Analysis of the stress components in a textile composite reinforcement

Abstract

When a continuum approach is considered for textile reinforcements, the internal loads are described by a stress tensor. The mechanical behaviour of the textile material is very much dependent on the fiber directions, and the frames defined from warp and weft directions are preferred to write the stress components. The exterior loads in these frames permit to define tensile and pure shear states. Nevertheless, these frames are generally not orthogonal. The relationships between the exterior loads and the different stress components are analyzed in the present paper, and, in particular, the relationship between direct stress components and longitudinal loads on one hand, and shear stress components and transversal loads on the other hand. When dealing with textile materials, the exterior loads in the direction of the fibres and transverse to the fibres define the pure tensile and pure shear state. It is shown that the covariant stress component matrix is diagonal in a pure tensile loading and that the first mixed direct stress components are equal to zero in a pure shear loading. In these cases, the direct relationship between the stresses and the loadings are given. This is applied to the cases of the picture frame test, the biaxial tensile test or of a combined tension-shear test.

Keywords

Composite textile reinforcements mechanical behaviour stress components picture frame biaxial tension

Introduction

Liquid composites moulding (LCM) technologies can be combined with dry fibre preform manufacturing technologies.^1–3 Textiles need to be draped on a shape that is frequently double curved and requires planar shear deformations of the reinforcement. The draping of this textile preform is often delicate and some software packages have been developed to simulate this forming process and avoid trial-and-error developments.^4–13 These simulations and in particular those that are based on the finite element method require the knowledge and a model of the mechanical behaviour of textile reinforcements during forming.

Textile materials are made of fibres often gathered in yarns that are woven, braided and stitched. The resulting mechanical behaviours are therefore specific. They are strongly anisotropic and above all some relative sliding is possible between fibres and yarns. Some stiffnesses are substantially reduced, in particular, in-plane shear and bending rigidities. These properties are used during the draping process, much to its advantage. In addition, these specificities may call the continuum nature of the textile material into question. Textiles can be seen as a set of beams in contact/friction (one per fibre).^14–16 However, the very large number of fibres in a preform restricts the use of this approach. Other approaches have been proposed at the yarn scale (mesoscopic approach).^17–22 Nevertheless, the currently most used approach consists in assuming that the sliding between the fibres is small enough so the textile material can be considered as a continuum.^{7,9,12,23–28} In the present paper, the textile composite reinforcement is considered as a continuum. In this case, the internal loads within the material are then represented by a stress tensor. Taking the very strong anisotropy of a textile reinforcement into account, the basis defined by the fibre directions (warp and weft in case of a woven fabric for instance) are preferred to express the stress tensor components and the mechanical behaviour of the textile material. These fibre directions do not remain perpendicular during the reinforcement deformation. Consequently, there are different variances for the stress components. The relationships between these components and the exterior applied loads are not straightforward. The objective of the present paper is to relate the different components of the stress tensor to the loading on the textile reinforcement. In particular it is shown in which cases there is a direct relationship between a tensile load and a direct stress component and between a transverse load and a shear stress component.

In the case of textile materials, the exterior loads define the pure tensile loading cases (no transverse loads) and the pure shear loadings (no tension in the yarns). It is shown that in the case of pure tension, the contravariant stress component matrix is such as the shear stress components are equal to zero. This is not the case for other basis. The expression of the direct stress in function of the tensile load is given. In the case of pure in-plane shear, the first mixed stress component matrix is such as the direct stress components are equal to zero. This is not the case for other basis. The expression of the shear stress in function of the transverse load is given.

The direct relationships between exterior loads and stress components are important to analyse the tensile and in-plane shear tests on textile composites reinforcements. Three specific tests used for textile materials are analysed: picture frame, biaxial tensile test and combined test. The different stress components (i.e. the different variances) are related to the load prescribed on the device.

Relationship between loads and stress components for a 2D continuous material

Definitions of the elementary load components on the elementary surfaces

A 2D continuous medium is considered. The position of a point M of the domain is defined by two curvilinear material coordinates $ξ^{1}$ and $ξ^{2}$ (Figure 1). These coordinates define the material covariant vectors at point M:

{\underline{g}}_{1} = \frac{\partial \underline{OM}}{\partial ξ^{1}} and {\underline{g}}_{2} = \frac{\partial \underline{OM}}{\partial ξ^{2}}

(1)

In the present paper, vectors are denoted with simple underlining and second order tensors with double underlining.

Figure 1.

Elementary load components on the elementary surfaces the normals of which are ${\underline{n}}_{1}$ , ${\underline{n}}_{2}$ , ${\underline{n}}^{1}$ and ${\underline{n}}^{2}$ .

The associated contravariant vectors ${\underline{g}}^{α}$ are such as:

{\underline{g}}_{α} \cdot {\underline{g}}^{β} = δ_{α}^{β}

(2)

In the sequel, α, β, λ and µ are indices taking the values 1 or 2. $δ_{α}^{β}$ is the Kronecker symbol.

The unit normal vectors ${\underline{n}}_{1}$ , ${\underline{n}}_{2}$ , ${\underline{n}}^{1}$ and ${\underline{n}}^{2}$ are defined from these vectors:

{\underline{n}}_{α} = \frac{{\underline{g}}_{α}}{‖ {\underline{g}}_{α} ‖} and {\underline{n}}^{α} = \frac{{\underline{g}}^{α}}{‖ {\underline{g}}^{α} ‖}

(3)

The upper and lower index has been kept to point out the association between ${\underline{n}}_{α}$ and ${\underline{g}}_{α}$ but ${\underline{n}}^{α}$ and ${\underline{n}}_{α}$ are not conjugated that is to say: ${\underline{n}}_{α} . {\underline{n}}^{β} \neq δ_{α}^{β}$ .

The elementary surfaces corresponding to ${\underline{n}}_{1}$ , ${\underline{n}}_{2}$ , ${\underline{n}}^{1}$ and ${\underline{n}}^{2}$ are, respectively, denoted ${d S}_{n_{1}}$ , ${d S}_{n_{2}}$ , ${d S}_{n^{1}}$ and ${d S}_{n^{2}}$ (Figure 1). The elementary loading vectors applied to these elementary surfaces are related to the Cauchy stress tensor $\underline{\underline{σ}}$ :

{\underline{dF}}_{n_{α}} = (\underline{\underline{σ}} \cdot {\underline{n}}_{α}) {d S}_{n_{α}} and {\underline{dF}}_{n^{α}} = (\underline{\underline{σ}} \cdot {\underline{n}}^{α}) {d S}_{n^{α}}

(4)

The elementary force vector ${\underline{dF}}_{n_{α}}$ has two components, the first one, denoted ${d T}_{n_{α}}$ , on the normal ${\underline{n}}_{α}$ in the material direction α, the second one, denoted ${dR}_{n_{α}}$ , in the perpendicular direction ${\underline{n}}^{3 - α}$ (Figure 1):

\begin{matrix} {\underline{dF}}_{n_{α}} = {d T}_{n_{α}} {\underline{n}}_{α} + {dR}_{n_{α}} {\underline{n}}^{_{3 - α}} \\ = {d T}_{n_{α}} \frac{{\underline{g}}_{α}}{‖ {\underline{g}}_{α} ‖} + {dR}_{n_{α}} \frac{{\underline{g}}^{_{3 - α}}}{‖ {\underline{g}}^{_{3 - α}} ‖} \end{matrix}

(5)

Remark

3-α is equal to 2 if α = 1 and equal to 1 if α = 2. Consequently, ${\underline{n}}_{α}$ is perpendicular to ${\underline{n}}^{3 - α}$ .

Similar components of the elementary load ${\underline{dF}}_{n^{α}}$ on the surface ${d S}_{n^{α}}$ are considered along the normal ${\underline{n}}_{α}$ and its perpendicular ${\underline{n}}^{3 - α}$ :

\begin{matrix} {\underline{dF}}_{n^{α}} = {d T}_{n^{α}} {\underline{n}}_{α} + {dR}_{n^{α}} {\underline{n}}^{_{3 - α}} \\ = {d T}_{n^{α}} \frac{{\underline{g}}_{α}}{‖ {\underline{g}}_{α} ‖} + {dR}_{n^{α}} \frac{{\underline{g}}^{_{3 - α}}}{‖ {\underline{g}}^{_{3 - α}} ‖} \end{matrix}

(6)

{d T}_{n^{α}}

and

{d T}_{n_{α}}

are named ‘longitudinal elementary loads’;

{dR}_{n^{α}}

and

{dR}_{n_{α}}

are ‘transverse elementary loads’.

