Paper Number

14TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS 7-10 June 2010, Budapest, Hungary

Paper ID: 598-ECCM14

CONSTITUTIVE MODELLING OF COMPOSITE MATERIALS WITH FULL 3D ORIENTATION OF REINFORCEMENT F. Stig1, S. Hallström1* 1

Kungliga Tekniska Högskolan (KTH), Dept of Aeronautical and Vehicle Engineering, SE 100 44 Stockholm, Sweden *[email protected]

Abstract True 3D weaving, incorporating dual shedding of warp yarns, is a new textile principle that inherently generates fabrics with yarns oriented in three orthogonal directions. It enables netshape production of textile beam preforms with great flexibility in cross-sectional shape. This paper describes modelling activities aiming at characterising the constitutive behaviour of composite materials containing 3D woven reinforcements. The architecture of such materials differs from that off conventional textiles since it contains reinforcement strands in three orthogonal directions. Mechanical modelling can thereby not be performed using classical laminate theory or other methods based on 2D (i.e. plate) theory. A method for generating the internal geometry of 3D reinforced composites is presented, including details of the strand paths and strand cross-section shapes on a meso scale. The main step in the proposed novel approach is an explicit finite element simulation. In the algorithm, the strand cross section is first reduced in size to avoid strand interpenetration, and then the strand volume is increased using an internal pressure and a full contact algorithm until the desired strand volume fraction is reached. The algorithm combines commercial and non-commercial codes and may be employed to analyse composites with any type of 3-dimensional reinforcement, such as weaves or braids. 1 Introduction The aerospace industry is more and more focusing on new materials and manufacturing methods in its strive to reduce weight and manufacturing cost of aircraft. This has led to an increased interest in 3D reinforced composites, since they enable modular design and direct use in stiffeners, stringers and joining elements [1]. However, in order for such new design concepts to gain acceptance, the mechanical properties of these novel materials must be known, either through experiments or modelling. This paper focuses on a modelling approach, and one of the key issues when modelling a textile composite is the geometric description of its reinforcement. Previous work in this area indicates the need to develop a robust modelling framework for 3D reinforced textile composites. According to Wu [2] there is need for models that take weave style, impregnation and post-curing into account. Larve et. al [3] state that the complexity of new preforms has prevented development of a robust analysis methodology. This paper aims at setting forth a framework for the modelling chain of 3D reinforced composites in general, and composites containing 3D woven reinforcement in particular, starting with the non-trivial task of modelling the internal geometry of representative volume elements (RVE). The elastic properties of a composite material are to a large extent governed by its micro 1



structure [4]. Chapman and Whitcomb [5] concluded that small changes in the strand architecture can have an substantial effect on the predicted moduli. Consequently, adequate descriptions of the internal geometry of the dry preform are needed in order to predict the mechanical properties of the composite material. This is not a trivial task in the general case, and even less so for 3D reinforced preforms due to their complex internal geometry. Furthermore, the dry fabric preform is deformed when placed in a mould for injection moulding. Then the yarn geometry, i.e. its trajectory and cross section shape, must be known after the preform is compacted in the mould [6]. There exist many models for describing the geometry of a fabric unit cell, however, the internal geometry must be known beforehand from experimental observations which are difficult and time consuming to conduct. Simple geometrical shapes are often used which do not necessarily capture all the geometrical features, see e.g. [7–9]. For instance it is commonly assumed that strand cross-section shapes are constant, which Hivet and Boisse [10] showed experimentally is not always the case. Mahadik et al. [11] performed an experimental study to obtain the internal structure of a 3D angle interlock fabric. The strand crimp angles and the size and shape of the resin channels were investigated during increasing compaction. They concluded that compaction had a significant effect on both crimp angles and the size and shape of resin pockets, emphasising the need to include compaction when analysing textile composites. Models also exist that are able to predict the internal geometry beforehand, however none is able to cope with the complex shape of a 3D woven structure and most of them only work for non-compacted weaves. Hivet and Boisse [10] developed a model for generating the internal geometry of dry 2D weaves with varying strand cross section, and no strand inter-penetration. The inherent drawback of such a model is the high number of parameters that have to be experimentally determined, and hence the predictive value is reduced. Lomov et. al [12] presented a textile pre-processor which is a part of the software package WiseTex [13]. The geometry pre-processor utilises the principle of minimum energy to calculate the strand trajectory and strand cross section shape. To the authors’ knowledge, the 3D-weave textile architecture, as depicted in Fig. 1, is not yet implemented in WiseTex. Sherburn et. al [14] and Sagar et. al [6] have also used the principle of minimum energy to generate models of the internal geometry of textiles. The model presented in [6] was limited to plain 2D weaves, while Sherburn et. al [14] state that 3D textiles with crimped vertical weft strands are not meshable in the software they use, making it less suitable for 3D woven yarn architectures. The proposed finite element (FE) method has a number of advantages. The model does not require geometrical parameters being measured on a sample, but merely that the user specifies the size of the RVE, yarn thickness and lengths. The nominal strand trajectories and their mutual interlacing constraints obviously need to be defined. Then a simulation converges towards a state where both strand paths and local variations in cross sectional shape are determined. The strength in the approach is that the number of measured parameters is kept to a minimum. The proposed model is able to predict the internal geometry in a first step, and then use it in a subsequent step, either in a mechanical model or in flow simulations.

