# Shear deformation characteristics under tensile stress during forming of tubular braided fabrics

Jinlei Li, Nahière Hamila, Gildas L'Hostis, Peng Wang

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Jinlei Li, Nahière Hamila, Gildas L'Hostis, Peng Wang. Shear deformation characteristics under tensile stress during forming of tubular braided fabrics. Composites Part A: Applied Science and Manufacturing, 2023, 167, pp.107458. 10.1016/j.compositesa.2023.107458. hal-05339153

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https://hal.univ-brest.fr/hal-05339153v1

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# Shear deformation characteristics under tensile stress during forming of tubular braided fabrics

Jinlei Li¹, ², Nahiène Hamila³, Gildas L'Hostis¹, ², Peng Wang ¹, ², *

¹University of Haute-Alsace, Ensisa, Lpmt, F-68000 Mulhouse, France

²University of Strasbourg, France

³ENI Brest, IRDL, UMR CNRS 6027, F-29200, Brest, France

* Corresponding author. Tel.: +33 3 89 33 66 48. Fax: +33 3 89 33 63 39

E-mail address: peng.wang@uha.fr

## Abstract:

Tubular braids with a hollow structure are widely used in the manufacturing of textile composites. As the dominant behavior during the forming of such fabrics, their mechanical responses under an axial tensile load have been investigated. Their complex behavior consisted of three main parts: non-homogenous in-plane shearing, pure yarn extension, as well as a combination of in-plane shearing and yarn extension. A “locking structure” was proposed as a sign of the structural damage and an analytical model was developed to describe the relationships of shear angle/shear moment and shear angle/shear load based on unit cells. The non-uniform shear deformation field, depending only on the global tensile deformation and the axial position, was well demonstrated by the analytical approach. We believe that the experimental and analytical approaches presented herein will improve the understanding of braided fabric forming and help advance the numerical simulation of this process.

**Keywords:** A. 3-Dimensional reinforcement; E. Forming; In-plane shearing; Braiding
Nomenclature:

<table><tr><td>r</td><td>The radius of any circle in the axial direction of the specimen (mm)</td><td>l</td><td>The side length of the unit cell (mm)</td></tr><tr><td>W</td><td>The original diameter of the specimen (mm)</td><td>γ</td><td>The shear angle (°)</td></tr><tr><td>L</td><td>The original length of the specimen (mm)</td><td>θ</td><td>Half of the angle between two neighboring yarns during the test (°)</td></tr><tr><td>u</td><td>The tensile displacement of the specimen (mm)</td><td>β/2</td><td>The initial braiding angle (°)</td></tr><tr><td>b<sub>0</sub></td><td>The minimum radius on the mean plane of the deformed tubular braid (mm)</td><td>F(γ)</td><td>The load acting on the specimen (N)</td></tr><tr><td>z</td><td>The axial coordinates of the specimen (mm)</td><td>M<sub>s</sub>(γ)</td><td>The shear moment per unit area (N/mm)</td></tr><tr><td>n</td><td>The number of the unit cells in a circumference</td><td>F<sub>sh</sub>(γ)</td><td>The shear load on a unit cell (N/mm)</td></tr></table>
# 1. Introduction

Composite materials have widespread use in the aerospace and automotive fields due to their excellent properties. Braided composites present many interesting advantages, such as structural variability, short preparation cycles, the possibility of being designed according to a variety of application conditions...[1], and these advantages play an important role in reducing the weight of components with complex shapes. As the classical reinforcements of braided composites, tubular fabrics are largely employed as braided fuel pipes, rocket launch tubes, automobile structural parts, medical instruments, sports equipment, etc. thanks to their unique shapes and first-rate structural integrity, design flexibility, damage tolerance and low manufacturing costs (Fig. 1) [2–4].

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Main applications of tubular braid-reinforced composites. (a) Propeller shaft in a helicopter, (b) medical equipment and (c) sports equipment.</figcaption></figure>

As the first stage in the manufacturing of textile composites, forming is normally a complex process in which multiple deformation modes are combined, in particular 3D fabrics forming, such as for a braided fabric with a tubular structure. Four important mechanical stresses of the fabric are experienced during the forming of fiber-reinforced composites: tensile stress, in-plane shear, compression and bending stress [4–6]. Moreover, torsional stresses can be added to the tubular
structure. By contrast, the in-plane shear and tensile stress play a more important role in textile fabrics forming [7–9].