Relationship between the Cauchy stress tensor components and the elementary load components

The covariant, contravariant and mixed components of the Cauchy stress tensor in the frames defined by the above vectors are considered:

\begin{matrix} \underline{\underline{σ}} = σ_{α β} {\underline{g}}^{α} \otimes {\underline{g}}^{β} = σ^{α β} {\underline{g}}_{α} \otimes {\underline{g}}_{β} \\ = σ_{α}^{β} {\underline{g}}^{α} \otimes {\underline{g}}_{β} = σ_{β}^{α} {\underline{g}}_{α} \otimes {\underline{g}}^{β} \end{matrix}

(7)

In this paper, whatever the variance $σ_{α α}$ will be given the name ‘direct stress’ and $σ_{α β}$ (α ≠ β) will be named ‘shear stress’. This is a standard vocabulary for instance used in Abaqus manuals.²⁹

The elementary load components ${d T}_{n_{α}}$ , ${dR}_{n_{α}}$ , ${d T}_{n^{α}}$ and ${dR}_{n^{α}}$ can be related to the stress components. For instance for the component ${d T}_{n_{α}}$ :

{d T}_{n_{α}} = (\underline{\underline{σ}} \cdot {\underline{n}}_{α}) \cdot {\underline{n}}_{α} {d S}_{n_{α}} = \frac{σ_{α α}}{‖ {\underline{g}}_{α} ‖^{2}} {d S}_{n_{α}}

(8)

In the same way, denoting $g_{α β} = {\underline{g}}_{α} \cdot {\underline{g}}_{β}$ and $g^{α β} = {\underline{g}}^{α} \cdot {\underline{g}}^{β}$ ,

{dR}_{n_{α}} = σ_{λ α} \frac{g^{λ (3 - α)}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖} {d S}_{n_{α}}

(9)

and

\begin{matrix} {d T}_{n^{α}} = σ_{α λ} \frac{g^{α λ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖} {d S}_{n^{α}} and \\ {dR}_{n^{α}} = σ_{λ μ} \frac{g^{α μ} g^{λ (3 - α)}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖} {d S}_{n^{α}} \end{matrix}

(10)

This leads to the following expressions of the stress components:

σ_{α β} = g_{α β} \frac{{d T}_{n_{β}}}{{d S}_{n_{β}}} + \frac{{dR}_{n_{β}}}{{d S}_{n_{β}}} \frac{‖ {\underline{g}}_{β} ‖}{‖ {\underline{g}}^{3 - β} ‖} δ_{α}^{3 - β}

(11)

The same approach can be used for the other variances of the Cauchy stress tensor. They are given in Table 1.

Table 1.

Components of the Cauchy stress tensor as functions of elementary load components

Elementary loads $\underline{\underline{σ}}$ Components	$\frac{{d T}_{n_{α}}}{{d S}_{n_{α}}}$	$\frac{{dR}_{n_{α}}}{{d S}_{n_{α}}}$	$\frac{{d T}_{n^{α}}}{{d S}_{n^{α}}}$	$\frac{{dR}_{n^{α}}}{{d S}_{n^{α}}}$
$σ_{λ μ} {\underline{g}}^{λ} \otimes {\underline{g}}^{μ}$	$\frac{σ_{α α}}{‖ {\underline{g}}_{α} ‖^{2}}$	$σ_{λ α} \frac{g^{λ (3 - α)}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$	$σ_{α λ} \frac{g^{α λ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖}$	$σ_{λ μ} = \frac{g^{α μ} g^{λ (3 - α)}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$
$σ^{λ μ} {\underline{g}}_{λ} \otimes {\underline{g}}_{μ}$	$σ^{λ μ} \frac{g_{α μ} g_{α λ}}{‖ {\underline{g}}_{α} ‖^{2}}$	$σ^{(3 - α) μ} \frac{g_{α μ}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$	$σ^{λ α} \frac{g_{α λ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖}$	$\frac{σ^{(3 - α) α}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$
$σ_{λ}^{μ} {\underline{g}}^{λ} \otimes {\underline{g}}_{μ}$	$σ_{α}^{μ} \frac{g_{α μ}}{‖ {\underline{g}}_{α} ‖^{2}}$	$σ_{λ}^{μ} \frac{g_{α μ} g^{λ (3 - α)}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$	$\frac{σ_{α}^{α}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖}$	$σ_{λ}^{α} \frac{g^{λ (3 - α)}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$
$σ_{μ}^{λ} {\underline{g}}_{λ} \otimes {\underline{g}}^{μ}$	$σ_{α}^{λ} \frac{g^{α λ}}{‖ {\underline{g}}_{α} ‖^{2}}$	$\frac{σ_{α}^{3 - α}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$	$σ_{μ}^{λ} \frac{g^{α μ} g^{α λ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖}$	$σ_{μ}^{3 - α} \frac{g^{α μ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$

The inverse relationships, that is the elementary load components as functions of the stress components are given in Appendix 1.

Remark

When substituting equation (8) in equation (9), another form of equation (11) is obtained for $σ_{21}$ . Even if these two expressions seem not to be equal, it is shown in Appendix 3 that they are so.

Direct relationships

Among the Cauchy stress tensor components given in Table 1, some give direct relationships between the longitudinal elementary loads and the direct stress components. That means that these loads only depend on these components and reciprocally. Others give direct relationships between shear stress components and transverse elementary load components. Nevertheless, no component set gives a direct relationship between both direct and shear stress components and longitudinal and transverse elementary load components.

The following section illustrates the above.

Direct stress components only depending on longitudinal loads

Among the Cauchy stress tensor components given in Table 1, two give a direct relationship between the longitudinal elementary load and the direct stress component. Considering the normal ${\underline{n}}_{α}$ and the covariant stress components $\underline{\underline{σ}} = σ_{α β} {\underline{g}}^{α} \otimes {\underline{g}}^{β}$ . Equation (8) leads to:

σ_{α α} = \frac{{d T}_{n_{α}}}{{d S}_{n_{α}}} ‖ {\underline{g}}_{α} ‖^{2}

(12)

The direct stress component in a given direction is directly linked to the longitudinal force in the same direction. The corresponding shear stress component is the following:

σ_{α β} = g_{α β} \frac{{d T}_{n_{β}}}{{d S}_{n_{β}}} + \frac{{dR}_{n_{β}}}{{d S}_{n_{β}}} \frac{‖ {\underline{g}}_{β} ‖}{‖ {\underline{g}}^{3 - β} ‖}

(13)

It is linked to both longitudinal and transverse force components. Consequently, with such a choice of material normal direction and variance, direct stress components are directly linked to longitudinal force components but shear stresses are linked to both longitudinal and transverse forces. Furthermore, considering the normal ${\underline{n}}^{α}$ and the mixed stress components $\underline{\underline{σ}} = σ_{α}^{β} {\underline{g}}^{α} \otimes {\underline{g}}_{β}$ gives:

σ_{α}^{α} = \frac{{d T}_{n^{α}}}{{d S}_{n^{α}}} ‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖

(14)

Here again, the direct stress component only depends on the force component in the same direction, but the expression of the shear stress component is more complicated (see Table 1).

With another variance, direct stress components are related to both longitudinal and transverse loads (and the other longitudinal loads are related to both normal and shear stresses).

Equations (12) and (14) are direct relationships between direct stress components and longitudinal forces. But in both cases, the corresponding shear stress components do not only depend on transverse forces.

If the elementary longitudinal load is equal to zero, the direct stress components are also null and a direct relationship is obtained between the transverse loads and shear stresses:

\begin{matrix} if {d T}_{n^{3 - α}} = 0, σ_{α}^{3 - α} = \frac{‖ {\underline{g}}^{3 - α} ‖}{‖ {\underline{g}}^{α} ‖} \frac{{dR}_{n^{3 - α}}}{{d S}_{n^{3 - α}}} and \\ if {d T}_{n_{3 - α}} = 0, σ_{α (3 - α)} = \frac{‖ {\underline{g}}_{3 - α} ‖}{‖ {\underline{g}}^{α} ‖} \frac{{dR}_{n_{3 - α}}}{{d S}_{n_{3 - α}}} \end{matrix}

(15)

Shear stress components only depending on transverse loads

A similar discussion can be held looking for a direct relationship between shear stresses and transverse forces. Among the Cauchy stress tensor components given in Table 1, two give a direct relationship between the transverse elementary load and the shear stress component.