2



2 The nominal geometry of a 3D weave A 3D weave is produced using the 3D weaving process which is characterised by dual directional shedding operations that fully interlace the warp yarns with both horizontal and vertical weft yarns, according to the principles of 3D weaving set forth by Khokar [15]. The textile architecture is schematically illustrated in Fig 1. The RVE of a plain 3D weave consists of four warp yarns, four horizontal and four vertical weft yarns. In this paper, dry yarns in a weave are referred to as yarns, while impregnated yarn bundles within a composite are referred to as strands.

Figure 1. A plain 3D weave, seen from the weave direction (left) and a side view (right). The weave consists of warp (W), horizontal (HW) and vertical weft (VW) yarns. Images are generated in TexGen [16]

3 Algorithm Developing a 3D model of a 3D weave RVE with a certain degree of packing involves several challenges. The first and foremost problem is that of strand inter-penetration. The softwares WiseTex and TexGen both have capabilities to treat interpenetrating strands. Both utilise schemes where either springs [14] or beam [17] elements are used to penalise the over-closure between strands. A different approach is chosen in this paper. First the nominal weave shape is generated but with relatively thin strands corresponding to a reduced volume fraction of strands defined as

vs

Vs Vtot

(1)

where Vs is the total strand volume and Vtot is the total RVE volume. The volume fraction of strands is reduced enough to ensure that no strand inter-penetration occurs. The strands are then modelled as tubes that are forced to expand using an inflow of fluid under increasing pressure until the desired volume fraction of strands is reached. This enables the strand cross sections to grow where there is space available, e.g. between the horizontal and vertical wefts, but not elsewhere. During the finite element (FE) analysis the interaction between the strands is governed by a general contact formulation. The proposed algorithm involves the use of both commercial and non-commercial codes as well as a script for reading and generating the different input files needed to export and import the model between the softwares as outlined in Fig. 2.

3



Figure 2. The algorithm of the proposed modelling framework.

Step 1 In the first step a nominal geometry with the lower vs is created in the geometry preprocessor TexGen [16], which has a Python interface. In TexGen the strand paths are defined using spline functions between designated master nodes. A strand cross section is given at each master node, and the change in cross section between master nodes is also described by splines [16]. Previous work with composites containing 3D woven reinforcements indicate that the cross section of the warp strands varies significantly along their trajectories, while the weft strands have a more constant elliptical shape in the RVE. The strand cross section is however here assumed to be circular, avoiding to impose any predefined ellipticity on the cross sections. The input data needed to create the nominal geometry is the strand thickness and length, and the size of the RVE domain. This means that the strand crimp, defined by Pierce [7] as

crimp

Lstrand

(2)

