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Draping of plain woven carbon fabrics over a double-curvature mold
Composites Science and Technology 92 (2014) 64–69
Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Draping of plain woven carbon fabrics over a double-curvature mold Hongling Yin a, Xiongqi Peng a,⇑, Tongliang Du a, Zaoyang Guo b,⇑ a b
School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200030, China Department of Engineering Mechanics, Chongqing University, Chongqing 400044, China
a r t i c l e
i n f o
Article history: Received 18 November 2013 Received in revised form 10 December 2013 Accepted 12 December 2013 Available online 26 December 2013 Keywords: A. Fabrics/textiles C. Anisotropy C. Modelling Draping C. Finite element analysis (FEA)
a b s t r a c t This paper investigates the complex fiber reorientation and redistribution of plain woven carbon fabrics draping over a double-curvature mold through experimental and numerical approaches. Firstly, uni-axial tensile, bias-extension and picture-frame tests are carried out to obtain the material properties of the woven carbon fabrics. Material parameters for a previously developed simple anisotropic hyperelastic model are accordingly obtained. Secondly, draping experiments are implemented at room temperature using samples with different original fiber orientation. Deformed fabric boundary profiles and local shear angle variations are recorded and analyzed. The draping process is then simulated by using the anisotropic hyperelastic model taking into account material and geometric non-linearity. Very good agreement between simulation and experimental results is obtained. The ultimate goal of the investigation is to develop a design tool for the numerical simulation and processing optimization of woven composites forming. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction With energy exhausting and increasing environment concern, lightweight structural components are emergent in auto industry. Application of carbon fiber composite in auto industry will be a main trend. Among them, woven carbon fabrics have the great potential as a viable alternative to metal sheet for providing high-strength and low-weight products with affordable cost. However, fundamental knowledge and systematical understanding of their potential manufacturing processes are essential for the introduction of woven fabric reinforced composites into high-volume products. Draping of woven fabrics is particular important in carbon fiber composite material products manufacturing [1], by providing preform for resin transfer molding or directly participating in thermo-stamping of prepregs. Draping of woven carbon fabrics over double curvature mold could lead to complex fiber redistribution and reorientation, which in turns deeply influence the mechanical properties of the finished part. There are two methods to understand the draping process: numerical simulation and trial/error experimental approach. As there are immense number of composite materials, possible fabric woven architectures, and changes in tooling conditions, the development of a forming process is expensive and complex by the trial/error experimental method. The design of forming process and property prediction of complex double curvature composite
⇑ Corresponding authors. Tel.: +86 13916390469 (X. Peng). E-mail addresses: [email protected] (X. Peng), [email protected] (Z. Guo). 0266-3538/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2013.12.013
components for improvement of lightweight structural can be effectively achieved by the numerical approach. Forming simulation of woven fabric over double curvature mold is a challenge job because of the inhomogeneous structures, anisotropic characteristics of fabrics and different kinds of mechanisms involved [2]. For this reason, many researchers proposed different approaches and models for simulation of forming textiles. They can be classified into kinematic, discrete and continuum mechanics based models. Kinematic models are geometrical mapping algorithms which do not consider friction effects, loads or boundary conditions [3,4]. These models are well developed and implemented in commercial software packages, such as PAM-FORM. In discrete models, single yarns are modeled as springs connecting to mass elements at cross points [5–7]. In continuum mechanics based models, the fabric is modeled as a flexible and thin shell, which reflects the macro-mechanical and meso-mechanical fabric behavior [8–14]. Uni-axial and/or bi-axial tensile [10,15], pictureframe test [16–19] and bias-extension test [10,20] are required to estimate material parameters. Phenomenological continuum models which take into account the viscous or visco-plastic response of the woven textile have also been developed [21–24]. In this paper, draping of plain woven carbon fabric over a double curvature mold is investigated via experimental and numerical approaches. Forming experiments are implemented using samples with different original fiber orientation. Deformed fabric boundary profiles and local shear angles are recorded and analyzed. A simple anisotropic hyperelastic material model [25,26] which defines the macro mechanical characteristics of fabric is used to simulate the draping process. The model has been turned out to be very
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H. Yin et al. / Composites Science and Technology 92 (2014) 64–69
successful for the draping simulation of woven structures on a benchmark double-doom [26]. Determination of the input parameters for the material model is achieved by various textile testing methods [27], including uni-axial tensile, bias-extension and picture-frame tests. 2. Material testing and draping experiment 2.1. Material Plain woven carbon fabrics are selected in current study because they have a higher density of interlaced yarns, which influence the fiber undulation greatly and result in a worse formability. If the plain woven fabric works out on draping over a double curvature mold, other patterns can be taken for granted. The carbon plain woven fabric is shown in Fig. 1. The fabric properties are listed in Table 1. 2.2. Material testing Various material tests including uni-axial tensile, bias-extension and picture-frame are carried out for the carbon woven fabric to obtain its basic mechanical characteristics. The material testing set-ups are shown in Figs. 2 and 3. In uni-axial tensile and biasextension tests, the fabric is cut into 115 mm 230 mm samples along the 0° and 45° of the yarn orientation, respectively. In the picture-frame test, the frame size is 235 mm 235 mm, fabric is cut into 180 mm 180 mm square shape. Each test is done with three samples. The averaged load–displacement results for the three tests are shown in Figs. 4–6, respectively. It must be pointed out that bending is a very important characteristic for fabric, especially when dealing with springback and wrinkling prediction for fabric reinforced composite forming [28]. Since the present study focuses on the drapability of carbon fabric over double-curvature mold, bending test of the carbon fabric is ignored. The bending stiffness of the fabric is invoked by delineating transverse shear stiffness in the proceeding simulation with ABAQUS.
