1. Introduction

In the composite industry, the shearing behaviour of dry woven plays a crucial role in fabric formability when doubly curved surfaces must be covered [1-9]. The ability of fabric to shear within a plain enables it to ﬁt three-dimensional surfaces without folds [10-12].

It has been proved that shear rigidity can be calculated from the tensile properties along a 45° bias direction. Bias Extension tests are simple to perform and provide reasonably repeatable results [13-14]. Extensive investigations have been carried out on the textile fabric in Bias Extension test [15]

The tests were conducted simply using two pairs of plates, clamping a rectangular piece of woven material such that the two groups of yarns are orientated ±45° to the direction of external tensile force. The ratio between the initial length and width of the specimen is deﬁned as aspect ratio:

λ = l₀/w₀ (see Figure 1a).

In the case of λ =2, the deformed conﬁguration of the material can be represented by Figure1b, which includes seven regions. Triangular regions C adjacent to the ﬁxture remain undeformed, while the central square region A and other four triangular regions B undergo shear deformation [16-17].

The present chapter focuses on numerical analysis of Bias Extension test using an orthotropic hyperelastic continuum model of woven fabric.

In the first, analytical responses of the Bias Extension test and the traction test on 45° are developed using the proposed model. Strain and stress states in specimen during these tests are detailed.

Figure 1.
Kinematic of Bias Extension test, a: Initial state, b: Deformed state

In the second, the proposed model is implanted into Abaqus/Explicit to simulate the Bias Extension test of three aspect ratios.

Exploiting numerical results, we studied the effect of the ratio between shearing and traction rigidities on homogeneities of stress and strain in the central zone of three Finite Element Models (FEM).

2. The proposed hyperelastic model

One of significant characteristics of the woven structure is the existence of two privileged material directions: warp and weft. We considered that the fabric is a continuous structure having two privileged material directions defined by the two unit tensors M₁ and M₂ as follows:

M1=M→1⊗M→1;

M2=M→2⊗M→2E1

WhereM→1andM→2are two unit vectors carried by two yarns directions. The sign indicate the tensor product. In the reference configuration, these privileged material directions are supposed to be orthogonal and they are defined by g₁ and g₂ presented by Equation 2.

g1=g→1⊗g→1,

g2=g→2⊗g→2E2

In Lagrangian formulation, the hyperelastic behavior is defined by the strain energy function W(E) depending of Green-Lagrange tensor components [18-21].

The second Piola Kirchhoff stress tensor S derives is presented in Equation 3:

S=∂W∂EE3

The physical behaviour is completely defined by the choice of W(E). The woven structures is very thin, we are interested more particularly in plane solicitations (plane stress or strain) in the plan (g→1,g→2). We supposed that W(E) is an isotropic function of variables (E, g₁, g₂). Using the representation theorems of isotropic functions, strain energy function W(E) depends of invariants:

gi:E,gi:E2,tr(E3)

(i=1..2)E4

We choose following invariants to present the strain energy function:

W(E)=W(I1,I2,I12)E5

Where

Ii=gi:E (i=1..2) ;

I12=12(g1E:Eg2)1/2E6

I_i measured elongations along directionsg→i. I₁₂ measured the sliding in the plane (g→1,g→2) witch is the angle variation between warp and weft direction. Components Egijof E in the reference system (g→1,g→2), are defined as follows:

Ii=Egij=12(δ12−1) ,i=1..2,I12=|Eg12|=12δ1δ2|cos(θ)|E7

δ1andδ2 are yarns extensions (ratio between deformed and initial lengths) along directions ofg→1and g→2. θ is the angle between M→1and M→2.