Considering the normal ${\underline{n}}^{α}$ and the co-variant stress components $\underline{\underline{σ}} = σ^{α β} {\underline{g}}_{α} \otimes {\underline{g}}_{β}$ leads to:

σ^{(3 - α) α} = \frac{{dR}_{n^{α}}}{{d S}_{n^{α}}} ‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖

(16)

In addition, considering the normal ${\underline{n}}_{α}$ and the mixed stress components $\underline{\underline{σ}} = σ_{β}^{α} {\underline{g}}_{α} \otimes {\underline{g}}^{β}$ gives:

σ_{α}^{3 - α} = \frac{{dR}_{n_{α}}}{{d S}_{n_{α}}} ‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖

(17)

The other shear stress components are related to both transverse and longitudinal loads (and the other transverse loads are related to both shear and direct stresses).

If the shear stress component considered in equations (16) and (17) are only depending on the transverse elementary load, the corresponding direct stress components are depending on both longitudinal and transverse elementary loads. If the elementary transverse force is equal to zero, shear stress components will be null and a direct relationship is obtained between longitudinal loads and direct stress components:

\begin{matrix} if {dR}_{n^{α}} = 0, σ^{α α} = \frac{‖ {\underline{g}}^{α} ‖}{‖ {\underline{g}}_{α} ‖} \frac{{d T}_{n^{α}}}{{d S}_{n^{α}}} and \\ if {dR}_{n_{α}} = 0, σ_{α}^{α} = \frac{{d T}_{n_{α}}}{{d S}_{n_{α}}} \end{matrix}

(18)

Direct dependency of stress components on loads

It is interesting to have direct stress components that only depend on elementary longitudinal loads dT, and shear stress components only depending on transverse loads dR.

Two different cases can be distinguished

In the case of combined loadings: only two choices of variances enable to obtain a direct relationship either between the longitudinal load components and direct stresses (equations (12) and (14)) or between the transverse load components and transverse stresses (equations (17) and (18)). Unfortunately, these variances do not correspond to the same frame and consequently, there is no tensorial basis in which the normal stress components only depend on longitudinal loads and shear stress only depends on transverse loads.

In the case of a longitudinal loading ( $\underline{dR} = \underline{0}$ ): two choices of components ( $\underline{\underline{σ}} = σ_{α β} {\underline{g}}^{α} \otimes {\underline{g}}^{β}$ and $\underline{\underline{σ}} = σ_{α}^{β} {\underline{g}}^{α} \otimes {\underline{g}}_{β}$ ) enable to obtain both null transverse stresses and a direct relationship between the longitudinal load components and the direct stresses (equations (17) and (18)). In the case of a pure shear loading ( $\underline{d T} = \underline{0}$ ): two choices of components ( $\underline{\underline{σ}} = σ^{α β} {\underline{g}}_{α} \otimes {\underline{g}}_{β}$ and $\underline{\underline{σ}} = σ_{β}^{α} {\underline{g}}_{α} \otimes {\underline{g}}^{β}$ ) enable to obtain both null direct stresses and a direct relationship between the transverse load components and the transverse stresses (equations (12) and (14)). Thus, the frames and the corresponding stress components suited to analyze a pure longitudinal and a pure transverse loading are not the same. Furthermore, the frames suited to analyze a pure longitudinal or pure transverse loading are different from those suited to analyze a combined loading. In any case, there is no tensorial basis in which the direct stress components only depend on longitudinal loads and shear stresses only depend on transverse loads.

Other stress tensors

The developments are presented here for the Cauchy stress tensor, but similar calculations can be performed for the first and second Piola-Kirchhoff stress tensors. The same type of expressions and conclusions are obtained and briefly presented in Appendix 2.

Textile composite reinforcement

Introduction

The specific mechanical behaviour of fibrous materials is mainly due to their micro-structure. They are made of fibres that can be loaded in tension in their direction. Transverse loads, contact and friction between the fibres are also possible, but contrarily to standard continuum materials, relative sliding of the fibres is possible. This leads to a strong anisotropy and to weak values for some stiffnesses, especially for in-plane shear and bending rigidities.^30,31 The tensile stiffness is preponderant over the others. The frames defined by the fibre directions are preferred to express the stress tensor components and the mechanical behaviour of textile materials. These directions are also important to define the pure tensile state for which there are only tensions in the fibres and the pure shear state for which there is no tension in the fibres.

A consistent identification of the homogenized behaviour law requires the knowledge of the relationship between loads and stresses. It is important to precisely define which stress components are related to the prescribed exterior load. This is done below in the case of the picture frame test, biaxial tensile test and combined test.

The textile material considered here has two yarn directions. For instance, it can be a woven fabric or a biaxial Non Crimp Fabric (NCF). The bending stiffness of the material is small enough to make a membrane assumption.

Referring to the notations used in the section Relationship between loads and stress components for a 2D continuous material and Figure 2, the two fibre directions are orientated along ${\underline{n}}_{α} = \frac{{\underline{g}}_{α}}{‖ {\underline{g}}_{α} ‖}$ .

Figure 2.

Basis, normals and elementary loads in the case of a 2D textile reinforcement.

Tensile and in-plane shear loading on a fabric

Let us consider a textile material with two fibre directions that is only submitted to tensions in the fibres. The elementary section ${d S}_{n^{α}}$ of normal ${\underline{n}}^{α}$ is parallel to the fibres (3-α) (Figure 3(a)). Consequently, these fibres, submitted only to tension, do not exert any load on this section. The load on the section ${d S}_{n^{α}}$ is only due to the tension in the fibres α, ${\underline{dF}}_{n^{α}} = {d T}_{n^{α}} {\underline{n}}_{α}$ (Figure 3(a)). A pure tensile loading is characterized by

{dR}_{n^{α}} = 0 (α = 1, 2)

(19)

Figure 3.

(a) Pure tensile loading, (b) Pure in-plane shear loading.

The loads ${d T}_{n^{α}}$ are the tension in the fibres α.

(Remark: ${dR}_{n^{α}} = 0$ imposes ${dR}_{n^{3 - α}} = 0$ taking equation (17) into account)

In the case of a pure in-plane shear loading, the tensions in the fibres are equal to zero (Figure 3(b)):

{d T}_{n^{α}} = 0 (α = 1, 2)

(20)

In any case, because section ${d S}_{n^{α}}$ of normal ${\underline{n}}^{α}$ is parallel to fibres (3-α), the tensions of these fibres do not contribute to ${\underline{dF}}_{n^{α}}$ . ${d T}_{n^{α}}$ is the tensile load in fibres α. ${dR}_{n^{α}}$ is the in-plane shear load.

This is specific to the textile materials for which the tensions in the warp and weft fibres are well defined as the interior load in the fibre direction. In a pure tensile state, there are only loads in the fibre directions (equation (19)). In a pure shear state, the tensions are equal to zero (equation (20)).

Cauchy stress components

The relationship between elementary load components and covariant, contravariant and mixed components of the Cauchy stress tensor can be established from the section Relationship between loads and stress components for a 2D continuous material and can be particularized for pure uni- or biaxial tensile and in-plane shear loadings.

The aim of the present section is to specify in which basis the stress components should be written to perform the identification of the fabric behaviour from standard mechanical tests. Tensile and in-plane shear loadings are considered below.

Pure tensile loading

A pure tensile load is such as ${dR}_{n^{α}} = 0$ (equation (19)). In order to identify a direct relationship between a direct stress component and longitudinal strain in the fibre direction, it is necessary to use a basis for the stress components in which a pure tensile load corresponds to zero shear stress components. It has been shown in the Relationship between loads and stress components for a 2D continuous material section that the only component of the stress tensor that is null when the transverse load is null is the contravariant component $σ^{(3 - α) α}$ (equation (16)). The corresponding direct components of the Cauchy stress tensor is then:

σ^{α α} = \frac{{d T}_{n^{α}}}{{d S}_{n^{α}}} \frac{‖ {\underline{g}}^{α} ‖}{‖ {\underline{g}}_{α} ‖} and σ^{(3 - α) α} = 0

(21)

In the case of a pure tensile test, only the use of the contravariant components of the Cauchy stress tensor enables to have the shear stresses null and the normal stresses directly related to the longitudinal loads.