two parts along a symmetry plane to get the left and right wingtips for an airplane. The experimental part is scaled down to 70%. The scaled part has an overall size of 105 mm in length, 46 mm in width and 40 mm in height with a thickness of 1 mm. The mold and a position coordinate system which is marked on the die are shown in Fig. 8. As shown in Fig. 9, rectangular samples with an outlined boundary size of 320 mm 160 mm are prepared with painted points for measuring shearing angles after forming over the double-curvature mold. The principle of choosing points on the sample is to cover the maximum and minimum shear angles. Samples are with two types of fiber orientations: the fiber yarn directions are 0° and 45° with respective to the axes of the positions coordinate system in Fig. 8. During the forming, the center of samples, point 3 shown in Fig. 9, is set exactly on the center of the mold, and the sample edges are parallel with the mold edges, respectively. After forming, a piece of transparent cellophane of 320 mm 160 mm with square grids of 10 mm 10 mm is placed on the formed fabric as an auxiliary to measure its deformation, as shown in Fig. 10. Photos of the formed fabrics over the mold are taken and then analyzed with image processing software. Data points of contour lines are produced to measure shear angles at special points using a digital protractor. 3. A simple anisotropic hyperelastic model for textile fabrics [26] A strain energy function W is defined to characterize the mechanical property of textile fabrics with weft and warp yarns represented by original fiber directional unit vectors a0 and b0, respectively, i.e.,
W ¼ WðC; a0 ; b0 Þ
ð1Þ
T
where C = F F is the right Cauchy-Green deformation tensor, F = @x/@X is the deformation gradient tensor, X represents the position of a material particle in the original configuration, while x is the position of the corresponding particle in the current configuration. Alternatively, the strain energy function could be expressed in terms of invariants Ii,
WðC; a0 ; b0 Þ ¼ WðIi Þ:
2.3. Draping experiment To cover most design complexities in industrial parts, a wingtip part with double curvature geometry, as shown in Fig. 7, is adopted to the experimental forming. The part was originally made of aluminum by deep-drawing. The formed part can be cut into
ð2Þ
By decoupling fiber extension energy with in-plane shearing energy and taking into account the fact that there is no matrix for dry fabrics, the strain energy function can be decomposed as,
W ¼ WðC; a0 ; b0 Þ ¼ W F þ W FF
ð3Þ
Fig. 1. Balanced plain carbon woven fabric used in the experiments. Table 1 Parameters of the dry balanced plain woven carbon fabric. Yarn space (mm) 2.0
Area density (g/m2) 200
Yarn thickness (mm) 0.25
Yarn width (mm) 1.3
E modulus (GPa) 240
Cell size (mm mm) 3.0 3.0
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H. Yin et al. / Composites Science and Technology 92 (2014) 64–69
Bias-extension Test 20
Load F (N)
15
10
5
0
0
5
10
15
20
25
30
35
40
Displacement (mm)
Fig. 2. Uni-axial tensile and bias-extension tests on plain woven carbon fabric.
Fig. 5. Averaged load–displacement curve in bias-extension test.
Picture Frame Test 300 250
Load (N)
200 150 100 50 0
0
50
100
150
Displacement (mm) Fig. 6. Averaged load–displacement curve in picture-frame test.
Fig. 3. Trellising test on plain woven carbon fabric.