The second Piola Kirchhoff stress tensor S can be written as:

S=∂W∂I1g1+∂W∂I2g2+∂W∂I1212I12(g1Eg2+g2Eg1)E8

A simplified hyperplastic model is proposed. It is based on following assumptions:

The coupling between I₁₂ and I_i is neglect,
The strain energy function W(E) is expressed by Equation 9:

W=12k1I12+12k2I22+k12I1I2+k3I122E9

This leads to the constitutive equation:

S=(k1I1+k12I2)g1+(k2I2+k12I1)g2+k3(g1Eg2+g2Eg1)E10

So k₁ and k₂ presented tensile rigidities in yarns directions. k₁₂ described the interaction between two groups of yarns. k₃ presented the shearing rigidity of woven.

The relation between components S_gij of second Piola Kirchhoff stress tensor S and E_gij of Green Lagrange strain tensor E in the baseg→i can be presented by one of flowing expressions

[Sg11Sg22Sg12]=[k1k120k12k2000k3][Eg11Eg22Eg12]E11

[Eg11Eg22Eg12]=[c1c120c12c2000c3][Sg11Sg22Sg12]E12

Where:

c1=k2k1k2−k122, c2=k1k1k2−k122, c3=1k3, c12=−k12k1k2−k122E13

2.1. Out-axes tensile test: Tensile test on 45°

In tis parts the proposed hyperelastic model is used to study the mechanical behaviour during the out-axes tensile test of the dry woven.

Out-axes tensile test is a tensile test exerted on a fabric but according to a direction which is not necessarily warp or weft directions [22]. In the case of anisotropic behavior stress and strains tensors have not, in general, the same principal directions. During this test, the simple is subjected to a shearing. Particular precautions must be taken to ensure a relative homogeneity of the test [23].

We considered a tensile test along a directionE→1 forming an angle ψ₀ with orthotropic direction g→i (Figure.2).

Figure 2.
Kinematics of Out-axes tensile test, a: Reference configuration, b: Deformed configuration.

In the basee→i, components of the second Piola Kirchhoff tensor S and the Gradient of transformation tensor F are as follows [23]

S/e→i=[S000] , F/e→i=[f1f1γ0f2]E14

Where:

f1=LL0; f2=BB0;γ=f2f1tg(ξ)E15

LetP=FS0 where F is the tensile force and S_o is the initial cross section of the specimen. P is related to S by:

P=FS0=f1SE16

The components of the Green–Lagrange strain tensor E, in the basee→i, are as follows:

E/e→i=[E11E12E12E22]E17

Where

2E11=f12−1; 2E22=f22−1+f12γ2; 2E12=f12γE18

The response of the model presented by Equation 8 for this solicitation can be summarised as follows:

P=f1E11C(ψ0); E22=−ν(ψ0)E11; E12=g(ψ0)E11E19

Where:

C(ψ0)=c1cos4(ψ0)+c2sin4(ψ0)+12(c3+c12)sin2(2ψ0)ν(ψ0=[(2c3+2c12−c1−c2)sin2(2ψ0)−c12]4C(ψ0)g(ψ0)=sin(2ψ0)[c1cos2(ψ0)−c2sin2(ψ0)−(c3+c12)cos(2ψ0)]2C(ψ0)E20

The tensile test on 45° is a particular case of out-axes tensile tests where ψ₀=45). To replacing ψ₀ by 45° , Equation 20 became like the following:

C45=12k3+k1+k2+2k124(k1k2−k122),ν45=[(2c3+2c12−c1−c2)−c12]4C45,

g45=−c22C45E21

S₁ and S₂ are respectively the maximum and the minimum Eigen values of Piola Kirchhoff tensor S.In Tensile test on 45° , Equation 14 shows that:

S2S1=0E22

The expression of the applied force F is deducted from Equation 16:

F=PS0=2k3S0f1(f12−1)(k1k2−k122)2k1k2−2k122+k3(k1+k2−2k12)E23

For a balanced woven (k₁=k₂=k) where the interaction between yarns is neglected (k₁₂=0), the expression of F became:

F=k3kS0f1(f12−1)k+k3E24

The ratio between the minimum and the maximum Eigen values of Green Lagrange tensor E., in the tensile test with 45°, is given by Equation 25:

E2E1=−k−k3k3+kE25

2.2. Bias extension test

To explainer the pure shearing test of woven fabric, it has been noted that woven cloths in general deform as a pin-jointed-net (PJN) [24-28]. Yarns are considered to be inextensible and fixed at each cross-over point, rotating about these points like it is shown in Figure 3.