As one of the essential parameters of tubular braided fabrics, the braiding angle (see Fig. 2a) strongly influences the mechanical behavior of braided reinforcements and composite parts. Phoenix [10] confirms that the increase in braiding angle would cause the tensile strength of braided composite materials to decrease. Bias-extension tests have demonstrated that the tensile behavior of non-orthogonal flat braids is asymmetrical, whereas that of woven fabrics is symmetrical [5]. It is in fact the presence of the braiding angle and the special three-dimensional structure that cause the tensile behavior of the tubular braided fabric to be more complicated than that of woven fabrics and flat braids. The importance gaining more understanding of the mechanical behaviors of tubular braided fabrics has been raised, but this topic has yet to be completely documented in the literature [11–14].

The radial contraction and deformation mode of two types of tubular braided fabrics was studied under uniaxial tensile loading [15]. The tensile response of tubular braids with an elastic core was investigated. According to Amit Rawal et al. [16], the braiding angle is the most important design parameter for varying the extensibility of braid–core systems. In addition, the unit cell model was used to build a new type of three-dimensional integrated multilayer fiber-reinforced composite tubular structure and its torsional damage behavior was explored [17].

Jerry Ochola et al. [18] revealed that the radial contraction depends on the axial elongation in tensile tests of tubular braids. They highlighted the change in the angle between yarns during the test, however, they have yet to point out the non-homogeneity of the changes in this angle due to the coupling of tensile stress and shearing. The mechanism of in-plane shearing during tensile stress within a tubular structure needs to be described. Following the studies of the in-plane shear characteristic of the 2D textile fabrics, the rotation of the yarns due to shearing stress during the forming is becoming increasingly recognized by researchers [19–22]. The in-plane shearing has an important effect on the drape of the fabric and is the main deformation mode to obtain double curved shapes [5]. Under shear deformation, the
fabric structure and orientation of the yarns play a major role in predicting the fabric properties, especially when it comes to characterizing the mechanical properties [23]. In-plane shear tests have confirmed that when the shear angle of intersecting yarns reaches a critical value (named the “locking angle”), wrinkling of the fabric will occur [24–29]. These wrinkles normally cause a negative effect on the mechanical performance of the final composite part.

The present paper has identified the shear deformation characteristics of tubular braided fabrics during tensile testing. The coupling phenomenon between the elongation and rotation of the yarns was explored. In addition, based on a unit cell, an analytical model was developed to predict the radial contraction and the angles between the yarns when the tubular braided fabric was subjected to tension. The changes in shear angle and shear load were considered in the proposed model in order to completely interpret the in-plane shear behavior during the forming of the tubular braided fabric.

## 2. Materials and methods

### 2.1 Materials

The flax/polyamide 12 commingled yarns with a linear density of 500 tex and 1.78 mass ratio (64% flax yarns / 36% PA12 yarns) used in this study were provided by Schappe Technique. The tested tubular braided fabric is shown in Fig. 2b and their main properties are listed in Table 1. The total number of yarns in the tubular fabric was 96. The braiding angle (β/2) shown in Fig. 2a is defined as the angle between the axial direction and the yarn.

<caption>**Table 1**</caption>

The main properties of the tested tubular braided fabric.

<table><thead><tr><th colspan="2">Parameters</th></tr></thead><tbody><tr><td>Type of fabric</td><td>Biaxial twill 2-2</td></tr><tr><td>Area density (g/m²)</td><td>376 ± 5</td></tr></tbody></table>
<table><tr><td>Thickness (mm)</td><td>2.1</td></tr><tr><td>Diameter (mm)</td><td>50</td></tr><tr><td>Braiding angle (β/2)</td><td>55°</td></tr><tr><td>Number of yarns per cm</td><td>4.2</td></tr></table>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Tested tubular braided fabric.</figcaption></figure>

## 2.2 Tensile testing of the tubular braided fabric

The experimental setup for the tensile testing is shown in Fig. 3. Both ends of the specimen were fixed in the tensile machine by clamps, and a stretching force was applied at a uniform speed (10 mm/min) along the axial direction of the fabric. The length of the specimen was 40 mm excluding the gripped zones. It was essential to leave an appropriate length of fabric for clamping to ensure that no slippage occurred between the tows of the specimen and the clamps during the test [30]. To accurately analyze the mechanical response during the tensile
testing of the tubular braided fabrics, a 3D scanning equipment with two cameras, shown in Fig. 3, was employed to monitor the test progression in real time and record the changing state of the sample with the stretching. Consequently, the extension of the yarns and the change of the angle between them could be determined thanks to a number of markers related to the non-homogenous deformation shown in Fig. 4.