Uni-axial Tensile Test 200 Fig. 7. Wingtip model.
Load (N)
150
where WF is the contribution from fiber stretches, and WFF is the shearing energy resulting from fiber–fiber interaction. The fiber extension energy WF can be simply characterized by fiber stretch ratios, i.e.,
100
W F ¼ W F ðC; a0 ; b0 Þ ¼ W Fa ðIa4 Þ þ W Fb ðIb4 Þ 50
0
W Fa
ð4Þ
W Fb
where and represent the weft and warp yarn fiber stretch energies, respectively. Ia4 and Ib4 are defined as, 0
0.5
1
1.5
2
2.5
3
Displacement (mm) Fig. 4. Averaged load–displacement curve in uni-axial tensile test.
Ia4 ¼ a0 C a0 ¼ ðka Þ2 ; Ib4 ¼ b0 C b0 ¼ ðkb Þ2
ð5Þ
where ka and kb are the stretch ratios of the weft and warp fiber yarns, respectively. Introducing Iab 7 to represent the shearing angle between the two fiber yarns:
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H. Yin et al. / Composites Science and Technology 92 (2014) 64–69
Fig. 11. Finite element model for woven fabric forming.
Fig. 8. Punch and die with position coordinate system.
Fig. 9. Forming sample with outlined boundary and painted points.
Fig. 12. Shear angle distribution of formed fabrics (a) 0° sample and (b) 45° sample.
12 a b Iab a0 C b0 a0 b0 : 7 ¼ Du ¼ u u 0 I 4 I 4
ð6Þ
The fiber–fiber interaction energy can be quantitatively represented by Iab 7 ; i.e., ab W FF ¼ W FF ab ðI 7 Þ:
ð7Þ
Then the second Piola–Kirchhoff stress tensor is S = 2oW/@C. The Cauchy stress tensor r, is given by r = J1FSFT, i.e., 2 3 FF F @W 1 a 2 @W Iab þ a0 b0 @Iabab a aþ a a 7 6 7 I @I 4 4 7 1 7 r ¼ I3 2 6 6 7 FF F FF 4 5 @W b @W ab @W ab ab 1 1 ffi ða b þ b aÞ 2 @Ib Ib I7 þ a0 b0 @Iab b b þ @Iab pffiffiffiffiffi a b 4
4
7
7
I4 I4
ð8Þ
Following the same material parameter determination procedure demonstrated in [30] and using the uni-axial tensile testing results shown in Fig. 4, the tensile strain energy function for the fabric is obtained as,
4 3 2 W Fa ðIa4 Þ ¼ k1 Ia4 1 þ k2 Ia4 1 þ k3 Ia4 1 Fig. 10. Fabrics formed over a double curvature mold: (a) 0° sample and (b) 45° sample.
ð9Þ
where k1 = 3.0e5, k2 = 1630.13, k3 = 1.15 with a unit of MPa. For the balanced plain weave composites, W Fb ðIb4 Þ has the same functional
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H. Yin et al. / Composites Science and Technology 92 (2014) 64–69
Fig. 13. Shear angles at marked points (a) 0° sample and (b) 45° sample.
format as that of W Fa ðIa4 Þ .The shearing strain energy function is obtained from the bias-extension testing results shown in Fig. 5 as a polynomial function of Iab 7 ,
W FF ðI7 Þ ¼ k4 ðI7 Þ4 þ k5 ðI7 Þ3 þ k6 ðI7 Þ2
ð10Þ
where k4 = 0.47, k5 = 0.318, k6 = 0.0842 with a unit of MPa. Adding Eqs. (9) and (10) gives the total strain energy for the woven fabric undergoing general deformation status. Accordingly, a user material subroutine UANISOHYPER_INV for commercial FEM software ABAQUS/Standard is designed for the woven fabric to implement numerical simulation on the forming experiment in the preceding section. It must be pointed out that the hyperelastic formulation used in the present study is a simple and primitive one but still capturing
the major characteristic of woven fabric during draping. Naturally, there is coupling between fiber yarn extensions and shearing. Biaxial tension test [15] may be needed for introducing this coupling into the hyperelastic model. Besides, the current model is only suitable for thin fabric. Hyperelastic model for 3D fabric can be found in [29]. The stability and ellipticity analysis [30] of the present model along with the coupling between fiber yarn extensions and shearing will be the future work. 4. Draping simulation Fig. 11 shows the finite element model for the woven fabric forming over the wing-tip mold. In the model, the punch, die and binder are modeled as rigid bodies. The rectangular plain woven
Fig. 14. Boundary profiles of formed fabrics (a) 0° sample and (b) 45° sample.
