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Composites: Part A 35 (2004) 637–643 www.elsevier.com/locate/compositesa
A simplified finite element model for draping of woven material S.B. Sharma, M.P.F. Sutcliffe* Department of Engineering, Cambridge University, Cambridge CB2 1PZ, UK Received 29 August 2003; revised 16 February 2004; accepted 24 February 2004
Abstract A model for draping of woven fabric is described which is intermediate to the two main approaches currently used, kinematic modelling and comprehensive finite element modelling. The approach used is a simple unit-cell finite element model, with the fabric modelled by a network of simple truss elements, connected across the ‘diagonals’ by soft elements to mimic the shear stiffness of the material. Bias extension tests on dry fabric are used both to calibrate and validate the model. The model is then used to simulate draping of an ellipsoid. The capability of the model is illustrated by construction of a simple process optimisation map for this geometry. q 2004 Elsevier Ltd. All rights reserved. Keywords: A. Fabrics/textiles; Carbon fibre; C. Finite element analysis; E. Forming; Draping
1. Introduction Woven fabrics have been increasingly used in composite material applications because of their attractive forming characteristics and mechanical properties. To a first approximation the material undergoes pure shear during draping, with the tows rotating freely at the crossover points—the pin-jointed net (PJN) approximation [1 – 4]. Advantages of this approach (termed ‘kinematic’ modelling) are simplicity and speed of calculation, making it suitable for commercialisation [5]. However, it is likely to be increasingly inaccurate for components which do not have uniformly curved surfaces, and ignores fabric properties which are known empirically to play an important role. For example, while it has been shown to model the broadbrush deformation for a helmet component, it is not able to model tow slippage or mimic details around the ‘ear’ features on this component [6,7]. It is expected that in-plane membrane forces and normal tool contact forces developed during draping will influence the changes in the tow architecture and hence change the forming response and final material properties [8]. Biaxial tests have been reported by Boisse et al. [9] and Sharma et al. [10] to account for the effects of membrane stresses on fabric deformation. * Corresponding author. Tel.: þ 44-1223-332996; fax: þ 44-1223332662. E-mail address: [email protected] (M.P.F. Sutcliffe). 1359-835X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2004.02.013
To model these effects it is necessary to use a more sophisticated approach. An alternative approach uses a comprehensive finite element (FE) method to model the material behaviour during draping. For example ESI PAM/FORM [11] uses several layers of 2D elements to represent the fabric. This program requires several input parameters such as the blank holder pressure, fabric shear stiffness, tool velocity, ply/tool and ply/ply friction coefficients. It has been successfully used to simulate the interaction of the woven ply with the tool as it is deformed [12]. Although the method overcomes most of the problems associated with the kinematic model, it requires significant computer power and an experienced engineer to drive the software. Examples of research focusing on this approach are listed in Refs. [9,11–15]. The aim of the present work is to provide a drape model intermediate to the above two approaches. An earlier proofof-concept study by the authors described how a kinematic drape simulation could be adapted to include membrane stresses [16]. However, an alternative approach, described in this paper, has been pursued because of the greater flexibility and ease of exploitation that it allowed. This ‘intermediate drape model’ is based on standard FE algorithms, thus exploiting their ability to model complex boundary conditions as well as their user-friendly input and output interfaces. The model uses a novel unit-cell implementation of the fabric, with material properties derived from standard tests. The bias extension test is
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used both to extract appropriate material shear properties from a ‘gauge section’ where the PJN approximation is good, and additional material parameters from consideration of regions near the grips where the PJN model is less accurate. The model is implemented in an FE code and applied to draping of an ellipsoid.
Conversion between the element stiffness EL and the corresponding effective modulus E0L of the fabric along the tow direction is straightforward, requiring comparison of forces over the unit cell. The effective modulus E0L is defined by E0L ¼
2. Unit cell model Fig. 1 shows the undeformed square unit cell of side length D used to represent the material. The four outer elements model the tows while the diagonal element endows the cell with shear stiffness. The four tow elements are connected at the corners via pin joints, to allow the trellising action observed in practice. All elements are modelled using two-noded 3D truss elements of area A: The strain 1 in both types of element follows the Green – Lagrange definition of strain used by the MARC.MSC FE implementation adopted [17,18] u 1 u 2 1¼ þ ð1Þ 2 D0 D0 Here u and D0 are the corresponding extension and initial length of an element. With this strain formulation and assuming small extensions of the tow elements, the strains are equal and opposite for the two diagonals of the unit cell, so that the same material response can be used for the diagonal element, irrespective or whether it is in compression or tension. Because it is difficult to measure the thickness of fabric material accurately, estimates of the applied stress are not straightforward. Instead we follow textile practice and characterise the applied loads using forces per unit length, or ‘line forces’. This can be separated into components NL and NS acting along the tows and in the shear direction, respectively [10]. The tow elements are given an appropriate constant stiffness EL to allow for decrimping or tow slip during deformation, by fitting the simulation to the measured bias extension deformation response, as described in Section 3.
