# Helical buckling in textile cord-elastomer composites under compressive loadings: Experiments and simulations

Gianluca Auteri, Jan Neggers, Sophie Charpin, Kablan Guy-Delphin
Agniman, Damien Durville

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## To cite this version:

Gianluca Auteri, Jan Neggers, Sophie Charpin, Kablan Guy-Delphin Agniman, Damien Durville. Helical buckling in textile cord-elastomer composites under compressive loadings: Experiments and simulations. International Journal of Mechanical Sciences, 2025, 302, pp.110529. 10.1016/j.ijmecsci.2025.110529. hal-05398064

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# Helical buckling in textile cord-elastomer composites under compressive loadings: Experiments and simulations

Gianluca Auteri<sup>a,b, ID</sup>,*, Jan Neggers<sup>a ID</sup>, Sophie Charpin<sup>b</sup>, Kablan Guy-Delphin Agniman<sup>a,b ID</sup>, Damien Durville<sup>a ID</sup>

## ARTICLE INFO

### Keywords:

Composite materials

Mechanical behavior

Structural instability

Helical buckling

X-ray tomography

Finite element modeling

## ABSTRACT

In this paper, the study of helical buckling is extended, for the first time, to slender structures with a complex architecture: textile cords. This phenomenon is investigated experimentally through *in-situ* X-ray tomography compression tests and numerically through filament-scale finite element simulations, which are then compared with the experimental results. Additionally, a parametric study is performed on the matrix stiffness, exploring configurations beyond the experiment's domain. The results highlight that the buckling amplitude is inversely related to the stiffness of the matrix and the twist of the cord. These findings are discussed in relation to the theory for buckling in single filaments. The consequences of the buckled configuration on strain distribution at the filament scale are analyzed, highlighting how helical buckling influences load redistribution within the cord. In this sense, this work could open the field towards a conscious design of textile cords around the buckling point.

## 1. Introduction

Textile cords are complex filament assemblies often used to reinforce elastomeric matrices. The terminology includes a variety of architectures, typically defined by the nature of filaments (Polyethylene terephthalate (PET), Nylon, Aramid, etc.), the number of filaments and yarns and their twist levels [1]. A common example is the two-yarn cord shown in Fig. 1. These structural characteristics influence how the cords interact with the surrounding matrix, dictating their mechanical behavior under load.

When embedded in elastomeric matrices, textile cords provide high tensile rigidity while keeping the overall bending stiffness low, an essential balance in engineering systems that require both strength and flexibility. They are often placed side by side in industrial applications, forming what is known as a textile ply. Their use spans a wide range of sectors. In pneumatic tires, cords provide the primary structural reinforcement. They carry tensile loads from inflation and road contact while allowing the tire to flex [1,2]. Recent studies have emphasized the role of the cord architecture both for handling and ride comfort performances [3] and on durability under cyclic loads [4]. Another category of reinforcement for industrial rubber products are steel cords [5]. However, textile reinforcements distinguish themselves

also by their lightweight nature, making them the preferred choice for strengthening the elastomeric matrix of passenger car and aircraft tires [6,7]. Textile cords are also embedded in elastomeric hoses to resist internal pressure without excessive radial expansion. Hose performance benefits from the flexibility and hoop strength imparted by the cords. Studies have confirmed that the interaction between reinforcing fibers and the rubber matrix plays a key role in overall hose performance, particularly in resisting internal loads and torsional effects [8,9]. In conveyor belts, the textile carcass ensures the longitudinal and transverse rigidity of the belt to guarantee its shape during the movement, absorbs the impact of the material when loaded and maintains the mechanical integrity of the belt over prolonged use [10,11]. In air springs, such as those used in rail vehicle suspensions, textile cords are embedded in a rubber matrix to support vertical loads. These cord-rubber composite systems are exposed to complex stress states, axial compression, internal pressure, torsion, and bending, making them susceptible to fatigue and damage [12].

The behavior of textile cords, whether alone or embedded in a matrix, is widely characterized for the most common loading conditions: traction [13–15] and bending [16,17]. More comprehensive

<footnote><sup>a</sup> Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS - Laboratoire de Mécanique Paris-Saclay, 3 rue Joliot-Curie, Gif-sur-Yvette, 91190, France</footnote>

<footnote><sup>b</sup> Michelin, Ladoux Research Center, 23 Place des Carmes Déchaux, Clermont-Ferrand, 63040, France</footnote>

<footnote>* Corresponding author at: Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS - Laboratoire de Mécanique Paris-Saclay, 3 rue Joliot-Curie, Gif-sur-Yvette, 91190, France.

*E-mail address:* gianluca.auteri@centralesupelec.fr (G. Auteri).</footnote>
<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Textile cord-elastomer composite. Representation of the composite sample consisting of a two-yarn textile cord embedded in an elastomer block. The filaments of each yarn are highlighted with different colors. Initial configuration numerically obtained through simulation of the assembly process with the in-house finite element software *Multifil*. Visualization through Paraview.</figcaption></figure>

models focus on the repartition of stresses when rupture of filaments begins under traction loadings [18]. While most of the studies remain limited to the macroscopic analysis, a few studies investigate the mechanical response down to the filament scale [19,20]. Nevertheless, particular industrial cases may require the composite to work under axial compressive loadings [21]. This loading condition has been the focus of some studies, as it can damage the cord in fatigue [22–24]. However, no study specifically addresses the filament scale response during compression. This aspect is non-trivial, as the local behavior is highly heterogeneous and depends on the local arrangement of filaments. Furthermore, once a certain compression level is reached, the onset of buckling adds further complexity to the response.

Textile cords are inherently slender structures; thus, their behavior within a composite exhibits strong analogies to beam-like structures embedded in a surrounding medium. It has been documented experimentally that a single elastic fiber rod, when embedded in an elastomer matrix, exhibits a specific type of buckling in a helical shape [25]. The same behavior has also been observed for other rod-like structures with similar constraints, such as plant roots growing in soil [26], coil tubing in oil-field operations [27–30] and nanowires in an elastomeric substrate for stretchable electronics [31]. Also in additive manufacturing, recent experiments have shown that helical buckling might take place during the initial stage of extrusion of the fused polymer filament [32]. In the biological field, helical buckling has been observed in DNA structures subjected to compressive forces [33,34] and is the cause of the length limit of filopodia (long membranes supported by cytoskeletal filaments) [35]. In fluid mechanics, helical buckling has been studied for flexible structures under a viscous compressive loading [36,37].

Elastic instabilities of slender structures are increasingly seen as opportunities to enhance structural and functional aspects [25,27,38,39]. Broadening comprehension of the buckling mechanisms of embedded textile cords can pave the way for a conscious design around the buckling functioning point. This can then be exploited to trigger, reversibly and on-demand, certain properties of the composite. Many examples of the functional use of buckling are present in the literature. Plant roots use helical buckling to transform longitudinal forces into lateral forces, increasing the support by the surrounding medium to generate a greater force at the tip [26]. A similar mechanism has been used in the design of a mechanical clutch for a needle insertion device, where locking due to helical buckling is exploited to transmit force from the filament (catheter/guide wire) to the needle [40]. In biology, microtubules can bear large levels of compressive force, assuming complex curved buckled shapes, because of the lateral reinforcement

provided by the elastic cytoskeleton [41]. In stretchable electronics, as reported by Chen et al. [31], fragile and stiff components placed on an elastomeric substrate are deliberately pre-compressed to induce fundamental buckling modes. They found that this configuration increases deformability and resistance to various loadings. This study also shows that the buckling effect helps release compressive strain and converts it into smaller bending strains: specifically, the helical buckling configuration uniformly distributes the local strain along the slender structure, avoiding stress concentration zones, which minimizes the local strain level under a given global compressive strain.