In order to reduce the experimental error when identifying the yarns' elongation, two intersecting yarns and pastes corresponding to markers at the intersection were chosen. The only criterion for selecting the markers was to ensure that they were positioned on the same single yarn. Considering the possible influence of yarn shearing, 5 marks were made on the fabric to obtain the length variation of two intersecting yarns and take the average value. For this, asymmetric markers were deemed to be more representative. The yarns' extension could be recorded during the test but not at the beginning, and in the course of the experiment the extension eventually led to damage of the sample's structure (Fig. 4d).

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Experimental device.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Tubular fabric specimen and markers, (a) initial state, (b) deformed state with 50% deformation, (c) deformed state with 75% deformation and (d) failure state.</figcaption></figure>

After the tensile testing, the tubular fabrics eventually took on a radially contracted shape. At the beginning of the test, the intersecting yarns had enough rotational space to be accompanied by a pure shear behavior under the tensile load, and the radial contraction thus strongly depended on the variation of the shear angle. The study of radial shrinkage of tubular fabrics relies on their 3D structure and several methods can be used to remodel this: mathematical models and related algorithms to establish fabric simulation models [31], X-ray tomography to remodel the representative volume element [32–34], and 3D scanning to quickly reconstruct the model to experimental analysis.

3D scanning is an emerging technology that can convert objects into digital forms in spatial dimensions for rapid imaging [35]. Taking 3D images with a camera to capture the coordinates (x, y, z) of an object and regenerate the point clouds on the surface of the object has become a popular technique for characterizing textile deformation, and includes
measurements of various strains during the textile reinforcements forming [36], reconstruction of wrinkling [37,38]; measurement of tow buckling [39], etc. In the present study, to vividly describe the profile shape of the deformed fabric during the test, Structure From Motion (SFM) was used complemented with of the software CloudCompare to increase the quality of modeling [40].

SFM is an algorithm that reconstructs a real three-dimensional scene using two-
dimensional images from various angles in the collected space. SFM automatically extracts
camera parameters from images, finds overlapping scenes in the input images, and performs
feature extraction. When the feature points of the image are obtained, they are matched by
calculating the Euclidean distance between the feature points of two images [41]. After
finding the image pairs that match the required number of feature points, for each image, the
matching pair is optimized and improved by the RANSAC algorithm [42]. The feature points
are passed on in a chain in such matching pairs, and they are always detected to form a
trajectory. At the same time, by estimating the internal and external parameter matrices, the
coordinates of 3D points are obtained by triangulation.

Based on the reference points, i.e., the points in the non-moving parts, CloudCompare
allows scaling of the model to obtain the real coordinates of the reconstructed point cloud in
the 3D scene according to the actual geometric size. It also supports the output of coordinates
for subsequent processing [43]. The original point-clouds exported by the SFM method
include quite a significant amount of noise points. To increase the quality of the 3D
reconstruction, post-processing was carried out in CloudCompare. The noise filtration was
able to remove the isolated points from the raw point clouds. Actually, the sizes of the
processed point clouds generated via the SFM algorithm were relative and had to be
calibrated to the real dimensions of the sample [40]. Errors were allowed in the calibration
process. By modeling the known physical objects, the dimensions of the model obtained by
the mentioned SFM and CloudCompare software were compared with the actual dimension
and the modeling accuracy was found to be closed to 99.32%.

Fig. 5 shows the evolution of this post-processing for a deformed tubular braided fabric in the axial tensile test. The new point-clouds in Fig. 5c reproduced the essential features of the sample in Fig. 5a after eliminating the isolated points (the noise) in the raw point clouds (Fig. 5b).

