Impact of Defects on Tensile Properties of Ancient and Modern Egyptian Flax Fibers: Multiscale X-Ray Microtomography and Numerical Modeling

Abstract

Flax fibers, while offering numerous benefits, are susceptible to mechanical weakening due to the presence of kink-bands within their structure. The novelty of this study lies in linking mechanical behavior to fiber morphology and defects at multiple scales by utilizing X-ray microtomography to generate detailed 3D images of elementary flax fibers, enabling the creation of accurate finite element (FE) models for analysis. Aging reduces flax fibers’ strength, so both modern and ancient fibers were analyzed to understand their structural evolution over time. Static X-ray microtomography images were converted into 3D FE models for tensile simulations, and tensile tests provided essential properties for numerical modeling. Morphological analysis for both fiber types revealed that kink-bands contain multiple pores oriented ~45° to the fiber/lumen axis, with ancient fibers showing higher porosity (5.6%) and kink-band density (20.8 mm⁻¹) than modern fibers (3.3% and 16.6 mm⁻¹). SEM images confirmed that the intricate lumen and kink-bands lead to fiber failure under tensile loading. Numerical analysis highlighted higher stress concentrations at the kink-band region, particularly at pores in the kink-band region, which can initiate cracks and lead to rupture.

1. Introduction

Composite materials are replacing metals, ceramics, and wood due to their specific superior properties. Typically made with carbon, glass, and aramid fibers, they are expensive, non-recyclable, non-biodegradable, and pose health risks [1]. Plant fibers like flax and hemp offer an eco-friendly alternative for reinforcement [2,3]. However, defects occurring during the extraction process or authentic to the plant growth conditions can reduce their mechanical properties [4,5]. To optimize their use, it is crucial to understand the relationship between these structural defects and mechanical properties at multiple scales [6], including elementary and bundle fibers. To address this, a combination of experimental and numerical analysis was applied in this study for both elementary and bundle flax fibers. Aging can reduce the strength and stiffness of flax fibers, impacting the durability and performance of composites [7]. Research on fiber aging examines how durability and degradation affect structural changes over time [8,9]. This study uses ancient Egyptian flax fibers to explore how structural morphology evolves and investigates the effects of defects on mechanical responses.

Elementary flax fibers have a complex structure composed of multiple cell wall layers, including the primary cell wall (PCW) and the secondary wall (S), which is further divided into S1, S2, and S3 layers. The S2 layer, being the thickest, significantly contributes to the fiber’s strength and stiffness due to its microfibrils oriented nearly parallel to the fiber axis [10]. The central hollow cavity, or lumen, runs through the length of the fiber, varying in size and shape [11]. Flax fiber bundles are composed of multiple elementary fibers bound together by middle lamella, which is composed of pectin and hemicellulose [12,13]. This structure varies in regions with defects, which can undermine reinforcing capabilities. Kink-bands are areas with irregular cellulose microfibril arrangement that disrupt mechanical properties [14]. Despite dislocations in up to 20% of hemp fiber cell walls, they showed no significant impact on tensile properties [15]. However, a negative correlation was found between kink-band area and Young’s modulus, with flax fibers over 75 GPa typically having less than 15% kink-band density [16]. Defects in elementary fibers cause significant shifts in cellulose orientation (30°–40°) and increased porosity, especially in the S2 layer [17,18,19]. These issues lead to microfibril angle deviations and localized pore formation, which act as rupture points and affect mechanical properties [18,20]. Despite experimental studies at the elementary scale, only a few research efforts include numerical analysis using finite element (FE) models of the fiber’s real 3D structure [21,22].

