# Improved hyperelastic model for simulating the forming of biaxial braided fabrics

Jinlei Li, Nahière Hamila, Eduardo Guzman-Maldonado, Gildas L'Hostis, Peng Wang

► To cite this version:

Jinlei Li, Nahière Hamila, Eduardo Guzman-Maldonado, Gildas L'Hostis, Peng Wang. Improved hyperelastic model for simulating the forming of biaxial braided fabrics. Composite Structures, 2023, 321, pp.117263. 10.1016/j.compstruct.2023.117263. hal-05339415

HAL Id: hal-05339415

https://hal.univ-brest.fr/hal-05339415v1

Submitted on 30 Oct 2025

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# Improved hyperelastic model

## for simulating the forming of biaxial braided fabrics

Jinlei Li<sup>1, 2</sup>, Nahière Hamila<sup>3</sup>, Eduardo Guzman-Maldonado<sup>4</sup>, Gildas L'Hostis<sup>1, 2</sup>, Peng Wang<sup>1, 2, *</sup>

**Abstract:**

The complex deformation behavior during preforming of composite reinforcements can be effectively predicted by numerical simulation. Most numerical studies generally concern orthogonally woven fabrics. The present work proposes a hyperelastic constitutive model to characterize the anisotropic mechanical properties of non-orthogonal braided reinforcements to simulate the preforming process. Based on the deformation modes, the strain energy was defined by corresponding specific invariants. The coefficients of the hyperelastic constitutive law were identified in the case of non-orthogonal fabrics thanks to the establishment of an energetic model between the second Kirchhoff stress tensor and the tensile load. The constitutive law and energetic model were then validated by means of comparison with the bias-extension test and hemispherical forming test.

**Keywords:**

Textile composites; Braided fabrics; Numerical simulation; Hyperelastic model; Forming

<footnote n="1"><sup>1</sup>University of Haute-Alsace, ENSISA, LPMT, F-68000 Mulhouse, France</footnote>

<footnote n="2"><sup>2</sup>University of Strasbourg, France</footnote>

<footnote n="3"><sup>3</sup>ENI Brest, IRDL, UMR CNRS 6027, F-29200, Brest, France</footnote>

<footnote n="4"><sup>4</sup>Innovamics, 69190 Saint-Fons, France</footnote>

<footnote>* Corresponding author. Tel.: +33 3 89 33 66 48. Fax: +33 3 89 33 63 39</footnote>

<footnote>E-mail address: peng.wang@uha.fr</footnote>
<table><tr><td>w</td><td>strain energy</td><td>D</td><td>diagonal length of the rhombus in the pure shear zone before deformation (mm)</td></tr><tr><td>S</td><td>second Piola-Kirchhoff stress tensor</td><td>L<sub>c</sub></td><td>theoretical side length of the rhombus in the pure shear zone before deformation (mm)</td></tr><tr><td>C</td><td>right Cauchy-Green deformation tensor</td><td>L<sub>0</sub></td><td>sample length (mm)</td></tr><tr><td>F</td><td>deformation gradient tensor</td><td>l</td><td>sample width (mm)</td></tr><tr><td>X</td><td>position of a material particle in the referential configuration</td><td>u</td><td>tensile displacement of the sample (mm)</td></tr><tr><td>x</td><td>position of a material particle in the current configuration</td><td>Z<sub>A</sub></td><td>area of zone A (mm<sup>2</sup>)</td></tr><tr><td>J</td><td>Jacobian matrix</td><td>Z<sub>B</sub></td><td>area of zone B (mm<sup>2</sup>)</td></tr><tr><td>M<sub>1</sub></td><td>first principal direction in the initial configuration</td><td>Z<sub>C</sub></td><td>area of zone C (mm<sup>2</sup>)</td></tr><tr><td>M<sub>2</sub></td><td>second principal direction in the initial configuration</td><td>F(γ)</td><td>load acting on the specimen (N)</td></tr><tr><td>I<sub>elongi</sub></td><td>tension invariants</td><td>S<sub>αβ</sub></td><td>components of the PK2 stress tensor</td></tr><tr><td>I<sub>sh</sub></td><td>in-plane shear invariant</td><td>h</td><td>sample thickness (mm)</td></tr><tr><td>γ</td><td>shear angle (°)</td><td>P<sub>ext</sub></td><td>external power (J)</td></tr><tr><td>β/2</td><td>initial braiding angle (°)</td><td>P<sub>int</sub></td><td>internal power (J)</td></tr><tr><td>α</td><td>angle between two interlaced yarns (°)</td><td>V<sub>0</sub></td><td>sample volume (mm<sup>3</sup>)</td></tr></table>

**Nomenclature:**

**1. Introduction**

Textile composites have played an irreplaceable role in the aerospace and automotive industries due to their great potential in the production of complex composite parts and weight reduction [1,2]. To manufacture composite parts with double curvature and complex shapes, one can apply preforming of dry reinforcement in Liquid Composite Molding Processes (LCM) [3]. However, this a complicated physical stage including complex deformability behaviors [4]. Due to the characteristics of the reinforcement and manufacturing parameters
(friction, speed, pressure, etc), the preforming stage of the fabric is also accompanied by various defects [5]. On the other hand, when the fabric is draped out of the expected complex shape on a double curvature mold, this will cause fiber redistribution and redirection [6–8], thus affecting the mechanical properties of the component. In this case, numerical simulation has been proposed and become an alternative to the "trial and error" processes in order to predict the conditions and optimize the manufacturing parameters [9].