H. Yin et al. / Composites Science and Technology 92 (2014) 64–69
carbon fabric has the same dimension of 320 mm 160 mm with a thickness of 0.25 mm as in the forming experiment. It is uniformly meshed with 4 mm 2 mm S4R shell elements being assigned with the designed user material subroutine UANISOHYPER_INV. A constant 50 N force is imposed on the binder to clamp the fabric to prevent wrinkling. The total punch stroke is set to be 60 mm. To improve convergence of numerical simulation, a soft-contact formulation with a friction coefficient of 0.3 is assigned on contacts between the fabric and tooling. Fig. 12 shows the simulation result of deformed fabric with shear angle distribution for the 0° and 45° samples, respectively, under a punch stroke of 60 mm. The large shear deformation areas appear in the flange of the larger curvature surface area for both samples. The maximum shear angle is 54.4°and 55.7°for the 0° and 45° samples respectively. Although the maximum shear angle of the 45° sample is larger, its shear angle variation is obviously smoother than that of the 0° woven fabric. This indicates that the formability of the 0° woven fabric is worse than that of the 45° fabric when forming the wingtip part. Fig. 13 presents the comparison between numerical and experimental shear angles of marked points in Fig. 9. As shown in Fig. 13, overall, for both cases, the predicted shear angles are in a good agreement with experimental ones. Experimental results show that wrinkles appear for both cases at points 12 and 14, which have shear angles smaller than 34° However, the picture-frame test shows that the critical shear angle of the woven carbon fabric is about 35°. Hence, the critical shear angle cannot be used as an solely indicator for wrinkling criterion. The mechanism of wrinkling needs to be further investigated. Fig. 14 shows the top views of deformed boundary profiles of both fabrics draping over the double curvature mold. The experimental deformed boundary profiles for both cases have a good repeatability. Large discrepancy on the deformed boundary profile is found between the 0° and 45° fabrics. As demonstrated in Figs. 10 and 12–14, fiber orientation plays an important role in forming double-curvature parts. As shown in Fig. 14, numerically predicted boundary profiles have good agreement with the experimental ones. The companions on shear angle and deformed boundary profile between numerical results and experimental data indicate that the simple anisotropic hyper-elastic model is effective in characterizing the nonlinear material behavior of the woven carbon fabric in forming over a double curvature mold. 5. Conclusions Draping of a plain woven carbon fabric over a double curvature mold is investigated numerically and experimentally in this paper. A previously developed simple anisotropic hyper-elastic model is used to characterize the material behavior of the woven carbon fabric under complex deformation modes. Material parameters for the constitutive model are obtained by using experimental uni-axial tensile and bias-extension testing results of the fabric. Very good agreements on shear angle and deformed boundary profile between experimental data and numerical results are obtained. It is shown that fiber orientation plays an important role in forming double-curvature parts. Although the present work has demonstrated the validity of the material model on predicting the complex fabric redistribution and reorientation during draping over double curvature mold, further investigations on the influence of fabric lay-up and bending stiffness are necessary to make a design tool for practical application of the model. Acknowledgements The supports from the National Nature Science Foundation of China (11172171 and 11272362), Ph.D. Programs Foundation of