Figure 3.
Kinematics Pure shear a: Reference configuration, b: Deformed configuration

During the Bias Extension test, the pure shearing occurred in the central zone A and the shear angle is defined by Equation 26:

φ=π2−θ=π2−2acos(D+d2D)E26

The Gradient of Transformation tensor F is presented by Equation 27:

F/Ei,ei=[f100f2]=[cos(φ2)+sin(φ2)00cos(φ2)−sin(φ2)]E27

Using the proposed model, components S_ij,E_ij of the second Piola Kirchhoff stress and Green Lagrange strain tensors are given, in the base e→i, as follows:

S/e→i=S(φ)[100−1]where

S(φ)=12k3sin(φ)E28

E/ei=E(φ)[100−1] where

E(φ)=12sin(φ)E29

Thus

S2S1=−1E30

And

E2E1=−1E31

Where S₁ and S₂ are respectively the maximum and the minimum Eigen values of the second Piola Kirchhoff tensor S and E₁ and E₂ are respectively the maximum and the minimum Eigen values of Green Lagrange tensor E.

The internal power per unit of volume in zone A is defined by Equation 32:

ωa=SA:E ˙A=2S(φ)E˙(φ)=14k3sin(2φ)φ˙E32

To calculate to internal power per unit of volume in zone B we replace φ by φ2 in Equation 32:

ωb=SB:E ˙B=2S(φ2)E˙(φ2)=18k3sin(φ)φ˙E33

Figure 4.
Kinematic of Bias Extension Test, a: initial configuration, b: deformed configuration

The total internal power in the specimen is given by Equation 34:

Pint=Va.ωa+Vb.ωbE34

Where Va and Vb are respectively the initial volume in zones A and B defined as follows

Vb=e0w02Va=e0(Dw0−w022)=e0Dw0−12VbE35

The External power is defined as:

Pext=F.d˙=12FDf2φ˙ E36

The equality between internal and external powers allows to determinate the expression of applied force F given by Equation 37:

F= e0w0f1k3sin(φ)[1+14(λ−1)cos(φ)(1−2cos(φ))]E37

Where λ=L0w0 is the aspect ratio.

3. Numerical simulation of Bias Extension test

In this section, we simulated the Bias Extension test (BE) using the hyperelastic proposed model implanted into Abaqus/Explicit thought user material subroutine (VUMAT). Out put of the VUMAT are stress components of Cauchy tensor projected in the Green-Nagdi basis, component of the second Piola Kirchhoff tensor S, and the Green Lagrange tensor E projected in(g→1,g→2). We can also drew curves of Fore versus displacement.

The fabric is modelling by rectangular part meshed by continuum element (M3D4R).The boundary condition of model is presented in Figure 5a.

[29-30] compared the numerical results for the biased mesh and the aligned mesh and they proved that by using the biased mesh (Figure 5b), where the fibres are run diagonally across the rectangular element, neither the deformation profile nor the reaction forces are predicted correctly, for this we used the aligned mesh (Figure 5c).

Figure 5.
FEM mesh for the Bias Extension simulation, a: boundary condition of FEM, b: biased mesh, c: aligned mesh.

In order to simplify the problem, we used a balanced woven (k₁=k₂=k=700 N/mm²) and we ignored the interaction between extension in yarns direction (k₁₂=0). The analysis is done for three different FEM with the same thickness of 0.2mm. Dimensions of FEM are presented in table 1.