where Lstrand is the strand length and the length of the RVE, is implicitly defined as well. However, the details of the strand trajectories are not part of the input. From these input data, the positions of the master nodes and their cross section area may be obtained. TexGen exports the reduced nominal geometry as a CAD file. Step 2 The CAD file is imported into a mesh generator, where a periodic surface mesh is created after some tidying of the CAD-file. Each imported strand is denoted a master strand. The shape of the RVE of a 3D woven fabric is very complex. A rectangular box is chosen as the boundary of the RVE in this paper, with the consequence that strands from neighbouring RVEs will enter and leave the rectangular RVE. These strands are included in the model by copying and translating the corresponding master strands, creating slave strands, see Fig. 2. For each master warp strand, three slave warp strands are copied and inserted as illustrated in Fig. 3a. The weft strand inside the RVE are divided into two sections and copied periodically as exemplified for two horizontal weft strands in Fig. 3b. Finally, end caps are added to the strand tubes to seal the cavities.

4



Figure 3. The master strands inside the RVE are marked with "M" and their respective slave strands marked with "S".

Step 3 In the third step, all nodes and elements from the mesh generator are first imported to a Python script and organised as objects. The slave strands’ deformation is then coupled to their respective master strands in order to ensure periodic contact conditions in step 4. A sorting algorithm finds the master-slave node pairs. All master nodes that are not on any RVE boundary, i.e. are interior nodes, are coupled with their corresponding nodes on all of their respective slave strands using linear multi-point constraints. Since the geometry on the boundary must be periodic during and after the simulation, the boundary master nodes on one side of the RVE must not only be coupled to their corresponding nodes on all their slave strands but also to the node on the opposite side of the RVE, and corresponding slave nodes. However, a node cannot be both a slave and a master node at the same time, which in practice means that a master node on a warp strand boundary, has a total of seven slave nodes, see Fig. 4.

Figure 4. A strand master node on a RVE boundary, and its corresponding slave nodes.

Since a yarn of fibres is transversely isotropic by nature, the tubes in the model must be anisotropic and hence a material orientation for each element must be assigned. This is done by first locating where on the tube an element is situated, and then assigning to it the orientation given by the local tangent of the strand trajectory spline defined in TexGen (see step 1). The material in the end caps of the tubes is assumed to be isotropic, and has to be compliant enough not to affect the cross-section shape of the tube during the simulation. Finally, in order to apply a pressure inside the tubes and expand them to their target volume, hydrostatic pressure elements are created using the existing nodes. This is done for all master strands. The coupling between the tube shell elements and the hydrostatic elements are ensured by the fact that they share the same nodes. 5



Step 4 The actual simulation is performed using Abaqus Explicit [18]. A general contact algorithm with a “hard” contact interaction model, which minimises the overclosure between the contact surfaces, governs the interaction between all strands. The friction between the tubes in the contact formulation is set to zero. To ensure that the final result does not contain any interpenetrating tubes, and to simplify volume meshing in a subsequent step, a small gap between two tubes in contact is preferred. The size of this gap may be altered by either changing the shell thickness or the contact surface thickness in Abaqus. Increasing the shell thickness, however, also stiffens the strands which affects the strand cross-section shape more than merely changing the contact surface thickness. In a real fabric the strands come in contact where they interlace, hence the distance is zero and a smaller gap would be preferred. On the other hand, if the gap is too small it leads to numerical problems when meshing the entire RVE with volume elements. In practice, this trade-off is governed by how many degrees of freedom (DOF) the user can accept and what quality of the elements is required. As mentioned previously, the desired vs is the stop criterion for the explicit FE-simulation and it is calculated using the target global volume fraction of fibre vf and the target fibre volume fraction of fibre inside a strand vsf as

vf vsf

vs

(3)

Hence, in addition to the geometrical input variables, vf and vsf are also needed to complete the modelling. The pressure inside the tubes are linearly increased until the target vs is reached, at a rate low enough to ensure that no inertia effects will influence the final result. In order to make sure that the relative growth of the warp and weft strands is similar, the ratio between the shell thickness, h, and the radius, r, of the tubes is the same for different tubes:

rweft

rwarp

hweft

hwarp

.