In the previous study, we reported an analysis of modern Egyptian flax fiber bundles using in situ tensile testing, X-ray microtomography, and numerical analysis [23]. The innovative aspect of the present study lies in analyzing the actual morphology of fibers and constructing a real 3D model of elementary fibers for tensile simulations. Additionally, the study examines the structural evolution and mechanical response of fibers using ancient samples. However, this study does not include experimental testing for ancient fiber bundles. The reason is the degradation of the middle lamella, which prevented us from obtaining sufficiently long fiber bundles. Instead, a shorter fiber length of 50 µm was used for numerical analysis. A significant degradation of non-cellulosic cell wall polymers, ensuring cohesion between cellulose fibrils, was observed [24,25], and this decrease in parietal components can affect the mechanical behavior of flax fibers. The study aims to assess how kink-bands affect the strength of both modern and ancient Egyptian flax fibers at multiple length scales. Understanding the mechanical properties of ancient Egyptian flax is crucial for evaluating modern flax as a viable alternative to synthetic materials in composites, particularly in enhancing its durability. Examining the mechanical properties of ancient and modern flax fibers is important, as it offers insights into how material properties have evolved over time. Understanding these differences allows us to evaluate the effects of structural degradation. Additionally, this comparison sheds light on the durability and performance of ancient fibers in various applications, which can contribute to the development of sustainable materials in modern contexts. The objectives of this study include: 1) analyzing the morphology of ancient flax fibers by considering the modern flax fibers as a reference, with a focus on defect structures and the impact of aging on the ancient fibers; and 2) performing tensile simulations on the real 3D fiber structures derived from static tomography images to investigate the consequences of fiber morphology under tensile loading. The main goal is to investigate how kink-bands impact tensile properties by examining stress concentrations in these regions.

2. Materials and Methods

2.1. Materials

Modern elementary flax fibers (MF) of the Eden variety were sourced from fabric at the Institut Français d’Archéologie Orientale (IFAO) in Cairo. Ancient elementary flax fibers (AF), ancient bundle flax fibers (ABF) which is composed of three elementary fibers, were obtained from a flax fabric preserved at the Louvre Museum in Paris (inv. E 13593G). Radiocarbon dating (Laboratoire de Mesure du Carbone 14, SacA70167, CEA-Saclay, Gif-sur-Yvette, France) dates this fabric to between 4324 and 4053 BCE, corresponding to the First Intermediate Period and the early Middle Kingdom.

2.2. X-Ray Microtomography

Inline phase-contrast microtomography scans of the samples were performed in the second ANATOMIX hutch at a 200 m distance from the X-ray source. Two different setups were used for a robust solution. In the first setup, the elementary fiber and bundle were attached to the ends of support needles, which were subsequently fixed to the sample holder. A camera CMOS ORCA Flash 4.0 V2 (Hamamatsu, Hamamatsu city, Japan) was used in full-frame unbinned mode (2048 × 2048 pixels, physical pixel size 6.5 µm) coupled with a 20 × M Plan APO (Mitutoyo, Kawasaki, Japan) objective to reach a pixel size of 0.325 µm. The sample-detector distance was set at 7 mm. X-ray radiographs were taken over a range of 180° (standard geometry) with 2000 projection angles per tomography scans at 300 ms exposure. A Paganin filter [26] with a length of 15 pixels (Paganin unsharp filter length: 2 pixels; Paganing unsharp filter coefficient 0.5) and flat field correction for ring artefacts were used. In the second setup, samples were glued to cardboard with a 10 mm gauge length and mounted on a tensile test system. Unlike the dynamic images used in previous study [23], this study used only static images for analysis and model development. A 40 keV white beam was employed, and a 10 × M Plan APO objective was used, achieving a 0.65 µm pixel size. Filters included brazed CVD diamond with Au coatings (10 µm) and a 100 µm Cu filter. The scan geometry was 180° with 2000 projections per scan, and reconstruction was performed using PyHST2 software Version 2021c with Paganin correction.

The reconstructed 3D volumes were analyzed using FIJI software (Version 1.54) [27]. Three-dimensional segmentation of the fibers and bundle volumes was performed to separate voids from the solid phase. This involved converting images from 32-bit to 8-bit, filtering, rotating, applying thresholds, selecting Regions of Interest (ROI), and image segmentation. Finally, a flooding procedure was used to distinguish fibers and porosity from the background [23]. The volumetric analysis of these fibers was performed to understand the morphology of the fibers. The shape factor was calculated from the ratio between the major and minor axes of the particle’s fitted ellipse, and was determined using transversal cross-sections of the fibers. The porosity content was determined for each transverse cross-section of the fiber using Equation (1). The mean porosity content was determined as the average of the cross-sectional porosity values along the fiber.

(1)$Porosity \mathrm{\%}=100\times {(\Sigma V}_P/{\Sigma V}_f)$

where $V_P$ represents the voxel counts of porosities and $V_f$ represents the voxel counts of solid phase within the fiber.