In the preforming process, the asymmetry and anisotropy of the fabric's structure increase the difficulty of numerical simulation. Hence, it is particularly vital to choose a constitutive model to accurately reflect the mechanical properties of fabrics and predict the anisotropic behavior of textile composites due to the complex reorientation and redistribution of the yarns [10]. A continuous model was put forward and proven to be effective in analyzing the preform from a macroscopic perspective and predicting the deformation behavior of fabrics during the forming process [11–15]. The continuous model includes the orthogonal constitutive model and non-orthogonal constitutive model [16]. The simulation results of the non-orthogonal model are closer to the experiment than those of the orthogonal model [17]. The large in-plane shear deformation of two-dimensional (2D) woven fabrics in the stamping process was simulated by a non-orthogonal constitutive model [12]. Xu et al. [18] proposed a non-orthogonal model and applied it to the study of the macroscopic behavior of complex fabric rubber seals. Han et al. [19] simulated the drapability and wrinkle behavior of uncured woven fabric prepreg using a non-orthogonal constitutive VUMAT code and a lay-up shell model. In addition, the non-orthogonal models have been applied to the analysis of the mechanical properties of 3D angle-interlock woven composites [20].

Based on the theories of continuum mechanics, hyperelastic models have been extensively studied and applied to simulate the macroscopic mechanical behavior of fabrics [2, 9, 10, 21–23]. Unlike traditional elastic models, the constitutive relations of hyperelastic models are
represented by strain energy and invariants instead of stiffness matrices. Different
deformation modes that appear in the fabric during the forming process were considered, such
as tension, in-plane shear, bending, etc. [24]. Charmetant et al. [25,26] proposed a
hyperelastic model in order to simulate the large deformation behavior of a 2D fabric and a
3D interlocking fabric with large thicknesses by taking into account 4 deformation modes of
the yarn and 6 deformation modes of the fabrics, respectively. The benchmark double dome
forming was simulated by a simple hyperelastic constitutive model [10]. Schäfer et al. [27]
proposed a new hyperelastic model considering biaxial coupling of interwoven fiber bundles
in a woven fabrics, normal shear coupling of positive and negative strains, and in-plane
compressive behavior. Based on the hyperelastic model, a visco-hyperelastic counterpart was
developed to simulate the thermoforming of thermoplastic composites by considering the
influence of temperature [9]. In addition, other models such as the hypoelastic model [12,28–
30] and semi-discrete shell finite elements [5] have also been put forward.

Nevertheless, the above studies were limited to the case of woven fabrics where the weft and
warp directions were perpendicular to each other and very few investigations have been
devoted to non-orthogonally woven fabrics. According to ref [31], non-orthogonal
reinforcements are being considered for certain aerospace and wind turbine components due
to their superiority. Consequently, a simulation method based on stress result shell elements
has been introduced and a picture frame test of non-orthogonal NCF reinforcement has been
simulated. Peng and Guo [10] proposed an invariant (I7) for the in-plane shear at an arbitrary
initial angle, but it has not been applied to non-orthogonal materials.

The objective of this paper was thus to first improve a non-orthogonal hyperelastic
constitutive model to simulate the forming process of non-orthogonal composite braided
fabrics, taking into account the tensile and shear effects of the fibers. The shear invariant
proposed by this model can be used to characterize the shear deformation of fabrics with
arbitrary braiding angles. The theoretical relations were introduced to obtain the in-plane
shear component for the hyperelastic model for fabrics with arbitrary braiding angles. In order
to prove the validity of the model and shear coefficients, the bias extension test and
preforming test of non-orthogonal braided fabrics were simulated. The numerical simulation
results are in good agreement with the experimental results, which effectively verified the
correctness of each theoretical model.

## 2. Hyperelastic modeling for non-orthogonal fabrics

2.1 Description of non-orthogonal fabrics.

Weaving, braiding and knitting are the main methods to produce textile preforms [32]. Among them, weaving is primarily used to create woven fabrics (two interlaced yarns that are orthogonal in the initial position). The braiding technology can obtain fabrics with arbitrary braiding angles (between 5° and 85°) by setting specific braiding process specifications and mandrel parameters [33–35]. The fabrics obtained by the braiding machine have a three-dimensional tubular structure, and needs to be cut and opened along their axial direction to become two-dimensional flat single-layer fabrics, as presented in Fig. 1. Braided fabrics are flexibly used in advanced composite parts due to their variable braiding angles. Compared with woven fabrics, braided fabrics present non-symmetrical structures in two directions, so the mechanical properties of the fabric should be defined simultaneously for different directions (axial direction and transversal direction) to achieve a comprehensive grasp of the in-plane shear and tensile behaviors.
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Acquisition of 2D braided fabrics and structural diagram in two directions: (a) three-dimensional tubular braided structure, (b) the axial direction and (c) the transversal direction.</figcaption></figure>

## 2.2 Specific physical invariants for non-orthogonal fabrics

To analyze the mechanical behavior of textile composite reinforcements, different hyperelastic models have been proposed describing the various structural types of fabrics. These models are based on formulations of physical invariants to portray the deformation modes by definition of strain energy. The deduction of the general form and details of the hyperelastic model is presented in Appendix A.