In addition to experimental observations, several analytical and theoretical models have been developed to describe the helical buckling of rods embedded in elastic or viscoelastic media. These approaches offer parametric insights into how factors such as geometry, stiffness contrast, confinement, and pre-strain influence the buckling onset and resulting morphology. At the macroscale, models based on reinforced buckling theory, which account for the presence of surrounding elastic supports, have been successfully applied to predict the helical buckling of embedded single filaments [26,41]. Similarly, continuum mechanics approaches based on strain energy minimization have been developed to describe the helical instability of nanowires in elastomeric substrates [31]. A widely used theoretical framework in this context is the Winkler foundation model, which treats the surrounding medium as an array of linear springs. This model has been shown to accurately predict both the buckling wavelength and mode shapes for a thin, stiff beam supported by a softer elastic substrate [25]. Further developments include theoretical models for rods confined within rigid tubes, which provide specific predictions for various geometric configurations of buckling coils [27,42,43].

While analytical models provide valuable physical intuition and closed-form solutions, they are often limited in handling complex boundary conditions, contact, and material heterogeneities. To address these limitations, computational modeling, particularly through finite element methods (FEM), has become increasingly important. FEM-based simulations allow for a more realistic representation of the nonlinear interactions between filaments and their confining environments, which may include elastomeric matrices, tubular geometries, or other boundary constraints. Notably, simulations using shear-deformable linear beam elements for the filament, coupled with 3D continuum elements for the surrounding medium, have been shown to consistently reproduce experimental helical buckling behaviors [25,31]. Several specialized numerical techniques have been developed to overcome the challenges posed by mechanical instabilities. For example, Kapitaniak et al. [44] introduced a novel beam element capable of capturing nonlinear torsional effects and coupling between axial, lateral, and torsional deformations. Goyal et al. [45] extended Kirchhoff rod theory to incorporate tension-torsion coupling, enabling simulation of complex 3D behaviors such as loop formation. To analyze post-buckling behavior in confined geometries, a dynamic relaxation method has been proposed to more efficiently find stable buckled configurations under axial compression inside cylindrical tubes [29,46–48]. In the context of buckling in compressive flows, Chakrabarti et al. successfully modeled filament deformation using both Euler–Bernoulli beam theory and a Hookean spring network approach. Additionally, Chelakkot et al. [37] reproduced similar helical buckling phenomena using molecular dynamics simulations, offering a particle-based perspective on the instability mechanisms.

In parallel, significant advancements have been made in the last decades in the simulations of textile structures. As these materials gain increasing industrial relevance, accurate numerical modeling becomes essential. Simulating such assemblies presents unique challenges, requiring dedicated algorithms capable of handling numerous contact interactions, large deformations, material nonlinearities, and complex geometries. Specific finite element frameworks have been developed for a wide range of fibrous structures, including woven materials [49–51], braided fabrics [15,17,52], fiber-based scaffolds for biomechanics [53],
textile cords [20,54] and entangled fiber networks [55]. Significant attention has been given to the description of complex textile structures geometries, with studies focusing on an analytical geometrical description [56] and other studies, when the structures are too complex to be described through idealized geometric patterns, focusing on the accurate modeling of the manufacturing processes [54,57]. A notable drive behind the advancement of fiber assembly simulations has also come from computer graphics, with the work of Bertails et al. [58] bridging the gap between visually realistic simulations for animation and physically accurate models suitable for mechanical prototyping. More recently, Crassous [59] introduced a Discrete Element Method tailored for simulating assemblies of frictional fibers, offering an alternative to classical continuum approaches.

The aim of this paper is to extend the study of helical buckling instability, so far investigated primarily in single filaments, to textile cords, which are complex multi-filament assemblies. The objective is to apply the analysis of this physical phenomenon to a more intricate, engineered structure, and in doing so, to explore the mechanical con-sequences it induces at the scale of the assembly. Due to the complex architecture, one of the challenges of studying these structures is to be able to distinguish the various mechanisms occurring at different scales. Considering the helical buckling as a possible response helps to rationalize the behavior of embedded textile cords under compressive loadings. This study considers a specific two-yarn PET textile cord, with each yarn containing multiple filaments twisted together. A schematic representation of the cord is shown in Fig. 2.a. The two yarns, together with their center-lines are shown in red and blue, while the black line represents the trajectory of the center of the cord. When a textile cord-elastomer composite is compressed, different buckling modes activate simultaneously to absorb the loading. The most evident response is the opening of the two yarns, separating from each other and creating a void between them. This is shown in Fig. 2.b, with the two yarn lines moving away from each other and the cord barycenter remaining unaltered. Another possible response is the more standard Euler buckling [60], shown in Fig. 2.c and leading to a curved trajectory of the cord center-line. The third possibility listed is the helical buckling (Fig. 2.d), which results in a helical trajectory of the cord barycenter. It should be pointed out that the formation of a regular 3D buckling configuration is here exclusively related to the compression of an embedded textile cord, to which no end moments are applied and without any residual torsion. Hence, it has to be distinguished from another class of helicoidal configuration of slender structures, created through the simultaneous application of an external moment and compressive load [61–63]. Similarly, it should be separated from helical patterns that form due to the presence of a natural curvature in the buckling system, as in the case of a suspended curly hair [64] or due to complex coupled loading conditions, as in the case of arteries [65]. Additionally, this work should not be seen as overlapping with classical studies on fiber-scale rearrangement or migration in twisted yarns, such as those by Morton & Yen (1952) [66], Hearle et al. (1965) [67] and Riding (1964) [68]. These seminal contributions established fundamental understanding of how fibers redistribute within yarns during twisting, primarily focusing on equilibrium configurations and geometrically induced migration phenomena. In contrast, even if migration phenomena are well captured during the assembly process simulation, the present study addresses a different physical mechanism, the helical buckling instability, observed in textile cords embedded in an elastomeric matrix under compressive loading. While both involve fiber motion, the driving forces and mechanical context differ: here, the deformation arises not from twist-induced packing effects but from global structural instability triggered by external loading. To our knowledge, this is among the first studies to characterize such buckling phenomena in complex, multi-filament cord architectures, where the interplay between fiber geometry and matrix compliance governs the instability phenomenon.

The presence of multiple interacting mechanisms, combined with the complex entanglement of filaments, makes the problem challenging to address analytically. Therefore, this study begins with an experimental investigation using in-situ X-ray tomography compression tests, followed by a numerical analysis based on finite element simulations. Section 2 outlines the experimental work. The methodology, including sample fabrication, in-situ tomography testing, and post-processing to extract buckling trajectories, is detailed in Section 2.1. The results are presented in Section 2.2, where the buckling behavior at various compressive strains is shown for both a single cord embedded in rubber and a textile ply. Two design parameters are varied in the experiments: the stiffness of the elastomer and the twist of the cord. Section 3 aims to reproduce the helical buckling behavior of textile cords through numerical simulations. This is achieved using the in-house finite element code Multifil [54], whose underlying theory and simulation procedures for textile assemblies are presented in Section 3.1. The results of the simulations are discussed in Section 3.2, where the experimental cases are replicated numerically to enable direct comparison. This section also explores additional configurations beyond those tested experimentally. The outcomes are critically analyzed in light of theoretical models and literature on helical buckling in single filaments. Section 3.3 further leverages the simulations to investigate the impact of buckling on strain distribution at the filament scale. Finally, Section 4 summarizes the main findings, discusses the limitations of the study, and outlines potential future directions.