MEF	Length(mm)	Width(mm)	Aspect ratio: λ
1	100	50	2
2	150	50	3
3	200	50	4

Table 1.

Dimensions of samples

This analysis is realised on four values of the ratio between shearing and tensile rigidities (k3k=0.007,0.02,0.1,0.3,1) along three paths in FEM (see Figure 6).

The first path is longitudinal line in the middle of FEM. It joined zones A and C, the second path is along the yarn direction and the third path is transversal middle line Flowing results are illustrated for a displacement of 10% of initial length.

Figure 6.
Different paths used in analysis

The deformed mesh with the contour of the Green Lagrange shear strain is shown in Figure7. We noticed that appearance of three discernible deformation zones of the Bias Extension test in three FEM. No significant deformation occurred in zone C. The main mode of deformation in zone A is the shearing. The most deformation of the fabric occurs in this zone.

In to order to study homogeneities of stress and strain states, we compared the analytical and the numerical results of strain and stress along three paths of Figure (6).

Figure 7.
Deformed mesh with contour of Green Lagrange shear strain E₁₂ for k3k=0.007 and U₁=40mm.

3.1. Strain state

Figure 8 shows the variation of the maximum principal E₁ of Green Lagrange along the first path. We noticed that E₁ is symmetric with regard to the centre of the FEM. For the higher value of ratio of rigidities (k3k=1), E₁ is homogenous and it conformed to the predicted value in the case of isotropic elastic material. To decreasing the ratio of rigidities (k3k), the central zone characterised by the higher value of E_1. En addition, we observed the appearance of two zones where the strain is not more important. In the first hand, to comparing with the analytical value of E₁ in the central zone, the numerical values of E₁ is closely to that predicted in the Bias Extension test for the few shearing rigidity. Zones C coincided with ends of the path where the deformation was not more significant. In another hand, we remarked that in the central zone of the path, the deformation is not homogenous especially in FEM1and FEM2. For more analyse the strains state in FEM, Figure 9 presented the evolution ofE2E1, along the first path. It is clear that to decreasingk3k, the value of E2E1tends to (-1) in three FEM. This proved that, in spite of the low displacement, the deformation in Bias Extension test is influenced by the ratio between shearing and tensile rigidities of the woven.

3.2. Stress state

Comparing the numerical and the analytical values of S2S1, we determinate the stress state in different FEM for an displacement of 10% along the first path.

Figure10 show that to decreasingk3k, the value of S2S1decrease but never achieved (-1).

Indeed, if this simulation is interpreted like a Bias Extension test, S2S1should be verifying Equation 30 in the central zone. However the ratio of principal strain is approximately equal to 0. So it is conforming to Equation 22, and the stress state is the traction state.

In addition to varying the value ofk3k, we evaluated the ratio of strain versus the ratio of stress in the central element of FEM. In Figure12, it can be noticed that in FEM1, to reducing the value ofk3k, E2E1tend to (-1) and it conformed to the predicted value by Equation 31 for a few values ofk3k. But S2S1 have a negative value and it remain different to (-1). In FEM2, it was visibly that S2S1stayed proximity null for different value of k3k thus it verified Equation 22 but E2E1tend to (-1) for few values ofk3k. In FEM3, it was clear that for few value ofk3k,E2E1tend to (-1), but the S2S1had positive values. Consequently, the shearing deformation in Bias Extension test depends of the ratio of rigidities between shearing and tensile, but the stress state is always the tensile stress.

Figure 8.
Variation of Maximum principal of Green Lagrange strain E₁ along the path1

Figure 9.
Variation of E2E1along the path 1

Figure 10.
Variation of S2S1along path 1.

Figure 11.
Variation of S2S1versusE2E1 along path 1.

3.3. Angle between yarns

In this section, we compared between the numerical and the predicted values of the angle between yarns, along the first path.

