th>γ<sub>UD</sub><sup>r</sup> [%]</th><th>n<sub>UD</sub> [-]</th></tr></thead><tbody><tr><td>S1/S2</td><td>4.75 ± 0.10</td><td>1.10 ± 0.01</td><td>6.61 ± 0.25</td></tr><tr><td>S3/S4</td><td>6.57 ± 0.13</td><td>0.90 ± 0.02</td><td>7.27 ± 0.39</td></tr><tr><td>S5</td><td>3.78 ± 0.16</td><td>1.17 ± 0.03</td><td>5.40 ± 0.07</td></tr><tr><td>S6</td><td>4.01 ± 0.06</td><td>1.24 ± 0.02</td><td>6.27 ± 0.10</td></tr><tr><td>S7</td><td>3.60 ± 0.06</td><td>1.25 ± 0.03</td><td>4.04 ± 0.08</td></tr><tr><td>S8</td><td>3.33 ± 0.06</td><td>1.14 ± 0.03</td><td>4.02 ± 0.1</td></tr><tr><td>S9</td><td>3.80 ± 0.13</td><td>1.14 ± 0.03</td><td>4.36 ± 0.12</td></tr></tbody></table>

**Table 5.** The values of the compressive strength ($\sigma_{UD}^{fail}$) range from a minimum of 1200 MPa to a maximum of 1700 MPa, with the exception of the S3 sample, for which a significantly higher compressive strength of approximately 1800 MPa is observed. The associated standard deviations are found to be well below 6% (coefficient of variation), with the exception of the S5 pair on the bagging side, which exhibits an 8% coefficient of variation. It is notable that all pairs exhibit failure values that are lower on the bagging side than on the mould side, with the exception of pair S3.

The estimated values of compressive strength are reported in the same Table 5. As reported by Stewart and Poursartip [29], the vast majority of research utilising the methodology proposed by Yurgartis [18] interprets the standard deviation of the fibre misalignment distribution $\phi_{std}$ as analogous to the initial fibre misalignment $\phi_0$ considered in

the Budiansky and Fleck [33]'s model in Eq. (2). The same methodology is employed in this section. In the case of prepreg tape composites, which are at stake in this study, cured in autoclaves at high pressures, fibre misalignment is very low and is three-dimensional in shape. To the best of our knowledge, for these very materials and associated very low values of fibre misalignment (standard deviation of ~1°), there is no clear experimental evidence that only the out-of-plane misalignment controls the instability leading to catastrophic failure. As outlined in Table 1, the composites community does not hold a unanimous opinion on this matter, with some individuals favouring in-plane values, others out-of-plane values, and some the 3D maximum values. This is, for example, different from results reported in studies on NCF composites with far higher fibre misalignment values (more than 5°) [30,31]. For this reason, we chose to take $\phi_{std}$ as the maximum value between the in-plane (IN) and out-of-plane (OP) values to select the highest defect severity which influences the onset of micro-buckling.

The comparison between measured and estimated values of compressive strength is shown in Table 5. We had the option of reporting a number comparing both values in percentage terms. However, we considered this to be somewhat reductive. For this reason, in the following, we have opted for a graphical comparison that takes into account the sources of variability.

Fig. 5 compares the measured ($\sigma_{UD}^{fail}$) and estimated ($\sigma_{UD}^{crit}$) values of compressive strength. A bisection line has been included to facilitate comparison between the two estimates, as well as dashed lines within ±10% difference. For the five pairs S5 to S9, the comparison is favourable for the mean values, which are within ±10% difference (except for S8 on the mould side, which is underestimated). Pairs S1 and S2, corresponding to Se84LV/IM2C with two different manufacturers, exhibit mean values or scatter values that are also within ±10% difference. However, the correlation is of lower quality compared to the pairs S5 to S9 mentioned above, and the estimated values are slightly overestimated. It is also worthwhile to consider the pairs S3 and S4, corresponding to Se84nano/IM2C with two different manufacturers. Similarly, pair S3 on the bagging side exhibits a comparable pattern. The results demonstrate that the experimental results are sometimes slightly underestimated and other times overestimated by the estimations. The largest discrepancies are observed for S4 and S3 (mould side), which also exhibit the highest scatter in $\sigma_{UD}^{crit}$, as illustrated in Fig. 6. The reasons for this will be investigated. For further details on the graphical comparisons between the experimental results and the estimations of compressive strength, please see the Supplemental material.
<figure><img src="image_10.png" /><figcaption>Fig. 5. Comparison between experimental and estimated values of compressive strength. Open and closed symbols refer to bag and mould sides, respectively. The bisecting line is in black colour and dashed lines are within ±10% of this line.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 6. Comparison of the compressive strength measured from experiments to the numerical predictions thanks to initial misalignment measurement (average and deviation) and shear behaviour of the UD.</figcaption></figure>