The reorientation of the fabric fibers during the forming process is key to obtaining complex shapes. In the initial configuration, there are two privileged directions of fabric (as shown in Fig. 2), $M_1$ and $M_2$, respectively. The symmetry group for such a material is characterized by the structural tensors:
$$ \underline{M}_{ij} = \underline{M}_i \otimes \underline{M}_j \quad i, j = 1, 2 \qquad (1) $$

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Schematic diagram of the two principal directions in a unit cell in the initial and current configurations.</figcaption></figure>

Let us assume that the deformation modes of 2D braided fabrics during formation are tension along the principal directions and shear between two threads; and that these are independent of each other. The total strain energy of the fabric can be written as the summation of the corresponding strain energies of these independent deformation modes [2]:

$$ w(\underline{C}) = w_{\text{elong1}}(I_{\text{elong1}}(\underline{C}, \underline{M}_{\underline{11}})) + w_{\text{elong2}}(I_{\text{elong2}}(\underline{C}, \underline{M}_{\underline{22}})) + w_{\text{sh}}(I_{\text{sh}}(\underline{C}, \underline{M}_{\underline{12}})) \quad (2) $$

where w is the strain energy, C is the right Cauchy-Green strain tensor, M is the structural tensors.

According to Eq. 2, the stress results become:

$$ \underline{S} = 2 \frac{\partial w}{\partial \underline{C}} = 2 \left( \frac{\partial w_{\text{elong1}}}{\partial I_{\text{elong1}}} \cdot \frac{\partial I_{\text{elong1}}}{\partial \underline{C}} + \frac{\partial w_{\text{elong2}}}{\partial I_{\text{elong2}}} \cdot \frac{\partial I_{\text{elong2}}}{\partial \underline{C}} + \frac{\partial w_{\text{sh}}}{\partial I_{\text{sh}}} \cdot \frac{\partial I_{\text{sh}}}{\partial \underline{C}} \right) \quad (3) $$

where $I_{\text{elong1}}$, $I_{\text{elong2}}$, $I_{\text{sh}}$ are the invariants of elongation and in-plane shear.
The invariants along two privileged directions reflect the elongation of the yarn. As it is not affected by the braiding angle, it is the same as the tension invariants for woven fabrics and can be expressed as:

$$I_{elongi} = \ln(\sqrt{I_{4i}}) \quad i=1,2 \qquad (4)$$

$$\text{where } I_{4i} = \underline{C} : \underline{M}_{ii} \quad i=1,2$$

The derivative of the elongation invariant concerning $\underline{C}$ is given by:

$$\frac{\partial I_{elongi}}{\partial \underline{C}} = \frac{1}{2I_{elongi}} \underline{M}_{ii} \quad i=1,2 \qquad (5)$$

The shear angle is a visual representation of the shear deformation mode and is the basis for calculating the shear strain energy. The definition of the shear invariant depends on the shear angle, expressed as $I_{sh} = \cos(\alpha) - \cos(\beta)$, as seen in Fig. 2. For woven fabrics, the shear invariant is equal to zero in the initial state. In non-orthogonal materials, due to the arbitrariness of the braiding angle, the shear invariant characterizes the angle variation between the principal directions, and the braiding angle of the fabric needs to be taken into account (Eq. 6). This is completely different from woven fabrics, which is also the perspective of this work.

$$I_{sh} = \frac{I_{412}}{\sqrt{I_{41}I_{42}}} - \underline{M}_1 \cdot \underline{M}_2 \qquad (6)$$

Consequently, the derivative of the shear invariant concerning $\underline{C}$ is given by Eq. 7.

$$\frac{\partial I_{sh}}{\partial \underline{C}} = \frac{1}{2\sqrt{I_{41}I_{42}}} (\underline{M}_{12} + \underline{M}_{21}) - \frac{1}{2} \frac{I_{sh1}}{I_{41}} \underline{M}_{11} - \frac{1}{2} \frac{I_{sh1}}{I_{42}} \underline{M}_{22} \qquad (7)$$

$$\text{where } I_{412} = \underline{C} : \underline{M}_{12}, I_{sh1} = \frac{I_{412}}{\sqrt{I_{41}I_{42}}}$$
### 3. Strain energy and material parameters identification

Now that the expressions for the physical invariants have been presented, each contribution to the strain energy can be expressed, in terms of the respective physical invariants, as a polynomial function. The identification of the polynomial coefficients (material parameters) is achieved by fitting simple tests, such as the uniaxial and/or biaxial tension test [12,36], the picture frame test [37–40] and the bias extension test [41].

#### 3.1 Identification of elongation strain energy and shear strain energy

The strain energy associated with the tension of 2D braids is closely related to the elongation of the yarns. Consider that the elongation behavior in the two main directions is the same, but that each contribution is independent of the other. The strain energy can be expressed as an even function of the type:

$$w_{\text{elong 1}} = \sum_{i=1}^{n} c_i (I_{\text{elong 1}})^{2i} \qquad w_{\text{elong 2}} = \sum_{i=1}^{n} c_i (I_{\text{elong 2}})^{2i} \qquad (8)$$

where $c_i(i = 1, 2, ...)$ are the tension strain energy coefficients, and the identification of which can be performed directly from a uniaxial tensile test in one of the principal directions. In this paper, the coefficients are obtained by fitting the force-displacement curve of the single yarn under uniaxial tensile test. According to Eq. 9, the partial derivative of tensile strain energy with respect to the invariant is:

$$\frac{\partial w_{\text{elongi}}}{\partial I_{\text{elongi}}} = \sum_{i=1}^{N} 2i \cdot c_i (I_{\text{elongi}})^{2i-1} \qquad (9)$$