## 2. Experiments

In this section, the helical buckling phenomenon is observed experimentally. Some composite samples, containing one or more textile cords, are compressed *in-situ* in an X-ray tomograph. The images are post-processed up to the extraction of the center-line trajectory of the cord. This curve is further analyzed to provide evidence of helical buckling and its evolution with compression.

### 2.1. *Experimental setup*

Textile cord-elastomer composite samples are fabricated ad-hoc for this study. The first type of sample consists of a single cord (diameter ~ 0.65 mm) embedded in a block of elastomer of dimensions 15 mm × 6 mm × 6 mm. The second one consists of a textile ply, made of 6 cords side by side, embedded in a block of rubber of dimensions 15 mm × 12 mm × 12 mm. This second configuration is relevant as cords are placed close together in industrial applications and may eventually interact. The two types of samples are shown in Fig. 3.

The manufacturing process is identical for the two cases: the cord (or ply) is put in a mold between two layers of uncured elastomer; the cord is carefully centered and kept straight by the application of a tension of ~ 15 N per cord; the mold is then put in a hot press to allow the curing of the elastomer and then cooled down; each sample is then extracted and a thin layer of sample is removed from the two extremities using a diamond wire saw; furthermore, a layer of resin is applied to fill any remaining shape imperfections; finally, a glass plate is applied on the two surfaces while keeping the sample in a slight compression (~ 2 N) between two plates. Particular attention is given to the last steps to obtain a sufficiently planar surface, hence guaranteeing a uniform compression on the sample ends. The total length of the sample is measured once again at the end of the process, in order to precisely impose the required strain during the *in-situ* test.

The textile reinforcement chosen for this study is a 2-yarn cord, made of 720 PET filaments, with a filament diameter of 19.21 µm and a cord diameter of ~ 0.6 mm. In total, 4 single-cord type samples are manufactured, varying the twist of the textile cord (290 tr/m and 420 tr/m) and the stiffness of the matrix ($E_{eq} \approx 5$ MPa (soft) and $E_{eq} \approx 100$ MPa (stiff)). For the textile ply version, one sample is manufactured in the low twist and soft matrix configuration.
<figure><img src="image_6.png" /><figcaption>Fig. 2. Buckling modes of an embedded textile cord. Conceptual representation of the main architectural buckling modes under compression. In black, red, and blue, respectively, the trajectories of the center of the cord and the centers of the two yarns. (a) Undeformed state. (b) Opening of the yarns. (c) Euler buckling. (d) Helical buckling.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. Textile cord-elastomer composite samples for in-situ compression tests. On the left, the sample consists of a single textile cord embedded in an elastomer block; this configuration corresponds to the representation of Fig. 1. On the right, the sample consists of a textile ply (6 cords placed side by side), embedded in an elastomer block.</figcaption></figure>

The purpose of the experiment is to observe the evolution of the textile cord center-line trajectory as compression is applied. Su et al. [25] used an experimental framework with two digital cameras to reconstruct the 3D trajectory of the buckling of a single rod in a transparent elastomeric matrix. In this study, the sample is kept as close as possible to the in-service condition of the composite. Because of the determination to keep the carbon-black-filled rubber mix, in-situ X-ray tomography has been judged to be the most appropriate methodology. The tomograph used is the ZEISS Versa 520, equipped with the in-situ mechanical stage DEBEN CT5000, which has been modified to host a 500 N load cell for improved accuracy. A good trade-off between image quality and scan speed has been obtained using 1601 projections and a resolution of 10.94 µm. Scans are performed at each loading step from 0% to 5% global compressive strain. In particular, two scans are performed at 0% (1 N contact load), followed by a scan at each 0.5% strain increment, up to 3%, and every 1%, up to 5%, finally the sample is unloaded and an extra scan at 0% is made. After each loading step, the sample is kept at its position for 5 min before starting the scan, to allow the relaxation of the elastomer.

The 3D volumes are reconstructed from 2D projection images using the FDK method [69]. Fig. 4 shows an example of a cross-section of the reconstructed volume. To avoid boundary effects, 25% of the total sample length is discarded from both sides before the image analysis. The cord volumes are separated from the other image components using image segmentation. This is performed through an in-house code that exploits the Max-Lloyd algorithm [70]. For the textile ply samples, an extra step is performed, which consists of separating the volumes of the different cords from each other. This is achieved through iterative morphological operations on the cross-sections of the binary volume,

after segmentation. The volume of one cord is highlighted in red in Fig. 4, while Fig. 5.a shows its 3D view.

For each slice, the barycenter is calculated, hence its trajectory. To properly analyze and measure the helical buckling phenomenon, it is essential to separate it from the Euler buckling, which may occur simultaneously, often triggered by any slight tilt, that could inevitably be present in experiments. For this purpose, a second-order polynomial is interpolated to the trajectory curve (Fig. 5.a). Subtracting the interpolated polynomial from the center-line trajectory curve (Fig. 5.b) allows isolating the helical buckling phenomena, removing other perturbations.

Moreover, the second-order interpolated polynomial is used to compute the average inclination angle $\alpha$ (Eq. (1)) and the average curvature $\kappa$ (Eq. (2)) of the cord, as

$$ \cos \alpha = \bar{u} \cdot \bar{z} \qquad (1) $$

and

$$ \kappa = \frac{\|r' \times r''\|}{\|r'\|^3}, \qquad (2) $$

where $\bar{u}$ is the vector between the two extremes of the polynomial curve, $\bar{z}$ is the vertical axis vector, $r'$ and $r''$ are respectively the first and second order derivative of the 2nd order polynomial curve [71]. On the contrary, for what concerns the “opening” mechanism, its consequences are not expressed on the cord center-line trajectory, so no correction is needed. From now on, the analysis will focus on the study of the cord center-line trajectory after correction. For visualization purposes, a b-spline interpolation of order 3 is applied to the curve, to eliminate small amplitude noise attributed to the reconstruction and segmentation steps. Fig. 6 shows different views of the center-line trajectory, in gray before b-spline interpolation and in blackşi colors after. To quantitatively measure the buckling amplitude, an ellipse is fitted on the top view of the helix using the image moments approach [72], a technique widely used in computer vision. All details about this methodology are provided in Appendix A. This approach offers an interpretable estimate of the buckling helix amplitude and orientation. The fitted ellipses are shown in orange, for different cases, in Fig. 6. Once the orientation of the ellipse is determined, it is possible to define the two principal coordinates as the two orthogonal directions corresponding to the major and minor diameters of the ellipse. Regarding the resolution of the center-line trajectory, it has to be pointed out that it is extracted from the barycenter of a group of approximately 2000 pixels, hence the resolution of the curve is:

$$ Res_{traj} = \frac{\text{Voxel size}}{\sqrt{\# \text{ pixels per slice}}} = \frac{10.94 \ \mu\text{m}}{\sqrt{2000}} \approx 0.25 \ \mu\text{m}. \qquad (3) $$

However, it is expected that the reconstruction and segmentation procedures introduce some additional uncertainty, making it so that the actual resolution will be larger than the theoretical one.