Pure in-plane shear

A pure in-plane shear loading is such as ${d T}_{n^{α}} = 0$ (equation (20)). In order to have direct components equal to zero, the mixed components $\underline{\underline{σ}} = σ_{α}^{β} {\underline{g}}^{α} \otimes {\underline{g}}_{β}$ are considered (equation (14)). The following expressions are obtained:

σ_{3 - α}^{α} = \frac{{dR}_{n^{α}}}{{d S}_{n^{α}}} \frac{‖ {\underline{g}}^{α} ‖}{‖ {\underline{g}}^{3 - α} ‖} and σ_{α}^{α} = 0

(22)

In the case of a pure in-plane shear test, only the use of the first mixed components of the Cauchy stress tensor enables to have the direct stresses null and the shear stresses directly related to the transverse loads.

Determination of the stress tensor components from standard experimental tests

Pure shear test: picture frame

Presentation and assumptions

In-plane shear is the main deformation mode for textile materials because it is necessary to drape the fabric on a double curved surface. The in-plane shear mechanical behaviour has been intensively studied, mainly using picture frame tests and bias extension tests.^32–44 A picture frame test is considered in Figure 4. A hinged frame with four bars of equal length is assembled in a tensile testing machine. A tensile force is applied across diagonally opposing corners of the picture frame rig, causing the picture frame to move from an initially square configuration into a rhomboid. This principle enables to obtain a uniform pure shear loading if experiments are carefully performed in the frames defined in Figure 4.

Figure 4.

Instrumented picture frame test. (a) Shear anglre = 0^° (b) Shear anglre = 30^°.

The following relationships can then be written:

{d T}_{n^{α}} = 0 and {dR}_{n^{α}} = cte at any point of the specimen

(23)

Identification of the stress state

F is the load exerted by the tensile device on the picture frame, and R is the resultant of the transverse loads exerted by the fabric on a frame arm (Figure 5).

Figure 5.

Frame and specimen parameters.

Referring to the Textile composite reinforcement section, the only stress variance that leads to a zero direct stress component is the mixed one $σ_{α}^{β}$ for which $σ_{α}^{α} = 0$ and:

σ_{3 - α}^{α} = \frac{{dR}_{n^{α}}}{{d S}_{n^{α}}} \frac{‖ {\underline{g}}^{α} ‖}{‖ {\underline{g}}^{3 - α} ‖} and σ_{α}^{3 - α} = \frac{{dR}_{n^{3 - α}}}{{d S}_{n^{3 - α}}} \frac{‖ {\underline{g}}^{3 - α} ‖}{‖ {\underline{g}}^{α} ‖}

(24)

The fixed frame $({\underline{G}}_{1}, {\underline{G}}_{2})$ is defined from the initial fibre directions and from the initial fabric dimensions: ${\underline{G}}_{1} = L \underline{X}$ and ${\underline{G}}_{2} = L \underline{Y}$ (Figure 5).

Let us consider the curvilinear coordinates $ξ_{1}, ξ_{2} \in [0, 1] \times [0, 1]$ , in the current configuration, with a shear angle equal to γ. ${\underline{g}}_{1}$ and ${\underline{g}}_{2}$ are, respectively, the image of ${\underline{G}}_{1}$ and ${\underline{G}}_{2}$ (Figure 5). $(x, y)$ are the current coordinates of any point M of the fabric in the (X,Y) frame, consequently:

\begin{matrix} x = ξ_{1} LCos \frac{γ}{2} + ξ_{2} LSin \frac{γ}{2} and \\ y = ξ_{2} LCos \frac{γ}{2} + ξ_{1} LSin \frac{γ}{2} \end{matrix}

(25)

\begin{matrix} {\underline{g}}_{1} = \cos \frac{γ}{2} {\underline{G}}_{1} + \sin \frac{γ}{2} {\underline{G}}_{2} and \\ {\underline{g}}_{2} = \cos \frac{γ}{2} {\underline{G}}_{2} + \sin \frac{γ}{2} {\underline{G}}_{1} \end{matrix}

(26)

The corresponding contravariant vectors are such as:

\begin{matrix} {\underline{g}}^{1} = \frac{1}{L^{2} \cos γ} (\cos \frac{γ}{2} {\underline{G}}_{1} - \sin \frac{γ}{2} {\underline{G}}_{2}) and \\ {\underline{g}}^{2} = \frac{4}{L^{2} \cos γ} (\cos \frac{γ}{2} {\underline{G}}_{2} - \sin \frac{γ}{2} {\underline{G}}_{1}) \end{matrix}

(27)

The summation of ${dR}_{n^{α}}$ leads to:

σ_{3 - α}^{α} = R_{n^{α}} e w_{α} for α = 1 and 2

(28)

and so:

[\underline{\underline{σ}}]_{{g^{1}, g_{2}}} = (\begin{matrix} 0 & \frac{R_{n^{2}}}{e w_{2}} \\ \frac{R_{n^{1}}}{e w_{1}} & 0 \end{matrix})

(29)

A normalized non orthogonal frame is defined in order to ease the physical interpretation of the stresses:

\begin{matrix} \underline{\underline{σ}} = σ_{α}^{β} ({\underline{g}}^{α} \otimes {\underline{g}}_{β}) = (σ_{α}^{β})_{N} ({\underline{n}}^{α} \otimes {\underline{n}}_{β}) with \\ (σ_{α}^{β})_{N} = σ_{α}^{β} ‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{β} ‖ \end{matrix}

(30)

It leads to:

(σ_{3 - α}^{α})_{N} = \frac{R_{n^{α}}}{e w_{α} \cos γ}

(31)

The static equilibrium of the frame enables to write the loads exerted on the frame arms as a function of the shear moment. Introducing the force exerted by the tensile device, one can obtain:

\begin{matrix} [\underline{\underline{σ}}]_{{{\underline{g}}^{1}, {\underline{g}}_{2}}} = \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e} (\begin{matrix} 0 & \frac{1}{w_{2}} \\ \frac{1}{w_{1}} & 0 \end{matrix}) and \\ [\underline{\underline{σ}}]_{{{\underline{n}}^{1}, {\underline{n}}_{2}}} = \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e \cos γ} (\begin{matrix} 0 & \frac{1}{w_{2}} \\ \frac{1}{w_{1}} & 0 \end{matrix}) \end{matrix}

(32)

If the sample is square, w₁= w_2, the mixed components for the Cauchy stress tensor are identical:

\begin{matrix} σ_{1}^{2} = σ_{2}^{1} = \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e w} and \\ (σ_{1}^{2})_{N} = (σ_{2}^{1})_{N} = \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e w \cos γ} \end{matrix}

(33)

The previous result gives the mixed component of the shear stress (σ₂¹)_N as a function of the shear angle.

An example is presented in Figure 6. The fabric is a Twintex® plain weave⁴² composed of glass fibres (60%) and polypropylene fibres (40%) (743 g/m²). The sample is a cross shape specimen 200×200 mm², with 1.2 mm thickness (Figure 6(b)).

Figure 6.

Tangential stress measured in a picture frame test. (a) Twintex® plain weave (b) Cross specimen (c) Shear Load F in function of the shear angle (d) Normalized tangential stress (σ²₁)_N.

Biaxial tensile test

Presentation and assumptions

The tensile behaviour of woven fabrics is non-linear and biaxial because of the weaving and of the related crimp.^45–48 Different load-strain curves are obtained when the warp-weft deformation ratio Kt is fixed. They are experimentally obtained using specific devices for biaxial tensile tests such as shown in Figure 7.

Figure 7.

Biaxial tensile device and cross shape specimen.

The test consists in imposing different strain ratios in the two yarn directions of the fabric cross sample. The yarn directions remain constant and there is no in-plane shear strain. The strains are generally measured by a DIC system.^43,47 Figure 8 shows the biaxial tensile curves for the Twintex plain weave. Stresses and strains are assumed to be uniform on the sample. The fabric thickness (e) is assumed to be constant because of the small value of the longitudinal strains.

Figure 8.

Warp and weft tensions in function of the biaxial strain ratio $Kt = \frac{ɛ_{warp}}{ɛ_{weft}}$ (a) Curves network for T_x (b) Curves network for T_y.