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Ministry of Education of China (20130073110054) and the Scientific and Technological Innovation Foundation of Shanghai, China (12521102203) are gratefully acknowledged. References [1] Wang J, Page JR, Paton R. Experimental investigation of the draping properties of reinforcement fabrics. Compos Sci Technol 1998;58(2):229–37. [2] Laroche D, Vu-Khanh T. Forming of woven fabric composites. J Compos Mater 1994;28(18):1825–39. [3] Van Der Weeen F. Algorithms for draping fabrics on doubly-curved surfaces. Int J Numer Methods Eng 1991;31(7):1415–26. [4] Vu-Khanh T, Liu B. Prediction of fibre rearrangement and thermal expansion behaviour of deformed woven-fabric laminates. Compos Sci Technol 1995;53(2):183–91. [5] Gelin JC, Cherouat A, Boisse P, Sabhi H. Manufacture of thin composite structures by the RTM process: numerical simulation of the shaping operation. Compos Sci Technol 1996;56(7):711–8. [6] Cherouat A, Billoët JL. Mechanical and numerical modelling of composite manufacturing processes deep-drawing and laying-up of thin pre-impregnated woven fabrics. J Mater Process Technol 2001;118(1):460–71. [7] Ben Boubaker B, Haussy B, Ganghoffer J. Discrete models of woven structures. Macroscopic approach. Compos Part B: Eng 2007;38(4):498–505. [8] Yu WR, Pourboghrat F, Chung K, et al. Non-orthogonal constitutive equation for woven fabric reinforced thermo-plastic composites. Compos Part A: Appl Sci Manuf 2000;33(1):1095–105. [9] Xue P, Peng XQ, Cao J. A non-orthogonal constitutive model for characterizing woven composite. Compos Part A: Appl Sci Manuf 2003;34(2):183–93. [10] Peng XQ, Cao J. A continuum mechanics-based non-orthogonal constitutive model for woven composite fabrics. Compos Part A: Appl Sci Manuf 2005;36(6):859–74. [11] Boisse P, Zouari B, Gasser A. A mesoscopic approach for the simulation of woven fibre composite forming. Compos Sci Technol 2005;65(3–4):429–36. [12] Zouari B, Daniel JL, Boisse P. A woven reinforcement forming simulation method. Influence of the shear stiffness. Comput Struct 2006;84(5–6):351–63. [13] Badel P, Vidal-Sallé E, Boisse P. Large deformation analysis of fibrous materials using rate constitutive equations. Comput Struct 2008;86(11–12):1164–75. [14] Charmetant A, Vidal-Salle E, Boisse P. Hyperelastic modelling for mesoscopic analyses of composite reinforcements. Compos Sci Technol 2011;71(14):1623–31. [15] Buet-Gautier K, Boisse P. Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements. Exp Mech 2001;41(3):260–9. [16] Lussier D, Chen J. Material characterization of woven fabrics for thermoforming of composites. J Thermoplast Compos Mater 2002;15(6):497–509. [17] Peng XQ, Cao J. A dual homogenization and finite element approach for material characterization of textile composites. Compos Part B: Eng 2002;33(1):45–56. [18] Zhu B, Yu TX, Tao XM. An experimental study of in-plane large shear deformation of woven fabric composite. Compos Sci Technol 2007;67(2):252–61. [19] Zhu B, Yu TX, Tao XM. Large shear deformation of E-glass/polypropylene woven fabric composites at elevated temperatures. J Reinf Plast Compos 2009;28(21):2615–30. [20] Zhu B, Yu TX, Tao XM. Large deformation and slippage mechanism of plain woven composite in bias extension. Compos Part A: Appl Sci Manuf 2007;38(8):1821–8. [21] Merodio J. On constitutive equations for fiber-reinforced nonlinearly viscoelastic solids. Mech Res Commun 2006;33(6):764–70. [22] Merodio J, Rajagopal KR. On constitutive equations for anisotropic nonlinearly viscoelastic solids. Math Mech Solids 2007;12(2):131–47. [23] Matsuda T, Nimiya Y, Ohno N, Tokuda M. Elastic–viscoplastic behavior of plain-woven GFRP laminates: homogenization using a reduced domain of analysis. Compos Struct 2007;79(4):493–500. [24] Guiot B, Nadot-Martin C, Dragon A. A multiscale (visco)-hyperelastic modelling for particulate composites. Mech Res Commun 2006;33(4):441–9. [25] Peng XQ, Guo ZY, Harrison P. A simple anisotropic fiber reinforced hyperelastic constitutive model for woven composite fabrics. Int J Mater Form 2010;3(1):723–6. [26] Peng XQ, Guo ZY, Du TL, Yu WR. A simple anisotropic hyperelastic constitutive model for textile fabrics with application to forming simulation. Compos Part B: Eng 2013;52:275–81. [27] Cao J, Akkerman R, Boisse P, et al. Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results. Compos Part A: Appl Sci Manuf 2008;39(6):1037–53. [28] Boisse P, Hamila N, Vidal-Salle E, Dumont F. Simulation of wrinkling during textile composite reinforcement forming. Influence of tensile, in-plane shear and bending stiffnesses. Compos Sci Technol 2011;71(5):683–92. [29] Charmetant A, Orliac JG, Vidal-Salle E, Boisse P. Hyperelastic model for large deformation analyses of 3D interlock composite preforms. Compos Sci Technol 2012;72(12):1352–60. [30] Merodio J, Neff P. A note on tensile instabilities and loss of ellipticity for a fiber-reinforced nonlinearly elastic solid. Arch Mech 2006;58(3):293–303.
Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Draping of plain woven carbon fabrics over a double-curvature mold Hongling Yin a, Xiongqi Peng a,⇑, Tongliang Du a, Zaoyang Guo b,⇑ a b
School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200030, China Department of Engineering Mechanics, Chongqing University, Chongqing 400044, China
a r t i c l e
i n f o
Article history: Received 18 November 2013 Received in revised form 10 December 2013 Accepted 12 December 2013 Available online 26 December 2013 Keywords: A. Fabrics/textiles C. Anisotropy C. Modelling Draping C. Finite element analysis (FEA)
a b s t r a c t This paper investigates the complex fiber reorientation and redistribution of plain woven carbon fabrics draping over a double-curvature mold through experimental and numerical approaches. Firstly, uni-axial tensile, bias-extension and picture-frame tests are carried out to obtain the material properties of the woven carbon fabrics. Material parameters for a previously developed simple anisotropic hyperelastic model are accordingly obtained. Secondly, draping experiments are implemented at room temperature using samples with different original fiber orientation. Deformed fabric boundary profiles and local shear angle variations are recorded and analyzed. The draping process is then simulated by using the anisotropic hyperelastic model taking into account material and geometric non-linearity. Very good agreement between simulation and experimental results is obtained. The ultimate goal of the investigation is to develop a design tool for the numerical simulation and processing optimization of woven composites forming. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction With energy exhausting and increasing environment concern, lightweight structural components are emergent in auto industry. Application of carbon fiber composite in auto industry will be a main trend. Among them, woven carbon fabrics have the great potential as a viable alternative to metal sheet for providing high-strength and low-weight products with affordable cost. However, fundamental knowledge and systematical understanding of their potential manufacturing processes are essential for the introduction of woven fabric reinforced composites into high-volume products. Draping of woven fabrics is particular important in carbon fiber composite material products manufacturing [1], by providing preform for resin transfer molding or directly participating in thermo-stamping of prepregs. Draping of woven carbon fabrics over double curvature mold could lead to complex fiber redistribution and reorientation, which in turns deeply influence the mechanical properties of the finished part. There are two methods to understand the draping process: numerical simulation and trial/error experimental approach. As there are immense number of composite materials, possible fabric woven architectures, and changes in tooling conditions, the development of a forming process is expensive and complex by the trial/error experimental method. The design of forming process and property prediction of complex double curvature composite
⇑ Corresponding authors. Tel.: +86 13916390469 (X. Peng). E-mail addresses: [email protected] (X. Peng), [email protected] (Z. Guo). 0266-3538/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2013.12.013
components for improvement of lightweight structural can be effectively achieved by the numerical approach. Forming simulation of woven fabric over double curvature mold is a challenge job because of the inhomogeneous structures, anisotropic characteristics of fabrics and different kinds of mechanisms involved [2]. For this reason, many researchers proposed different approaches and models for simulation of forming textiles. They can be classified into kinematic, discrete and continuum mechanics based models. Kinematic models are geometrical mapping algorithms which do not consider friction effects, loads or boundary conditions [3,4]. These models are well developed and implemented in commercial software packages, such as PAM-FORM. In discrete models, single yarns are modeled as springs connecting to mass elements at cross points [5–7]. In continuum mechanics based models, the fabric is modeled as a flexible and thin shell, which reflects the macro-mechanical and meso-mechanical fabric behavior [8–14]. Uni-axial and/or bi-axial tensile [10,15], pictureframe test [16–19] and bias-extension test [10,20] are required to estimate material parameters. Phenomenological continuum models which take into account the viscous or visco-plastic response of the woven textile have also been developed [21–24]. In this paper, draping of plain woven carbon fabric over a double curvature mold is investigated via experimental and numerical approaches. Forming experiments are implemented using samples with different original fiber orientation. Deformed fabric boundary profiles and local shear angles are recorded and analyzed. A simple anisotropic hyperelastic material model [25,26] which defines the macro mechanical characteristics of fabric is used to simulate the draping process. The model has been turned out to be very
65
H. Yin et al. / Composites Science and Technology 92 (2014) 64–69
successful for the draping simulation of woven structures on a benchmark double-doom [26]. Determination of the input parameters for the material model is achieved by various textile testing methods [27], including uni-axial tensile, bias-extension and picture-frame tests. 2. Material testing and draping experiment 2.1. Material Plain woven carbon fabrics are selected in current study because they have a higher density of interlaced yarns, which influence the fiber undulation greatly and result in a worse formability. If the plain woven fabric works out on draping over a double curvature mold, other patterns can be taken for granted. The carbon plain woven fabric is shown in Fig. 1. The fabric properties are listed in Table 1. 2.2. Material testing Various material tests including uni-axial tensile, bias-extension and picture-frame are carried out for the carbon woven fabric to obtain its basic mechanical characteristics. The material testing set-ups are shown in Figs. 2 and 3. In uni-axial tensile and biasextension tests, the fabric is cut into 115 mm 230 mm samples along the 0° and 45° of the yarn orientation, respectively. In the picture-frame test, the frame size is 235 mm 235 mm, fabric is cut into 180 mm 180 mm square shape. Each test is done with three samples. The averaged load–displacement results for the three tests are shown in Figs. 4–6, respectively. It must be pointed out that bending is a very important characteristic for fabric, especially when dealing with springback and wrinkling prediction for fabric reinforced composite forming [28]. Since the present study focuses on the drapability of carbon fabric over double-curvature mold, bending test of the carbon fabric is ignored. The bending stiffness of the fabric is invoked by delineating transverse shear stiffness in the proceeding simulation with ABAQUS.