## 2.2. Experimental results

In this section, the results from in-situ tomography experiments are presented. The cord center-line trajectories are analyzed and the amplitude of helical buckling is quantified.
<figure><img src="image_8.png" /><figcaption>Fig. 4. Tomography image of embedded textile cords. A slice of the 3D volume scan of the composite. In red, the segmented volume of one cord.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Postprocessing steps. From the tomography image, to the extraction of the barycenter of the trajectory of the cord. (a) Cord volume with its centerline trajectory and an interpolated second-order polynomial. (b) Cord volume and its centerline trajectory corrected for the tilt angle and global curvature by subtracting the interpolated polynomial.</figcaption></figure>

### 2.2.1. Single cord

Firstly, experiments on single cords are performed, to assess the presence and evolution of helical buckling during compression. Four samples in total are analyzed, combining two levels of twist of the cord (290 tr/m and 420 tr/m) and two elastomers with different stiffnesses ($E_{eq} \approx 5$ MPa (soft) and $E_{eq} \approx 100$ MPa (stiff)). Fig. 6 shows the center-line trajectories (3D view, top view and orthogonal views) for the low twist case, in the presence of a soft and stiff matrix, at 5% compressive strain. The use of orthogonal views allows for a clearer visualization of its 3D shape. Moreover, it highlights the nature of 3D helical buckling, originating from the superposition of two orthogonal planar buckling modes activated simultaneously [25]. The helical buckling amplitude is monitored during the compression using the diameters of the fitted ellipses on the top view of the cord center-line trajectory (Fig. 6).

Fig. 7 shows, for all cases, the evolution of the fitted ellipse average diameter, together with the major and minor axes. Once again, the two diameters are shown to highlight the evolution of the two perpendicular modes forming the 3D buckling. If the buckling loads of two modes become comparable, the helix configuration is triggered, then it continues to grow, with the two modes interacting due to geometrical non-linearities. Still, a 2D planar mode may be triggered first, changing the configuration of the system as a consequence of non-linearity, which leads to the activation of an orthogonal mode to end up eventually in a 3D configuration [25,36]. The cords embedded in the soft mix develop a higher amplitude and longer period helical buckling with respect to the stiff mix ones. This can be observed both in Fig. 6 qualitatively and in Fig. 7 looking at the trends for the amplitude. The same trend is present between low and high-twist samples, with the first developing more buckling than the second one. Overall, the curves are characterized by a stable phase up to around 4%, followed by a rapid increase, when present, of the buckling amplitude. To estimate

<table><caption>Table 1
Average cord tilt angle for different elastomers and twist levels.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Tilt angle [°]</th></tr><tr><th>Soft elastomer</th><th>Stiff elastomer</th></tr></thead><tbody><tr><td>290 tr/m</td><td>0.96</td><td>1.6</td></tr><tr><td>420 tr/m</td><td>1.1</td><td>1.9</td></tr></tbody></table>

<table><caption>Table 2
Average cord global curvature for different elastomers and twist levels.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Global curvature [1/m]</th></tr><tr><th>Soft elastomer</th><th>Stiff elastomer</th></tr></thead><tbody><tr><td>290 tr/m</td><td>12.3</td><td>10.5</td></tr><tr><td>420 tr/m</td><td>3.9</td><td>6.9</td></tr></tbody></table>

the uncertainty in the measurement of the helical buckling diameter, two independent scans were conducted at the initial state, under a contact load of 1 N. In Fig. 7 they correspond to the two points at 0% compression. Although they represent the same mechanical state, the scans were independently processed through all stages: image acquisition, volume reconstruction, segmentation, curve extraction, filtering, and ellipse fitting. Across all tested samples, the difference in the final fitted ellipse diameter between these two acquisitions remained below 6%. To assess the reversibility of the buckling phenomenon, the sample was loaded to 5% global compressive strain and subsequently unloaded, after which a new scan was performed. In all cases, the amplitude of the buckled shape returned to its initial value within the same 6% error margin observed between independent scans at zero compression. This level of error is considered acceptable by the authors given the small dimension of the sample and the complexity of the post-processing workflow. More importantly, it is deemed sufficient for the aims of this study, which are to provide the first experimental observation of the helical buckling phenomenon in textile assemblies and to establish a semi-quantitative comparison with numerical simulations. It should be also noted that, at 0% compressive strain, the experimental helix amplitude is non-zero. We speculate that this is due to the shrinking of the elastomer, taking place during the cooling of the sample, which in turn applies a certain amount of pre-compression to the textile cord. This is confirmed by the lower initial helix on the stiff rubber case, which has a lower coefficient of thermal expansion.

The tilt angles and global curvatures are reported respectively in Tables 1 and 2. These values do not present a significant and monotone evolution during the compression, therefore the average values are reported. Inevitably, a slight angle is left in the sample during the sample preparation. The authors believe that the presence of a tilt can induce the cord to take on Euler buckling (expressed by the curvature), reducing the proportion of helical buckling.

### 2.2.2. Textile ply

The second type of sample consists of a textile ply (6 cords side by side), embedded in an elastomer matrix. For this case, only one configuration (290 tr/m, soft rubber) is analyzed. The goal is to have a sample more representative of a composite used in industrial applications, where cords often come in plies. Fig. 8 shows the segmented tomography volumes of the 6 cords, together with their center-line
<figure><img src="image_10.png" /><figcaption>Fig. 6. Cords centerline trajectories after helical buckling. 3D, top and side projections of cord barycenter trajectory in principal coordinates for different matrix stiffness at 5% compressive strain obtained experimentally (EXP) and by numerical simulation (SIM). (a) 290 tr/m - Soft matrix - EXP. (b) 290 tr/m - Soft matrix - SIM. (c) 290 tr/m - Stiff matrix - EXP. (d) 290 tr/m - Stiff matrix - SIM.</figcaption></figure>

trajectories (in black). A certain tilt is visible for the cords at 0% deformation, visually amplified by the non-equal axis scale. Consequently, at 5% compressive strain, part of the compressive energy is diverted to Euler rather than helical buckling.

Following the same methodology as for the single cord experiment, the curves are corrected with respect to tilt and Euler buckling, to isolate the helical buckling effects. The top view of the helices and their fitted ellipses are shown in Fig. 9 for different levels of compression.

A certain difference is noticeable between the buckling amplitudes of the 6 different cords. This difference can also be seen in Fig. 10, where the monitoring of the ellipse average diameter is shown for the 6 cords.

An explanation for this difference can be provided by looking at the tilt and global curvature of the cords. A direct proportionality exists

between the inclination angle of the cord and the global curvature (Fig. 11.a). The smaller the inclination angle, the smaller the global curvature, and thus the greater the helical buckling amplitude (Fig. 11.b). In other words, the compression energy is expressed by the cord through different modalities (e.g. opening, Euler buckling, helical buckling and potentially others). If one of these mechanisms is preferred (e.g. the tilt angle induces Euler buckling) the proportion of the others decreases and vice-versa.

Then, in Fig. 10, the results of the helix amplitude monitoring are compared between the textile ply case and the corresponding single cord configuration. The helical buckling appears to happen in the same proportions between the two cases, both for the final amplitude and the global trend. A slight difference is noticeable on the initiation point, slightly anticipated towards 3% for the textile ply case. However, this
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<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 7. Evolution of the helical buckling diameter with the compression for the different cases. (a) 290 tr/m - Soft matrix - EXP. (b) 290 tr/m - Stiff matrix - EXP. (c) 290 tr/m - Soft matrix - SIM. (d) 290 tr/m - Stiff matrix - SIM. (e) 420 tr/m - Soft matrix - EXP. (f) 420 tr/m - Stiff matrix - EXP. (g) 420 tr/m - Soft matrix - SIM. (h) 420 tr/m - Stiff matrix - SIM.</figcaption></figure>

is considered to be within sample to sample variance. There is no evidence of different phenomena due to possible interactions between cords close to each other. This confirms the possibility of referring to the single cord case when studying the compression of these composites, as this variation from the real composite does not affect its behavior.

### 3. Numerical simulations

In this section, the finite element simulations' theoretical framework is presented. The results from experiments are compared with the numerical ones. Then, a parametric study is performed on the matrix stiffness, exploring configurations beyond the experiment's domain.