Identification of the stress state

The angle between warp and weft directions is $(\frac{π}{2} - γ)$ (Figure 9). In most biaxial tensile tests, the warp and weft directions are perpendicular and γ = 0. An angle γ ≠ 0 is considered in this section in order to be more general and to prepare for the next section (tension+in-plane shear). The initial length in direction α is denoted L_α. Let us introduce a fixed frame $({\underline{G}}_{1}, {\underline{G}}_{2})$ , which corresponds to the initial fibre directions and such as ${\underline{G}}_{1} = L_{1} \underline{X}$ and ${\underline{G}}_{2} = L_{2} \underline{Y}$ where X and Y are unit vectors (Figure 9). As the angle γ is a constant, the non-orthogonal frame (X, Y) defines the current fibre orientations.

Figure 9.

Initial and current state of a fabric under biaxial loading.

The length $L_{α}$ becomes $ℓ_{α}$ in the current configuration. Let us consider the curvilinear coordinates $ξ_{1}, ξ_{2} \in [0, 1] \times [0, 1]$ . ${\underline{g}}_{1}$ and ${\underline{g}}_{2}$ are, respectively, the transformed vectors of ${\underline{G}}_{1}$ and ${\underline{G}}_{2}$ (Figure 9):

{\underline{g}}_{1} = ℓ_{α} \underline{X} and {\underline{g}}_{2} = ℓ_{α} \underline{Y}

(34)

so that:

‖ {\underline{g}}_{α} ‖ = ℓ_{α} and ‖ {\underline{g}}^{α} ‖ = \frac{1}{ℓ_{α} \cos γ}

(35)

One can write for any point M of the fabric:

\underline{OM} = ξ_{1} \frac{l_{1}}{L_{1}} \underline{G} + ξ_{2} \frac{l_{2}}{L_{2}} {\underline{G}}_{2}

(36)

Let us denote T₁ and T₂ the macroscopic tensile loads measured on the biaxial tension device (Figure 9). One can define the global tensile load as the sum of the local tensile loads:

T_{α} = \int_{S_{n^{α}}} {d T}_{n^{α}}

(37)

The stress tensor components that will be considered are those that lead to simple relationships with the internal loads. Equation (21) gives a direct relationship between the tensions and the contravariant components of the Cauchy stress tensor. Assuming that ${d T}_{n^{α}}$ is constant on the specimen leads to:

T_{α} = σ^{α α} \frac{‖ {\underline{g}}_{α} ‖}{‖ {\underline{g}}^{α} ‖} e ℓ_{3 - α}

(38)

where

S_{n^{α}} = e ℓ_{3 - α}

is the surface on which the tension is applied.

That leads to:

[\underline{\underline{σ}}]_{{{\underline{g}}_{1}, {\underline{g}}_{2}}} = (\begin{matrix} \frac{T_{1}}{e ℓ_{1}^{2} ℓ_{2} \cos γ} & 0 \\ 0 & \frac{T_{2}}{e ℓ_{2}^{2} ℓ_{1} \cos γ} \end{matrix})

(39)

The Cauchy stress components matrix in the covariant basis is diagonal as expected. Direct components are simply deduced from the measured tensions in the yarn directions. Those contravariant components are given in a non-unit frame $({\underline{g}}_{1}, {\underline{g}}_{2})$ . The stress tensor is also expressed in the unit frame $({\underline{n}}_{1}, {\underline{n}}_{2})$ :

[\underline{\underline{σ}}]_{{{\underline{n}}_{1}, {\underline{n}}_{2}}} = (\begin{matrix} \frac{T_{1}}{e ℓ_{2} \cos γ} & 0 \\ 0 & \frac{T_{2}}{e ℓ_{1} \cos γ} \end{matrix})

(40)

When γ=0, the obtained result logically corresponds to the standard form for two orthogonal networks. An example is given on Figure 10 for an orthogonal equibiaxial loading on the Twintex® plain weave. (The axial strain in warp and weft directions are equal.) The sample considered is a 50×50 mm² cross sample, with a 1.2 mm thickness.

Figure 10.

Warp tension and associated contravariant normal stresses $(σ^{11})_{N}$ as a function of $ɛ_{11}$ .

Combined shear and tensile loading

In the present section, in-plane shear and tension are simultaneously applied to the sample. The components that are considered are those identified in the Relationship between loads and stress components for a 2D continuous material section to give direct relationships between tension and direct components on one hand and between transverse forces and shear components on the other hand. This type of loading can be generated using the instrumented picture frame, presented in the Pure shear test: picture frame section and Figure 4 or an instrumented bias test.⁴⁹ The sample is supposed to be square.

Identification of the stress state

From the Textile composite reinforcement section, direct relationships between stresses and loads can be written for tensile and shear loading, respectively:

\underline{\underline{σ}} = σ^{α β} {\underline{g}}_{α} \otimes {\underline{g}}_{β} {dR}_{n^{α}} = \frac{σ^{(3 - α) α}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖} {d S}_{n^{α}}

(41)

\underline{\underline{σ}} = σ_{α}^{β} {\underline{g}}^{α} \otimes {\underline{g}}_{β} {d T}_{n^{α}} = \frac{σ_{α}^{α}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖} {d S}_{n^{α}}

(42)

If the Cauchy stress components have to be expressed using the same variance, either the contravariant or the mixed components can be used

Contravariant components

If the choice is made to work with the contravariant components, one can obtain for the transverse components:

σ^{21} = \frac{R_{1} ‖ {\underline{g}}^{1} ‖ ‖ {\underline{g}}^{2} ‖}{S_{n^{1}}} and σ^{12} = \frac{R_{1} ‖ {\underline{g}}^{1} ‖ ‖ {\underline{g}}^{2} ‖}{S_{n^{2}}}

(43)

As shown in Appendix 1, the direct stress components are less simple and depend on R_α:

\begin{matrix} σ^{11} = \frac{‖ {\underline{g}}^{1} ‖}{‖ {\underline{g}}_{1} ‖} \frac{T_{1}}{S_{n^{1}}} - \frac{g_{12} ‖ {\underline{g}}^{1} ‖ ‖ {\underline{g}}^{2} ‖}{‖ {\underline{g}}_{1} ‖^{2}} \frac{R_{1}}{S_{n^{1}}} and \\ σ^{22} = \frac{‖ {\underline{g}}^{2} ‖}{‖ {\underline{g}}_{2} ‖} \frac{T_{2}}{S_{n^{2}}} - \frac{g_{21} ‖ {\underline{g}}^{1} ‖ ‖ {\underline{g}}^{2} ‖}{‖ {\underline{g}}_{2} ‖^{2}} \frac{R_{2}}{S_{n^{2}}} \end{matrix}

(44)

Using equation (35), for a square sample, the contravariant Cauchy stress components become:

\begin{matrix} σ^{21} = σ^{12} = \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e w^{3} \cos^{2} γ} and \\ σ^{α α} = \frac{T_{α}}{e w^{3} \cos γ} - \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2}) \sin γ}{\sqrt{2} e w^{3} \cos^{2} γ} \end{matrix}

(45)

where w is the dimension of the square piece of fabric resulting from the tensile loading.

Introducing the normalized stresses in the same way as for equation (30):

\begin{matrix} (σ^{21})_{N} = (σ^{12})_{N} = \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e w \cos^{2} γ} and \\ (σ^{α α})_{N} = \frac{T_{α}}{e w \cos γ} - \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2}) \sin γ}{\sqrt{2} e w \cos^{2} γ} \end{matrix}

(46)

With such variance, if the shear stress component is simple and directly related to the transverse forces, the direct longitudinal components of the Cauchy stress tensor can be calculated but are more complicated and depend on both tension T_α and shear load F.

Mixed components

The same work can be done for the mixed components:

σ_{1}^{1} = \frac{T_{1}}{e w \cos γ} and σ_{2}^{2} = \frac{T_{2}}{e w \cos γ}

(47)

\begin{matrix} σ_{2}^{1} = \frac{T_{1} \tan γ}{e w} + \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e w} and \\ σ_{1}^{2} = \frac{T_{2} \tan γ}{e w} + \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e w} \end{matrix}

(48)

As expected, with such variance, the conclusions are different.