(4)

The ratio between the pressure, p, and the cross section area, A, is also kept consistent:

pweft

pwarp

Aweft

Awarp

.

(5)

4 Validation of geometry modelling approach To validate the proposed methodology, a sample 3D weave is woven. The sample weave’s cross-section has to be large enough to ensure that it contains RVEs in the interior of the weave, i.e that are unaffected by surface effects. The benchmark weave contains 8x8 rows and columns of 12k warp yarns, 8 horizontal and 8 vertical 6k weft yarns per weave cycle. An RTM mould with a square cross-section is used to manufacture the composite sample using vinyl ester. The sample is sectioned, polished and photographed. The photographs are used to verify that the deformed geometry from the geometry modelling corresponds to the actual internal geometry of the composite. Input data to the model is compiled and presented in Table 1. The fibre volume fraction is 6



estimated from fibre weight before, and composite weight after, manufacturing, and dry yarn thicknesses from yarn data sheets. The photographs are used to estimate the strand crimp and global volume fraction of strand, vs. Warp Weft Entire model

Af [mm2] 0.46 0.23 -

Crimp [%] 1.055 1.000 -

vf [-] 0.40

vs [-] 0.61

RVE-size [mm3] 11.0 x 2.5 x 2.5

Table 1 Input data to the geometry simulation

Two material models are used in the FE-simulation. One for the elements along a strand trajectory (strand elements), which are modelled using a lamina material model, and one for the elements on the tube end caps, which are assumed to be isotropic. For the strand elements the ratio between the Young’s moduli in the strand direction and the transverse direction should be sufficiently high to ensure that the strands are not deformed in the strand direction, altering the crimp. The elements on the end caps need to be compliant enough to not influence the simulated geometrical shape of the strand elements. The used material parameters are given in Table 2. Strand elements

EL/ET EL/GLT EL/GTT

End cap elements

EL/Eend caps

LT

1000 2000 2000 0.3 20000

Table 2 Material data for the geometry simulation, subscript L and T denotes longitudinal and transversal direction respectively.

The stiffness data is given as ratios since the absolute values are insignificant. The input data is entered into the model, and steps 1-4 are executed. View cuts from Abaqus are generated and overlaid on photographs of sections from the sample. The strand cross-sections of the sample composite are traced with white lines where appropriate, to enhance visibility in the presentation. A comparison between the simulated internal geometry and the composite sample geometry along a cut in the weave direction is presented in Fig. 5. The simulated warp strand trajectories match the sample geometry almost perfectly. The positions of the weft strands are not clearly visible in Fig. 5, which makes the comparison of the weft strands difficult in this view.

Figure 5. Photograph of the sample composite and modelling results on a cut along the weave direction.

7



Results from two cuts perpendicular to the weave direction are presented in Figs. 6 and 7. In Fig. 6, the cut is made where the warp strands interlace with the horizontal weft strands, and in Fig. 7 the cut is made in between sets of horizontal and vertical weft strands. The sample and the simulated strand geometries are in good agreement also at these sections. As can be seen, and expected, the cross-section shapes are less regular in the composite sample than in the model.

Figure 6. Photograph of the sample composite and modelling results on a cut perpendicular to the weave direction where the warp interlaces with the horizontal weft. White lines have been added to highlight the true strand boundaries, for enhanced visibility.

Figure 7. Photograph of the sample composite and modelling results on a cut perpendicular to the weave direction in between the horizontal and vertical weft. White lines have been added to highlight the true strand boundaries, for enhanced visibility.

4 Conclusion The basis of all constitutive modelling is an accurate geometry description of the modelled material structure, which is one of the major challenges when a FE approach is chosen for modelling of composite materials. This paper takes a novel approach to generate the internal geometry. The method is based on a simplified and reduced nominal strand geometry, which is subsequently enlarged using pressure under general contact conditions. A comparison with a sample material confirms that the proposed methodology can successfully be used for modelling of a complex textile reinforcement geometry.