In the last part, the consequences of helical buckling on the distribution of strains at the filament scale are analyzed.

#### 3.1. Finite element model

The complex mechanical response of textile assemblies originates from mechanisms occurring at the filament scale. Advanced homogenized macro-scale simulations, taking into account the inclination of the filaments inside the cord, can accurately reproduce the overall mechanical response [73]. However, they inherently fail to capture the filament-to-filament interactions responsible for more intricate responses. Therefore, investigating the helical buckling instability phenomenon in a textile cord requires simulations at the filament level.
<figure><img src="image_19.png" /><figcaption>Fig. 8. Helical buckling in a textile ply. (a) Cord volumes and trajectories of the 6 cords in the ply at 0% and 5% compressive strain.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 9. Helical buckling in a textile ply. Top view of the helical buckling trajectories at different compression levels for a textile ply.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 10. Helical buckling evolution as a function of compression for a textile ply. Trends for the 6 cords of the ply, their average and comparison with the single cord case.</figcaption></figure>

To this end, the in-house code Multifil, specifically developed for the simulation of complex textile assemblies, is employed. Multifil has been

widely used in previous studies for both filament-scale and mesoscale simulations of textile and metal assemblies. It has been applied to the simulation of textile cords [20,74] and metal cables [75] for tire use, metal spiral strands [76,77] and braided ropes [15,17] for marine applications, woven textile fabrics [50,51], textile scaffolds for biomedical purposes [53], and superconducting cables [78]. The theoretical foundations of this software are detailed in the work of Durville [54]. Nevertheless, a brief overview is provided here to support a better understanding of the numerical simulations presented in this study.

In Multifil, each filament is discretized using three-node quadratic elements, modeled as 1D kinematically enriched beams that capture cross-sectional deformations within a large deformation framework. The kinematical beam model describes the position of each particle in the beam as a first-order Taylor expansion with respect to its transverse coordinates. As a consequence, a full 3D Green–Lagrange strain tensor is derived, which allows the use of a 3D elastic constitutive law. A 3D overlapping non-conforming mesh of linear hexahedral elements represents the matrix. The coupling between the filaments and the matrix is enforced at nodes selected at the periphery of the cord through penalization of the relative displacement between a particle located on the outline of the beam cross-section at this node and the corresponding overlapping matrix particle. Efficient modeling of contact–friction interactions among a large number of filaments requires specialized algorithms, particularly given the large relative displacements between filaments that can cause contacts to appear or disappear at any time. Contact detection relies on constructing an intermediate geometry, an average of nearby filament segments, which approximates the actual contact geometry. Discrete contact elements are then defined on this geometry by associating pairs of material points from each filament predicted to come into contact. Since this process is predictive and inherently nonlinear, contact elements are frequently updated during the simulation. Normal contact is handled with a regularized penalty formulation, while tangential friction is modeled using a regularized version of Coulomb’s law. An implicit solver is used to solve the global nonlinear problem under quasi-static assumptions. The different nonlinearities present in the global problem are treated with different approaches. In particular, the determination of contact elements and their associated normal directions, both dependent on the evolving displacement field, cannot be directly linearized and are thus handled using two nested fixed-point loops. Within these loops, once the contact elements and normal directions are fixed, remaining nonlinearities, such as contact status, sliding behavior, finite strains and geometric effects, are solved using a Newton–Raphson method. Convergence is assessed using a relative residual threshold criterion. The material constitutive law’s parameters, both for the filaments and the elastomer, are obtained experimentally. A tabbed non-linear elastic behavior law is used for the textile filaments. The law is obtained directly from the traction test of a bundle of straight filaments. The matrix follows a Mooney–Rivlin [79,80] hyperelastic law based on the strain energy density function $\Psi$ expressed as a function of the modified invariants $\tilde{I}_1$ and $\tilde{I}_2$, in turn written as a function of the invariants of the left Cauchy–Green tensor:

$$ \Psi = C_{10}(\tilde{I}_1 - 3) + C_{01}(\tilde{I}_2 - 3) + \frac{k}{2}(\det(F) - 1)^2. \quad (4) $$

The constants $C_{10}$, $C_{01}$ are determined experimentally by fitting the Mooney–Rivlin strain energy model to stress–strain data from pure shear tests [81,82]. The last term accounts for compressibility by penalizing the variation of volume, mathematically expressed as the deviation of the determinant of the deformation gradient $F$ from unity. Rubber incompressibility is determined according to the carbon black content of the elastomer [83,84]. Concerning the helical buckling phenomenon, other authors report no or few differences between linear elastic and hyperelastic constitutive relations for the modeling of the matrix [31], with some studies even relying solely on linear elastic properties [25,85,86]. In this work, although the hyperelastic behavior is not critical for the phenomenon under investigation, the
<figure><img src="image_22.png" /><figcaption>Fig. 11. Global curvatures and inclination angles in textile ply samples. (a) Global curvature with respect to inclination angle. (b) Global curvature with respect to helix amplitude.</figcaption></figure>

Mooney-Rivlin formulation is adopted to remain consistent with standard industrial practices. The material law curves and coefficients are not disclosed here for confidentiality reasons, in view of their industrial relevance. For the filaments, a linear elastic modulus is estimated to be approximately 5 GPa. For the elastomeric matrices, the equivalent elastic modulus spans from 5 to 100 MPa, as shown in Fig. 14. These are highly incompressible, with Poisson's ratio exceeding 0.499.

A peculiarity of textile cords is that the filament arrangement is not known a priori. On the contrary, it is strongly dependent on the manufacturing process. The first step of the latter is the yarn twisting. Industrially, this consists of applying a twist to a bundle of disordered straight filaments. The fact that this twist is applied locally while the bundle passes over a roller creates the so-called migration phenomenon, a particular filament arrangement that consists of filaments migrating from the cord center to the periphery of the bundle and vice-versa [66-68]. The work of Sibellas et al. [87] highlighted the complexity of filament paths due to the migration phenomena, suggesting that an analytical geometrical description would be inappropriate for an accurate representation. For this reason, the strategy chosen here is to reproduce the textile cord assembly procedure numerically, thereby naturally capturing the filament migration phenomena. In a previous study, textile assemblies generated using this numerical method were compared with real textile constructions. High-resolution X-ray tomography was used to scan actual cords and reconstruct the volume of each filament. The distribution of filament orientations in the real cords closely matched with the numerically generated ones, with few differences depending on the initial disorder in the filament bundle prior to assembly. This validation was confirmed across multiple textile constructions [74]. The second step of the assembly process consists of twisting two yarns together with a twist equal but opposite to the one applied during the single yarn twisting. To complete the composite model, the cord must be coupled with the surrounding matrix. A distinctive feature of this class of composites is that the elastomer does not penetrate the inside of the cord. Consequently, coupling is enforced only on the filaments located at the periphery of the cord, while the stiffness of the matrix inside the cord is set to zero. A dedicated algorithm identifies the filaments on the cord's exterior. Notably, these filaments vary along the cord length due to the migration phenomenon. Therefore, the external contour is determined on a series of discretized cross-sections along the cord. Fig. 12 presents a full visualization of the numerical sample. The matrix is meshed with 3D elements; the filaments are modeled using beam elements; and the coupling elements, constituted of pairs of material particles, one

attached to the beam surface and the other to the matrix, are shown in white.