The in-plane shear mechanical behavior of the material is determined by using a bias-extension test or a picture frame test. The shear strain energy is expressed by a polynomial function;

$$w_{\text{sh}} = \sum_{i=1}^{n} k_i (I_{\text{sh}})^{2i} \qquad (10)$$

In the formula, $k_i(i = 1, 2, ...)$ are the shear coefficients of the strain energy. The derivative with respect to the shear invariants is given by Eq. 11:
$$ \frac{\partial w_{sh}}{\partial I_{sh}} = \sum_{i=1}^{n} 2i \cdot k_i (I_{sh})^{2i-1} \qquad (11) $$

## 3.2 Theoretical relations between global tensile load and second Piola-Kirchhoff shear stress
Determining fabric shear parameters has been widely discussed in the literature [10,42] and most of the investigations used the picture frame test to carry out the identification. For the picture frame test, the fabric is in a pure shear state under the assumption that there is no sliding between yarns and that all the power created by the external force is converted into shear strain energy. Consequently, it becomes easier to obtain shear parameters. During frame testing, misalignment can easily be introduced during clamping, which induces spurious tension during testing. At the same time, the fibers on the boundary of the picture frame are not purely sheared due to the clamping [38].

In contrast, attempting to obtain the shear parameters of materials by the bias-extension test is more accurate. It is worth noting that the pure shear behavior in the bias-extension test only occurs in a partial area of the fabric, so it is a challenge to identify the shear parameters. In this section, based on an energetic approach, the second Piola-Kirchhoff (PK2) shear stress was directly related to the traction load of the bias-extension test.

In the initial configuration, the fabric was not deformed, and the principal directions in a unit cell can be represented in the global coordinate system, as shown in Fig. 2.

$$ \underline{M}_1 = \sin\left(\frac{\beta}{2}\right) \underline{e}_{10} + \cos\left(\frac{\beta}{2}\right) \underline{e}_{20} \qquad (12) $$

$$ \underline{M}_2 = -\sin\left(\frac{\beta}{2}\right) \underline{e}_{10} + \cos\left(\frac{\beta}{2}\right) \underline{e}_{20} \qquad (13) $$

When the fabric was stretched, the fibers became reoriented, and the components of the principal directions were thus:

$$ \underline{M}'_1 = \sin\left(\frac{\beta - \gamma}{2}\right) \underline{e}_{10} + \cos\left(\frac{\beta - \gamma}{2}\right) \underline{e}_{20} \qquad (14) $$
$$ \underline{M}'_2 = -\sin\left(\frac{\beta-\gamma}{2}\right) \underline{e}_{10} + \cos\left(\frac{\beta-\gamma}{2}\right) \underline{e}_{20} \quad (15) $$

From the kinematics of the bias-extension test one can obtain the deformation gradient $\underline{F}$:

$$ \underline{F} = \frac{1}{\sin\left(\frac{\beta}{2}\right)} \cdot \sin\left(\frac{\beta}{2} - \frac{\gamma}{2}\right) \underline{e}_{10} \otimes \underline{e}_{10} + \frac{1}{\cos\left(\frac{\beta}{2}\right)} \cdot \cos\left(\frac{\beta}{2} - \frac{\gamma}{2}\right) \underline{e}_{20} \otimes \underline{e}_{20} \quad (16) $$

When the braided angle is 90°, the deformation gradient is expressed as:

$$ \underline{F} = \sqrt{2} \cdot \sin\left(\frac{\pi}{4} - \frac{\gamma}{2}\right) \underline{e}_{10} \otimes \underline{e}_{10} + \sqrt{2} \cdot \cos\left(\frac{\pi}{4} - \frac{\gamma}{2}\right) \underline{e}_{20} \otimes \underline{e}_{20} \quad (17) $$

where $(\underline{e}_{10}, \underline{e}_{20})$ is the global coordinate system and $\underline{e}_{20}$ follows the tensile direction. From the deformation gradient, the right Cauchy Green strain tensor $\underline{C} = \underline{F}' \underline{F}$ and its derivative with respect to time $\dot{\underline{C}}$ are:

$$ \left\{ \begin{aligned} \underline{C} &= \left( \frac{\sin\left(\frac{\beta-\gamma}{2}\right)}{\sin\left(\frac{\beta}{2}\right)} \right)^2 \underline{e}_{10} \otimes \underline{e}_{10} + \left( \frac{\cos\left(\frac{\beta-\gamma}{2}\right)}{\cos\left(\frac{\beta}{2}\right)} \right)^2 \underline{e}_{20} \otimes \underline{e}_{20} \\ \dot{\underline{C}}(t) &= \frac{1}{2} \cdot \dot{\gamma}(t) \cdot \sin(\beta-\gamma) \left( \frac{-1}{\sin^2\left(\frac{\beta}{2}\right)} \underline{e}_{10} \otimes \underline{e}_{10} + \frac{1}{\cos^2\left(\frac{\beta}{2}\right)} \underline{e}_{20} \otimes \underline{e}_{20} \right) \end{aligned} \right. \quad (18) $$

At the same time, for the woven fabrics:

$$ \left\{ \begin{aligned} \underline{C} &= (1 - \sin(\gamma)) \underline{e}_{10} \otimes \underline{e}_{10} + (1 + \sin(\gamma)) \underline{e}_{20} \otimes \underline{e}_{20} \\ \dot{\underline{C}}(t) &= \dot{\gamma}(t) \cdot \cos(\gamma) (-\underline{e}_{10} \otimes \underline{e}_{10} + \underline{e}_{20} \otimes \underline{e}_{20}) \end{aligned} \right. \quad (19) $$

where $\gamma$ is the shear angle.