# 4. Discussion

## 4.1. Clustering of highly misaligned fibres: fibre waviness

The fibre misalignment values ($\phi_{std}$) for pairs S3 and S4 (Se84 nano-resin and IM2C fibres) are approximately equal, with a mean value of 0.9° on the mould side and 1.0° on the bagging side (Table 3). However, the compressive strength values obtained for S3 and S4 exhibit a discrepancy of up to approximately 300 MPa. The sole information yielded from the analysis of the fibre misalignment distribution is its standard deviation, denoted as $\phi_{std}$. Nevertheless, the spatial configuration of the fibres may influence the compressive performance, as previously demonstrated [40]. Consequently, further investigation into this aspect is warranted. The spatial distributions of fibre misalignment (OP) are presented for three different pairs in Fig. 7.

S1 is used as a reference (Se84 LV), S3 is the specimen with the highest measured compressive strength, and S4 is the specimen with the poorest comparison between measured and estimated compressive strength values. Fibres exhibiting an initial misalignment less than the standard deviation are represented by a green colouring (representing 68% of the fibres). Fibres exhibiting an initial misalignment between the standard deviation and twice the standard deviation are represented by blue colouring, which encompasses 27% of the fibres. Fibres exhibiting a misalignment exceeding twice the standard deviation are represented in red (5% of the fibres).

The following section focuses on the spatial distribution of potential clusters of highly misaligned fibres (red and blue zones) as illustrated in Fig. 7. With regard to S1 and S3, the red zones are observed to be distributed in a relatively uniform manner throughout the width and height of the most loaded unidirectional ply. These are found in
<caption>**Table 5**</caption>

Compressive strength values $\sigma_{\text{UD}}^{\text{fail}}$ and associated compressive failure strains $\epsilon_{\text{UD}}^{\text{fail}}$ for all resin/fibre pairs on both bagging and mould sides [left]. Estimated values of compressive strength $\sigma_{\text{UD}}^{\text{crit}}$ [right]. The variations in the failure strains and strengths are directly related to one standard deviation of the experimental results. Those of the estimated values of the compressive strength are derived from the variation of fibre misalignment, from Table 3.

<table><thead><tr><th>Property</th><th>Pair</th><th>Side</th><th>ε<sub>UD</sub><sup>fail</sup><br/>[%]</th><th>σ<sub>UD</sub><sup>fail</sup><br/>[MPa]</th><th>σ<sub>UD</sub><sup>crit</sup><br/>[MPa]</th></tr></thead><tbody><tr><td rowspan="4">S1</td><td>Mould</td><td>1.08 ± 0.06</td><td>1500 ± 82</td><td>1820 ± 147</td></tr><tr><td>Bag</td><td>1.00 ± 0.04</td><td>1406 ± 66</td><td>1637 ± 154</td></tr><tr><td>Mould</td><td>1.09 ± 0.03</td><td>1514 ± 41</td><td>1663 ± 152</td></tr><tr><td>Bag</td><td>1.07 ± 0.06</td><td>1479 ± 57</td><td>1630 ± 112</td></tr><tr><td rowspan="4">S2</td><td>Mould</td><td>1.18 ± 0.09</td><td>1689 ± 57</td><td>2226 ± 257</td></tr><tr><td>Bag</td><td>1.30 ± 0.05</td><td>1819 ± 85</td><td>1935 ± 202</td></tr><tr><td>Mould</td><td>1.08 ± 0.02</td><td>1575 ± 42</td><td>2280 ± 214</td></tr><tr><td>Bag</td><td>1.02 ± 0.04</td><td>1504 ± 41</td><td>2068 ± 123</td></tr><tr><td rowspan="2">S3</td><td>Mould</td><td>1.04 ± 0.02</td><td>1374 ± 18</td><td>1275 ± 108</td></tr><tr><td>Bag</td><td>1.00 ± 0.06</td><td>1330 ± 109</td><td>1332 ± 121</td></tr><tr><td rowspan="2">S4</td><td>Mould</td><td>1.11 ± 0.03</td><td>1350 ± 56</td><td>1241 ± 221</td></tr><tr><td>Bag</td><td>1.09 ± 0.03</td><td>1330 ± 84</td><td>1293 ± 152</td></tr><tr><td rowspan="2">S5</td><td>Mould</td><td>1.33 ± 0.04</td><td>1399 ± 44</td><td>1281 ± 60</td></tr><tr><td>Bag</td><td>1.28 ± 0.07</td><td>1379 ± 41</td><td>1398 ± 134</td></tr><tr><td rowspan="2">S6</td><td>Mould</td><td>1.24 ± 0.04</td><td>1395 ± 54</td><td>1162 ± 56</td></tr><tr><td>Bag</td><td>1.03 ± 0.07</td><td>1208 ± 67</td><td>1261 ± 59</td></tr><tr><td rowspan="2">S7</td><td>Mould</td><td>1.00 ± 0.07</td><td>1363 ± 79</td><td>1380 ± 45</td></tr><tr><td>Bag</td><td>0.86 ± 0.05</td><td>1199 ± 80</td><td>1278 ± 62</td></tr></tbody></table>