NL 1L
ð2Þ
where 1L is the strain along the tow direction. Since each tow element carries a load equal to the product of the applied load NL and the side length D of the unit cell, Hooke’s law for the element gives EL ¼
NL D A1L
ð3Þ
so that the tow element stiffness EL corresponding to a fabric effective modulus E0L is given by EL ¼
E0L D A
ð4Þ
The effect of changes in geometry (e.g. large deformations) on the tow elastic modulus are not included, nor is the small contribution from the shear element to stretching along the tow directions. The diagonal element is given a non-linear stress strain response sS ¼ f ð1S Þ; which is implemented here using a specified stress – strain curve within the elastic – plastic option of MARC. This curve is derived from the material response Nx ¼ f1 ð1x Þ; measured along a bias direction (see Section 3), where Nx is the nominal line load applied in the bias direction (i.e. applied force divided by original specimen width) and 1x is the corresponding strain in the bias direction. Again the element and corresponding material responses can be related by considering forces on the unit cell. As the unit cell is pin-jointed, the only resistance to elongation along the bias direction is provided by the shear element. Equating pffiffi the external force applied to the unit cell of initial width 2D with the force in the shear element gives pffiffi N x 2D ¼ As s ð5Þ By using a nominal line load formulation, changes in unit cell width do not need to be considered in the above expression. Hence, the element response sS ¼ f ð1S Þ is related to the material response Nx ¼ f1 ð1x Þ via a simple scale factor pffiffi 2D f ð1 Þ ss ¼ ð6Þ A 1 s
Fig. 1. Unit cell model of material behaviour.
The ‘bracing’ nature of this element effectively gives the material a non-linear shear stiffness. The exact form of material response Nx ¼ f1 ð1x Þ used for the shear elements in the simulation is given in Fig. 4, derived from the material response of the ‘gauge section’ of a bias extension test. In practice the area of each element was set to unity,
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and the appropriate stiffness derived for the corresponding mesh density and material properties using Eqs. (3) and (6).
3. Measurement of material parameters for a dry fabric Fabric tests are described in this section which are used to estimate values for the tow and shear springs of the unit cell model. Material property data for the simulation are based on measurements on Tenax HTA 6k carbon fibre dry fabric, with 5% binder, in a five-harness satin weave, manufactured by Hexcel Composites. Picture frame and bias extension tests have been widely used to characterise the shear response of dry fabric, prepregs and thermoplastic fabrics, and results from the two types of test have been compared recently by Sharma et al. [10] and Harrison et al. [19]. In a picture frame test the fabric is clamped in a square frame and deforms uniformly as the frame is deformed into a rhombus shape. This test provides a direct strain measurement but the stress calculation is less reliable due to unknown tension in the fabric. Although the picture frame test would provide suitable shear data [19], here we use instead the bias extension test. In this test the fabric specimen is elongated along the bias directions (i.e. ^ 458 to the warp and weft directions), as shown in the sketch of Fig. 2. The specimen has initial length L ¼ 200 and width W ¼ 75 mm and the grips are moved apart at a constant relative velocity V ¼ 60 mm/min. Further details of the bias extension tests used in the present work are given in Ref. [10]. The bias extension test was chosen for the work because the axial tow stresses are well characterised, allowing extraction of a suitable axial tow stiffness, and indeed are probably more representative of those present during draping. The nonuniformities in shear arising in this test were treated using suitable image analysis, and in fact provided an opportunity to extract suitable material properties from different regions of the specimen. Tests on the material at 60 and 100 mm/min produced similar results, indicating that an effect of binder on the shear response was small. Although some influence of binder cannot be ruled out, it is expected to be relatively small, certainly in comparison with the effect of matrix in prepreg material. The fabric showed some
Fig. 2. Sketch of the bias extension test.
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asymmetry in shear (see for example the deformed specimen, Fig. 6). However, the effect appeared to be relatively minor and has not been considered in detail. If the deformation is modelled using pin-jointed analysis then the constraints at the grips lead to rigid triangular regions at the two ends of the specimen. The apex of these triangles is termed here the ‘stationary point’, as it is supposed using the pin-jointed analysis to remain stationary with respect to the grip. The central gauge section of the specimen undergoes uniform shear, while there is a triangular ‘half-shear’ region towards the ends of the specimen. The material response for this gauge section is used to find an appropriate stress –strain curve for the diagonal element in the unit-cell model (Section 2). The deformation during the test of a grid marked on the central 50 £ 50 mm2 gauge section is recorded photographically and these images are used to measure the shear angle u in the mid-section. The longitudinal strain again follows the Green –Lagrange definition [17], given by: 1x ¼
ux ‘0
þ
1 ux 2 2 ‘0
ð7Þ
where ux is the gauge section extension measured from the images and ‘0 is the gauge section length at the start of the test. The nominal applied line force Nx is given by dividing the measured force F by the initial specimen width W0 Nx ¼ F=W0
ð8Þ
The force– displacement response for the bias extension specimen, over the whole length of the specimen, is shown in Fig. 3, showing significant stiffening (lock-up) at a displacement of around 60 mm. The corresponding nominal line force– strain response, derived now from measurements only over the gauge section, is given in Fig. 4. This shows an initial low stiffness region, followed by a large increase in stiffness at a strain of about 0.5, corresponding to a shear deformation angle of about 408. The relationship plotted in Fig. 4 is used, via the scaling factor of Eq. (6), to give the material properties for the diagonal elements in the FE model.
Fig. 3. Typical force–displacement curve over the whole specimen length for a bias-extension test.
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Fig. 4. Nominal line force–strain response taken from the gauge section of a bias extension test. This function is used, via the scaling factor given in Eq. (6), to describe the mechanical response of the shear element of the unit cell. The dashed line shows a linear stiffness of 15 kN/m, to allow comparison with the stiffness found appropriate for the tow elements.