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Example of three-dimensional reconstruction: (a) photo of the sample, (b) raw point cloud, (c) processed point cloud.</figcaption></figure>

## 2.3 Mechanical responses in the tensile test

As presented in Fig. 4, tensile stretching of a tubular braided fabric leads principally to shearing deformation and extension of the yarns. The rotation of the yarns related to the shearing deformation on the plane of the unit cell is shown schematically in Fig. 6. This shearing deformation can be quantified by a shear angle. It is evident from Fig. 4 that the shear angle is not uniform in all of the unit cells, which is similar to what is seen for a flat braid [5] and to the bias-extension of a woven fabric [7,20,44] presenting different shear angle distributions, i.e., for which the shear angle is not unique.

From the bias-extension test, it was found that the in-plane shearing was divided into three zones, i.e., the no-shear zone, the pure-shear zone and the semi-shear zone. The unit cells in
the same zone had identical shear angles. Compared to the bias-extension test of flat fabrics,
tensile testing of tubular fabrics led to the shear angles being identical only if the unit cells
were on the circumference of the same height (i.e., had the same Z position), as shown in Fig.
6b. Therefore, the shearing deformation described by the shear angles is related to the tensile
displacement in the axial direction. During a tensile test, the fabric is subjected to axial
tension, and the intersecting yarns rotate and approach each other along the stretching
direction [45–48], thereby causing a decrease of the angle between the yarns in the axial
direction, resulting in a shearing behavior.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Schematic representation of the shearing deformation (a) on the plane of a unit cell and (b) on the whole structure of the braided fabric.</figcaption></figure>

The shearing deformation on the plane of the unit cell is shown schematically in Fig. 6a.
The shearing angle is one of the essential parameters during the forming of textile fabrics. To
describe how the shearing mechanism is coupled with the extension of the tubular fabrics
and then optimize the pre-forming process of such a 3D structure, the analytical models need
to be based on experiments to ascertain the relationship between tensile displacement and radial shrinkage, and subsequently determine the changes in shear angles and shear moments. Consequently, it is necessary to establish a kinematic and mechanical model to portray how the shear angle is connected to the tensile displacement [7,20,49,50] and also to describe the shear moment as a function of the shear angle [7,20,49,51].

### 2.3.1 Prediction of the shear angle

Fig. 7 gives a schematic display of the whole tubular specimen and unit cell deformations under tensile stress. The tubular fabric undergoes radial contraction under tensile displacement [52,53] and the yarns become reoriented. In this case, the shear angles are generated distinctly on different zones corresponding to the Z position of the unit cell. The shear angles increase with the tensile displacement. To determine the shear angles, the axisymmetric fabric can first be projected on a plane as shown in Fig. 7, after which the radial contraction leads to the fabric taking on the shape of a parabola as described in Eq. 1.

$$r = \sum_{i=0}^{N} b_i \cdot z^{2i} \qquad (1)$$

Here, $z$ and $r$ are axial and radial coordinates, respectively, and $N$, $b_i$, $i$ are constants. To simplify the calculation, $N$ can be suggested as 1 which will be confirmed by experiments. Consequently, Eq. 1 can be expressed as:

$$r = \frac{2(W - 2b_0)}{(u + L)^2} \cdot z^2 + b_0 \qquad (2)$$

where $u$ is the tensile displacement, $b_0$ represents the minimum radius of the deformed tubular fabric (the radius on the mean plane), and $L$ and $W$ symbolize the original length and diameter of the fabric, respectively.
<figure><img src="image_8.png" /><figcaption>Fig. 7. The tubular braided fabric under a tensile load and a deformed unit cell in the i^th^ circumference.</figcaption></figure>

Based on Eq. 3, the radial contraction at any position in the axial direction and the corresponding circumferential radius can be obtained since there is an assumption of no network slippage at the cross-over points of the yarns [20,30]. Therefore, the i^th^ unit cell can be represented as the research object (Fig. 7) to calculate the shear angle as shown in the following equations.

$$ \gamma_i = \beta - 2\theta_i \quad (3) $$

$$ \sin \theta_i = \frac{a_i}{l} \quad (4) $$

In the i^th^ unit cell, $\gamma_i$ is the shear angle, $a_i$ and $l$ represent half of the diagonal length and the side length of the cell, respectively.