Introducing the normalized stresses (equation (30)):

(σ_{1}^{1})_{N} = \frac{T_{1}}{e w \cos^{2} γ} and (σ_{2}^{2})_{N} = \frac{T_{2}}{e w \cos^{2} γ}

(49)

\begin{matrix} (σ_{2}^{1})_{N} = \frac{T_{1} \sin γ}{e w} + \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e w \cos γ} and \\ (σ_{1}^{2})_{N} = \frac{T_{2} \sin γ}{e w} + \frac{F (\cos \frac{γ}{2} - \sin \frac{γ}{2})}{\sqrt{2} e w \cos γ} \end{matrix}

(50)

Stress state in the case of a combined loading

Figure 11 presents experimental results for a coupled shear and tension test on the Twintex® plain weave. These results have been obtained performing a picture frame test on an initially tensed sample.⁵⁰ The tensile system enables to measure and control the tension evolution during the test so that any kind of combined loading can be imposed³⁹ (Figure 4). The real shear angle is measured directly on the sample using optical measurements through a marker tracking software. In the case presented here, the protocol consists in applying a pretension on the sample before it is sheared, its lengths remaining constant. The resulting tensions in the warp and weft directions and the frame load F are measured by the load sensors. Due to the limit conditions imposed by the picture frame, tensions in the yarns change with the shear angle.⁵⁰ Figure 11 depicts the two tensions and the frame response as a function of the shear angle. The previous results enable to plot the stress components directly related to the measured loads, which are the mixed normal stresses and the contravariant transverse stresses (Figure 12). On the other hand, it is also possible to express all the stress components of the same variance so that the complete stress tensor is given (Figure 13) in any variance. But there is no simple physical interpretation of the latter values.

Figure 11.

Results of a coupled shear/tension test using an instrumented picture frame. (a) frame load (b) warp tension (c) weft tension.

Figure 12.

Stresses components directly related to the measured loads. (a) Contravariant Tangential stress (b) Mixed direct stresses.

Figure 13.

Other components of the stress tensor for the variances used Figure 12. (a) Mixed tangential stress (b) Contravariant direct stresses.

A consistent stress state has been obtained for a combined loading. Direct components only depending on the tensile state, and tangential components only depending on the shear state have been calculated, so that the stress state in the case of a combined loading is physically related to the loading.

Conclusion

The global loads measured during a picture frame test, a biaxial tensile test and a combined shear/tension test have been related to the stress components in the specimen. In the case of a picture frame test, the matrix of the mixed stress components in null on the diagonal. The shear stress components have also been given in function of the load on the picture frame. In the case of a biaxial tensile test in two non-orthogonal directions, the matrix of the contravariant stress components is diagonal and the two direct stresses have been related to the two tensile loads on the device. These results have been extended to the case of a combined shear/tensile test.

As the loads on the picture frame or tensile machine are crucial quantities measured during the test, their relation with the stress components are of major interest. It is the objective of the present paper to determine what the stress components are for which the shape of the component matrix is specific with only direct stress in tension test and shear stress in shear test. The relationships with the exterior loads have been given.

This work covers stresses and loads. Our next step will be to expand to strain fields in the specimen and to constitutive behaviour tensors.

Footnotes

Funding

This research was supported in part by the French Agency for Research (ANR) in the scope of the project LCM3M.

Conflict of Interest

None declared.

Appendix 1

Elementary load components according to the normal choice as functions of the Cauchy stress tensor components in the different basis (covariant, contravariant and mixed) Table A1.

Elementary loads as functions of the Cauchy stress tensor components

Elementary loads $\underline{\underline{σ}}$ Components	$\frac{{d T}_{n_{α}}}{{d S}_{n_{α}}}$	$\frac{{dR}_{n_{α}}}{{d S}_{n_{α}}}$	$\frac{{d T}_{n^{α}}}{{d S}_{n^{α}}}$	$\frac{{dR}_{n^{α}}}{{d S}_{n^{α}}}$
$σ_{λ μ} {\underline{g}}^{λ} \otimes {\underline{g}}^{μ}$	$\frac{σ_{α α}}{‖ {\underline{g}}_{α} ‖^{2}}$	$σ_{λ α} \frac{g^{λ (3 - α)}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$	$σ_{α λ} \frac{g^{α λ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖}$	$σ_{λ μ} = \frac{g^{α μ} g^{λ (3 - α)}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$
$σ^{λ μ} {\underline{g}}_{λ} \otimes {\underline{g}}_{μ}$	$σ^{λ μ} \frac{g_{α μ} g_{α λ}}{‖ {\underline{g}}_{α} ‖^{2}}$	$σ^{(3 - α) μ} \frac{g_{α μ}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$	$σ^{λ α} \frac{g_{α λ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖}$	$\frac{σ^{(3 - α) α}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$
$σ_{λ}^{μ} {\underline{g}}^{λ} \otimes {\underline{g}}_{μ}$	$σ_{α}^{μ} \frac{g_{α μ}}{‖ {\underline{g}}_{α} ‖^{2}}$	$σ_{λ}^{μ} \frac{g_{α μ} g^{λ (3 - α)}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$	$\frac{σ_{α}^{α}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖}$	$σ_{λ}^{α} \frac{g^{λ (3 - α)}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$
$σ_{μ}^{λ} {\underline{g}}_{λ} \otimes {\underline{g}}^{μ}$	$σ_{α}^{λ} \frac{g^{α λ}}{‖ {\underline{g}}_{α} ‖^{2}}$	$\frac{σ_{α}^{3 - α}}{‖ {\underline{g}}_{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$	$σ_{μ}^{λ} \frac{g^{α μ} g^{α λ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}_{α} ‖}$	$σ_{μ}^{3 - α} \frac{g^{α μ}}{‖ {\underline{g}}^{α} ‖ ‖ {\underline{g}}^{3 - α} ‖}$

Appendix 2

If the covariant and contravariant vectors in the initial configuration are denoted ${\underline{G}}_{α}$ , ${\underline{G}}^{α}$ , we can introduce the unit vectors along the material directions in the initial configuration: (B1)

{\underline{N}}_{α} = \frac{{\underline{G}}_{α}}{‖ {\underline{G}}_{α} ‖}, {\underline{N}}^{α} = \frac{{\underline{G}}^{α}}{‖ {\underline{G}}^{α} ‖}

The elementary loads in the initial configuration on the surface elements ${d S}^{0} {\underline{N}}_{α}$ and ${d S}^{0} {\underline{N}}^{α}$ and their components can then be, respectively, denoted ${\underline{{dF}^{0}}}_{N_{α}}$ , ${dR}_{N_{α}}^{0}$ , ${d T}_{N_{α}}^{0}$ and ${\underline{{dF}^{0}}}_{N^{α}}$ , ${dR}_{N^{α}}^{0}$ , ${d T}_{N^{α}}^{0}$ so that: (B2)

{\underline{{dF}^{0}}}_{N_{α}} = {d T}_{N_{α}}^{0} \frac{{\underline{G}}_{α}}{‖ {\underline{G}}_{α} ‖} + {dR}_{N_{α}}^{0} \frac{{\underline{G}}^{3 - α}}{‖ {\underline{G}}^{3 - α} ‖}

The elementary loads in the actual configuration brought back to the initial surface elements $d S {\underline{N}}_{α}$ and $d S {\underline{N}}^{α}$ and their components can then be, respectively, denoted ${\underline{dF}}_{N_{α}}$ , ${dR}_{N_{α}}$ , ${d T}_{N_{α}}$ and ${\underline{dF}}_{N^{α}}$ , ${dR}_{N^{α}}$ , ${d T}_{N^{α}}$ so that: (B3)

{\underline{dF}}_{N_{α}} = {d T}_{N_{α}} \frac{{\underline{G}}_{α}}{‖ {\underline{G}}_{α} ‖} + {dR}_{N_{α}} \frac{{\underline{G}}^{3 - α}}{‖ {\underline{G}}^{3 - α} ‖}

The relationships between the first and second Piola-Kirchhoff tensor components and the elementary loads can then be calculated and are presented on Tables B1 and B2. Table B1.