As mentioned in the literature [43], the fabric consists of three different partitions (A, B and C) during the bias-extension test. Zone A namely “no-deformed zone”, Zone B called “semi-shear zone” and Zone C is “pure shear zone”, as shown in Fig. B1 in Appendix B. Assuming that the only mode of deformation in the specimen is pure shear, the external power due to
machine forces can be related to the internal power due to shear forces in each zone. However, the only deformation mode involved in this balance are zones B and C, since zone A is not deformed-. In a quasi-static tensile test, the load (F) on the specimen is related to the second Piola-Kirchhoff stress according to the principle of equal internal ($P_{int}$) and external power ($P_{ext}$).

$$F(\gamma)\dot{u} = P_{int}^{(B)} + P_{int}^{(C)} = \frac{1}{2} \int_{\Omega_0} S : \dot{C} dV_0 \quad (20)$$

By integrating the energy on the initial effective surfaces of the specimen, the relationship between the external force and the components of the PK2 tensor can be deduced:

$$F(\gamma) = \frac{1}{4} \dot{\gamma}(u) \cdot h \cdot \left( \sin(\beta - \gamma) \cdot Z_A \left( -\frac{S_{11}(\gamma)}{\sin^2(\frac{\beta}{2})} + \frac{S_{22}(\gamma)}{\cos^2(\frac{\beta}{2})} \right) + \sin(\beta - \frac{\gamma}{2}) \cdot Z_B \left( -\frac{S_{11}(\frac{\gamma}{2})}{\sin^2(\frac{\beta}{2})} + \frac{S_{22}(\frac{\gamma}{2})}{\cos^2(\frac{\beta}{2})} \right) \right) \quad (21)$$

where $S_{11}(\gamma)$ and $S_{22}(\gamma)$ are diagonal stress components in the pure shear zone of the specimen, $S_{11}(\frac{\gamma}{2})$ and $S_{22}(\frac{\gamma}{2})$ are stress components in the semi-shear zone, and $Z_A$ and $Z_B$ are the initial surface area of zones A and B. $u$ is the tensile displacement of the sample,

$$\dot{\gamma}(u) = \frac{1}{L_c \cdot \sin(\frac{\alpha}{2})} \cdot h \quad \text{h is the thickness of the sample. The specific geometric relationship can be found in Appendix B.}$$

# 4. Validation of the hyperelastic law

## 4.1 Bias-extension test for non-orthogonal fabrics

### 4.1.1 Sample preparation

The proposed models were examined by using braided fabrics of flax/PA12 commingled yarns listed in Table 1. The biaxial braided fabrics were all produced by overbraiding on the braiding looms provided by the Gemtex laboratory [44,45]. It should be stated that it was
necessary to carefully cut the fabric into a single layer of samples for the bias-extension test.

A fair amount of attention was also required to master the braiding angle and avoid fiber slippage or local pull-out.

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

The main physical properties of the tested braided fabrics.

<table><thead><tr><th>Parameters</th><th>Flax/PA12</th></tr></thead><tbody><tr><td>Type of braid</td><td>Biaxial twill 2-2</td></tr><tr><td>Yarn density (tex)</td><td>500</td></tr><tr><td>Braiding angle (β/2)</td><td>35°</td></tr><tr><td>Yarn thickness (mm)</td><td>1.03</td></tr><tr><td>Yarn width (mm)</td><td>1.15</td></tr><tr><td>Number of yarns per cm</td><td>4.2</td></tr><tr><td>Area density of the fabric (g/m²)</td><td>376 ± 5</td></tr></tbody></table>

### 4.1.2 Experimental setup

A series of bias-extension tests were performed on a rectangular sample to obtain the material parameters for the in-plane shear behavior. As mentioned in Section 2.1, to obtain a more complete mechanical behavior and validate the proposed theoretical model, two “Bias-extension” type tests were carried out in the axial and transversal directions to apply stress on the specimen in negative and positive shear, respectively [26].

The bias-extension test of non-orthogonal fabrics was conducted on a standard MTS system at a loading rate of 50 mm/min. The relevant parameters of the textile materials are listed in Table 1. It is worth noting that the braiding angle (β/2) of the fabric was 35°, and that the fabric structure was asymmetrical in the axial and transversal directions. So, the angle of the interlaced yarn is 70° and 110° in the axial and transversal directions respectively. According to the requirements of Eq. B9 in Appendix B, the parameters whose aspect ratio met the test requirements were obtained. The aspect ratios had to be greater than 2.9 in the axial direction and 1.4 in the transversal direction, and the specific parameters are shown in Table 2. Meanwhile, a camera was used to observe the deformation of the fabric in real-time during
the experiment. In order to reduce the error, each test is repeated five times to obtain an average data.