The radius of the circle in which the i^th^ cell is located can be defined as $r_i$, and $a_i$ can thus be expressed as:

$$ a_i = \frac{\pi r_i}{n} \quad (5) $$
$$n = \frac{\pi \cdot W}{2 \cdot l \cdot \sin(\frac{\beta}{2})} \qquad (6)$$

where n presents the number of the unit cells in a circumference. This number is a constant in the initial state and does not change in the deformed state.

The shear angle $\gamma_i$ of the i-th unit cell during the extension of the tubular braided fabric can be obtained by Eqs. 7-8.

$$\theta_i = \text{Arc sin} \left[ \left( \frac{4 \cdot (W - 2b_0)}{W(u+L)^2} \cdot z_i^2 + \frac{2b_0}{W} \right) \cdot \sin\left(\frac{\beta}{2}\right) \right] \qquad (7)$$

$$\gamma_i = \beta - 2 \arcsin \left[ \left( \frac{4 \cdot (W - 2b_0)}{W(u+L)^2} \cdot z_i^2 + \frac{2b_0}{W} \right) \cdot \sin\left(\frac{\beta}{2}\right) \right] \qquad (8)$$

Here, $z_i$ is the Z coordinate of the center of the i-th cell.

It is clear that the shear angle is identical in all unit cells on the same circumference (with the same Z positions). Therefore, the shear angle as a function of the tensile displacement can be given as:

$$\gamma = \beta - 2 \arcsin \left[ \left( \frac{4 \cdot (W - 2b_0)}{W(u+L)^2} \cdot z_i^2 + \frac{2b_0}{W} \right) \cdot \sin\left(\frac{\beta}{2}\right) \right] \qquad (9)$$

### 2.3.2 Shear moment and shear load

As presented previously, the shear deformation is the domestic deformation mode during the tensile testing of a tubular braided fabric, and one can neglect the compaction on the lateral sides of the yarns and the extension of the fibers which normally occurs after the “locking shear angle”. Consequently, the power from the clamping load can be considered to dissipate mainly in the shearing zones. Regarding the non-homogenous shear deformation in the tubular structure, the prediction of the shear moment is focused on the unit cells that have the same Z. Considering a continuum structure, the dissipation of energy during the tensile stress can be described by Eq. 10.
$$F(\gamma) \cdot u = \int_{-(u+L)/2}^{(u+L)/2} 2\pi \cdot r \cdot M_s(\gamma) \cdot \gamma \, dz$$

(10)

where $M_s(\gamma)$ is the shear moment per unit area, $F(\gamma)$ is the clamping force, and $u$ is the rate of the quantity $u$. As a result, the shear moment $M_s(\gamma)$ is a function of the shear angle and depends on $Z$ in the axial direction as detailed in Eq. 11.

$$M_s(\gamma) = \frac{F(\gamma)}{2\pi \int_{-(u+L)/2}^{(u+L)/2} \left[ \frac{2(W-2b_0)}{(u+L)^2} \cdot z^2 + b_0 \right] \cdot \frac{16(W-2b_0) \cdot \sin\left(\frac{\beta}{2}\right) \cdot z^2}{W(u+L)^3 \sqrt{1 - \left[ \left( \frac{4 \cdot (W-2b_0)}{W(u+L)^2} \cdot z^2 + \frac{2b_0}{W} \right) \cdot \sin\left(\frac{\beta}{2}\right) \right]^2}} \, dz} \quad (11)$$

As the tangential load along the side of a fabric rhomboid element with unit cell dimensions (Fig. 6), the shear load has been described in [20], and depends on the shear moment of the fabric element (cf. Eq. 12). Consequently, this load on a unit cell of the tubular braided fabric can be expressed by Eq. 13. On this basis, the numerical integration algorithm is employed to calculate the shear load during the tensile test. This is mainly achieved by employing MATLAB to call the integral function to implement.