large zones of lower misalignment values, indicated by the blue and green colours. In contrast, S4 exhibits the presence of prominent green zones, characterised by low fibre misalignment values. It can therefore be surmised that larger areas of highly misaligned fibres (blue and red colours) exist. The aforementioned clusters of fibre misalignment values will henceforth be referred to as fibre waviness, i.e. localised areas of high fibre misalignment.

Takahashi et al. [41] investigated the impact of fibre misalignment frequency and spatial distribution on material properties. They observed that a clustering effect led to a 7% reduction in compressive strength. The objective of what follows is to propose a quantitative analysis of fibre waviness by estimating the density of clusters per image. The number of fibres in each cluster, the size of the largest cluster and the standard deviation of the cluster size distribution are determined by measuring the fibre clusters. The measurement of cluster size is conducted in two stages. Initially, fibres exhibiting an initial misalignment exceeding the standard deviation $\phi_{\text{std}}$ are identified (these are represented by fibres of red and blue colours in Fig. 7). Subsequently, the cluster size is computed by counting the fibres within the circle, which misalignment is higher than $\phi_{\text{std}}$ and are contained within the circle. The radius of the circle was selected as five times the diameter of the fibres, based on a qualitative observation and comparison of the fibres' distribution (Fig. 7).

The results are presented in Fig. 8 for selected pairs (S1, S3, S4 and S6) for both the mould and bagging sides. Pairs S1 and S6 exhibit cluster sizes between 1 and 2% of the total number of fibres in an image, irrespective of the image in question, which is to say, along the width of the specimen. This is relatively homogeneous. In contrast, pairs S3 and S4 demonstrate a markedly disparate cluster size, with low and high percentages along the width of the specimen. S3 exhibits the largest cluster sizes while S4 exhibits the smallest ones. This result indicates that the defect distribution for the S4 pair is more organised at the local level, while exhibiting identical defect variation ranges ($\phi_{\text{std}}$) to the other samples.

Based on the results of the plastic micro-buckling model of Drapier et al. [42, Fig. 10], a more organised imperfection distribution leads to a reduction in compressive failure strains in ranges (from 1.6% to 1.1% on failure strain, -30%). This equates the discrepancy between the estimated compressive strength and the corresponding experimental

data for S4 (see Table 5). There is an imbalance in the distribution of stress, which results in neighbouring fibres being subjected to significant misalignment, thereby leading to premature failure at the global level. In the case of the largest cluster sizes, the arrangement of the randomness in the fibres' misalignment on the bagging side is more consistent for pair S3, which results in higher failure strains.<fnref n="1" />

## 4.2. Initial fibre misalignment as a fitting parameter

In the work of Budiansky and Fleck [33] the initial fibre misalignment, designated as $\phi_0$, is not directly measured but rather inferred from comparisons with experiments. In many cases, $\phi_0$ values are found to be around 2° for polymer matrix composites. However, they cite the experimental work of Yurgartis [18] who obtained a standard deviation of 1.9° for his fibre misalignment distribution. In this study, the values of initial fibre misalignment $\phi_0$, corresponding to the standard deviation $\phi_{\text{std}}$ of our measurements following Yurgartis [18]'s method, are significantly smaller than approximatively 1°. This reflects the significant advancements made over the past three decades in the production of autoclave-cured CFRP at elevated pressures.

