18TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS
MACRO-SCALE MODELLING OF DISCONTINUOUS FIBRE COMPOSITES 1
R. Luchoo1, L. T. Harper1*, M.D. Bond1, A. Dodworth2, N.A. Warrior1 Division of Materials, Mechanics and Structures, The University of Nottingham, UK 2 New Technology Department, Bentley Motors Ltd, Crewe, UK * Corresponding author (
[email protected])
Keywords: Discontinuous, Random, Carbon fibre, Macro modelling, Mixed-dimensional 1 Introduction Bentley Motors Ltd are developing a fully automated process for producing discontinuous, long fibre composites for structural applications [1]. Fibre architectures can be tailored locally, using variable fibre length, bundle size, level of fibre alignment and fibre volume fraction (VF) [1, 2]. In-plane tensile stiffness and strength values have been shown to approach those of quasi-isotropic noncrimp fabric. Mechanical characterization of these materials however, is challenging, due to high levels of local heterogeneity and the compounding effects of the aforementioned parameters. This causes uncertainty for designers and is preventing wider commercialization. An accurate numerical model is therefore essential if these materials are to be adopted for structural applications, reducing the cost of experimental testing programmes and enabling optimal designs.
2 Background The primary obstacle in developing a representative numerical model is the requirement to model substantially larger geometric volumes for the random fibre architecture. Unlike textile reinforcements, there is no repeating unit cell; therefore a representative volume element is adopted which can be of structural dimensions. This proves particularly difficult at higher fibre volume fractions (~50%) due to the computational expense and the complexity of the numerical algorithms required to successfully deposit bundles without selfintersection. It is also difficult to generate quality meshes at higher volume fractions for finite element modelling. Whilst fibres should not intersect or make contact, a minimum separation distance is
required for 3D geometries to ensure finite elements do not become distorted. Forcing a gap between fibres can lead to incorrect failure strengths, as the model does not capture local failure of the matrix or fibre-matrix interface. The local stress concentration factor increases as the gap size decreases between fibres and this is heavily influenced by neighbouring fibre pairs in high volume fraction composites [3]. Computationally inexpensive 2D models [4] have previously been developed, using 1D linear beam elements to represent fibre bundles. It is possible to analyse large volumes (200×200×3mm) and predictions are reported to be within ~10% and ~15% for tensile modulus and strength respectively [1]. Simulations are suited to in-plane tension due to the 2D fibre architecture, but cannot capture the effects of fibre crimp which can dominate the compressive properties. Commercial software (Digimat®) extends the beam model to 3D and introduces a degree of fibre tortuosity. Architectures are not representative of automated processes as the curvature of each fibre is independently induced, which limits the packing efficiency and results in low fibre volume fractions (<20%). Pan [5] has made progress in explicitly modelling the fibre geometry in 3D offering improved accuracy, yet the volume fractions are still limited to 35% due to fibre jamming during random sequential fibre adsorption. In addition, this approach causes local changes to the fibre distribution and does not capture the effects of compaction and fibre redistribution in areas of high volume fraction. Reproducing fibre architectures from automated deposition processes (Fig. 1) has been demonstrated in [6, 7] using image analysis. However, little consideration has been given to the influence of the
moulding process which affects the final level of crimp, tortuosity and lateral movement of fibre to areas of lower volume, depending on whether tooling is matched or single-sided. These are important features, since they enable higher and more realistic global volume fractions to be created. It was previously shown in [7] that the in-plane tensile modulus of the composite can be affected by up to 20% when the level of out of plane curvature is increased from a planar 2D distribution.
framework. The model is applicable to a number of meso-scale discontinuous fibrous materials, such as ASMC, P4, DCFP and BRAC3D [1, 2, 9-12], and can be expanded to receive fibre positional data from process-driven models such as in [13, 14]. An electromagnetic force-directed algorithm has been implemented, which has been combined with a number of deposition, compaction, intersection and spline interpolation algorithms to provide a noncontacting network of fibres, randomly distributed and smoothly interpolated within 3D space. 4.1 Force direction algorithm
Fig. 1 A discontinuous fibre preform from an automated deposition process (prior to consolidation)
3 Scope of paper A geometrical modelling schema is presented to produce representative discontinuous fibre architectures for downstream prediction of mechanical properties. A force-directed attractionrepulsion algorithm is used to redistribute the fibres as pressure is applied from the matched mould tool, in order to achieve realistic fibre volume fractions. Constituent materials are modelled at the meso-scale using a mixed-dimension modelling technique [8] with a view to explore which approach offers the best solution in terms of accuracy and computational efficiency. Two modelling strategies are examined by representing fibres as either 1D truss elements (case 1) or 2D shell elements (case 2), constrained within 3D continuum elements (matrix).