$$M_s(\gamma) = F_{sh} \sin(\beta - \gamma) \quad (12)$$

$$F_{sh}(\gamma) = \frac{F(\gamma)}{2\pi \cdot \sin(\beta - \gamma) \cdot \int_{-(u+L)/2}^{(u+L)/2} \left[ \frac{2(W-2b_0)}{(u+L)^2} \cdot z^2 + b_0 \right] \cdot \frac{16(W-2b_0) \cdot \sin\left(\frac{\beta}{2}\right) \cdot z^2}{W(u+L)^3 \sqrt{1 - \left[ \left( \frac{4 \cdot (W-2b_0)}{W(u+L)^2} \cdot z^2 + \frac{2b_0}{W} \right) \cdot \sin\left(\frac{\beta}{2}\right) \right]^2}} \, dz} \quad (13)$$
### 3. Results and discussion

#### 3.1 Verification of the hypothesis

Several circles were marked on the tubular samples in their initial state to verify that no sliding took place between the yarns at the cross-over points. As shown in Fig. 4, the circles remained plane and perpendicular to the tubular axis throughout the test, which means that the circles were deformed only in the radial direction.

As described in section 2, the establishment of the theoretical shear model first considered the profile change during the stretching process of the fabric. Thanks to the Digital Image Correlation (DIC) method, the deformed profiles were reconstructed by using SFM and are shown in Fig. 8. During the acquisition of the images, the surface of the samples was sufficiently and evenly illuminated without the appearance of reflections. To ensure the accuracy, a full-range image stream including 72 images from all angles of the specimen was used to reconstruct the 3D point clouds.

It was found that the radial shrinkage of the fabric increased with the tensile displacement. The deformed tubular fabric maintained a symmetrical shape. Fig. 9a-9c present the comparisons between the measured lateral profiles and the ones obtained theoretically from Eq. 1 at a deformation of 25, 50 and 75 %, respectively. The good agreement found validated the hypothesis of the lateral profile and the expression of the theoretical profile curve provided by Eq .1.
<figure><img src="image_9.png" /><figcaption>Fig. 8. The tubular fabric at 25, 50 and 75% deformation.</figcaption></figure>

<figure><img src="image_10.png" /></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Comparisons between measured and theoretical approaches with regard to the lateral profile of the deformed tubular sample.</figcaption></figure>

## 3.2 Tensile curves of the tubular braided fabrics

Fig. 10 presents the tensile load vs. the global deformation of tubular braided samples in axial extension tests. The curves can be divided into three parts before reaching the maximum load. In the 1st part, the load increased relatively slowly as a function of the tensile deformation. It was believed that this phenomenon concerned the pure in-plane shearing of each unit cell. At 60% deformation, the load started to rise rapidly due to the effects of extension and lateral compaction of the yarns. Between the 60 and 75% deformation, in-plane shearing was the dominating deformation mode: when the strong contact appeared on the lateral sides of the yarns, the lateral compaction blocked the yarns' rotation. Consequently, the coupling of extension and lateral compaction of the yarns generated a rapid rise of the load. This significant increase started to fade at 125% deformation when the first damage to the yarns was observed near the clamps.
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Tensile load vs. global deformation for tubular braided samples.</figcaption></figure>

Fig. 11 shows the shear angle and elongation of the yarn as a function of the fabric's global deformation. As can be seen, there was hardly any extension of the yarns before 75% deformation and the shear angle augmented rapidly in this pure shearing zone (Zone 1). Between 75 and 125% deformation, denoted "Zone 2" in Fig. 11, the shear angle increased slowly due to the impediment of the yarns' lateral compaction. In contrast, a significant extension of the yarns was noted. After 125% deformation (in Zone 3), the shear angle barely changed. From this moment, the deformed structure was completely locked and was denoted the "locking structure". Compared with plane textile fabrics, it was not appropriate to define it as a "locking angle", as a "locking angle" is not uniform in deformed tubular fabrics. In Fig. 11, the "locking angle" was close to 60° in the zone of 5 mm from the central line of the fabric (zone B in Fig. 12). The results displayed in Fig. 11 also explain the phenomena extracted from the tensile load/global deformation curve in Fig. 10.
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Shear angle and elongation of the yarn vs. the fabric's global deformation during a tensile test (observed at 5 mm from the central line of the fabric).</figcaption></figure>