Using the proposed model, the numerical angle between yarns is given by the following expression:

θN=arcos((2Eg12)2(2Eg22+1)(2Eg11+1))E38

In the case of the Bias Extension test, the predict angle between yarns in the central zone A is given by Equation 39:

θB=2arcos(D+d2D)E39

The predict angle between yarns in the Tensile test in 45° is given by Equation 41:

θT=arcos(E11(1−ν45)(E11+E22+1)2)E40

Figure 12.
Comparison between Numerical and Predicted angles between yarns along the path 1 in FEM1.

Figure 12 demonstrated that the value of the angle between yarns was not uniform in the central zone of the FEM and it was not null in ends of the path1. For three FEM, the numerical angle between yarns tend to verify the predict angle (solid line) in the Bias Extension test for the lower value ofk3kThis is another reason to justify the influence of the ration of rigidities on the shearing deformation of woven.

3.4. Elongation of yarns

Under the pin-joint assumption for trellising deformation mode, the edge length of the membrane element should remain unchanged during the deformation; thus the Green Lagrange stretch Eg₁₁ and Eg₂₂ should be null in Bias Extension test:

Eg11=Eg22=0E41

In Tensile test on 45°, warp and weft yarns are submitted respectively to Green Lagrange deformations Eg11and Eg22as follows:

Eg11=12(|F.g→1|2−1)=k1−k122(k1k2−k122)E11C45E42

Eg22=12(|F.g→2|2−1)=k2−k122(k1k2−k122)E11C45E43

In the case of balanced fabric without coupling between elongations in yarns directions, the warp and weft yarns are submitted to the same elongation:

So

Eg11=Eg22=EE44

Where

E=k3k+k3E11E45

In Figure 13, we compared numerical stretch deformation along the second path and the predicted elongation in yarn direction.

In the first hand, we noticed that the numerical elongation was not null. It became more important by increasing the value ofk3k in all FEM. In another hand, numerical value of elongation is closely conforming to the expected value in the tensile test in 45 for different values ofk3k in all FEM. This analysis provided that during Bias Extension test, yarns are subjected a few elongation. These stretches depend of the value of the ratio between shearing and tensile rigidities of woven. Same previous analyses are taken also along the third vertical path (Path3) and same results are verified.

Figure13 represented the evolution of S2S1versus E2E1 along the third path. Like the first path, for few values ofk3k, the shearing is the utmost deformation. But in all cases, the Bias Extension test is characterized by the tensile state.

Figure 13.
Variation of E_g11 along the path 2in FEM1.

4. Conclusion

In this work, an orthotropic hyperelastic model test of woven fabric is developed and implanted into Abaqus/explicit to simulate Bias-Extension at low displacement. The analysis of numerical answers along longitudinal and transversal middle paths, proved, in the first hand, that to decreasing the ratio between shearing and tensile rigidities, the state deformation became to be conform to that predicted by the proposed model in the Bias Extension test for all FEM. In another hand, the angle between yarns tends to verify the predicted angle during the Bias Extension test. Although the stress state, is conform to the expected analysis of Traction test on 45°. The analysis of Green Lagrange stretching strain in the yarns direction, demonstrated that there was an elongation of yarns during test for different shearing rigidity. This elongation was exactly conforming to the predicted analytical elongation in the Traction test in 45°. Curves of Force versus displacement of the Traction test in 45° applied to of the central zone A is closely to the numerical answers. We are able to adjust both curves by coefficients of adjustment.

This study allowed to verify analytical hypothesis adopted to interpret the Bias Extension test. The comparison between in Bias Extension test, the shearing deformation depends of the ratio between shearing and tensile rigidities of fabric. In Spite of the low displacement, this test presented always a stress state.

Alejandro Acevedo-Malave

Chapters authored

Related collaborators

Konstantin Afanasiev

Donghong Gao

Lars Linsen

Roman Makarchuk

Vladimir Molchanov

Petar Dobrev

Stephan Rosswog

Paul Rosenthal

Tran Van Long

Jin Sun