Compressive tests are simulated under the same conditions as the *in-situ* experiments. A 5 mm cord is embedded and coupled with a 6 mm × 6 mm × 5 mm elastomer block, as shown in Fig. 12. The elastomer matrix is discretized using 27000 linear hexahedral 3D elements, while the 720 filaments forming the cord are each modeled with 50 kinematically enriched quadratic beam elements. A mesh convergence analysis was conducted by independently varying the mesh densities of both the matrix and the filaments. Further details are available in Appendix B. For the investigated phenomenon, the helical buckling instability, the selected mesh provides results within 5% of the most refined mesh, ensuring a good balance between accuracy and computational efficiency. A uniform incremental displacement is applied to the opposite longitudinal faces at both ends. Specific boundary conditions are applied to the filaments at both ends of the sample, in order to leave enough freedom for them to rearrange, while preventing them from detwisting, both at yarn and cord levels. To achieve this, transverse displacements of individual filaments are left free, while their average rotation and average transverse displacements are locked both at the yarn and cord levels. This ensures that boundary effects remain negligible.

## 3.2. Simulation results

Fig. 6 shows a comparison of the helical buckling phenomena at 5% global composite compression, obtained experimentally and numerically for the two cases with soft and stiff matrices for a twist of 290 tr/m. Globally, the simulations accurately reproduce the helical buckling phenomena. The amplitudes and wavelengths of the buckling are comparable between simulations and experiments and the same trends are observed between the different cases. Fig. 7 compares the monitoring of the buckling amplitude during compression between experiments and simulations. In general, a good agreement is found between the two, both for the final amplitudes and buckling onset. As in the experiments, simulations present a stable phase followed by a rapid growth of the buckling helix. During the first phase, a non-null slope is present in the simulations, while in the experiments this is not visible, probably hidden by the error band. It should be pointed out that the thermal part of the manufacturing process is neglected in simulations. As a consequence, the pre-compression of the cord, observed in the experiments because of the shrinking of the elastomer during cooling, is not present in simulations. Furthermore, the trajectories obtained
<figure><img src="image_23.png" /><figcaption>**Fig. 12.** Numerical sample for compressive tests. Full visualization of the numerical sample of the textile cord-elastomer composite used for compression tests. The matrix, meshed with 3D elements, the filaments, modeled with beam elements, and the coupling points between them (shown in white) are all visible. Visualization produced using ParaView.</figcaption></figure>

experimentally generally present more irregular shapes than the ones obtained by simulations. Irregular helical profiles are also found in [25] for embedded single filaments. Brangwynne et al. [41] speculate that some regions of the structure may not undergo buckling even when others do and attribute it to local weak spots in the surrounding medium. Indeed, imperfections in the matrix or in the textile-elastomer interface, are characteristics which are not included in the numerical model and may be the cause of the irregularity of the helix present in experimental results. Both in experiments and simulations, it can be noted that the amplitudes at which helical buckling is manifested in textile cords appear to be significantly inferior with respect to the scale of the phenomenon in single beams [25]. We speculate that this is because a significant part of the compression energy is actually used by the opening mechanism between the yarns.

Moreover, Fig. 13 provides a qualitative comparison of a cross-section of the cord at 5% compression obtained experimentally and numerically. The cross sections are taken at the mid-length of each sample, and the observed differences in fiber orientation are due to phase shifts between samples, despite having the same twist. In both figures, it is possible to observe the matrix on the outside, the textile filaments constituting the cords and the void formed between the yarns. In the experimental image, the distinction can be done thanks to the difference in gray level; the area belonging to the cord and its void are highlighted for easier visualization. In the simulation image, the filaments are easier to distinguish; for completeness, their axial strain value is also reported. The key insight is that the configuration of the filaments and the amount of opening between the yarns are comparable between the experimental and numerical tests. Even if qualitative, this consideration is essential to make sure that simulations are aligned with experiments.

Having validated the consistency of numerical simulations with experimental results, the interest is to numerically explore different configurations. A parametric study is done on the elastomer stiffness. Still using Mooney-Rivlin hyper-elasticity law, different simulations are performed while varying the equivalent Young's modulus ($E_{eq} = 6(C_{01} + C_{10})$) from 3 MPa to 100 MPa, keeping the proportion between the two coefficients constant. The results, reported in Fig. 14, show a rapid decrease in the helical buckling amplitude when increasing the equivalent elastomer modulus up to 30 MPa, followed by a stabilization of the buckling amplitude for stiffer elastomers. The trend is validated for two twist levels (290 tr/m and 420 tr/m) of the textile cord. Increasing the twist of the cord, leads to smaller buckling amplitudes for softer matrices. However, the same values are found for higher values of support stiffness.

Moreover, it is possible to measure the buckling period by fitting a sine function to the orthogonal projection of the 3D buckling curve. Fig. 15.a shows a linear correlation between buckling amplitude and period, for the case at 290 tr/m.

The decay of buckling amplitude and frequency as a function of matrix rigidity has been reported in several experimental studies across various systems [25,26,31,36,41]. This trend can be understood using analytical models. In particular, the Winkler foundation model [25] and related energetic formulations [25,26,31,41] provide a theoretical basis for this behavior. For a rod to buckle within a surrounding elastic medium, it must deform the matrix, incurring additional energy costs. As a result, configurations that minimize matrix displacement, such as shorter wavelength buckling, are energetically favored. This is quantified by the following energy balance [41]:

$$E = \frac{k}{2} \int dx(u'')^2 - \frac{f}{2} \int dx(u')^2 + \frac{\alpha}{2} \int dx(u)^2, \quad (5)$$

where $u(x)$ is the transverse deflection of the rod, $k$ is its bending stiffness, $f$ is the applied axial load, and $\alpha$ is a parameter proportional to the elastic modulus of the surrounding matrix. Minimizing this total energy leads to the characteristic buckling wavelength:

$$\lambda = 2\pi \left(\frac{k}{\alpha}\right)^{1/4}, \quad (6)$$

which clearly shows that increasing matrix stiffness ($\alpha$) results in a decrease in buckling wavelength and so amplitude. These equations are not directly used in our simulations, but they serve as theoretical tools to qualitatively interpret the observed trends across experiments and numerical results.

Though derived for a single filament, this model reflects key physics relevant to multi-filament textile cords. Twist reduces the effective bending stiffness $k$ by misaligning filaments from the main axis, which increases the resistance to buckling. Thus, the twist effect is indirectly incorporated into the analytical framework. However, the actual architecture of textile cords introduces further complexities, such as yarn separation or opening, that could contribute additional energetic terms. These effects suggest directions for future refinement of the model.

In addition, Fig. 15.b shows that the helix aspect ratio decreases globally towards unity as the matrix stiffness increases, but with fluctuations at low stiffness. This behavior is again consistent with theory: at low stiffness, the energy difference between competing orthogonal buckling modes is small, allowing coexistence of 2D and 3D configurations. As the matrix becomes stiffer, one mode dominates, and more distinct 3D modes emerge. Such mode competition and transitions at specific stiffness ratios have been analytically predicted [25,36], and explain the observed deviations from the global trend.

Furthermore, in the present analysis, the length of the textile cord-elastomer composite sample was limited to a maximum of 10 mm. This constraint was imposed in experiments to comply with the physical dimensions of the in-situ compression device and to ensure sufficient image resolution, and numerically to limit computational costs. Naturally, this raises the question of whether the helical buckling phenomenon consistently reproduces itself regardless of sample length. To explore this aspect, a parametric numerical study was conducted by varying the sample length from 5 mm to 25 mm. The results reveal a non-monotonic variation in the helical buckling diameter across different lengths, with a maximum deviation of 4% between samples, indicating a relatively weak dependence of the buckling response on sample length. This finding aligns with the work of Chakrabarti et al. [36], who showed that, in the case of a single filament, the helix radius remains independent of the filament length.