4 Modelling schema A versatile numerical modelling program has been developed in C# within the Microsoft .NET 3.5
Birbil and Fang [15] demonstrated a global optimisation method for a network of nodes using an attraction-repulsion mechanism. Each point is electromagnetically charged (with charge Q), repelling any neighbouring point in close proximity under Coulomb’s inverse square law (eqn. 1). Dispersion of the points can be controlled by connecting certain nodes via springs, causing attraction forces under Hooke’s law (eqn. 2), to produce a structured set of points, positioned in harmony. The objective of this approach is to obtain a spatially optimised distribution of points, which is commonly used for aesthetics in tree diagrams as well as many energy based physics applications. Its application has been extended to the development of this model by generating spatially optimised fibre architectures from a collection of charged nodes in Euclidean space. Note that all nodes in the current program are of equal charge (Q1 = Q2) reducing eqn. 1 to eqn. 3. FREPULSION = kREPULSION(Q1Q2/r2)
(1)
where kREPULSION = Coulomb force constant, r = The separation distance between two nodes FATTRACTION = −kATTRACTIONδL
(2)
where kATTRACTION = Spring constant δL = Change in natural spring length FREPULSION = kREPULSION/r2
(3)
4.2 Nodal network generation For case (1), the spline of the truss is created from a series of charged nodes, defined as sister nodes, distributed along the length of the fibre to provide cylindrical trusses, as detailed in Fig. 2. Each node
18TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS
along the fibre axis is connected by a sister spring (stiffness KSISTER), providing an attraction force to maintain the original longitudinal fibre length and prevent fibre elongation. Initial fibre positions on the X-Y deposition plane (Fig. 2 - inset image) are retained by connecting each sister node to two additional nodes fixed to each RVE cell surface. This relationship, classified by a connecting parentchild spring (where KPARENT-CHILD << KSISTER), enables both through-thickness and lateral displacement of fibres. Smooth fibre undulations are provided by interpolating a Catmull-Rom spline through the fibre sister nodes, where spline quality is controlled by a minimum distance (SMin) between sister nodes. Note the presence of two additional (dummy) sister nodes at each fibre end (Fig. 2) due to the minimum number of points required for spline interpolation as detailed in [16].
nodes in very close proximity to fold through π when attempting to reach an energy minima. F = −kAXIAL(ΘAXIAL – Θ)2 where natural spring angle ΘAXIAL = π kAXIAL = Axial rotational spring constant 4.3 Shell fibre creation For case (2), fibre shells are represented by four connected trusses to produce a rectangular fibre shell as detailed in Fig. 3. Similar intersection algorithms to case (1) have been used, however, in the event of an intersection on a longitudinal shell edge a twin relationship is created. An additional twin node, in addition to the intersection node, is created on the adjacent longitudinal shell edge, directly opposite (Fig. 3). Each node is connected via a twin spring KTWIN, where KTWIN ~ KSISTER, and prevents lateral spreading of the fibre (eqn. 5). While this mechanism is commonly witnessed in the manufacture of fibre reinforced composites, this effect is neglected in the current work to reduce complexity. Twin relationships are enforced for all sister nodes residing along a longitudinal fibre edge. An additional rotational spring is located at each sister node and connected to its corresponding twin and a neighbouring sister node (eqn. 6). This prevents fibre shear and the tendency for the shell to form a parallelogram and in some instances collapse when trying to reach an energy minima. F = -kTWIN(LTWIN - L)
Fig. 2 A truss fibre architecture, generated from a network of connected springs and electromagnetised nodes
In the event of an intersection between two fibres on the X-Y plane (Fig. 2 - inset image), an intersection node is created on each fibre at the X-Y intersection point, and connected via a parent-child spring to repel each fibre and prevent intersection. Remaining sister nodes are equidistantly redistributed along the fibre axis, based on SMin, after each inclusion of an intersection node. Fibre curvature and crimp is controlled by a rotational spring located at each sister node and connected to its’ neighbouring sister nodes. A squared relationship (eqn. 4) is required to permit moderate levels of fibre curvature yet prevent sister
(4)
(5)
where natural spring length (LTWIN) is the transverse fibre width F = −kTRANSVERSE(ΘTRANSVERSE – Θ)2
(6)
where natural spring angle, ΘTRANSVERSE = π/2 kTRANSVERSE = Transverse rotational spring constant Additional intersection detections are required in the event that a sister, twin or corner node intersects the shell surface of another fibre on the X-Y deposition plane. An internal node (Fig. 3) is tied to the intersected shell surface and replaces the RVE cell surface node residing behind the intersected shell. Note that all repulsion and attraction forces exerted on the internal node are redistributed to its neighbouring nodes on each longitudinal shell edge.