Elementary loads as functions of the first Piola-Kirchhoff stress tensor components

Elementary loads $\underline{\underline{P}}$ Components	$\frac{{d T}_{N}_{_{α}}}{{d S}_{N}_{_{α}}}$	$\frac{{dR}_{N}_{_{α}}}{{d S}_{_{_{α}}}}$	$\frac{{d T}_{N}^{α}}{{d S}_{N}^{α}}$	$\frac{{dR}_{N}^{α}}{{d S}_{N}^{α}}$
$P_{λ μ} {\underline{G}}^{λ} \otimes {\underline{G}}^{μ}$	$\frac{P_{α α}}{‖ {\underline{G}}_{α} ‖^{2}}$	$P_{λ α} \frac{G^{λ (3 - α)}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$	$P_{α μ} \frac{G^{α μ}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{α} ‖}$	$P_{λ μ} \frac{G^{α μ} G^{λ (3 - α)}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$
$P^{λ μ} {\underline{G}}_{λ} \otimes {\underline{G}}_{μ}$	$P^{λ μ} \frac{G_{α μ} G_{α λ}}{‖ {\underline{G}}_{α} ‖^{2}}$	$P^{(3 - α) μ} \frac{G_{α μ}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$	$P^{λ α} \frac{G_{α λ}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}_{α} ‖}$	$\frac{P^{(3 - α) α}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$
$P_{λ}^{μ} {\underline{G}}^{λ} \otimes {\underline{G}}_{μ}$	$P_{α}^{μ} \frac{G_{α μ}}{‖ {\underline{G}}_{α} ‖^{2}}$	$P_{λ}^{μ} \frac{G_{α μ} G^{λ (3 - α)}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$	$\frac{P_{α}^{α}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{α} ‖}$	$P_{λ}^{α} \frac{G^{λ (3 - α)}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$
$P_{μ}^{λ} {\underline{G}}_{λ} \otimes {\underline{G}}^{μ}$	$P_{α}^{λ} \frac{G^{α λ}}{‖ {\underline{G}}_{α} ‖^{2}}$	$\frac{P_{α}^{3 - α}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$	$P_{μ}^{λ} \frac{G^{α μ} G^{α λ}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}_{α} ‖}$	$P_{μ}^{3 - α} \frac{G^{α μ}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$

Table B2.

Elementary loads as functions of the second Piola-Kirchhoff stress tensor components

Elementary loads $\underline{\underline{S}}$ Components	$\frac{{d T}_{N_{α}}^{0}}{{d S}_{N_{α}}^{0}}$	$\frac{{dR}_{N_{α}}^{0}}{{d S}_{N_{α}}^{0}}$	$\frac{{d T}_{N_{α}}^{0}}{{d S}_{N_{α}}^{0}}$	$\frac{{dR}_{N^{α}}^{0}}{{d S}_{N^{α}}^{0}}$
$S_{λ μ} {\underline{G}}^{λ} \otimes {\underline{G}}^{μ}$	$\frac{S_{α α}}{‖ {\underline{G}}_{α} ‖^{2}}$	$S_{λ α} \frac{G^{λ (3 - α)}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$	$S_{α μ} \frac{G^{α μ}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{α} ‖}$	$S_{λ μ} \frac{G^{λ (3 - α)} G^{α μ}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$
$S^{λ μ} {\underline{G}}_{λ} \otimes {\underline{G}}^{μ}$	$S^{λ μ} \frac{G_{α μ} G_{α λ}}{‖ {\underline{G}}_{α} ‖^{2}}$	$S^{(3 - α) μ} \frac{G_{α μ}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$	$S^{α λ} \frac{G_{α λ}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{α} ‖}$	$\frac{S^{(3 - α) α}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$
$S_{λ}^{μ} {\underline{G}}^{λ} \otimes {\underline{G}}_{μ}$	$S_{α}^{μ} \frac{G_{α μ}}{‖ {\underline{G}}_{α} ‖^{2}}$	$S_{λ}^{μ} \frac{G_{α μ} G^{λ (3 - α)}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$	$\frac{S_{α}^{α}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{α} ‖}$	$S_{λ}^{α} \frac{G^{λ (3 - α)}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$
$S_{μ}^{λ} {\underline{G}}_{λ} \otimes {\underline{G}}^{μ}$	$S_{α}^{λ} \frac{G^{α λ}}{‖ {\underline{G}}_{α} ‖^{2}}$	$\frac{S_{α}^{3 - α}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$	$S_{μ}^{λ} \frac{G^{α λ} G^{α μ}}{‖ {\underline{G}}_{α} ‖ ‖ {\underline{G}}^{α} ‖}$	$S_{μ}^{3 - α} \frac{G^{α μ}}{‖ {\underline{G}}^{α} ‖ ‖ {\underline{G}}^{3 - α} ‖}$

Appendix 3

From equation (11), we have (C1)

σ_{α β} = g_{α β} \frac{{d T}_{n_{β}}}{{d S}_{n_{β}}} + \frac{{dR}_{n_{β}}}{{d S}_{n_{β}}} \frac{‖ {\underline{g}}_{β} ‖}{‖ {\underline{g}}^{3 - β} ‖} δ_{α}^{3 - β}

Then, (C2)

σ_{21} = g_{21} \frac{{d T}_{n_{1}}}{{d S}_{n_{1}}} + \frac{{dR}_{n_{1}}}{{d S}_{n_{1}}} \frac{‖ {\underline{g}}_{1} ‖}{‖ {\underline{g}}^{2} ‖} δ_{2}^{2}

(C3)

σ_{21} = g_{21} \frac{{d T}_{n_{1}}}{{d S}_{n_{1}}} + \frac{{dR}_{n_{1}}}{{d S}_{n_{1}}} \frac{‖ {\underline{g}}_{1} ‖}{‖ {\underline{g}}^{2} ‖}

From the combination of equations (8) and (9) we have (C4)

σ_{21} = \frac{{dR}_{n_{1}}}{{d S}_{n_{1}}} \frac{‖ {\underline{g}}_{1} ‖}{‖ {\underline{g}}^{2} ‖} - \frac{{d T}_{n_{1}}}{{d S}_{n_{1}}} \frac{‖ {\underline{g}}_{1} ‖^{2}}{‖ {\underline{g}}^{2} ‖^{2}} {\underline{g}}^{1} . {\underline{g}}^{2}

Demonstrating the compatibility of equations (C3) and (C4) then consists in establishing (C5)

g_{21} = - \frac{‖ {\underline{g}}_{1} ‖^{2}}{‖ {\underline{g}}^{2} ‖^{2}} {\underline{g}}^{1} . {\underline{g}}^{2}

Since g_i are the covariant vectors and gⁱ the contravariant vectors, we have the following properties: (C6)

(\overset{\land}{\underline{g_{i}}, \underline{g_{j}}}) = π - (\overset{\land}{\underline{g^{i}}, \underline{g^{j}}})

(C7)

(\overset{\land}{\underline{g^{i}}, \underline{g_{i}}}) = - (\overset{\land}{\underline{g^{3 - i}}, \underline{g_{3 - i}}})

(C8)

\frac{‖ \underline{g_{i}} ‖}{‖ \underline{g_{j}} ‖} = \frac{‖ \underline{g^{j}} ‖}{‖ \underline{g^{i}} ‖}

From these properties, we have (C9)

\underline{g^{1}} . \underline{g^{2}} = ‖ \underline{g^{1}} ‖ ‖ \underline{g^{2}} ‖ Cos (\overset{\land}{\underline{g^{1}}, \underline{g^{2}}}) = - ‖ \underline{g^{1}} ‖ ‖ \underline{g^{2}} ‖ Cos (\overset{\land}{\underline{g^{1}}, \underline{g^{2}}})

We obtain (C10)

\underline{g^{1}} . \underline{g^{2}} \frac{‖ \underline{g_{1}} ‖^{2}}{‖ \underline{g^{2}} ‖^{2}} = \frac{‖ \underline{g_{1}} ‖^{2}}{‖ \underline{g^{2}} ‖^{2}} ‖ {\underline{g}}^{1} ‖ ‖ {\underline{g}}^{2} ‖ Cos (\hat{\underline{g^{1}}, \underline{g^{2})}}

(C11)

\underline{g^{1}} . \underline{g^{2}} \frac{‖ \underline{g_{1}} ‖^{2}}{‖ \underline{g^{2}} ‖^{2}} = ‖ \underline{g_{1}} ‖^{2} \frac{‖ \underline{g^{1}} ‖}{‖ \underline{g^{2}} ‖} Cos (\hat{\underline{g_{1}}, \underline{g_{2}})} = ‖ \underline{g_{1}} ‖ ‖ \underline{g_{2}} ‖ Cos (\hat{\underline{g^{1}}, \underline{g^{2}})}

(C12)

\underline{g^{1}} . \underline{g^{2}} \frac{‖ \underline{g_{1}} ‖^{2}}{‖ \underline{g^{2}} ‖^{2}} = g_{12}

References

Advani

. Flow and rheology in polymeric composites manufacturing , Amsterdam: Elsevier, 1994.