The quantification of kink-bands was carried out by assessing the kink-band density, defined as the number of kink-bands per unit length of the fiber (mm) [28], and the average distance between kink-band was determined from tomographic images. Due to the degradation of the middle lamella, this measurement was not conducted for ancient bundle fibers, as only shorter lengths were obtained. Additionally, the porosity in the kink-band region was determined for both elementary and bundle fibers.

2.3. Tensile Testing

The tensile properties of elementary flax fibers to be incorporated into numerical models were determined through a tensile experiment utilizing the Dia-Stron LEX820 Extensometer (Dia-Stron Limited, Andover, UK) with a 20N load cell and a displacement rate of 1 mm/min. Tests were conducted in a controlled environment at 25 °C with 55% relative humidity. The fibers were extracted manually, placed in plastic tabs, and bonded with adhesive, which underwent exposure to ultraviolet light for 15 s to facilitate adhesive curing, maintaining a sample length of 4 mm. The dedicated software UvWin V4.2.6.3 served as an interface to both control and analyze the measurements obtained by the LEX820. The diameter of the fibers was determined using Fiber Dimensional Analysis system (FDAS) (Dia-Stron Ltd., Andover, UK). Prior to tensile testing, the fibers were observed under an optical microscope to determine the scale of the fiber (elementary or bundles), which further highlighted the kink-bands present in elementary fibers, as shown in Figure 1. Ten samples from MF were utilized. The extraction of ancient fibers proved challenging due to their degradation and brittleness. A total of 70 samples were prepared. However, due to the fragile nature of these fibers, several samples broke before tensile testing while being observed under the microscope. Additionally, some fibers were damaged during diameter measurements, and some failed during the installation of the sample in tensile equipment. As a result, only 5 samples out of 70 samples of ancient fibers were used for tensile testing, providing a success rate of 7%. The average and standard deviation of the tensile test results were calculated, along with the coefficient of variation, which represents the ratio of the standard deviation to the mean. After tensile testing, the fracture surface was examined by SEM, which was performed using a Quattro S ESEM (Thermo Scientific) microscope at INRAE, Nantes, France. Images were recorded using a low-vacuum detector to prevent sample deformation due to the use of high vacuum and without metallization. An acceleration voltage of 10 kV was used, the pressure was around 100 Pa at an environmental temperature, and the working distance ranged from 8 mm to 10 mm. Images were obtained in low-vacuum conditions.

2.4. Finite Element Computation

The static images captured from the X-ray microtomography were transformed into numerical models through a combination of surface tessellation using triangular elements and propagating tetrahedral elements in the core. FIJI Version 1.54 (https://fiji.sc/ (accessed on 1 December 2024)) and Simpleware ScanIP Version 2021.03 (https://www.synopsys.com/simpleware.html (accessed on 1 December 2024)) software were used to convert 3D images into 3D models. The meshed fibers exhibited degrees of freedom ranging from 5 × 10⁵ to 2 × 10⁶, with a mean volume element of 128 × 10⁻³ µm³ and 231 × 10⁻³ µm³. The meshed volume of elementary fibers featured a height of approximately 120 µm. For the bundles, the degrees of freedom were 6 × 10⁵, with a height of 50 µm. The total number of elements for elementary and bundle fibers ranged from 1 × 10⁵ to 4 × 10⁵. All computations were carried out using the structural mechanics module in COMSOL software (Version 5.6).

Five models were created: one from modern fiber with kink-bands (MF), one from ancient fiber with kink-bands (AF), a model with porosity artificially filled in modern fiber (MFF), an elliptical model derived from real ancient fiber geometry (GF), and finally, the ancient bundle fiber (ABF), composed of three elementary fibers glued together by a middle lamella with a kink-band. The elliptical model was chosen because the cross section of the elementary fiber closely resembled an elliptical shape [29]. The material was assumed to be linear, elastic, and isotropic, with an experimentally obtained Young’s modulus and a Poisson’s ratio of 0.2 [22], and the change in microfibril angle was not considered. In order to replicate the tensile conditions, the following boundary conditions were used: the lower surface of the fiber was clamped (i.e., displacement was equal to 0 in all directions at all nodes of the surface), and a displacement ‘$d$’ was applied in the z-direction to the top surface using the following equations:

(2)$Ux=Uy=Uz=0 For the lower surface$

(3)$Ux=Uy=Uz=0 For the upper surface$

The applied displacement ($d$) was 1.2 µm and 0.5 µm for the elementary and bundle fiber, which corresponded to 1% strain over the entire simulated length $L$. The average stress ($\sigma$) and strain values ($\epsilon \mathrm{\%}$) were determined using the Equations (4) and (5), where $Fz$ is the total nodal reaction force in the z-direction, $S$ is the surface of the filled fiber, $d$ is the imposed displacement, and $L$ is the length of the fiber. Given that the reaction forces at the nodes of both fiber edges had distinct transverse surface areas, the resultant stress was computed from the mean between these integrals (Equation (4)).