In this section, we adhere to the original concept proposed by Jelf and Fleck [43], Budiansky and Fleck [33] and consider the initial misalignment $\phi_0$ as a fitting parameter with the objective of aligning the estimated values with the experimental results. The results are presented in Table 6. Pairs S7 to S9 exhibit the lowest initial fibre misalignment required. Subsequently, pairs S1 and S2 (Se84LV resin) and S5 and S6 are considered. Conversely, the highest initial fibre misalignment is required for pairs S3 and S4 (Se84 nano resin).

Subsequently, a proportionality factor is calculated between the values of $\phi_0$ from Table 6 and the measured standard deviations $\phi_{\text{std}}$ from Table 3 for each pair. The resulting numbers are displayed in Fig. 9. It is notable that the mean factor to apply to the standard deviation for pairs S5 to S9 is close to one on both sides (bagging and mould). This observation reinforces the widely accepted recommendation to use the standard deviation as the initial misalignment for the compressive strength prediction [29,44].

Nevertheless, the coefficient to be applied to the standard deviation per type of fibre/resin pair differs for Se84 LV (1.2 to 1.5) and Se84 nano (1.2 to 1.9). In Fig. 9, the values of UD shear stiffness are superimposed. A correlation seems to exist between the UD shear stiffness and the correction factor to be applied to the standard deviation. This in accordance with the work of Launay [45]. It can be observed that as the shear stiffness increases, the factor in question also rises. In other words, a relatively small number of highly misaligned fibres is sufficient to reach the compressive limit for Se84 nano (S3, S4) when the stiffness of the resin increases. Conversely, a substantial number of misaligned fibres are necessary to attain the compressive strength of pairs S5 to S9. These observations reinforce the necessity to consider the arrangement of fibres, in addition to the shear response, in order to accurately assess the performance of different fibre/resin pairs.

## 4.3. Limitations and challenges

The experimental findings of this study pertain to compression failure, specifically focusing on bending tests. These tests facilitate the elimination of the majority of the experimental challenges encountered in conventional pure compression tests. However, it should be noted that the values obtained through these tests cannot be directly compared with those derived from compression tests, particularly in the case of pure unidirectional specimens. In addition to the aforementioned experimental challenges, it is important to acknowledge that

<footnote n="1">¹ The same data analysis has been for in-plane measurements. The conclusion was that the data did not contribute any additional information, and therefore they are not shown.</footnote>
<table><caption>**Table 6**
Computed initial fibre misalignment ranges (in degrees) which would be expected to get agreement between experiments and predictions of the compressive strength.</caption><thead><tr><th></th><th>S1</th><th>S2</th><th>S3</th><th>S4</th></tr></thead><tbody><tr><td>Mould</td><td>1.33 ± 0.13</td><td>1.30 ± 0.06</td><td>1.53 ± 0.09</td><td>1.72 ± 0.08</td></tr><tr><td>Bagging</td><td>1.49 ± 0.13</td><td>1.37 ± 0.10</td><td>1.36 ± 0.11</td><td>1.85 ± 0.09</td></tr><tr><td></td><td>S5</td><td>S6</td><td>S7</td><td>S8</td><td>S9</td></tr><tr><td>Mould</td><td>1.09 ± 0.04</td><td>1.34 ± 0.10</td><td>1.04 ± 0.08</td><td>0.82 ± 0.08</td><td>1.04 ± 0.14</td></tr><tr><td>Bagging</td><td>1.18 ± 0.19</td><td>1.37 ± 0.17</td><td>1.09 ± 0.07</td><td>1.13 ± 0.14</td><td>1.37 ± 0.18</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. 7.** Distribution of the fibres misalignment OP measurements for couples IM2C/Se84 LV (S1) IM2C/Se84 nano (S3/S4). Red colour refers to fibres with a misalignment values higher than twice the standard deviation of the fibre misalignment distribution, blue colour refers to fibres with values between one and twice the standard deviation, and green colour is used for values lower than the standard deviation.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 8.** Spatial distribution of the maximum cluster size (determined from Fig. 7) (in percentage of the total number of fibres) along the width (image number) of four-point bending specimens for selected pairs. Shaded zones are for rapid visualisation.</figcaption></figure>

the compressive strength of a unidirectional ply in the fibre direction does not constitute a material parameter. The structural effect [6,32], including the type of loading (compression vs. bending, for example) and the type of stacking of the laminate, as well as the misalignment of the fibres, which will depend on the manufacturing process and the thickness of the laminates, are responsible for this. For engineers accustomed to utilising values derived from pure compression tests on unidirectional materials or cross-laminates, it will imperative to establish a clear pathway between these two methodologies. The bending loading and the use of a particular thick composite laminate are

linked to uses in certain parts of the racing yachts field by engineers designing very thick parts and for which this situation (load, stack) is representative, allowing the direct use of these fracture values [45,46].