<table><caption>**Table 2**</caption><thead><tr><th>Stretch direction</th><th>Sample yarns</th><th>L<sub>0</sub> (length)<br>(mm)</th><th>l (width)<br>(mm)</th><th>D(mm)</th><th>r (ratio)<br>(L<sub>0</sub>/l)</th></tr></thead><tbody><tr><td rowspan="2">Axial Direction</td><td rowspan="2">Flax/PA12</td><td>200</td><td>50</td><td>128.59</td><td>4</td></tr><tr><td>300</td><td>50</td><td>228.59</td><td>6</td></tr><tr><td>Transversal<br>Direction</td><td></td><td>100</td><td>60</td><td>57.98</td><td>1.67</td></tr></tbody></table>

The main geometrical properties of the tested specimens.

In order to ensure a sufficient pure shear space of the fabric, the geometric ratios of 4 in the axial direction and 1.67 in the transversal direction were selected for the test. According to the definition of in-plane shear angle, the axial direction along the sample is the positive direction. So, for the bias-extension test in the axial direction of the sample, the load and displacement are taken as positive values; for the test in the transversal direction of the sample, the load and displacement are taken as negative values. Fig. 3 shows the evolution of the experimental force obtained for these two tests as a function of the displacement, and the calculation of the shear angle determined from Eq. B6 in Appendix B. However, only experimental data for a shear angle between -80° and 40° were used to characterize the material shearing. At this stage, there is enough shear space between the yarns. The yarns rotate with each other without being stretched, and the fabric undergoes pure shear deformation.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Results of flax/PA12 in different directions. (a) Experimental load and (b) theoretical shear angle</figcaption></figure>

### 4.1.3 In-plane shear parameter identification and simulation of the bias-extension test

The relation, resulting from the energy approach, between the load and the shear stress PK2 was used to obtain the shear parameters. The Levenberg–Marquardt algorithm is proposed to get a numerical solution for nonlinear minimization. According to Eq. 21, by using the Levenberg–Marquardt algorithm to interpolate the average experimental results of bias-extension tests (shown in Fig. 4) with geometric ratios, which were 4 in the axial direction and 1.67 in the transversal direction, the shearing coefficients in Eq. 10 could be determined, and are summarized in Table 3. Using this algorithm, only three sets of shear coefficients were obtained without overfitting problem.
<figure><img src="image_5.png" /><figcaption>Fig. 4. Identification of shear parameters for flax/PA12 in the axial and transversal directions.</figcaption></figure>

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

The shear coefficients of the strain energy

<table><thead><tr><th>Fabric types</th><th>Axial direction<br>(J / mm²)</th><th>Transversal direction<br>(J / mm²)</th></tr></thead><tbody><tr><td>Flax/PA12</td><td>k₁ = 0.000251, k₂ = 0.0563,<br>k₃ = 0.2086</td><td>k₁ = 0.0023, k₂ = 0.0019,<br>k₃ = 0.0011</td></tr></tbody></table>

Different directions (Table 2) of flax/PA12 were selected to simulate the bias-extension test. According to the experimental conditions, one end of the fabric was fixed and the other end was subjected to a displacement load along the tensile direction. The simulation results of the bias-extension test and a comparison with an experimental approach are shown in Fig. 5, which presents the distribution of the shear angles. According to the simulation results, the fabric was asymmetrical in the axial and transversal directions due to the non-orthogonal structure of the braided preform. The three different zones could be clearly seen in both types of fabrics (as mentioned in Appendix B: no deformation in zone A, semi-shear deformation in zone B and pure shear deformation in zone C).
<figure><img src="image_6.png" /><figcaption>(a) axial direction</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) transversal direction</figcaption></figure>

<caption>Fig. 5. In-plane shear angles during the bias-extension tests.</caption>

Eq. B6 gives the theoretical model for calculating the shear angle in the pure shear zone (zone C), and the theoretical value of the in-plane shear angle in the tensile process can be obtained by substituting the experimental displacement into the equation. Comparing the simulation results of the selected fabric in the axial and transversal directions with the theoretical model, the relation between the shear angle (in the pure shear zone) and
displacement could be determined and is portrayed in Fig. 6. The different ratios (r) of the
length to the width of the fabric were compared. The predicted shear angle using the shearing
parameters (Table 3) was in good agreement with its theoretical counterpart.

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Comparison of shear angle between theoretical and simulated values in (a) the axial direction (AD) and (b) the transversal direction (TD).</figcaption></figure>

## 4.2 Hemispherical forming for non-orthogonal fabrics

### 4.2.1 Experimental setup

Forming experiments with hemispherical punches were carried out on a specific preforming
device at room temperature, and the schematic diagram of the device is shown in Fig. 7a. The
device was able to provide a variation of forming conditions for the textile reinforcement in
order to achieve the desired double-curved shape. The flax/PA12 fabric was used in the
forming test, and was initially a 155mm×155mm square. The thickness of the fabric was 2.06
mm and the braiding angle was 35° (Fig. 7b). The main properties of the textile were the same
as those in Table 1. During the forming process of non-orthogonal braided fabrics, both
tensile and shear strain energies need to be considered. The coefficients used to characterize
the tensile strain energy and the shear strain energy are obtained by experiments. It is
noteworthy that the coefficients ($c_i$) of tensile strain energy obtained from the data of uniaxial
tensile tests of a single yarn (flax/PA12) and the value is 181. The coefficients ($k_i$) of shear
strain energy were obtained from bias-extension test as described in 4.1.3 (Table 3). The pressure required during the test was provided by pneumatic jacks. In order to compare the preforming results, draping experiments were conducted with 0.05 and 0.2 MPa pressure.