(4)$\sigma ={\displaystyle \frac{1}{2}}\iint {\displaystyle \frac{{Fz}_{(x=0)}}{S_{(x=0)}}}+\iint {\displaystyle \frac{{Fz}_{(x=d)}}{S_{(x=d)}}}$

(5) $\epsilon \mathrm{\%}=100 \ast ({\displaystyle \frac{d}{L}})$

3. Results and Discussion

3.1. Morphological Analysis

Figure 2 shows volumetric and cross-section images of MF and AF, highlighting the kink-band region, while Figure 3 shows similar images for ABF. Figure 2a,d provide a volumetric view in the XZ plane, revealing the central cavity lumen and the kink-band region (emphasized in the white box). The pores in the kink-band region (Figure 2b,c) are oriented at approximately 45° to the lumen axis [18,23]. Similarly, Figure 2e,f shows the lumen and pores in the kink-band region of AF, where discontinuities in the lumen and structural differences in the kink-band compared to MF are evident. AF showed a higher porosity in the kink-band region, reaching 5.6%, compared to MF, which had a porosity of 3.3% (Table 1). Additionally, the kink-band density was higher in AF (20.8 mm⁻¹) compared to MF (16.6 mm⁻¹), indicating a greater number of kink-bands in AF. Similarly, many defects were observed in same variety of ancient yarns compared to modern ones [24]. The average distance between kink-bands along the fiber length was greater in MF (114 µm) than in AF (77 µm). This structural difference could be due to the varying processing methods used during fiber extraction, as well as potential alterations in the structure of ancient fibers over time [30]. The pore orientation in the kink-band region for AF was also about 45° to the fiber/lumen axis. It can be observed that the pores were scattered and arranged concentrically around the lumen [18,23]. Figure 3a shows tomographic images of ABF in the YZ plane, emphasizing the kink-bands in the white box. As with AF, lumen discontinuities in elementary fibers of ABF are visible. The structure of kink-bands in the bundle was almost similar to modern fiber bundles [23]. Although the pores in the kink-band region of bundle fibers are not clearly visible, a small area in the fiber cross-section (Figure 3b) shows that the pore orientation was similar to that in MF and AF relative to the fiber/lumen axis. The porosity measured in the kink-band region for ABF was 2.6%.

Table 1 presents data extracted from tomography images. The lengths of the elementary fibers, MF and AF, were 120 µm, while the bundle fiber ABF was 50 µm. The shorter length of ABF was due to the degradation of the middle lamella in ancient bundles. The shape factor for both elementary and bundle fibers was approximately 1. The volume of elementary fibers was higher than that of bundle fibers. Among the elementary fibers, MF had a larger volume than AF, with a volume ratio of 9:1, which may suggest fiber degradation in AF. Table 1 also shows the porosity percentage, which includes both the lumen and pores in the kink-band region. This percentage was higher in ancient fibers compared to modern fibers. Figure 4 shows the porosity percentage along the 2D stacks of the fiber, i.e., along the length of the fiber, for MF, AF, and ABF. Variations in porosity along the fibers were evident for all three types. The largest peaks in the graph indicate regions with lumen and pores in the kink-band, confirmed by transverse cross-sections that correspond to the peak region, as shown in Figure 4. In bundle fibers, the presence of lumen, pores in the kink-band, and degradation of the middle lamella were observable and confirmed by transverse cross-sections. The graphs show that ABF has higher porosity for every slice, indicating that porosity is not only due to the lumen or pores in the kink-band, but also to the pores induced by the degradation of the middle lamella along the fiber.