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Fig. 4. A damping factor (D = 1x10e-5), applied to the magnitude component of each resultant vector, is required to prevent system instability arising from high node repulsion forces and also aids dynamic visualisation via the program GUI. Total system energy for a given iteration step is characterised by the summation of displacements for each node and iterative steps are performed until a user-specified minimum energy criterion is met.
4.4 Implementation A similar deposition algorithm to [7] is employed where fibres are assigned a random Cartesian coordinate and planar orientation in the X-Y plane. The relevant intersection algorithms are performed and a Euclidean nodal network is generated.
4.5 Compression The aforementioned algorithms have been implemented for two fibre architecture states, uncompressed and compressed, and the results are displayed in Fig. 5 after the energy minimum criterion has been met.
Fig. 3 A shell fibre architecture, created by connecting four trusses, and their corresponding nodal network
Fig. 4 The resultant vector as a product of all repulsion and attraction forces on a given node
Upon creation of the nodal network, the forcedirected algorithm is invoked and iteratively called until the dynamic system reaches a state of minimum energy. Within each iterative step, the net force exerted on each node is calculated, as a function of all forces (eqns. 2-6) caused by neighbouring nodes and connected springs, and thus provides a resultant Euclidean vector as shown in
Fig. 5 Uncompressed (Top) and compressed (Bottom) shell fibre architecture under the minimum energy criterion
The uncompressed state represents fibre architectures resulting from automated discontinuous performing techniques, prior to consolidation (Fig. 1). This can be a useful tool for characterising preform loft and B-surface waviness
18TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS
for a vacuum infusion part. Cell cavity height and hence distance between each pair of cell surface nodes, is significantly increased providing a sufficient volume for nodes to disperse through thickness and reach their positions of minimum energy. This results in fibres with significantly higher levels of fibre curvature when compared with the compressed state. The cavity height is reduced in the compressed state to simulate a closed moulding process. Reduced levels of fibre curvature and crimp are present as a result of significant fibre ‘flattening’ through reducing the distance between cell surface nodes to that of the moulding cavity height. Fig. 6 A compressed RVE (cropped) with a truss fibre architecture (Case 1)
5 RVE Generation An RVE of size 70x70x2mm, nominal fibre length 36mm and fibre aspect ratio 4.5, has been generated. Implementation of the developed algorithms is demonstrated in Fig. 6 for case 1 (1D trusses) and Fig. 7 for case 2 (2D Shells). Fibres were initially deposited over an area which was two fibre lengths longer and wider than the required RVE. This ensures that both bridging and ending fibres are captured within the RVE. A Liang-Barsky clipping algorithm has been modified for polygon clipping to produce the the cropped RVE architectures. A Delaunay meshing algorithm has been employed to manage meshing of irregular polygon surfaces. All algorithms have been successfully implemented producing an otpimised distibution of fibres through thickness, without fibre self-intersection, at a consolidation cavity height of 2mm (10mm uncompressed). The model provides a platform for introducing a larger number of moulding mechanisms. These include realistic lateral displacement by continuous detachment and reattachment of nodes as fibres laterally displace to areas of lower energy, as well as fibre spreading as compaction forces and fibre volume fractions are increased. Note that some degree of lateral displacement is permitted by parent-child spring extensions in the X-Y plane, but is restricted by the fixation of the surface nodes to the RVE surfaces.