<figure><img src="image_9.png" /><figcaption>Fig. 7. (a) Schematic diagram of the preforming test and (b) the flax/PA12 braided fabric sample.</figcaption></figure>

### 4.2.2 Simulation of the hemispherical preforming

The preforming process of hemispheric stamping was simulated by using the non-orthogonal hyperelastic model proposed in section 2. This model was implemented in a nonlinear explicit finite element code provided by Innovamics. In the draping simulation of the fabric, finite
element models of the fabric and the experimental setup were constructed. The blank was modeled by triangular membrane elements (the element size was 3 mm) and a penalty approach was used to model the contact. For the punch and die, contact triangle elements were used and the size of elements were 3mm. For the fabric, the membrane elements were used and the size was 2mm.

For the preforming simulation by the FE analysis, the geometry of the forming tool (punch, die and holder) is given in Fig. 8. The different pressures were uniformly applied to the holder and the friction coefficient between the fabric and the tool was set to 0.2. To improve the computational efficiency, only a quarter of the fabric was modeled to take into account the symmetry of the forming process. The entire draping simulation process is summarized as: the fabric is placed flat on the die, the pressure was applied on the blank to press the fabric and the velocity boundary condition was used to control the punch speed (50 mm/min) to drape the fabric.

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** The geometry of the forming tool in numerical simulation analysis.</figcaption></figure>

Fig. 9 shows the comparison between the deformed shape of a braided fabric predicted by the hyperelastic model and its experimental counterpart at 0.2 MPa pressure of the blank-holder. The two shapes exhibited decent consistency. It was found that the non-orthogonal braiding angles led to deformation differences of the fabric in the axial and transversal directions. The forming of the hemisphere of the braided fabric is asymmetric. The fabric
presented a draw-in phenomenon along the axial direction, accompanied by in-plane shear. Furthermore, small changes were observed in the transversal direction. At the same time, two significant intra-ply shear behaviors appeared in the axial direction, and the local fiber volume fraction in the shearing regions were higher than in other regions.

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Deformed braids after preforming under 0.2 MPa pressure of the blank-holder: (a) experimental approach and (b) numerical simulation.</figcaption></figure>
Under the pressure of 0.05 and 0.2 MPa, the deformed boundary profile of the simulated fabric was compared with the experiment, as shown in Fig. 10. According to a direct comparison of the area of maximum error, the simulation results were basically consistent with their experimental counterparts. The simulation could thus give a satisfactory portrayal of the preformed shapes, with errors ranging from approximately 0% to 12%, which could be reasonably accepted.

<figure><img src="image_12.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b)</figcaption></figure>
**Fig. 10.** Comparison of the deformed boundary profile under different preforming conditions:
(a) 0.05 MPa and (b) 0.2 MPa blank-holder pressure.

The shear angle is an important reference when it comes to predicting the wrinkling of the deformed part and the key to calculate the fiber volume fraction. Accurate prediction of the shear angle distribution is of great significance for fabric forming optimization. In order to obtain accurate shear angle results, the fabric was divided into 17 zones according to the shear angle distribution presented by the test, as shown in Fig. 11. The average value of the shear angle for each zone was computed and compared with the simulation results, as can be seen in Fig. 12. The maximum errors of shear angle at a pressure of 0.05 MPa and 0.2 MPa were 9.3% and 8.2%, respectively. The maximum shear angle occurred in the transition region between the double dome curve surface and the flat area of the fabric (zone 13). So, the effectiveness and accuracy of the numerical predictions on the shear angles were vindicated from the comparison. Furthermore, the measured shear angles and deformed profiles and simulation results showed excellent agreement establishing the reliability of the hyperelastic model.

<figure><img src="image_14.png" /><figcaption>(a) 0.05 MPa-Experiment</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) 0.05 MPa-Simulation</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>(c) 0.2 MPa-Experiment</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(d) 0.2 MPa-Simulation</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(a) blank-holder pressure of 0.05 MPa</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(b) blank-holder pressure of 0.2 MPa</figcaption></figure>

<caption>**Fig. 12.** Comparison of in-plane shears between experiment and numerical simulation approaches in 17 different zones.</caption>

**5. Conclusion**

The present paper proposed a new hyperelastic model to characterize the mechanical behavior of non-orthogonal braided fabrics during a preforming process. Tension and shear deformation modes were introduced to describe the relationship between strain energy and invariants. In particular, the shear invariant presented in this paper can be applied to braided fabrics with arbitrary angles and is not limited to woven fabrics. At the same time, a theoretical relation between the second Piola-Kirchhoff stress tensor and the tensile load was established to predict the shear coefficients in the hyperelastic model by utilizing the data of
the bias-extension test. To examine the validity of the model, the bias-extension test and hemispherical test of non-orthogonal braided fabrics were also simulated, and good agreement was found between the numerical simulation results and their experimental counterparts, thus proving the validity of the numerical model. Although the soundness of the hyperelastic model has been demonstrated herein, bias-extension tests are still required to determine the shear parameters. Moreover, further investigations on the influence of the bending stiffness are necessary to predict the development of wrinkles.

**Data availability statement**

The simulation and experimental data required to reproduce these studies can be shared on request.