3.2. Tensile Results

Stress–strain curves for modern and ancient elementary flax fibers are shown in Figure 5. Generally, the tensile behavior of fibers can be categorized into three types: Type one (TI) exhibited a linear and purely elastic tensile response. Type two (TII) showed two distinct linear section, and type three (TIII) showed a nonlinear section up to a threshold point, followed by an increase in the tangent modulus until failure. A similar type of tensile behavior (TIII) was also observed in [31], where the authors hypothesized that the nonlinear behavior was due to alignment of cellulose micro fibrils with the tensile axis, and this could be interpreted as an elasto-visco-plastic deformation, which happens specifically in the secondary cell wall (S2, thickest cell wall). This type of behavior was also observed for hemp [32]. Type III (TIII) is most commonly seen in ligno-cellulosic fibers and exhibits higher mechanical properties. However, this trend varies across studies. For example, Type I was identified as the predominant behavior for hemp [33], while Type III was found to be predominant for flax [34]. The extraction processes differed in these studies: [33] used water-retted hemp fibers, whereas [34] utilized scutched fibers. In the present study, all three types of curves were observed for both modern and ancient fibers. Type I and Type II were predominantly observed in modern fibers (MF), while Type III was predominantly observed in ancient fibers (AF). The difference in behavior between MF and AF may be attributed to the different extraction processes used for these fibers as well as the contribution of structural defects and morphological variations of the fibers, which is predominant.

Table 2 presents a summary of the tensile properties of both Modern Egyptian and Ancient Egyptian fibers. A significant difference is evident in the Young’s modulus and tensile strength between modern and ancient flax fibers, with modern fibers exhibiting higher strength and modulus compared to the ancient fibers. The lower properties observed in ancient fibers may result from potential long-term alterations in mechanical integrity or fiber polymer degradation [24,25]. Additionally, the variability in stiffness is higher in ancient fibers compared to modern fibers, possibly attributable to variations in microstructural parameters such as microfibril angle and porosity [11,17,19], alongside biochemical changes due to fiber degradation. Nevertheless, the elongation at break (%) appears to be nearly identical for both modern and ancient fibers. However, a high standard deviation in elongation at break (%) is observed in ancient fibers, along with a significant standard deviation in tensile strength with coefficient of variation of ~40% for both modern and ancient fibers, leading to variations in their properties. The post-treatment analysis of the fibers was conducted using SEM. Figure 6 displays the SEM images of MF and AF after tensile testing, revealing crack initiation and fiber failure at the kink-band regions for both fibers [4]. Additionally, the failed regions of the fibers reveal the presence of lumen at those specific locations, which may indicate the impact of cavities on fiber failure [22]. Kink-bands initiate cracks, and the deviation in microfibril angles at these regions may leads to shear failures in the fiber, ultimately causing its rupture. Additionally, pores in the kink-band region may contribute to fiber failure. Also, the presence of significant kink-bands can lower the tensile strength of the fibers [4,23], which can be attributed to the low properties of ancient fibers.

3.3. Numerical Results

Figure 7a and Figure 8a display the Z-direction displacement for all fibers, confirming boundary conditions: the bottom surface is clamped, and the top surface is displaced by 1% (1.2 µm for MF, MFF, AF, GF, and 0.5 µm for ABF). Comparing the Z-direction displacement field of fiber models to geometrical model, it is evident that the displacement field in AF is less homogeneous compared to MF, which may be due to the influence of structural changes in ancient fibers over time. Figure 7b and Figure 8b show the axial stress (σ33) distributions, with the elliptical model having uniform stress and other models showing heterogeneity. The maximum stress exceeds 1000 MPa, with higher concentrations in boundaries of kink-band regions highlighted in black boxes in Figure 7b and Figure 8b. The pores in the kink-band region are assumed to be located at the boundaries, especially within the S2/G layer, which may explain the higher stress in this area [23]. This behavior is further illustrated using longitudinal cross sections of the fibers. Increased stress concentration in the kink-band region can initiate cracks, leading to fiber breakage [4,35]. Despite the lumen and pores being artificially filled in MFF, stress heterogeneity is observed, especially in the kink-band region with high stress. This suggests that, in addition to the pores, structural alterations such as deviations in the MFA in this region also influence the stress distribution. Visualizations are in the XZ plane for elementary fibers and the YZ plane for the bundle to better show kink-band regions. Figure 8a and Figure 9 show transverse displacements (y and x) with lateral shrinkage due to z-direction elongation. Additionally, the fibers exhibit asymmetric shrinkage compared to the elliptical model, highlighting the geometry’s impact on uniaxial extension [22]. Figure 10a presents the average elastic modulus, matching the experimental values for MF and AF. It can be observed that ABF exhibited a lower modulus, likely due to higher porosity within the bundle (measured over a shorter length of 50 µm compared to 120 µm for elementary fibers), which may have resulted from the degradation of the middle lamella and structural changes over time. The calculated mean axial stress (σ33) ranged from 280 MPa to 450 MPa.