### 3.3 In-plane shear response

As discussed previously, the axial tensile test of the tubular braided fabric generates a variation of the angle between the yarns. This angle can be defined as the in-plane shear angle on a unit cell and is not homogenous along the tubular sample. Fig. 12 shows four selected zones on the tested fabric in the axial direction according to the relevant dimensions. Due to the symmetry, it was possible to monitor the angles between the yarns only on the upper half of the sample.
<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Schematic representation of the shear region division.</figcaption></figure>

Fig. 13 shows a comparison between the theoretical and experimental shear angles as functions of the global tensile deformation. The theoretical angles were calculated by using Eq. 8 and the experimental ones were obtained by DIC. The shear angles were found to be different in various distinct zones on the tubular sample. The central zone (zone A) consistently underwent the biggest shearing effect compared with the other zones. The zone closest to the clamps (zone D) had little rotation space, which led to it having the smallest shear angles. A good agreement (<10%) between theoretical and experimental approaches confirmed the developed analytical model, but this agreement lasted only up to 75% deformation or in the pure-shearing stage. From 75% global deformation, analytical and experimental results became increasingly divergent (30% difference at 120% deformation and 50% difference at 150% deformation). Moreover, at 75% global deformation, a notable shearing effect with a shear angle of almost 60° and radial shrinkage of 41% were observed in zone A. With the given braided fabric in the present study, 60° in-plane shearing deformation can be considered deep forming. In other words, it was possible to carry out deep forming while remaining in the pure-shearing stage of the tubular braided fabric in question.
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Shear angle *vs.* global tensile deformation: comparison between theoretical and experimental results.</figcaption></figure>

As presented in Fig. 13, the theoretical model provided a good prediction of the shear angle in the pure-shearing stage during the tensile testing of the tubular braided fabrics. Fig. 14 displays the shear loads as a function of the shear angles of the tubular braided samples under the axial tensile stress, with values obtained by Eq. 13. The four curves of shear load vs. shear angle corresponded to the four zones defined in Fig. 14. To obtain the same shearing deformation (i.e., the same shear angle), the various zones required different shear loads. There thus existed a local shearing behavior or shearing deformation field on the tubular fabric. Compared with the zones close to the clamps, the central zones were deformed more easily under the shearing effects. It can be also observed that the “locking angle” was not uniform in the zones and depended on the position in the axial direction (the Z position). Furthermore, when going from zone A to zone D, the maximum shear angle that could be reached gradually decreased, indicating that zone A had a larger shearing space than the other areas in the pure-shearing stage.
<figure><img src="image_16.png" /><figcaption>**Fig. 14.** Shear load vs. shear angle in the different zones of the tubular braided fabric.</figcaption></figure>

**4. Conclusion**

The forming of textile reinforcements is a complex process in which serval deformation
modes are coupled. When it comes to a tubular braided fabric with a hollow structure, the
axial tensile stiffness is much more significant than for instance the torsion, compression and
bending stiffnesses. Therefore, as the dominant behavior during forming of a tubular braided
fabric, the mechanical response of such a material under an axial tensile load was determined
and analyzed in the present study. This complex mechanical response was divided into three
main parts from the tensile load/deformation curve. Pure shearing was the first part and
generated the non-homogenous in-plane shear angles on the deformed tubular structure. A
combination of in-plane shearing and yarn extension was observed in the second part until the
lateral compaction of yarns completely blocked their rotation, and this phenomenon was
called a “locking structure”. In the last part, after the “locking structure”, yarn extension
became the dominant deformation mode, which finally led to structural damage.
When it comes to the forming deformation, it was believed to take place within the first part, i.e., in the pure-shearing zone. For this, an analytical model was developed to describe the in-plane shear behavior under axial tensile stress. The relationship between the shear angle/shear moment and shear angle/shear load was modeled on the unit cells. The theoretical and experimental results correlated well and the non-uniform shear deformation field (shear angle) could be aptly revealed by the analytical approach. The shear angle was found to depend only on the global tensile deformation and the axial position. Future work will be carried out to integrate the analytical model into the numerical simulation of braided fabric forming.

## Acknowledgment

The authors gratefully acknowledge: the financial support from the China Scholarship Council (CSC) (CSC project N°202006060054), the technical support from Sascha Krugl and Christian Pidancier from the LPMT laboratory during the development of the experimental set-up as well as the support from the Gemtex laboratory in developing the tubular braids.

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