two parts along a symmetry plane to get the left and right wingtips for an airplane. The experimental part is scaled down to 70%. The scaled part has an overall size of 105 mm in length, 46 mm in width and 40 mm in height with a thickness of 1 mm. The mold and a position coordinate system which is marked on the die are shown in Fig. 8. As shown in Fig. 9, rectangular samples with an outlined boundary size of 320 mm 160 mm are prepared with painted points for measuring shearing angles after forming over the double-curvature mold. The principle of choosing points on the sample is to cover the maximum and minimum shear angles. Samples are with two types of fiber orientations: the fiber yarn directions are 0° and 45° with respective to the axes of the positions coordinate system in Fig. 8. During the forming, the center of samples, point 3 shown in Fig. 9, is set exactly on the center of the mold, and the sample edges are parallel with the mold edges, respectively. After forming, a piece of transparent cellophane of 320 mm 160 mm with square grids of 10 mm 10 mm is placed on the formed fabric as an auxiliary to measure its deformation, as shown in Fig. 10. Photos of the formed fabrics over the mold are taken and then analyzed with image processing software. Data points of contour lines are produced to measure shear angles at special points using a digital protractor. 3. A simple anisotropic hyperelastic model for textile fabrics [26] A strain energy function W is defined to characterize the mechanical property of textile fabrics with weft and warp yarns represented by original fiber directional unit vectors a0 and b0, respectively, i.e.,
W ¼ WðC; a0 ; b0 Þ
ð1Þ
T
where C = F F is the right Cauchy-Green deformation tensor, F = @x/@X is the deformation gradient tensor, X represents the position of a material particle in the original configuration, while x is the position of the corresponding particle in the current configuration. Alternatively, the strain energy function could be expressed in terms of invariants Ii,
WðC; a0 ; b0 Þ ¼ WðIi Þ:
2.3. Draping experiment To cover most design complexities in industrial parts, a wingtip part with double curvature geometry, as shown in Fig. 7, is adopted to the experimental forming. The part was originally made of aluminum by deep-drawing. The formed part can be cut into
ð2Þ
By decoupling fiber extension energy with in-plane shearing energy and taking into account the fact that there is no matrix for dry fabrics, the strain energy function can be decomposed as,
W ¼ WðC; a0 ; b0 Þ ¼ W F þ W FF
ð3Þ
Fig. 1. Balanced plain carbon woven fabric used in the experiments. Table 1 Parameters of the dry balanced plain woven carbon fabric. Yarn space (mm) 2.0
Area density (g/m2) 200
Yarn thickness (mm) 0.25
Yarn width (mm) 1.3
E modulus (GPa) 240
Cell size (mm mm) 3.0 3.0
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H. Yin et al. / Composites Science and Technology 92 (2014) 64–69
Bias-extension Test 20
Load F (N)
15
10
5
0
0
5
10
15
20
25
30
35
40
Displacement (mm)
Fig. 2. Uni-axial tensile and bias-extension tests on plain woven carbon fabric.
Fig. 5. Averaged load–displacement curve in bias-extension test.
Picture Frame Test 300 250
Load (N)
200 150 100 50 0
0
50
100
150
Displacement (mm) Fig. 6. Averaged load–displacement curve in picture-frame test.
Fig. 3. Trellising test on plain woven carbon fabric.
Uni-axial Tensile Test 200 Fig. 7. Wingtip model.