To understand the stress distribution, we examined the stress evolution in the longitudinal cross-section of the fibers. Figure 10b shows the maximum axial and shear stress for all models. Figure 11 illustrates the axial stress (σ33) distribution. Modern, ancient, and porosity-filled fibers exhibit stress variations, while elliptical models display homogeneous stress distribution, indicating that defects like kink-bands and pores cause stress heterogeneities. The highest stress levels occur in pores within kink-band regions of modern and ancient fibers. For porosity-filled fibers, higher stress levels are observed on the surface, with reduced stress in filled pores. Figure 11b,c (σ22 and σ11) show symmetric stress distribution in elliptical and filled models compared to modern and ancient fibers, indicating the stress induced by defects in fiber models. Similarly, Figure 12a shows the axial stress distribution for bundles, highlighting higher stress in pores within kink-band regions. This suggests that pores in these areas can initiate cracks under tensile loading.

The shear stress distribution (σ12, σ13, σ23) for all models is illustrated in Figure 12b and Figure 13, highlighting heterogenous stress distributions along the fiber, specifically higher stress levels in defective regions and on surfaces, which were potentially caused by factors like porosity, kink-bands, and surface roughness. Similarly to strain components, i.e., displacement in the transverse direction, the values range from negative to positive, indicating the asymmetric shrinkage effect, which is due to the influence of the geometry in the surface on the effectiveness of the test to produce uniaxial extension. Comparing stress levels within the fibers (Figure 10b), we noticed higher axial stress (σ33) in the tensile direction across all models compared to other axial and shear stresses. Modern and ancient fibers exhibited the highest stress levels, reaching up to 2700 MPa and 1600 MPa, respectively, and the bundle fiber reached up to 600 MPa. Conversely, elliptical and filled fibers showed comparatively lower stress levels, indicating the impact of structural defects in the fibers. However, the mean stress values were insignificant in σ22 and σ11 compared to σ33, and the report will not delve further into them. Similarly, the shear stress related to the tensile direction (σ13 and σ23) was higher compared to σ12, and it will not be further addressed in the report.

The primary stress components, σ33, σ13, and σ23, were measured by plotting their maximum stress on transverse cross-sections spaced 20 µm apart in the XY plane (Figure 14 and Figure 15). The average axial stress of σ33 was lower in ancient fibers (reaching around 300 MPa) compared to modern (reaching around 460 MPa) and elliptical fiber models (Figure 14a). In modern fibers (MF and MFF), σ33 was higher than in ancient fibers (AF), particularly between 20 µm to 60 µm along the fiber’s length, which reflects the kink-band region (Figure 14b). For ancient fibers, higher stress was seen between 40 µm and 80 µm, which is also the kink-band region. The elliptical model showed constant stress around 450 MPa, except in the clamping region, where it reached 650 MPa. In bundle fibers (ABF), higher axial stress (σ33) was found between 30 µm and 40 µm along the fiber’s length, reaching around 720 MPa (Figure 15).

Longitudinal cross-sections ranging from 20 µm to 60 µm along the fiber’s length were analyzed by creating two lines: one encompassing the pores in the kink-band region and the other covering the non-porous region within the kink-band (Figure 16a). For modern fibers, the maximum axial stress σ33 corresponding to these lines is depicted in Figure 16b. Similarly, for modern filled fibers, longitudinal cross-sections from 20 µm to 60 µm along the fiber’s length were examined, lines were drawn, and the maximum axial stress σ33 is illustrated in Figure 17. Likewise, for ancient fibers, sections from 40 µm to 80 µm along the fiber’s length were studied, two lines were generated, and their maximum axial stress σ33 is shown in Figure 18. Notably, the highest axial stresses σ33 were observed in pores located within the kink-band region [36,37] for both modern and ancient fibers, reaching 2400 MPa and 1200 MPa, respectively, and this aligns with the ultimate stress obtained experimentally, suggesting that the pores in kink-band have potential to initiate the cracks under tensile loading. This behavior was consistent at the bundle scale as well (Figure 19), reaching 610 MPa. In the case of filled fibers, elevated stress levels were noted on the fiber surfaces in longitudinal cross-sections, reaching 600 MPa. The stress distribution appeared relatively constant, except on the surfaces or in the middle of the fibers, indicating areas where pore closure led to reduced stress levels reaching around 450 MPa.