8



In reality, the yarn tension in the loom, as well as friction, also affect the dry yarn trajectories and cross-section shapes. These properties may be implemented in the algorithm in the future if needed. However, good agreement is shown when only a few input parameters are used. In order to further validate the proposed methodology a parameter study will be performed to see how different material input parameters, and contact thicknesses affect the final strand trajectories and cross-section shapes. The obtained internal geometry may be used to create FE-models of RVEs to either calculate the materials constitutive properties, or to be used in permeability simulations. The 3D-woven preforms were supplied by Biteam AB, Sweden. Acknowledgement The presented work has been financially supported by the European Commission through the EU FP6 contract no AST5-CT-2006-030871 Modular Joints for Composite Aircraft Components, MOJO. References [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12]

[13]

[14]

Modular joints for composite aircraft components. www.projectmojo.eu Z. Wu. Three-dimensional exact modeling of geometric and mechanical properties of woven composites. Acta Mechanica Solida Sinica, 22(5):479 – 486, 2009. E. V. Larve, D. H. Mollenhauer, E. G. Zhou, T. Breitzman, and T. J. Whitney. Independent mesh method-based prediction of local and volume average fields in textile composites. Composites Part A: Applied Science and Manufacturing, 40(12):1880 – 1890, 2009. Special Issue: CompTest 2008. B.N. Cox and M.S. Dadkhah. Macroscopic elasticity of 3D woven composites. Journal of Composite Materials, 29(6):785 – 819, 1995. C. Chapman and J. Whitcomb. Effect of assumed tow architecture on predicted moduli and stresses in plain weave composites. Journal of Composite Materials, 29(16):2134– 2159, 1995. T.V. Sagar, P. Potluri, and J.W.S. Hearle. Mesoscale modelling of interlaced fibre assemblies using energy method. Computational Material Science, 28(1):49–62, 2003. F.T. Peirce. Geometry of cloth structure. Journal of the Textile Institute, 28(3):45–96, 1937. J.L. Kuhn and P.G. Charalambides. Modeling of plain weave fabric composite geometry. Journal of Composite Materials, 33(3):188 – 220, 1999. S. Z. Sheng and S.V. Hoa. Modeling of 3D angle interlock woven fabric composites. Journal of Thermoplastic Composite Materials, 16(1):45 – 58, 2003. G. Hivet and P. Boisse. Consistent 3D geometrical model of fabric elementary cell. application to a meshing preprocessor for 3D finite element analysis. Finite elements in analysis and design, 42:25–49, 2005. Y. Mahadik, K.A. Robson Brown, and S.R. Hallett. Characterisation of 3d woven composite internal architecture and effect of compaction. Composites Part A: Applied Science and Manufacturing, In Press, Accepted Manuscript, 2010. S.V. Lomov, A.V. Gusakov, G. Huysmans, A. Prodromou, and I. Verpoest. Textile geometry preprocessor for meso-mechanical models of woven composites. Composites Science and Technology, 60(11):2083 –2095, 2000. I. Verpoest and S. V. Lomov. Virtual textile composites software WiseTex: Integration with micro-mechanical, permeability and structural analysis. Composites Science and Technology, 65(15-16 SPEC ISS):2563–2574, 2005. M. Sherburn, A. Long, and A. Jones. Prediction of textile geometry using strain energy 9



[15] [16] [17]

[18]

minimisation. In Proceedings of the 1st World Conference On 3D Fabrics, 2008. N. Khokar. 3D-weaving: Theory and practice. Journal of the Textile Institute, 92(1):193–207, 2001. M. Sherburn. Geometric and Mechanical Modelling of Textiles. PhD thesis, The University of Nottingham, July 2007. S.V. Lomov, D.S. Ivanov, I. Verpoest, M. Zako, T. Kurashiki, H. Nakai, and S. Hirosawa. Meso-FE modelling of textile composites: Road map, data flow and algorithms. Composites Science and Technology, 67(9):1870 – 1891, 2007. Abaqus. http://www.simulia.com/.

10