Rate constitutive equations for computational analyses of textile composite reinforcement mechanical behaviour during forming

doi:10.1016/j.compositesa.2008.04.015

Composites Part A: Applied Science and Manufacturing

Volume 40, Issue 8, August 2009, Pages 997-1007

https://doi.org/10.1016/j.compositesa.2008.04.015 Get rights and content

Abstract

Textile composite reinforcements are made up of fibres. Consequently, their mechanical behaviour is a result of the possible sliding and the interactions between the fibres. When they are formed on double curved shapes, these fabrics are submitted to large strains, in particular large in-plane shear. Among the mechanical behaviour models for these textile reinforcements, continuous models are most commonly used for forming simulations because they can be used with standard finite elements. The objective of the present paper is to propose a continuous approach for textile reinforcement deformation analysis based on a rate constitutive equation specific to materials made of fibres. The objective derivative of this constitutive model is defined by the fibre rotation. This constitutive model is implemented in ABAQUS and can be used in most commercial F.E. software. The approach is extended to materials with two-fibre directions in order to perform simulations of woven fabric forming processes. A set of simulations of large deformations of textile composite reinforcements at the mesoscopic scale (deformation of a woven unit cell) and at the macroscopic scale (deep drawing) is presented to show the efficiency of the proposed approach.

Introduction

Process simulation is of great importance in the development of cost-effective manufacturing of composite structures. Among the different aspects of composite forming, the draping of textile reinforcement on a double-curvature surface is an important feature. For instance, the first stage of the resin transfer moulding (RTM) process consists in shaping a dry textile preform before resin injection [1], [2], [3].

There are two main reasons for simulating the deformation of textile reinforcements during a forming process. The simulation can give the conditions that will make the forming possible (for instance loads on the tools, initial orientation, type of material) and also can describe any possible defects after forming (wrinkles, porosities, yarn fractures). Secondly, and unique to fibre-reinforced composite forming analysis, is the need to know at any point the direction and density of the fibres after forming. These directions and densities can be significantly modified by the forming process in the case of a shaping on a double-curvature surface. This information is very important for structural analyses of the composite part in use (stiffness, damage, fatigue, …).

Some approaches have been developed for woven fabric draping simulations based on ‘kinematic models’ [4], [5], [6]. In these methods, the yarns of the fabric are assumed to be inextensible and the rotation between warp and weft directions is free. The knowledge of the shape on which the fabric is formed and sufficient initial kinematical conditions leads to the final positions of the yarns on the form. Several packages are commercially available. These methods are fairly efficient especially for hand draping, but they do not account for either the mechanical behaviour of the fabric or for the load boundary conditions. For an accurate simulation of a textile shaping, a finite element analysis of the process is necessary including the modelling of the contact, friction and above all the mechanical behaviour of the textile reinforcement during forming.

The mechanical behaviour of woven fabric is complex due to the interactions of the yarns and fibres. (The present work concerns continuous fibres.) It is a multi-scale problem. The interactions and the possible relative sliding at the meso-scale (scale of the yarns) and at the micro-scale (scale of the fibres) influence the macroscopic behaviour of the fabric. Despite multiple efforts in this field, there is no widely accepted model that describes accurately all of the important aspects of fabric mechanical behaviour. The main model kinds come from the multi-scale nature of textile. Some approaches have been recently proposed in which the components at meso-scale (yarns) or micro-scale (fibres) are modelled [7], [8], [9], [10], [11]. Nevertheless, because of the very large number of yarns or fibres in a composite part and their complex interactions, these approaches are (today) limited to small-domain analyses. If these approaches are used for complex part forming simulations, the mesoscopic modelling has to be coarse (usually rods and springs …) and the accuracy of the analysis decreases.

The alternative to these discrete approaches is to consider the textile reinforcement as a continuum. The textile made of fibres is not continuous at lower scales, but an equivalent continuous material can be considered at the macro-scale. The material behaviour model of this continuous material has to convey the very specific behaviour resulting from the textile reinforcement composition of yarns and fibres. A main advantage of the continuous approach is that it can be used within standard finite element methods. Most of the approaches proposed for textile deformation analyses, and especially for forming simulations, belong to this continuous model family [12], [13], [14], [15], [16], [17], [18], [19], [20].

Among continuous mechanical behaviours at large strains, many finite element works have adopted a hypo-elastic approach (or rate constitutive equation). In these constitutive laws, a Cauchy stress objective derivative is related to the strain rate by a constitutive tensor. Truesdell was the first to introduce these hypo-elastic models in modern and general form [21], [22], [23]. The objective derivative is the derivative for an observer fixed in a frame following the continuum in its deformation. As there are several possibilities for approaching this condition, there are several objective rates and several hypo-elastic models [24], [25], [26], [27], [28], [29], [30]. These rate constitutive equations are often used in finite element codes at finite strains [31], [32], [33], [34]. In the present work, an approach based on a rate constitutive equation is proposed for the simulation of textile-composite-reinforcement deformations. This approach only concerns the constitutive equation and can be implemented in commercial F.E. codes (for instance in the material user subroutine of ABAQUS).

The standard objective derivatives such as the Jaumann or Green–Naghdi rates cannot be used in the case of textile materials as they are based on average rotation of the continuum and do not follow the fibre direction. The approach proposed in this paper is based on a specific hypo-elastic model where the rotated frame is defined by the rotation of the fibre. This approach permits the tracking the fibre direction during the deformation and gives the ability to distinguish mechanical properties in the fibre direction and in the direction directions. This approach gives exact solutions to a set of four elementary tests significant for fibrous materials at large strains. It is not the case for other models that have been proposed for fabric forming and that are based on Green–Naghdi’s rate used in material user subroutines. The proposed approach is used to analyse the deformation of a carbon twill unit cell submitted to large in-plane shear. Finally, the method is extended to textile reinforcements with two-fibre directions as it is the case for most woven fabrics. This method allows performing macroscopic woven-reinforcement forming simulations. Hemispherical deep-drawing simulations are compared to experimental formings. The influence of the in-plane shear properties on wrinkle development is investigated.