4. Conclusions

X-ray microtomography revealed distinct internal structures in modern (MF), ancient (AF), and bundle fibers (ABF). AF had higher porosity at the kink-band region (5.6%) and a greater kink-band density (20.8 mm⁻¹) compared to MF, which showed 3.3% porosity and a kink-band density of 16.6 mm⁻¹. The average distance between kink-bands is longer in MF (114 µm) than in AF (77 µm). In both fiber types, kink-band pores were angled at approximately 45° to the fiber axis. ABF exhibited similar kink-band structures, but likely with more degradation. Ancient fibers exhibited higher porosity due to both inherent and degradation-induced pores.

This study examined the tensile characteristics of both modern and ancient fibers, revealing higher stiffness and strength in modern fibers compared to ancient ones. This disparity may elucidate the effects of aging of fibers in terms of structural evolution and fiber polymer degradation. Modern fibers have higher Young’s modulus and tensile strength compared to ancient fibers, which may be due to degradation and microstructural variability. SEM analysis reveals failure in both fiber types at kink-band regions, where cracks and ruptures are influenced by structural deviations and pores.

By utilizing a finite element model within the elastic range, the impact of the intricate structure of flax fibers on their tensile resistance was investigated. The calculated apparent modulus closely matched experimental values for elementary fibers, but was lower for bundle fibers. The numerical analysis revealed significant stress heterogeneity, especially in modern, ancient, and porosity-filled fibers, with higher stress levels observed in axial tensors near the kink-band region both volumetrically and in longitudinal cross-sections. Notably, the maximum axial stress (σ33) was found in pores within the kink-band region, with modern fibers (MF) exhibiting the highest stress at 2400 MPa, followed by ancient fibers at up to 1200 MPa, and bundle fibers at up to 600 MPa—values that align with the experimental ultimate stress. These high stress concentrations, particularly at pores in the kink-band region, suggest potential for crack initiation and rupture, whereas elliptical and filled fibers displayed more uniform stress distributions with lower maximum stresses, highlighting the impact of structural defects like pores and kink-bands in influencing stress distribution and fiber integrity under tensile loading. Additionally, apart from kink-bands and pores, stress induced by surface irregularities, particularly in porosity-filled fibers (MFF), was also noted. Overall, this study demonstrates both experimentally and numerically that kink bands can initiate cracks in fibers under tensile loading, affecting both individual and bundled fibers.

Author Contributions

Conceptualization, V.R., S.G., J.B. and A.B.; methodology, A.M., V.R., A.D., M.S., J.P., H.P. and T.W.; software, S.G. and V.R.; validation, S.G., V.R., J.B. and A.B.; formal analysis, S.G., J.B. and V.R.; investigation, V.R. and S.G; resources, A.B.; data curation, S.G. and V.R; writing—original draft preparation, V.R.; writing—review and editing, S.G., V.R., J.B., A.D., T.W., A.M., M.S., J.P., H.P. and A.B.; supervision, S.G., J.B. and A.B.; project administration, A.B.; funding acquisition, A.B. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data supporting the findings of this study are available upon simple request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Modern and ancient Egyptian flax fibers observed under optical microscope (polarized light top and bright field bottom), with some kink-bands present in fiber.

Figure 2. X-ray microtomography images of fibers: (a) 3D volumetric view of MF in XZ plane; (b) magnified view of kink-band region; (c) longitudinal cross section at kink-band region; (d) 3D volumetric view of AF in XZ plane; (e) magnified view of kink-band region; (f) longitudinal cross section at kink-band region.

Figure 3. X-ray microtomography images of ABF: (a) 3D volumetric view in YZ plane; (b) longitudinal cross section in YZ plane.

Figure 4. Porosity along the length of the fiber (emphasizing the transverse cross section at high peaks), obtained from microtomography data.

Figure 5. Stress–strain graph: (a) MF, (b) AF.

Figure 6. SEM images after tensile testing: (a,b) MF and (c,d) AF.

Figure 7. (a) Displacement in tensile direction (z) in the plane XZ. (b) Resulting axial stress (σ33) contour plot in the plane XZ.