The stereological technique utilised in this study for the measurement of fibre misalignment enabled the analysis of substantial quantities of fibres. In comparison with the majority of previous research that employed this technique [17], this study enhances the spatial and frequency representativeness through the utilisation of modern Machine Learning capabilities. A systematic comparison of this method with
<figure><img src="image_14.png" /><figcaption>**Fig. 9.** Evolution of the initial misalignment angle (expressed in terms of a proportionality factor of the measured standard deviation of the misalignment distribution) of each fibre/resin pair and side (bagging or mould) to match analytical predictions and experimental results. The evolution of the shear modulus is also presented for the discussion.</figcaption></figure>

alternative approaches proposed in the literature [15,30,31] would be of great use.

The analytical criterion for failure in compression employed in this study [32,33] is very limited in its utilisation of the extensive information associated with fibre misalignment, as it considers only a single parameter. The question of how to incorporate information regarding the full frequency distribution, spatial distribution, or even the amplitude or wavelength of the defects remains largely unaddressed in analytical models with clearly measurable parameters (see a recent study by Ueda et al. [3] for a proposition on some of these aspects). This represents a significant challenge for the composites community.

## 5. Concluding remarks

The objective of this study was to examine the impact of incorporating fibre misalignment features into the estimation of the compressive strength of CFRP, with the aim of establishing a correlation with experimental data. The present study focused on the examination of thick laminates with tape prepregs that underwent manufacturing from companies utilising autoclave curing and vacuum bagging, with the objective of identifying the characteristics of high-quality composite materials. The compressive strength values of unidirectional plies in these thick laminates were obtained through the utilisation of flexural tests on laminates, in conjunction with the implementation of appropriate identification methodologies involving nearly ten fibre/matrix pairs. It was necessary to distinguish between the mould and bagging sides, as different defect severities might be involved. The analytical criterion of Grandidier et al. [32] was selected for estimation purposes, as it is the only one that incorporates ease of use for engineering design, quantification of the different mechanistic contributions [33] and the structural effect [7]. All the relevant parameters were measured for each fibre/matrix pair. Specifically, the influence of initial fibre misalignment, which represents a single scalar parameter in the model proposed by Budiansky and Fleck [33] was quantified using a stereological method [18]. For each pair and each side of bending specimens, a dedicated in-house machine learning-based algorithm was employed to analyse more than 12,000 fibres, resulting in the generation of meaningful spatial and frequency distributions of fibre misalignment.

In the majority of instances, the correlation between the estimated and experimental results was found to be either satisfactory or, in some cases, excellent. The application of the standard deviation of the frequency distribution of fibre misalignment (found to be approximately 1°) was demonstrated to be a pertinent approach in these instances. It can therefore be concluded that the analytical criterion proposed

by Grandidier et al. [32] can be employed with a relatively high degree of confidence in engineering design. Nevertheless, in a limited number of cases, the correlation was found to be weak. The analysis of the spatial distribution of initial fibre misalignment demonstrated that in cases where the same fibre/matrix pair was utilised by different manufacturers, the formation of localised clusters of highly misaligned fibres (fibre waviness) or, conversely, the development of more uniform spatial distributions could result in disparate failure outcomes that the standard deviation is unable to account for. In the context of engineering design, the incorporation of a safety factor linked to a specific manufacturer, or more precisely, its degree of maturity with a specific fibre/matrix pair, may be necessary to utilise the model. Additionally, regular investigations of manufacturing defects may also be required.

## CRediT authorship contribution statement

P.-Y. Mechin: Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. M. Grabow: Writing – review & editing, Validation, Methodology, Investigation, Data curation. A. Launay: Writing – review & editing, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation. J.C. Grandidier: Writing – review & editing, Visualization, Validation, Supervision, Software, Investigation, Formal analysis, Data curation. Z. Zhang: Software, Investigation, Data curation. V. Keryvin: Writing – original draft, Visualization, Validation, Supervision, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

## 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.

## Acknowledgements

The authors are grateful to E. Fabing, H. Deplagne and R. Keswani for their assistance in automated image analysis. We thank also I. Pillin and E. Bois for her experimental help. The financial support of ANRT (CIFRE) for A. Launay's PhD grant, and of Région Bretagne and Conseil Départemental 56 for M. Grabow's PhD grant, is gratefully acknowledged.
# Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.compositesa.2025.108950.

## Data availability

Data will be made available on request.

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