### 3.3. Consequences of helical buckling

Local quantities, such as trajectories and strains at filament scale, are experimentally inaccessible. Using simulations, it is possible to monitor these quantities to get insights about the deformation modes
<figure><img src="image_24.png" /><figcaption>**Fig. 13.** Visual comparison of experimental and simulation results. Example of cross-sectional shape for (a) simulations (obtained through visualization in Paraview) and (b) experiments (a slice of the reconstructed X-ray tomography volume).</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>**Fig. 14.** Buckling amplitude as a function of matrix rigidity. The trend is reported for two torsion levels, curves are fitted on the data points for easier visualization.</figcaption></figure>

activated by helical buckling, hence the consequences on the distribution of strains at filament scale. The results presented in this section are related to the soft matrix, low-twist case, in which the helical buckling amplitude is the most pronounced.

The evolution of the behavior of the cord during compression is analyzed through Singular Value Decomposition (SVD) [88]. This allows the extraction of the dominant features of the dataset, allowing for a more efficient analysis and interpretation of the underlying patterns. The SVD analysis on the radial displacement of the filaments, taken with respect to the cord center, highlights that two modes describe 90.5% of the radial displacements of the filaments along the uniaxial compression test. Fig. 16.a visually shows the two modes, projected back to the initial configuration of the cord. In Fig. 16.b it is possible to follow the expression of the modes as a function of the global compressive strain imposed to the composite. The first mode, exhibiting a monotone growth, shows a positive radial displacement all along the cord, with the exception of values close to zero around the ply line. This pattern relates to the opening mechanism of the two yarns when the cord is compressed, as in Fig. 13. The second mode puts in evidence oscillations in the radial position of the filaments along the longitudinal coordinate. This deformation mechanism can be associated with helical buckling. As a matter of fact, the trend of mode 2, shown in Fig. 16.b exhibits a stable phase (excluding a stabilization in the beginning), where the mode is almost non-active, followed by a sudden growth at around 4%, which coincides with the helical buckling activation point.

To go further in the analysis, the strain level distribution on the filaments is analyzed. Specifically, the observed quantity is the minimum strain over the cross-section of each node, calculated using interpolation from the integration points. Therefore, this quantity results from

the combination of both uniaxial compression and bending of the filament. The oscillations on the radial displacement, just presented above, consequently have an impact on the strain distribution on the two yarns along the longitudinal coordinate. Fig. 17 shows the minimum strains separately for the two yarns, after helical buckling. The strains in both yarns vary globally in opposite phase as a function of the longitudinal coordinate, alternating zones of maximum discrepancy between the two yarns with zones where the strain level is the same. This phenomenon can be understood by analyzing the amplitude of the vector product of the normalized vector $\vec{V}_1$, between the center of the helix and the center of the cord, and the normalized vector $\vec{V}_2$, between the center of the cord and the center of one of the two yarns. The two vectors rotate, a priori, independently, $\vec{V}_1$ according to the period of the helix, $\vec{V}_2$ according to the twist of the cord. The part of the cord which is on the inner part of the helix will have the tendency to develop compressive strains while the opposite part will behave more in tension. When the two vectors are orthogonal ($||\vec{V}_1 \times \vec{V}_2|| = 1$), both yarns are on the inner and outer side of the helix, hence the strain of the two yarns is homogeneous (case A in Fig. 17). When the two vectors are parallel ($||\vec{V}_1 \times \vec{V}_2|| = 0$), one yarn is fully on the inside and the other on the outside of the helix, hence the maximum discrepancy (case B in Fig. 17). Similar fluctuations, taking place at the same period of the helical buckling, have been highlighted in [31]. We speculate that these fluctuations of the strain level can induce a non-uniform loading on the cord and lead to different damage mechanisms.

Next, the minimum strains over the cross-section of each node of the filaments are analyzed through a purely statistical analysis. Fig. 18, shows the histogram distribution of the minimum strains at different steps of global compression. A clear difference is visible on the histograms between the pre-buckling regime (1% to 4%) and the post-buckling regime (5%). These differences are synthesized in Fig. 18.b. The trend of the mean with global composite strain, puts in evidence a change of slope at around 4%, corresponding to the onset of buckling. After buckling, the slope of the mean becomes slightly more negative, suggesting an overall higher compressive state of the filaments. Another quantity, the skewness, measures the asymmetry of the distribution, indicating whether data points are concentrated more on the left or right side of the mean. The analyzed data set always exhibits a negative skewness, indicating that the left tail of the distribution is longer due to extreme values. In other words, some filaments present a particularly high compressive load with respect to the mean. The skewness increases in the pre-buckling regime, while the distribution regains some symmetry after the buckling point. Hence, the effect of helical buckling is to homogenize the loading between the filaments, increasing the average overall compressive state rather than concentrating the stresses on a sub-group of filaments. Based on that, it can be said that helical buckling may have a protective effect on the filaments. This consideration can also be found for the helical
<figure><img src="image_26.png" /><figcaption>**Fig. 15.** Other buckling characteristic quantities. (a) Buckling wavelength as a function of helical buckling amplitude. (b) Helix ratio as a function of matrix stiffness. The data are reported for the case of 290 tr/m, curves are fitted for better visualization of the trends. Centerline trajectories, side and top views, are reported to enhance clarity.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>**Fig. 16.** Single value decomposition on the radial position of the filaments along the compressive test. (a) 3D visualization of the first two modes (outer surface and longitudinal cut). (b) Evolution of the first two modes with the compressive strain applied to the composite.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>**Fig. 17.** Strain oscillations due to helical buckling. Oscillations in the strain values between the two yarns of the cord. The dashed line shows the cross product of the vectors $\vec{V}_1$ and $\vec{V}_2$ along the longitudinal axis of the cord. For visualization purposes the proportion of the ellipse with respect to the cord are increased in the figure.</figcaption></figure>

buckling of single wires. With respect to other in-plane buckling, the 3D helical buckling configuration is, in fact, the one that distributes the strain more uniformly, reducing the peaks of local strains [31]. The ability of the buckling configuration to alter the strain distribution among filaments could, once again, offer new opportunities during the design phase.

In the case of a single filament, Chakrabarti et al. [36] have shown that the helix radius is independent of the filament length. For this study, the length of the samples was limited to a maximum of 10 mm:

however, further analysis is needed to understand if the length may have a slight effect on the interactions of the different buckling mechanisms (opening, Euler, helical).

## 4. Conclusion

This work presents the first combined experimental and numerical study of 3D helical buckling in textile cords, extending the investigation of this instability from isolated filaments to multi-filament textile
<figure><img src="image_29.png" /><figcaption>**Fig. 18.** Filament scale strain statistical analysis. Analysis on the distribution of minimum strain along the compressive test. (a) Minimum strain histogram at different global compression steps. (b) Mean and skewness of the minimum strain distribution as a function of global compressive strain.</figcaption></figure>

assemblies embedded in elastomers. The helical buckling phenomenon, though coupled with the complex architecture of the cord, has been clearly identified and characterized. It is shown to occur in both single-cord and multi-cord (textile ply) configurations, and its formation and evolution during loading have been systematically analyzed. The results show that the buckling amplitude and wavelength are influenced by the matrix stiffness and the bending rigidity of the structure, confirming theoretical expectations.

A significant contribution of this study lies in demonstrating that numerical simulations can accurately reproduce helical buckling phenomena in such complex structures. This validation is particularly valuable as it confirms the ability of simulations to capture geometrically nonlinear instabilities in textile-reinforced composites.

From a structural mechanics perspective, entering the helical buckling domain induces a redistribution of deformation and compressive strain across filaments, hinting at a functional role of buckling in strain accommodation. This suggests a possible re-evaluation of buckling not as a failure mode, but as a design opportunity for textile reinforcements.