In principle it seems reasonable to estimate the fabric stiffness along the tow directions from appropriate tests of specimens loaded along this direction. Although this approach was tried, the values of tow stiffness derived were found to give a poor fit to the data. It is believed that there were two reasons for this; firstly that an appropriate stiffness should be taken from only the initial decrimping part of the load– displacement response (since this corresponds to the loads typically found during draping), while in fact much higher stiffnesses corresponding to higher loads and extension of the fibres were measured. Secondly, it is likely that fibre slip may play an important role in the bias extension tests used to validate the model. Since the conditions in the bias extension test are closer to draping conditions, the stiffness extracted from these tests, as described in Section 4, was preferred. 4. Validation using bias extension test Detailed analysis of the bias extension tests presented here [8] has shown that the behaviour of the central gauge section is well modelled using the PJN assumption, while there is significant deviation from this model at the ends of the specimen. While the gauge section provides the shear response of the material, the overall deformation of the specimen and the behaviour near the grips provides a way of extracting the tow stiffness and helping validate the overall model. To simulate the bias extension test, it is modelled using thepunit ffiffi cell approach, taking a unit cell with diagonal length 2D ¼ 5 mm. The material used both for the experiments and the simulation is the dry woven CFRP described above. The FE specimen has the same size as the real specimen (200 £ 75 mm2). Fig. 5(a) and (b) plot the variation with end displacement of the width and shear angle of the central gauge section. The simulation results are shown for three values of the tow stiffness, and compared with the PJN analysis. The simulations results approach the PJN analysis at sufficiently large values of tow stiffness E0L : Experimental results show
Fig. 5. Comparison between measured and theoretical change in specimen geometry as a function of overall specimen displacement U during a bias extension test, (a) gauge section width W, (b) shear angle u in the gauge section.
significant deviations from the PJN analysis at large deformations. These deviations are well modelled using a tow element stiffness E0L ¼ 15 kN/m. Further analysis shows that the central gauge section is effectively behaving in a pin-jointed manner [8], so that the observed deviations from the PJN analysis are due to the behaviour near the ends of the specimen (for example, the significant slippage observed in Fig. 6). A straight line with slope corresponding to this value of tow stiffness is plotted on the load – strain response of the diagonal elements Fig. 4, to allow a direct comparison between the stiffnesses of the tow and diagonal elements. This figure shows how, with a value of tow stiffness E0L ¼ 15 kN/m, the shear stiffness of the material is small compared with the tow stiffness until lock-up is approached at a strain of about 0.5. At that point the material exhibits significant shear resistance and departs from PJN behaviour. A more critical test of the model can be made by investigating the variation in shear angle along the length of the specimen. As indicated in Section 3 above, there is a region near the grips which is nearly rigid, while the central region undergoes nearly uniform shear. This is illustrated in the photo of the specimen after an elongation of 40 mm, Fig. 6. Although the shear response of the material has been extracted using the centre of the specimen, nevertheless
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Fig. 6. Bias extension specimen after an extension U ¼ 40 mm, illustrating the grid used for measuring deformations outside the gauge section.
the significant non-uniform deformation which occurs near the grips is sensitive to the in-plane response of the fabric and acts as a useful test of the overall material model. Fig. 7 compares measurements and predictions of the variation of shear angle along the centre line of the specimen after an extension U ¼ 40 mm. Experimental points are taken from the photographic image, Fig. 6, tracing individual tows in the deformed specimen. The scatter in the measurements reflects the difficulty in estimating the shear deformation from the photograph. In the centre of the specimen, at the right hand side of the graph, the shear deformation is nearly uniform. Near the grip on the left hand side of Fig. 7 there is very little deformation. Predictions using the PJN model give the dashed line rising abruptly at the ‘stationary’ point. Fig. 7 shows that inclusion of a more realistic material model, with again a tow element stiffness E0L ¼ 15 kN/m, allows a good prediction of the shear response through the specimen.
5. Application to draping over an ellipsoid
equal to 150 mm. Two geometric parameters associated with the ply orientation are considered, as illustrated in Fig. 8c. Rotation of the nominally ^ 458 ply relative to the axis of the ellipse is given by the ply orientation angle f; while some pre-shear in the fabric, perhaps generated by inplane forces prior to draping, is modelled by a pre-shear angle a: The material model for dry woven CFRP described above is again used for the drape simulation, with 1600 unit cells of side length D ¼ 15 mm. Two values of the tow axial stiffness E0L of 15 and 8000 kN/m are used. The sheet is held in place at its edges by springs, which here are used to model blank holder forces. These springs are given an effective stiffness of 200 kN/m. A Coulomb friction coefficient of 0.2 is used at all contacts. 5.2. Simulation results Fig. 9(a) shows the effect of tow stiffness on the variation of shear angle along the major axis of the ellipsoid. The arc length has been normalised by the half-perimeter of the ellipsoid, so that a value of one corresponds to the
The intermediate drape model described in this paper is able to take into account processing parameters (e.g. blank holder forces), taking advantage of standard FE capabilities, while being sufficiently rapid (around 350 s per simulation) to permit the many calculations needed when searching for optimum forming conditions. Here we illustrate the value of this approach by considering draping of an ellipsoid, again using an MSC.MARC implementation. 5.1. Description of drape model The geometry is sketched in cross-section in Fig. 8a, and as an isometric view of the deformed geometry in Fig. 8b. The ellipsoidal punch has depth 200 mm and major and minor axes of length 250 and 200 mm, respectively. The sheet of material, which is initially square with side length 600 mm, drapes around the rigid punch as it is driven upwards, making contact with the punch guide at the edges. The travel of the punch after first contact with the sheet is
Fig. 7. Comparison between measured and theoretical variation of shear angle u with position along the centre line of the bias extension specimen, U ¼ 40 mm.