Section snippets

Rate constitutive equations

The rate constitutive equations (or hypo-elastic laws) are often used in finite element analyses at large strains [31], [32], [33], [34]. User subroutines that can be implemented in codes such as ABAQUS to define the mechanical constitutive behaviour are written within this framework. A stress rate σ^∇ is related to the strain rate D by a constitutive tensor C. To avoid rigid body rotations that can affect the stress state, the derivative σ^∇, called the objective derivative, is the derivative

Textile composite reinforcement mechanical behaviour

Textile materials are made of fibres, which make their mechanical behaviour very specific. Relative sliding is possible between fibres (see Fig. 1a). The yarns are made of thousands of fibres and it is in general not possible to model each of them. The constitutive models that are introduced in the present paper are continuum models intended to describe the specific mechanical behaviour of the fibre bundle (Fig. 1b). As a first step and to simplify the presentation a single-fibre direction is

Objective derivative based on the fibre rotation

In the present paper an alternative to the standard Green–Naghdi or Jaumann objective derivatives is introduced [43]. The rate constitutive equation (1) is based on the rotation of the fibre f₁: ${\underset{̲}{\underset{̲}{σ}}}^{\nabla_{ϕ}} = \underset{̲}{\underset{̲}{\underset{̲}{\underset{̲}{C}}}} : \underset{̲}{\underset{̲}{D}}$ $with {\underset{̲}{\underset{̲}{σ}}}^{\nabla_{ϕ}} = \underset{̲}{\underset{̲}{Φ}} \cdot (\frac{d}{d t} ({\underset{̲}{\underset{̲}{Φ}}}^{T} \cdot \underset{̲}{\underset{̲}{σ}} \cdot \underset{̲}{\underset{̲}{Φ}})) \cdot {\underset{̲}{\underset{̲}{Φ}}}^{T}$ where Φ is the rotation of the fibre. It is shown in Appendix B that the derivative ${\underset{̲}{\underset{̲}{σ}}}^{\nabla_{ϕ}}$ is objective. The stress update (5) becomes $[σ^{n + 1}]_{f_{i}^{n + 1}} = [σ^{n}]_{f_{i}^{n}} + [C^{n + 1 / 2}]_{f_{i}^{n + 1 / 2}} [Δ ε]_{f_{i}^{n + 1 / 2}}$ From the transformation gradient F the current fibre direction can be

Use of Green–Naghdi’s frame associated to a change of basis

The work frame used in material subroutines of ABAQUS/Explicit is that of Green–Naghdi. Some authors suggest to use this frame to update stresses (Eq. (5)), i.e., to use the rotation Q = R from the polar decomposition in Eq. (2) [14], [17], [44]. Consequently, a basis change of the constitutive matrix is to be performed from the fibre frame (where the matrix has its specific form) to Green–Naghdi’s frame. The equation for stress update can be used in its form (5) with {e_i} Green–Naghdi’s frame.

Elementary tests

In the following tests the initial geometry is a unit cube. The stiffness in the fibre direction is equal to 35,400 MPa. All of the other rigidities are equal to zero (with Voigt notation): $[C]_{f_{i}} = [\begin{matrix} 35, 400 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$ The tests under consideration are in the plane $({\underset{̲}{e}}_{1}^{0}, {\underset{̲}{e}}_{2}^{0})$ , therefore only the stress components σ₁₁, σ₂₂, σ₁₂ will be considered.

The results are given for the approach developed in Section 4 using the rotated frame defined by the fibre rotation (denoted FF) and for the approach defined in

Macroscopic simulations of textile reinforcement forming

The previous sections discussed single-fibre direction materials, which is the case of unidirectional fabrics or mesoscopic simulations (Section 6.2). From a macroscopic point of view, woven reinforcements are generally made of two-fibre directions. To simulate textile forming processes, the present section introduces the extension of the previous material model to two-fibre direction materials. The suggested method consists of superimposing within the same material elementary volume the stress

Conclusions and prospects

A hypo-elastic approach has been developed for large strain analysis of textile composite reinforcements. It is based on an objective derivative defined from the fibre rotation. The approach is simple and can be implemented in any commercial F.E. software. Nevertheless, it must be underlined that its efficiency will depend on the quality of the identification of the constitutive matrix [C]. Especially, in the case of mesoscopic analyses (see Section 6.2 for instance), the transverse moduli

Acknowledgements

The work reported here has been carried out in the scope of the projects ITOOL (European Commission) and MACODEV (Rhône-Alpes region).

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Rate constitutive equations for computational analyses of textile composite reinforcement mechanical behaviour during forming

Abstract

Introduction

Section snippets

Rate constitutive equations

Textile composite reinforcement mechanical behaviour

Objective derivative based on the fibre rotation

Use of Green–Naghdi’s frame associated to a change of basis

Elementary tests

Macroscopic simulations of textile reinforcement forming

Conclusions and prospects

Acknowledgements

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Comput Mater Sci

Comput Mater Sci

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Compos Sci Technol

Compos Sci Technol

Composites A

History of the resin transfer moulding for aerospace applications

Composites A

Liquid composite molding

Algorithms for draping fabrics on doubly curved surfaces

Int J Num Meth Eng

A simulation of reinforcement deformation during the production of preform for liquid moulding processes

Proc Inst Mech Eng B, J Eng Manuf

Geometrical and mechanical draping of composite fabric

Eur J Comput Mech