**Acknowledgement.**

The authors gratefully acknowledge the financial support from the China Scholarship Council (CSC) (Project No. 202006060054).

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## Appendix A: The general form of the constitutive model

Hyperelastic materials refer to nonlinear elastic materials with large deformations. The stress work is only related to initial and current strain states, and is independent of deformation paths. Although friction between yarns consumes very little energy during deformation [10], the hyperelastic material can be approximately considered as to have no energy dissipation in the deformation process without considering unloading, and the deformation energy required by the material is equivalent strain energy. For this reason, the constitutive model of the hyperelastic material is deduced from the strain energy function, where the energy density is usually defined by:

$$w(\underline{C}) = \tilde{w}(I_1, I_2, \dots, I_n) \qquad (A1)$$

Consequently, the Lagrangian constitutive equation of a hyperelastic material can be written as:
$$ \underline{S} = 2 \frac{\partial w(\underline{C})}{\partial \underline{C}} = 2 \frac{\partial \tilde{w}}{\partial I_n} \frac{\partial I_n}{\partial \underline{C}} \qquad (A2) $$

where w is the strain energy, $I_n$ are the invariants, $\underline{S}$ is the second Piola-Kirchhoff stress tensor, and $\underline{C}$ is the right Cauchy-Green strain tensor. It can be directly represented by the deformation gradient tensor $\underline{F}$ as:

$$ \underline{C}(x,t) = \underline{F}^T \cdot \underline{F} \qquad (A3) $$

In continuum mechanics, the deformation gradient tensor is then:

$$ \underline{F}(x,t) = \frac{\partial \underline{x}}{\partial \underline{X}} \qquad (A4) $$

where $\underline{X}$ and $\underline{x}$ are the position of a material particle in the referential configuration and the current configuration, respectively.

Finally, the Cauchy stress tensor can be expressed as:

$$ \underline{\sigma} = \frac{1}{J} \underline{F} \cdot \underline{S} \cdot \underline{F}^T \qquad (A5) $$

where J is the Jacobian matrix.

## Appendix B: Geometric model in the initial state of the fabric

To study the bias-extension test, two hypotheses are proposed: one to assume that the preform has no yarn slippage during the initial stage of loading, and that the yarns in the fabric are inextensible. Under this condition, there will be three shear areas in the fabric, as shown in Fig.B1. The area A represents the non-deformed zone ($\gamma=0$), and no shearing took place in this zone. The area B represents the half shear zone ($\gamma/2$), in two interwoven yarns, where one of the yarns was free at both ends. Free yarns undergo in-plane shearing during tensile test. The area C represents the pure shear zone ($\gamma$), which appears in the middle part
of the preform. The interwoven yarns were free at both ends. Therefore, in the process of the preform bearing the load, the interwoven yarns in this region underwent complete shearing.

<figure><img src="image_20.png" /><figcaption>Fig. B1. The geometry of the fabric during a bias extension test: (a) initial state and (b) deformed state.</figcaption></figure>

Fig. B1(a) shows the undeformed stage of the fabric, the braiding angle (β/2) was unchanged since the fabric had not yet been stretched. With the increase of the load, the two interlaced yarns in zone C started to rotate, and β gradually decreased to α, causing the shear angle γ to be generated. This shear angle can be expressed as:

$$ \gamma = \beta - \alpha \qquad (B1) $$

In the pure shear zone, the value of α can be determined according to the fabric geometry.

$$ \cos\left(\frac{\alpha}{2}\right) = \frac{(D+u)}{2L_c} \qquad (B2) $$

Here, D and L_c are the diagonal length and theoretical side length of the rhombus of the pure shear zone (zone C) before deformation. The height of the triangle (L_β) in area A
remained unchanged after the deformation and can be expressed as $L_β = l/\tan(β/2)$. The diagonal length of the pure shear zone D can thus be computed as:

$$D = L_0 - L_β = L_0 - \frac{l}{\tan(\frac{β}{2})} \qquad (B3)$$

The relationship between the braiding angle (β) and $L_2$ is:

$$L_c = \frac{D}{2\cos(\frac{β}{2})} \qquad (B4)$$

According to Eq.B2 and Eq.B3:

$$L_c = \frac{1}{2\cos(\frac{β}{2})} \left( L_0 - \frac{l}{\tan(\frac{β}{2})} \right) \qquad (B5)$$

Therefore, the theoretical value of the shear angle is defined as:

$$γ = β - 2 \arccos \left[ \frac{u \sin(\frac{β}{2})}{L_0 \cdot \tan(\frac{β}{2}) - l} + \cos(\frac{β}{2}) \right] \qquad (B6)$$

In the initial configuration, the fabric was not stretched and had no shearing behavior. The areas of zones A, B and C are expressed as:

$$Z_A = \frac{l \cdot L_β}{2} = \frac{l^2}{2 \cdot \tan(\frac{β}{2})}; \quad Z_B = l \cdot L_β = \frac{l^2}{\tan(\frac{β}{2})}; \quad Z_C = l \cdot \left( L_0 - \frac{3}{2} \frac{l}{\tan(\frac{β}{2})} \right) \qquad (B7)$$

It is worth noting that the geometric relationship mentioned above is only correct when the aspect ratio of the fabric satisfies the following condition:

$$r = \frac{L_0}{l} \qquad (B8)$$

$$r \ge \frac{2}{\tan(\frac{β}{2})} \qquad (B9)$$