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Fig. 8. Details of ellipsoid drape simulation; (a) arrangement seen in crosssection, (b) isometric view of mesh, (c) definition of initial ply geometry and orientation.
base of the punch. For the lower value of stiffness of E0L ¼ 15 kN/m, corresponding to the value found appropriate from the bias extension tests, the peak shear angle occurs just outside the contact between the ellipsoid and the punch, taking a value of around 368. The effect of much stiffer tow stiffness is to increase the shear in the fabric, as stretching of the fabric becomes more difficult. Fig. 9(b) illustrates the effect of ‘pre-shear’ in the fabric, plotting the change in shear angle along both the major and minor axes of the ellipsoid. The case of no pre-shear corresponds to a ^ 458 layup, relative to the major axis of the ellipsoid. The case with a ‘pre-shear’ angle of 108 represents a uniform shear deformation of the fabric by 108 prior to draping. This might be associated with in-plane stretching prior to draping and is implemented simply by taking an initial ply orientation of ^ 408 relative to the major axis. Again the maximum shear angle occurs just outside the contact between the punch and the material. Application of 108 of pre-shear causes the position of maximum shear to switch from the major axis to the minor axis. Fig. 9(b) suggests that choosing a pre-shear angle of around 58 will lead to a value
Fig. 9. Effect of (a) tow stiffness and (b) pre-shear on the variation along the major and minor axes in the total shear in the material after draping an ellipsoid. The distance along the arc is normalised by the arc length from the centre to the edge of the ellipsoid.
of maximum shear angle below both the no-shear and 108 pre-shear cases. 5.3. Process optimisation Here we illustrate the value of the proposed FE model, by exploring the effect of ply orientation angle and initial preshear angle on the maximum shear induced when draping the ellipsoid. Fig. 10 shows a contour plot of the maximum shear angle in the component as a function of pre-shear and ply orientation angles. The maximum shear angle has a minimum when the bias direction is aligned along the major axis and with about 48 of pre-shear. Pre-shear has a greater influence on the maximum shear angle than does ply orientation angle. Similar process optimisation maps could equally be produced for other process parameters such as
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Company Ltd, MSC.Software, Saint-Gobain Vetrotex, BP Amoco and Hexcel.
References
Fig. 10. Process optimisation map; the contours heights shown on the figure give the maximum shear angle in a given simulation as a function of the ply orientation and pre-shear angle of that simulation.
blank holder forces or material properties (perhaps associated with changes in the pre-preg or with modifications to the mould temperature during forming).
6. Conclusions A simplified FE model is proposed to model draping of composite material. The unit cell consists of a network of pin-jointed stiff trusses, with shear stiffness introduced via diagonal bracing elements. Suitable material properties are extracted from bias extension tests measurements on a dry CFRP fabric. Measurements of the deformation in the central gauge section during a bias extension test are used to extract a suitable stress – strain curve for the diagonal elements. The stiffness of the trusses is estimated by fitting bias extension measurements over the whole specimen length. Predictions of the deformation of a bias extension specimen away from the gauge section are in good agreement with measurements, confirming the ability of the unit cell model to represent the effect of in-plane forces on shear deformation and tow slippage. It is expected that this approach would also be applicable to pre-preg woven material. In this case the flexibility of the model allows incorporation of the more sophisticated shear response needed for such fabrics, for example including rate-sensitive terms. The material response would need to be extracted from experiments or micromechanics-based models [20]. The model is used to predict the draping of an ellipsoid and to illustrate how it might be used for process optimisation. The combination of flexibility in the FE approach and speed of computation allows optimisation studies to be undertaken with only modest computing time.
Acknowledgements The authors gratefully acknowledge the contributions from the Universities of Nottingham and Leeds, EPSRC, BAE SYSTEMS, QinetiQ, ESI Software, Ford Motor
[1] Potter KD. The influence of accurate stretch data for reinforcements on the production of complex structural mouldings. Part I. Deformation of aligned sheets and fabrics. Composites 1979; 161–73. [2] Bergsma OK, Huisman J. Deep drawing of fabric reinforced thermoplastics. Proceedings of the CadComp Conference, Southampton; 1988. [3] Van West BP, Pipes RB, Keefe M. A simulation of the draping of bidirectional fabrics over arbitrary surfaces. J Text Inst 1990;81(4): 448–60. [4] Long AC, Rudd CD. A simulation of reinforcement deformation during the production of preforms for liquid moulding processes. J Engng Manufact Proc Inst Mech Engrs (Part B) 1994;(208):269– 78. [5] Laminate Modeler version 2001. MSC.Software Ltd. [6] Sutcliffe MPF, Sharma SB, Long AC, Clifford MJ, Gil RG, Harrison P, Rudd CD. A comparison of simulation approaches for forming of textile composites. Fifth Int ESAFORM Conf Mater Forming, Kracow, Poland 14– 17 April 2002;311–4. [7] Long AC, Gil RG, Clifford MJ, Harrison P, Sharma S, Sutcliffe MPF. Experimental analysis of fabric deformation during forming of textile composites. Fifth Int ESAFORM Conf Mater Forming, Kracow, Poland 14–17 April 2002;279– 82. [8] Chang SH, Sharma SB, Sutcliffe MPF. Microscopic investigation of tow geometry of a dry satin weave fabric during deformation. Compos Sci Technol 2003;(63):99– 111. [9] Boisse P, Gasser A, Hivet G. Analyses of fabric tensile behaviour: determination of the biaxial tension–strain surfaces and their use in forming simulations. Composites, Part A 2001;32:1395 –414. [10] Sharma SB, Sutcliffe MPF. Characterisation of material properties for draping of dry woven composite material. Composites, Part A 2003; 34:1167–75. [11] PAM/FORM. ESI International, France. [12] de Luca P, Lefebure P, Pickett AK. Numerical and experimental investigations of some press forming parameters of two fibre reinforced thermoplastics: APC2-AS4 and PEI-CETEX. Proc Fourth Int Conf FPCM-96 1996, Aberystwyth 7 –9 Sept 1996;. [13] Lebrun G, Denault J. Influence of the temperature and loading rate on the intraply shear properties of a polypropylene fabric. Proceedings of the American Society for Composites September 2000, Lancaster: Technomic; 2000. p. 659 –67. [14] Cartwright BK, de Luca P, Wang J, Stellbrink K, Paton R. Some proposed experimental tests for use in finite element simulation of composite forming. Proceedings of ICCM-12, Paris, France; 1999. [15] Mohammed U, Lekakou C, Dong L, Bader MG. Shear deformation and micromechanics of woven fabrics. Composites, Part A 2000;(31): 299–308. [16] Sharma SB, Sutcliffe MPF. Draping of woven composites: a progressive drape model. Plast Rubb Compos 2003;32:57–64. [17] Hibbitt HD, Marcal PV, Rice JR. A finite element formulation for problems of large strain and large displacement. Int J Solids Struct 1970;6:1069 –86. [18] MSC.MARC. vol. A. Theory and user information, version 2001. MSC.Software Ltd; 2001. [19] Harrison P, Clifford MJ, Long AC. Shear characterisation of woven textile composites. Proc 10th Eur Conf Compos Mater, ECCM-10, Bruges June 2002; Paper 280. [20] Harrison P, Clifford MJ, Long AC, Rudd CD. Constitutive modelling of impregnated continuous fibre reinforced composites: a micromechanical approach. Plast Rubb Compos 2002;31:76 –86.
A simplified finite element model for draping of woven material S.B. Sharma, M.P.F. Sutcliffe* Department of Engineering, Cambridge University, Cambridge CB2 1PZ, UK Received 29 August 2003; revised 16 February 2004; accepted 24 February 2004
Abstract A model for draping of woven fabric is described which is intermediate to the two main approaches currently used, kinematic modelling and comprehensive finite element modelling. The approach used is a simple unit-cell finite element model, with the fabric modelled by a network of simple truss elements, connected across the ‘diagonals’ by soft elements to mimic the shear stiffness of the material. Bias extension tests on dry fabric are used both to calibrate and validate the model. The model is then used to simulate draping of an ellipsoid. The capability of the model is illustrated by construction of a simple process optimisation map for this geometry. q 2004 Elsevier Ltd. All rights reserved. Keywords: A. Fabrics/textiles; Carbon fibre; C. Finite element analysis; E. Forming; Draping
1. Introduction Woven fabrics have been increasingly used in composite material applications because of their attractive forming characteristics and mechanical properties. To a first approximation the material undergoes pure shear during draping, with the tows rotating freely at the crossover points—the pin-jointed net (PJN) approximation [1 – 4]. Advantages of this approach (termed ‘kinematic’ modelling) are simplicity and speed of calculation, making it suitable for commercialisation [5]. However, it is likely to be increasingly inaccurate for components which do not have uniformly curved surfaces, and ignores fabric properties which are known empirically to play an important role. For example, while it has been shown to model the broadbrush deformation for a helmet component, it is not able to model tow slippage or mimic details around the ‘ear’ features on this component [6,7]. It is expected that in-plane membrane forces and normal tool contact forces developed during draping will influence the changes in the tow architecture and hence change the forming response and final material properties [8]. Biaxial tests have been reported by Boisse et al. [9] and Sharma et al. [10] to account for the effects of membrane stresses on fabric deformation. * Corresponding author. Tel.: þ 44-1223-332996; fax: þ 44-1223332662. E-mail address: [email protected] (M.P.F. Sutcliffe). 1359-835X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2004.02.013
To model these effects it is necessary to use a more sophisticated approach. An alternative approach uses a comprehensive finite element (FE) method to model the material behaviour during draping. For example ESI PAM/FORM [11] uses several layers of 2D elements to represent the fabric. This program requires several input parameters such as the blank holder pressure, fabric shear stiffness, tool velocity, ply/tool and ply/ply friction coefficients. It has been successfully used to simulate the interaction of the woven ply with the tool as it is deformed [12]. Although the method overcomes most of the problems associated with the kinematic model, it requires significant computer power and an experienced engineer to drive the software. Examples of research focusing on this approach are listed in Refs. [9,11–15]. The aim of the present work is to provide a drape model intermediate to the above two approaches. An earlier proofof-concept study by the authors described how a kinematic drape simulation could be adapted to include membrane stresses [16]. However, an alternative approach, described in this paper, has been pursued because of the greater flexibility and ease of exploitation that it allowed. This ‘intermediate drape model’ is based on standard FE algorithms, thus exploiting their ability to model complex boundary conditions as well as their user-friendly input and output interfaces. The model uses a novel unit-cell implementation of the fabric, with material properties derived from standard tests. The bias extension test is
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used both to extract appropriate material shear properties from a ‘gauge section’ where the PJN approximation is good, and additional material parameters from consideration of regions near the grips where the PJN model is less accurate. The model is implemented in an FE code and applied to draping of an ellipsoid.