Figure 7. (a) Displacement in tensile direction (z) in the plane XZ. (b) Resulting axial stress (σ33) contour plot in the plane XZ.

Figure 8. (a) Displacement in tensile (z) and transverse (y,x) directions in the plane YZ for ABF. (b) Resulting axial stress (σ33) contour plot in the plane YZ.

Figure 8. (a) Displacement in tensile (z) and transverse (y,x) directions in the plane YZ for ABF. (b) Resulting axial stress (σ33) contour plot in the plane YZ.

Figure 9. Displacement in transverse direction in the plane XZ: (a) y direction, (b) x direction.

Figure 9. Displacement in transverse direction in the plane XZ: (a) y direction, (b) x direction.

Figure 10. (a) Predicted elastic modulus (E) as a function of fibers. (b) Maximum axial and shear stress in longitudinal cross section of the fibers.

Figure 11. Axial stress contour plot in longitudinal cross section of the fibers in the XZ plane. (a) σ33, (b) σ22, (c) σ11.

Figure 11. Axial stress contour plot in longitudinal cross section of the fibers in the XZ plane. (a) σ33, (b) σ22, (c) σ11.

Figure 12. Stress contour plot in longitudinal cross section of the fiber bundle in the YZ plane. (a) Axial plot, (b) shear plot.

Figure 13. Shear stress contour plot in longitudinal cross section of the fibers in the XZ plane (a) σ12 (b) σ13 (c) σ23.

Figure 13. Shear stress contour plot in longitudinal cross section of the fibers in the XZ plane (a) σ12 (b) σ13 (c) σ23.

Figure 14. (a,b) Axial stress along transverse cross section of fibers with varying lengths (dash line represents the average ultimate stress) and (c,d) shear stress along transverse cross section of fibers with varying lengths.

Figure 14. (a,b) Axial stress along transverse cross section of fibers with varying lengths (dash line represents the average ultimate stress) and (c,d) shear stress along transverse cross section of fibers with varying lengths.

Figure 15. Maximum axial and shear stress along transverse cross section of fibers with varying lengths.

Figure 16. (a) Axial stress (σ33) distribution in longitudinal cross section of the modern fiber (line 1 excludes the pores and line 2 includes the pores); (b) resulting axial stress profile along two lines (1 and 2), with the black line representing average tensile strength.

Figure 16. (a) Axial stress (σ33) distribution in longitudinal cross section of the modern fiber (line 1 excludes the pores and line 2 includes the pores); (b) resulting axial stress profile along two lines (1 and 2), with the black line representing average tensile strength.

Figure 17. (a) Axial stress (σ33) distribution in longitudinal cross section of the modern fiber with artificially filled porosity; (b) resulting axial stress profile along two lines (1 and 2).

Figure 18. (a) Axial stress (σ33) distribution in longitudinal cross section of the ancient fiber (line 1 excludes the pores and line 2 includes the pores); (b) resulting axial stress profile along two lines (1 and 2), with the black line representing average tensile strength.

Figure 19. (a) Axial stress (σ33) distribution in longitudinal cross section of the ancient fiber (line 1 includes the pores and line 2 excludes the pores); (b) resulting axial stress profile along two lines (1 and 2).

Figure 19. (a) Axial stress (σ33) distribution in longitudinal cross section of the ancient fiber (line 1 includes the pores and line 2 excludes the pores); (b) resulting axial stress profile along two lines (1 and 2).

Table 1

Structural properties of elementary and bundle flax fibers.

FiberType	Length(µm)	AverageDiameter(µm)	ShapeFactor	Volume(µm3)	Porosity(%)	Porosity (%)at Kink-Band	Kink-BandDensity (mm−1)	Average DistanceBetween Kink-Bands (µm)
MF	120	18.5	1.01	1.89 × 105	1.4	3.3	16.6	114
AF	120	15.8	1.07	2.09 × 104	1.6	5.6	20.8	77
ABF	50	20.5	1.11	1.27 × 104	1.7	2.6	-	-

Table 2

Tensile properties of elementary flax fibers.

Sample	Number of Samples	Diameter (µm)	Young’s Modulus (GPa)	Elongation at Break (%)	Tensile Strength (MPa)
MF	10	17.4 ± 4.23	45.4 ± 13.2	2.91 ± 0.55	1321 ± 533
AF	5	16.1 ± 3.85	28.2 ± 19.9	3.07 ± 1.89	643 ± 267

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