Nonetheless, the present study is subject to some limitations. First, the analysis is restricted to a single loading cycle, and while the buckling diameter is observed to return to its initial value upon unloading, the detailed evolution of the buckling morphology during unloading was not investigated. As demonstrated by Liu et al. [28] for the case of a rod confined in a pipe, loading and unloading may follow qualitatively different paths, with loading characterized by sudden instability and unloading by a gradual return to the initial state. A dedicated investigation of unloading behavior in textile cords would be a valuable extension.

Second, no conclusions can yet be drawn regarding the repeatability of the helical buckling phenomenon under cyclic loading. Damage mechanisms, such as sliding or debonding at the textile-elastomer interface [89], or internal damage in the filaments or matrix, may alter the buckling response in subsequent cycles. This work specifically addresses the helical buckling behavior in complex textile assemblies, such as cords, representing a significant advancement compared to studies focused on isolated embedded filaments. Nonetheless, the investigation of this phenomenon under repeated loading and in the presence of damage mechanisms remains an open direction for future research.

Future work should focus on the stability and repeatability of the buckling response under cyclic loading and in the presence of damage. A deeper understanding of this behavior may open opportunities to control or exploit helical buckling in advanced composite structures, turning it into a viable design strategy rather than an unpredictable instability.

### CRediT authorship contribution statement

Gianluca Auteri: Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal

analysis, Data curation, Conceptualization. **Jan Neggers:** Writing – review & editing, Validation, Supervision, Project administration, Funding acquisition. **Sophie Charpin:** Writing – review & editing, Validation, Resources, Methodology. **Kablan Guy-Delphin Agniman:** Writing – review & editing, Validation, Methodology. **Damien Durville:** Writing – review & editing, Validation, Supervision, Software, Project administration, Funding acquisition.

### 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.

### Acknowledgments

This project has received funding in the framework of 'Conventions Industrielles de Formation par la Recherche (CIFRE), managed by the ANRT (Association Nationale de la Recherche et de la Technologie) and supported by the French tire manufacturer MICHELIN. The authors want to thank Simon Nuytten, Thierry Lacroix and François Bataille from Michelin R&D for the helpful discussions and support during the challenging phases of this research.

### Appendix A. Ellipse fitting

To quantitatively measure the helix amplitude, an ellipse is fitted on the top view Fig. 19, using image moments, following a standard

<figure><img src="image_30.png" /><figcaption>**Fig. 19.** Ellipse fitting.</figcaption></figure>
method for shape analysis in computer vision [72]. This method provides a compact and robust description of the overall geometry based solely on the 2D coordinates of the filament trajectory projected onto a plane. This procedure can be outlined in a few key steps:

* Given a set of 2D points, one can calculate the raw moments (up to the second order). Each moment $M_{ij}$ is computed as the sum over all points of the coordinate products $x_k^i \cdot y_k^j$ where $(x_k, y_k)$ are the coordinates of the kth point.

$$M_{ij} = \sum_{k=1}^{n} x_k^i \cdot y_k^j, \qquad (7)$$

where $(x_k, y_k)$ are the coordinates of the kth point in the set. These moments encode geometric information such as centroid, and spread of the point cloud.

* The centroid of the point set $(\bar{x}, \bar{y})$ is calculated using the first-order moments:

$$\bar{x} = \frac{M_{10}}{M_{00}}, \quad \bar{y} = \frac{M_{01}}{M_{00}}, \qquad (8)$$

where $M_{00}$ is the zero-th order moment (essentially the total number of points), $M_{10}$ and $M_{01}$ are the first order moments along the x and y axex respectively (each one representing respectively the sum of x-coordinates and of y-coordinates). The centroid $(\bar{x}, \bar{y})$ represents the geometric center of the projected helix shape.

* The second order central moments $\mu_{20}$, $\mu_{02}$, $\mu_{11}$, which are corrected for translation by subtracting the squared centroid terms, are calculated as follows:

$$\mu_{20} = \frac{M_{20}}{M_{00}} - \bar{x}^2, \quad \mu_{02} = \frac{M_{02}}{M_{00}} - \bar{y}^2, \quad \mu_{11} = \frac{M_{11}}{M_{00}} - \bar{x} \cdot \bar{y}. \qquad (9)$$

These capture the variance and covariance of the point cloud and are essential for determining the orientation and elongation of the distribution.

* They form the entries of the covariance matrix C, which encodes the point cloud's orientation and spread:

$$C = \begin{pmatrix} \mu_{20} & \mu_{11} \\ \mu_{11} & \mu_{02} \end{pmatrix}, \qquad (10)$$

* The eigenvalues $\lambda_1, \lambda_2$ of the covariance matrix correspond to the variances along the principal axes of the data, which relate directly to the squared axes of the fitted ellipse. Therefore, the major and minor diameters of the ellipse are computed as:

$$D_{max} = 4 \cdot \sqrt{\lambda_1}, \quad D_{min} = 4 \cdot \sqrt{\lambda_2}. \qquad (11)$$

* To obtain the orientation $\theta$ of the ellipse, the direction of the principal axis is computed from the first eigenvector of C and its orientation by calculating the angle between the first eigenvectors and the x-axis:

$$\tan \theta = \frac{e_{21}}{e_{11}}, \qquad (12)$$

where $e_{11}$ and $e_{21}$ are the x and y component of the first eigenvector.

This image-moment approach is a widely used method for describing the global shape of a 2D distribution, offering a quantitative and interpretable estimate of the buckling helix amplitude and orientation.

## Appendix B. Finite element simulations: mesh refinement study

To ensure the reliability and mesh independence of the finite element simulation results, a mesh refinement study was conducted. The simulations incorporate two different non-conforming meshes: quadratic beam elements for the filaments and 3D hexahedral elements for the matrix. The convergence criterion is the helical buckling amplitude, a key quantity in this study, expressed as a percentage relative to

<figure><img src="image_31.png" /><figcaption>Fig. 20. Convergence analysis trends. Results expressed in percentage with respect to the most refined mesh. (a) Beam elements. (b) Matrix 3D elements (width). (c) Matrix 3D elements (length).</figcaption></figure>
the result obtained with the most refined mesh. A 5% deviation from this reference has been arbitrarily defined, in the context of this paper, as an acceptable threshold for mesh convergence.

First, the convergence analysis focuses on the spatial discretization of the filaments. Textile cords are modeled using varying numbers of beam elements per filament, ranging from 10 to 80. Compression simulations are conducted with high-resolution, computationally expensive matrix meshes to isolate the effect of filament discretization. The results are shown in Fig. 20.a. A general trend of decreasing buckling amplitude with increasing mesh refinement is observed. For the coarsest discretization, the buckling amplitude is overestimated by 60% relative to the most refined solution. From 50 beam elements per filament onward, the buckling amplitude remains within the 5% threshold, and this configuration is therefore selected for subsequent simulations.

After establishing the appropriate discretization for the filaments, the study turns to the matrix mesh convergence. The influence of mesh density is investigated along two directions: across the width (transverse to the cord) and along the length of the matrix. Results are presented in Figs. 20.b and 20.c. In both directions, deviations up to 20% are observed in the coarser meshes. However, the buckling amplitude falls within the acceptable 5% deviation when the element densities reach 0.5 elements/mm across the width and 0.6 elements/mm along the length.

In summary, the filament discretization has a greater impact on the computed buckling amplitude than the matrix mesh. While coarse filament meshes can lead to deviations of up to 60%, the influence of matrix coarseness is more limited, with a maximum deviation of approximately 20%.

## Data availability

The data that has been used is confidential.

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