Conversion between the element stiffness EL and the corresponding effective modulus E0L of the fabric along the tow direction is straightforward, requiring comparison of forces over the unit cell. The effective modulus E0L is defined by E0L ¼
2. Unit cell model Fig. 1 shows the undeformed square unit cell of side length D used to represent the material. The four outer elements model the tows while the diagonal element endows the cell with shear stiffness. The four tow elements are connected at the corners via pin joints, to allow the trellising action observed in practice. All elements are modelled using two-noded 3D truss elements of area A: The strain 1 in both types of element follows the Green – Lagrange definition of strain used by the MARC.MSC FE implementation adopted [17,18] u 1 u 2 1¼ þ ð1Þ 2 D0 D0 Here u and D0 are the corresponding extension and initial length of an element. With this strain formulation and assuming small extensions of the tow elements, the strains are equal and opposite for the two diagonals of the unit cell, so that the same material response can be used for the diagonal element, irrespective or whether it is in compression or tension. Because it is difficult to measure the thickness of fabric material accurately, estimates of the applied stress are not straightforward. Instead we follow textile practice and characterise the applied loads using forces per unit length, or ‘line forces’. This can be separated into components NL and NS acting along the tows and in the shear direction, respectively [10]. The tow elements are given an appropriate constant stiffness EL to allow for decrimping or tow slip during deformation, by fitting the simulation to the measured bias extension deformation response, as described in Section 3.
NL 1L
ð2Þ
where 1L is the strain along the tow direction. Since each tow element carries a load equal to the product of the applied load NL and the side length D of the unit cell, Hooke’s law for the element gives EL ¼
NL D A1L
ð3Þ
so that the tow element stiffness EL corresponding to a fabric effective modulus E0L is given by EL ¼
E0L D A
ð4Þ
The effect of changes in geometry (e.g. large deformations) on the tow elastic modulus are not included, nor is the small contribution from the shear element to stretching along the tow directions. The diagonal element is given a non-linear stress strain response sS ¼ f ð1S Þ; which is implemented here using a specified stress – strain curve within the elastic – plastic option of MARC. This curve is derived from the material response Nx ¼ f1 ð1x Þ; measured along a bias direction (see Section 3), where Nx is the nominal line load applied in the bias direction (i.e. applied force divided by original specimen width) and 1x is the corresponding strain in the bias direction. Again the element and corresponding material responses can be related by considering forces on the unit cell. As the unit cell is pin-jointed, the only resistance to elongation along the bias direction is provided by the shear element. Equating pffiffi the external force applied to the unit cell of initial width 2D with the force in the shear element gives pffiffi N x 2D ¼ As s ð5Þ By using a nominal line load formulation, changes in unit cell width do not need to be considered in the above expression. Hence, the element response sS ¼ f ð1S Þ is related to the material response Nx ¼ f1 ð1x Þ via a simple scale factor pffiffi 2D f ð1 Þ ss ¼ ð6Þ A 1 s
Fig. 1. Unit cell model of material behaviour.
The ‘bracing’ nature of this element effectively gives the material a non-linear shear stiffness. The exact form of material response Nx ¼ f1 ð1x Þ used for the shear elements in the simulation is given in Fig. 4, derived from the material response of the ‘gauge section’ of a bias extension test. In practice the area of each element was set to unity,
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and the appropriate stiffness derived for the corresponding mesh density and material properties using Eqs. (3) and (6).
3. Measurement of material parameters for a dry fabric Fabric tests are described in this section which are used to estimate values for the tow and shear springs of the unit cell model. Material property data for the simulation are based on measurements on Tenax HTA 6k carbon fibre dry fabric, with 5% binder, in a five-harness satin weave, manufactured by Hexcel Composites. Picture frame and bias extension tests have been widely used to characterise the shear response of dry fabric, prepregs and thermoplastic fabrics, and results from the two types of test have been compared recently by Sharma et al. [10] and Harrison et al. [19]. In a picture frame test the fabric is clamped in a square frame and deforms uniformly as the frame is deformed into a rhombus shape. This test provides a direct strain measurement but the stress calculation is less reliable due to unknown tension in the fabric. Although the picture frame test would provide suitable shear data [19], here we use instead the bias extension test. In this test the fabric specimen is elongated along the bias directions (i.e. ^ 458 to the warp and weft directions), as shown in the sketch of Fig. 2. The specimen has initial length L ¼ 200 and width W ¼ 75 mm and the grips are moved apart at a constant relative velocity V ¼ 60 mm/min. Further details of the bias extension tests used in the present work are given in Ref. [10]. The bias extension test was chosen for the work because the axial tow stresses are well characterised, allowing extraction of a suitable axial tow stiffness, and indeed are probably more representative of those present during draping. The nonuniformities in shear arising in this test were treated using suitable image analysis, and in fact provided an opportunity to extract suitable material properties from different regions of the specimen. Tests on the material at 60 and 100 mm/min produced similar results, indicating that an effect of binder on the shear response was small. Although some influence of binder cannot be ruled out, it is expected to be relatively small, certainly in comparison with the effect of matrix in prepreg material. The fabric showed some
Fig. 2. Sketch of the bias extension test.