Rudd

Long

. Liquid molding technologies , Cambridge: Woodhead Publishing Limited, 1997.

Saouab

Bréard

Lory

. Injection simulations of thick composite parts manufactured by the RTM process. Compos Sci Technol 2001; 61: 445–451.

Van Der Ween

. Algorithms for draping fabrics on doubly curved surfaces. Int J Numer Methods Eng 1991; 31: 1414–1426.

De Luca

Pickett

. Numerical and experimental investigation of some press forming parameters of two fibre reinforced thermoplastics: APC2-AS4 and PEICETEX. Compos Part A 1998; 29: 101–110.

Cherouat

Borouchaki

Billoët

. Geometrical and mechanical draping of composite fabric. Eur J Comput Mech 2005; 14: 693–708.

Pourboghrat

Chung

. Non-orthogonal constitutive equation for woven fabric reinforced thermoplastic composites. Compos Part A 2002; 33: 1095–1105.

Pickett

. Review of finite element methods applied to manufacturing and failure prediction in composite structures. Appl Compos Mater 2002; 9: 43–58.

Peng

Cao

. A continuum mechanics-based non orthogonal constitutive model for woven composite fabrics. Compos Part A 2005; 36: 859–874.

10.

Skordos

Monroy

Sutcliffe

MPF

. A simplified rate dependent model of forming and wrinkling of pre-impregnated woven composites. Compos Part A 2007; 38: 1318–1330.

11.

Hamila

Boisse

Sabourin

. A semi-discrete shell finite element for textile composite reinforcement forming simulation. Int J Numer Method Eng 2009; 79: 1443–1466.

12.

Khan

Mabrouki

Vidal-Sallé

. Numerical and experimental analyses of woven composite reinforcement forming using a hypoelastic behaviour. Application to the double dome benchmark. J Mater Process Technol 2010; 210: 378–388.

13.

Allaoui

Boisse

Chatel

. Experimental and numerical analyses of textile reinforcement forming of a tetrahedral shape. Compos Part A 2011; 42: 612–622.

14.

Durville

. Numerical simulation of entangled materials mechanical properties. J Mater Sci 2005; 40: 5941–5948.

15.

Duhovic

Bhattacharyya

. Simulating the deformation mechanisms of knitted fabric composites. Compos Part A 2006; 37: 1897–1915.

16.

Miao

Zhou

Wang

. Mechanics of textile composites: micro-geometry. Compos Sci Technol 2008; 68: 1671–1678.

17.

Wang

Page

. Prediction of shear force using 3d non-linear fem analyses for a plain weave carbon fabric in a bias extension state. Finite Elem Anal Design 2002; 38: 755–764.

18.

Sagar

Potluri

Hearle

JWS

. Mesoscale modelling of interlaced fibre assemblies using energy method. Textile Composites Group. Comput Mater Sci 2003; 28: 49–62.

19.

Potluri

Perez Ciurezu

Ramgulam

. Measurement of meso-scale shear deformations for modelling textile composites. Compos Part A 2006; 37: 303–314.

20.

Creech

Pickett

. Meso-modelling of non-crimp fabric composites for coupled drape and failure analysis. J Mater Sci 2006; 41: 6725–6736.

21.

Badel

Vidal-Sallé

Boisse

. Large deformation analysis of fibrous materials using rate constitutive equation. Comput Struct 2008; 86: 1164–1175.

22.

Hivet

Boisse

. Consistent mesoscopic mechanical behaviour model for woven composite reinforcements in biaxial tension. Compos Part B 2008; 39: 345–361.

23.

Spencer

AJM

. Theory of fabric-reinforced viscous fluid. Compos Part A 2000; 31: 1311–1321.

24.

Dong

Lekakou

Bader

. Processing of composites: simulations of the draping of fabrics with updated material behaviour law. J Compos Mater 2001; 35: 138–163.

25.

King

Jearanaisilawong

Socrate

. A continuum constitutive model for the mechanical behavior of woven fabrics. Int J Solids Struct 2005; 42: 3867–3896.

26.

Ten Thije

RHW

Akkerman

Huetink

. Large deformation simulation of anisotropic material using an updated Lagrangian finite element method. Comput Methods Appl Mech Eng 2007; 196: 3141–3150.

27.

Badel

Gauthier

Vidal-Salle

. Rate constitutive equations for computational analyses of textile composite reinforcement mechanical behaviour during forming. Compos Part A 2009; 40: 997–1007.

28.

Aimène

Vidal-Sallé

Hagège

. A hyperelastic approach for composite reinforcement large deformation analysis. J Compos Mater 2010; 44: 5–26.

29.

Abaqus. Theory manual , Albany: Dassault Systèmes, 2009.

30.

Kawabata

. The standardization and analysis of hand evaluation, , 2nd edn. Osaka, Japan: The Textile Machinery Society of Japan, 1986.

31.

De Bilbao

Soulat

Launay

. Experimental study of bending behaviour of reinforcements. Exp Mech 2010; 50: 333–351.

32.

Prodromou

Chen

. On the relationship between shear angle and wrinkling of textile composite performs. Compos Part A 1997; 28: 491–503.

33.

McGuinness

ÓBrádaigh

. Characterisation of thermoplastic composite melts in rhombus-shear: the picture-frame experiment. Compos Part A 1998; 29: 115–132.

34.

Wang

Page

Paton

. Experimental investigation of the draping properties of reinforcement fabrics. Compos Sci Technol 1998; 58: 229–237.

35.

Sagar

Potluri

. Compaction modelling of textile preforms for composite structures. Compos Struct 2008; 86: 177–185.

36.

Potter

. Bias extension measurements on cross-plied unidirectional prepreg. Compos Part A 2002; 33: 63–73.

37.

Lebrun

Bureau

Denault

. Evaluation of bias-extension and picture-frame test methods for the measurement of intraply shear properties of PP/glass commingled fabrics. Compos Struct 2003; 61: 341–352.

38.

Lomov

Willems

Verpoest

. Picture frame test of woven composite reinforcements with a full-field strain registration. Text Res J 2006; 76: 243–252.

39.

Launay

Hivet

Duong

. Experimental analysis of the influence of tensions on in plane shear behaviour of woven composite reinforcements. Compos Sci Technol 2008; 68: 506–515.

40.

Harrison

Clifford

Long

. Shear characterisation of viscous woven textile composites: a comparison between picture frame and bias extension experiments. Compos Sci Technol 2004; 64: 1453–1465.

41.

Peng

Cao

Chen

. Experimental and numerical analysis on normalization of picture frame tests for composite materials. Compos Sci Technol 2004; 64: 11–21.

42.

Cao

Akkerman

Boisse

. Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results. Compos Part A 2008; 39: 1037–1053.

43.

Lomov

Boisse

de Luycker

. Full-field strain measurements in textile deformability studies. Compos Part A 2008; 39: 1232–1244.

44.

Wiggers

Harrison

Long

. Normalisation of shear test data for rate-independent compressible fabrics. J Compos Mater 2008; 42: 2315–2343.

45.

Kawabata

Niwa

Kawai

. The finite deformation theory of plain weave fabrics. Part I: the biaxial deformation theory. J Text Inst 1973; 64: 21–46.

46.

Buet-Gautier

Boisse

. Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements. Exp Mech 2001; 41: 260–269.

47.

Willems

Lomov

Verpoest

. Optical strain fields in shear and tensile testing of textile reinforcements. Compos Sci Technol 2008; 68: 807–819.

48.

Carvelli

Corazza

Poggi

. Mechanical modelling of monofilament technical textiles. Comput Mater Sci 2008; 42: 679–691.

49.

Abdiwi

Harrison

Guo

. Measuring the shear-tension coupling of engineering fabrics. AIP Conf Proceedings ESAFORM 2011 2011; 1353: 889–894.

50.

Hivet

Duong

. A contribution to the analysis of the intrinsic shear behaviour of fabrics. J Compos Mater 2011; 45: 695–717.