Fig. 7 A compressed RVE (cropped) with a shell fibre architecture (Case 2)
6 Mixed dimension analysis Both mixed dimension modelling strategies (cases 1 and 2) have been developed to evalute the trade-off between accuracy and computational efficiency. Independent fibre meshes (1D truss and 2D shell) for each case are constrained with 3D continuum elements using the *EMBEDDED ELEMENT technique in ABAQUS/Standard. This method provides a simple way of coupling two independent meshs via multi-point kinematic constraints to tie the fibres to the matrix, overcoming the need for complex intersection and meshing algorithms. Further to these advantages, it enables high fibre volume fractions (>50%) to be realised, with no restriction on fibre bundle aspect ratio.
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The next development step is to create a custom meshing algorithm with coincident nodes for both fibre and resin and create relevant tie constraints/relationships.
7 Conclusions A numerical model has been developed, offering the ability to characterise a number of meso-scale discontinuous fibre architectures. A variety of algorithms have been integrated to produce a network of electro-magnetised nodes, structured by an ordered set of springs, and provides a globally optimised distribution of non-contacting fibres for downstream modelling of mechanical results. RVEs are generated using a mixed dimension modelling technique where fibres are represented as either 1D trusses or 2D shells, constrained within a 3D continuum. Realistic fibre tortuosity is captured within the model and two fibre architecture states, uncompressed (preform) and compressed (consolidated), are simulated.
Acknowledgements This work was funded by Bentley Motors Ltd, as part of the RAYCELL project. The authors also wish to acknowledge the financial support of the IMechE and SAMPE UK.
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5. Y. Pan, L. Iorga, and A. Pelegri. Numerical generation of a random chopped fibre composite rve and its elastic properties. Composites Science and Technology, 2008. 68(13): p. 2792-2798. 6. L.T. Harper, T.A. Turner, N.A. Warrior, J.S. Dahl, and C.D. Rudd. Characterisation of random carbon fibre composites from a directed fibre preforming process: Analysis of microstructural parameters. Composites Part A: Applied Science and Manufacturing, 2006. 37(11): p. 2136-2147. 7. R. Luchoo, L.T. Harper, M.D. Bond, N.A. Warrior, and A. Dodworth. Three dimensional numerical modelling of discontinuous fibre composites for high performance applications. Plastics, Rubbers and Composites: Macromolecular Engineering, 2010. In Press. 8. J. Cuilliere, S. Bournival, and V. Francois. A meshgeometry-based solution to mixed-dimensional coupling. Elsevier, 2010. Computer-Aided Design(42): p. 509-522. 9. Hexcel. Product data sheet - carbon epoxy hexmc / c / 2000 / r1a. 2008, Hexel Corporation. 10. Menzolit. Preliminary data sheet - menzolit advancedsmc 1300. 2004. 11. Yla. Product bulletin - ms-4a. 2009, TenCate Advanced Composites USA Inc. 12. L.T. Harper, T.A. Turner, N.A. Warrior, and C.D. Rudd. Automated preform manufacture for affordable lightweight body structures. in 26th International SAMPE Europe Conference. 2005. Paris. 13. N.A. Warrior, C. Patel, and T.A. Turner. Advances in discontinuous composites. 2011, SAMPE: Paris. 14. H.C. Chen, N. Xi, S.K. Masood, Y. Chen, and J.S. Dahl. Development of automated chopper gun trajectory planning for spray forming. Industrial Robot, 2004. 31(3): p. 297-307. 15. S.I. Birbil and S. Fang. An electromagnetism-like mechanism for global optimization. Journal of Global Optimization, 2003. 25: p. 263-282. 16. E. Catmull and R. Rom. A class of local interpolating splines, in Computer Aided Geomtric Design. 1974, Academic Press, Inc.: Salt Lake City.