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asymmetry in shear (see for example the deformed specimen, Fig. 6). However, the effect appeared to be relatively minor and has not been considered in detail. If the deformation is modelled using pin-jointed analysis then the constraints at the grips lead to rigid triangular regions at the two ends of the specimen. The apex of these triangles is termed here the ‘stationary point’, as it is supposed using the pin-jointed analysis to remain stationary with respect to the grip. The central gauge section of the specimen undergoes uniform shear, while there is a triangular ‘half-shear’ region towards the ends of the specimen. The material response for this gauge section is used to find an appropriate stress –strain curve for the diagonal element in the unit-cell model (Section 2). The deformation during the test of a grid marked on the central 50 £ 50 mm2 gauge section is recorded photographically and these images are used to measure the shear angle u in the mid-section. The longitudinal strain again follows the Green –Lagrange definition [17], given by: 1x ¼
ux ‘0
þ
1 ux 2 2 ‘0
ð7Þ
where ux is the gauge section extension measured from the images and ‘0 is the gauge section length at the start of the test. The nominal applied line force Nx is given by dividing the measured force F by the initial specimen width W0 Nx ¼ F=W0
ð8Þ
The force– displacement response for the bias extension specimen, over the whole length of the specimen, is shown in Fig. 3, showing significant stiffening (lock-up) at a displacement of around 60 mm. The corresponding nominal line force– strain response, derived now from measurements only over the gauge section, is given in Fig. 4. This shows an initial low stiffness region, followed by a large increase in stiffness at a strain of about 0.5, corresponding to a shear deformation angle of about 408. The relationship plotted in Fig. 4 is used, via the scaling factor of Eq. (6), to give the material properties for the diagonal elements in the FE model.
Fig. 3. Typical force–displacement curve over the whole specimen length for a bias-extension test.
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Fig. 4. Nominal line force–strain response taken from the gauge section of a bias extension test. This function is used, via the scaling factor given in Eq. (6), to describe the mechanical response of the shear element of the unit cell. The dashed line shows a linear stiffness of 15 kN/m, to allow comparison with the stiffness found appropriate for the tow elements.
In principle it seems reasonable to estimate the fabric stiffness along the tow directions from appropriate tests of specimens loaded along this direction. Although this approach was tried, the values of tow stiffness derived were found to give a poor fit to the data. It is believed that there were two reasons for this; firstly that an appropriate stiffness should be taken from only the initial decrimping part of the load– displacement response (since this corresponds to the loads typically found during draping), while in fact much higher stiffnesses corresponding to higher loads and extension of the fibres were measured. Secondly, it is likely that fibre slip may play an important role in the bias extension tests used to validate the model. Since the conditions in the bias extension test are closer to draping conditions, the stiffness extracted from these tests, as described in Section 4, was preferred. 4. Validation using bias extension test Detailed analysis of the bias extension tests presented here [8] has shown that the behaviour of the central gauge section is well modelled using the PJN assumption, while there is significant deviation from this model at the ends of the specimen. While the gauge section provides the shear response of the material, the overall deformation of the specimen and the behaviour near the grips provides a way of extracting the tow stiffness and helping validate the overall model. To simulate the bias extension test, it is modelled using thepunit ffiffi cell approach, taking a unit cell with diagonal length 2D ¼ 5 mm. The material used both for the experiments and the simulation is the dry woven CFRP described above. The FE specimen has the same size as the real specimen (200 £ 75 mm2). Fig. 5(a) and (b) plot the variation with end displacement of the width and shear angle of the central gauge section. The simulation results are shown for three values of the tow stiffness, and compared with the PJN analysis. The simulations results approach the PJN analysis at sufficiently large values of tow stiffness E0L : Experimental results show
Fig. 5. Comparison between measured and theoretical change in specimen geometry as a function of overall specimen displacement U during a bias extension test, (a) gauge section width W, (b) shear angle u in the gauge section.
significant deviations from the PJN analysis at large deformations. These deviations are well modelled using a tow element stiffness E0L ¼ 15 kN/m. Further analysis shows that the central gauge section is effectively behaving in a pin-jointed manner [8], so that the observed deviations from the PJN analysis are due to the behaviour near the ends of the specimen (for example, the significant slippage observed in Fig. 6). A straight line with slope corresponding to this value of tow stiffness is plotted on the load – strain response of the diagonal elements Fig. 4, to allow a direct comparison between the stiffnesses of the tow and diagonal elements. This figure shows how, with a value of tow stiffness E0L ¼ 15 kN/m, the shear stiffness of the material is small compared with the tow stiffness until lock-up is approached at a strain of about 0.5. At that point the material exhibits significant shear resistance and departs from PJN behaviour. A more critical test of the model can be made by investigating the variation in shear angle along the length of the specimen. As indicated in Section 3 above, there is a region near the grips which is nearly rigid, while the central region undergoes nearly uniform shear. This is illustrated in the photo of the specimen after an elongation of 40 mm, Fig. 6. Although the shear response of the material has been extracted using the centre of the specimen, nevertheless
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Fig. 6. Bias extension specimen after an extension U ¼ 40 mm, illustrating the grid used for measuring deformations outside the gauge section.
the significant non-uniform deformation which occurs near the grips is sensitive to the in-plane response of the fabric and acts as a useful test of the overall material model. Fig. 7 compares measurements and predictions of the variation of shear angle along the centre line of the specimen after an extension U ¼ 40 mm. Experimental points are taken from the photographic image, Fig. 6, tracing individual tows in the deformed specimen. The scatter in the measurements reflects the difficulty in estimating the shear deformation from the photograph. In the centre of the specimen, at the right hand side of the graph, the shear deformation is nearly uniform. Near the grip on the left hand side of Fig. 7 there is very little deformation. Predictions using the PJN model give the dashed line rising abruptly at the ‘stationary’ point. Fig. 7 shows that inclusion of a more realistic material model, with again a tow element stiffness E0L ¼ 15 kN/m, allows a good prediction of the shear response through the specimen.
5. Application to draping over an ellipsoid
equal to 150 mm. Two geometric parameters associated with the ply orientation are considered, as illustrated in Fig. 8c. Rotation of the nominally ^ 458 ply relative to the axis of the ellipse is given by the ply orientation angle f; while some pre-shear in the fabric, perhaps generated by inplane forces prior to draping, is modelled by a pre-shear angle a: The material model for dry woven CFRP described above is again used for the drape simulation, with 1600 unit cells of side length D ¼ 15 mm. Two values of the tow axial stiffness E0L of 15 and 8000 kN/m are used. The sheet is held in place at its edges by springs, which here are used to model blank holder forces. These springs are given an effective stiffness of 200 kN/m. A Coulomb friction coefficient of 0.2 is used at all contacts. 5.2. Simulation results Fig. 9(a) shows the effect of tow stiffness on the variation of shear angle along the major axis of the ellipsoid. The arc length has been normalised by the half-perimeter of the ellipsoid, so that a value of one corresponds to the
The intermediate drape model described in this paper is able to take into account processing parameters (e.g. blank holder forces), taking advantage of standard FE capabilities, while being sufficiently rapid (around 350 s per simulation) to permit the many calculations needed when searching for optimum forming conditions. Here we illustrate the value of this approach by considering draping of an ellipsoid, again using an MSC.MARC implementation. 5.1. Description of drape model The geometry is sketched in cross-section in Fig. 8a, and as an isometric view of the deformed geometry in Fig. 8b. The ellipsoidal punch has depth 200 mm and major and minor axes of length 250 and 200 mm, respectively. The sheet of material, which is initially square with side length 600 mm, drapes around the rigid punch as it is driven upwards, making contact with the punch guide at the edges. The travel of the punch after first contact with the sheet is
Fig. 7. Comparison between measured and theoretical variation of shear angle u with position along the centre line of the bias extension specimen, U ¼ 40 mm.
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Fig. 8. Details of ellipsoid drape simulation; (a) arrangement seen in crosssection, (b) isometric view of mesh, (c) definition of initial ply geometry and orientation.
base of the punch. For the lower value of stiffness of E0L ¼ 15 kN/m, corresponding to the value found appropriate from the bias extension tests, the peak shear angle occurs just outside the contact between the ellipsoid and the punch, taking a value of around 368. The effect of much stiffer tow stiffness is to increase the shear in the fabric, as stretching of the fabric becomes more difficult. Fig. 9(b) illustrates the effect of ‘pre-shear’ in the fabric, plotting the change in shear angle along both the major and minor axes of the ellipsoid. The case of no pre-shear corresponds to a ^ 458 layup, relative to the major axis of the ellipsoid. The case with a ‘pre-shear’ angle of 108 represents a uniform shear deformation of the fabric by 108 prior to draping. This might be associated with in-plane stretching prior to draping and is implemented simply by taking an initial ply orientation of ^ 408 relative to the major axis. Again the maximum shear angle occurs just outside the contact between the punch and the material. Application of 108 of pre-shear causes the position of maximum shear to switch from the major axis to the minor axis. Fig. 9(b) suggests that choosing a pre-shear angle of around 58 will lead to a value
Fig. 9. Effect of (a) tow stiffness and (b) pre-shear on the variation along the major and minor axes in the total shear in the material after draping an ellipsoid. The distance along the arc is normalised by the arc length from the centre to the edge of the ellipsoid.
of maximum shear angle below both the no-shear and 108 pre-shear cases. 5.3. Process optimisation Here we illustrate the value of the proposed FE model, by exploring the effect of ply orientation angle and initial preshear angle on the maximum shear induced when draping the ellipsoid. Fig. 10 shows a contour plot of the maximum shear angle in the component as a function of pre-shear and ply orientation angles. The maximum shear angle has a minimum when the bias direction is aligned along the major axis and with about 48 of pre-shear. Pre-shear has a greater influence on the maximum shear angle than does ply orientation angle. Similar process optimisation maps could equally be produced for other process parameters such as
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Company Ltd, MSC.Software, Saint-Gobain Vetrotex, BP Amoco and Hexcel.
References
Fig. 10. Process optimisation map; the contours heights shown on the figure give the maximum shear angle in a given simulation as a function of the ply orientation and pre-shear angle of that simulation.
blank holder forces or material properties (perhaps associated with changes in the pre-preg or with modifications to the mould temperature during forming).
6. Conclusions A simplified FE model is proposed to model draping of composite material. The unit cell consists of a network of pin-jointed stiff trusses, with shear stiffness introduced via diagonal bracing elements. Suitable material properties are extracted from bias extension tests measurements on a dry CFRP fabric. Measurements of the deformation in the central gauge section during a bias extension test are used to extract a suitable stress – strain curve for the diagonal elements. The stiffness of the trusses is estimated by fitting bias extension measurements over the whole specimen length. Predictions of the deformation of a bias extension specimen away from the gauge section are in good agreement with measurements, confirming the ability of the unit cell model to represent the effect of in-plane forces on shear deformation and tow slippage. It is expected that this approach would also be applicable to pre-preg woven material. In this case the flexibility of the model allows incorporation of the more sophisticated shear response needed for such fabrics, for example including rate-sensitive terms. The material response would need to be extracted from experiments or micromechanics-based models [20]. The model is used to predict the draping of an ellipsoid and to illustrate how it might be used for process optimisation. The combination of flexibility in the FE approach and speed of computation allows optimisation studies to be undertaken with only modest computing time.
Acknowledgements The authors gratefully acknowledge the contributions from the Universities of Nottingham and Leeds, EPSRC, BAE SYSTEMS, QinetiQ, ESI Software, Ford Motor
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