Load (N)
150
where WF is the contribution from fiber stretches, and WFF is the shearing energy resulting from fiber–fiber interaction. The fiber extension energy WF can be simply characterized by fiber stretch ratios, i.e.,
100
W F ¼ W F ðC; a0 ; b0 Þ ¼ W Fa ðIa4 Þ þ W Fb ðIb4 Þ 50
0
W Fa
ð4Þ
W Fb
where and represent the weft and warp yarn fiber stretch energies, respectively. Ia4 and Ib4 are defined as, 0
0.5
1
1.5
2
2.5
3
Displacement (mm) Fig. 4. Averaged load–displacement curve in uni-axial tensile test.
Ia4 ¼ a0 C a0 ¼ ðka Þ2 ; Ib4 ¼ b0 C b0 ¼ ðkb Þ2
ð5Þ
where ka and kb are the stretch ratios of the weft and warp fiber yarns, respectively. Introducing Iab 7 to represent the shearing angle between the two fiber yarns:
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Fig. 11. Finite element model for woven fabric forming.
Fig. 8. Punch and die with position coordinate system.
Fig. 9. Forming sample with outlined boundary and painted points.
Fig. 12. Shear angle distribution of formed fabrics (a) 0° sample and (b) 45° sample.
12 a b Iab a0 C b0 a0 b0 : 7 ¼ Du ¼ u u 0 I 4 I 4
ð6Þ
The fiber–fiber interaction energy can be quantitatively represented by Iab 7 ; i.e., ab W FF ¼ W FF ab ðI 7 Þ:
ð7Þ
Then the second Piola–Kirchhoff stress tensor is S = 2oW/@C. The Cauchy stress tensor r, is given by r = J1FSFT, i.e., 2 3 FF F @W 1 a 2 @W Iab þ a0 b0 @Iabab a aþ a a 7 6 7 I @I 4 4 7 1 7 r ¼ I3 2 6 6 7 FF F FF 4 5 @W b @W ab @W ab ab 1 1 ffi ða b þ b aÞ 2 @Ib Ib I7 þ a0 b0 @Iab b b þ @Iab pffiffiffiffiffi a b 4
4
7
7
I4 I4
ð8Þ
Following the same material parameter determination procedure demonstrated in [30] and using the uni-axial tensile testing results shown in Fig. 4, the tensile strain energy function for the fabric is obtained as,
4 3 2 W Fa ðIa4 Þ ¼ k1 Ia4 1 þ k2 Ia4 1 þ k3 Ia4 1 Fig. 10. Fabrics formed over a double curvature mold: (a) 0° sample and (b) 45° sample.
ð9Þ
where k1 = 3.0e5, k2 = 1630.13, k3 = 1.15 with a unit of MPa. For the balanced plain weave composites, W Fb ðIb4 Þ has the same functional
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Fig. 13. Shear angles at marked points (a) 0° sample and (b) 45° sample.
format as that of W Fa ðIa4 Þ .The shearing strain energy function is obtained from the bias-extension testing results shown in Fig. 5 as a polynomial function of Iab 7 ,
W FF ðI7 Þ ¼ k4 ðI7 Þ4 þ k5 ðI7 Þ3 þ k6 ðI7 Þ2
ð10Þ
where k4 = 0.47, k5 = 0.318, k6 = 0.0842 with a unit of MPa. Adding Eqs. (9) and (10) gives the total strain energy for the woven fabric undergoing general deformation status. Accordingly, a user material subroutine UANISOHYPER_INV for commercial FEM software ABAQUS/Standard is designed for the woven fabric to implement numerical simulation on the forming experiment in the preceding section. It must be pointed out that the hyperelastic formulation used in the present study is a simple and primitive one but still capturing
the major characteristic of woven fabric during draping. Naturally, there is coupling between fiber yarn extensions and shearing. Biaxial tension test [15] may be needed for introducing this coupling into the hyperelastic model. Besides, the current model is only suitable for thin fabric. Hyperelastic model for 3D fabric can be found in [29]. The stability and ellipticity analysis [30] of the present model along with the coupling between fiber yarn extensions and shearing will be the future work. 4. Draping simulation Fig. 11 shows the finite element model for the woven fabric forming over the wing-tip mold. In the model, the punch, die and binder are modeled as rigid bodies. The rectangular plain woven
Fig. 14. Boundary profiles of formed fabrics (a) 0° sample and (b) 45° sample.
H. Yin et al. / Composites Science and Technology 92 (2014) 64–69
carbon fabric has the same dimension of 320 mm 160 mm with a thickness of 0.25 mm as in the forming experiment. It is uniformly meshed with 4 mm 2 mm S4R shell elements being assigned with the designed user material subroutine UANISOHYPER_INV. A constant 50 N force is imposed on the binder to clamp the fabric to prevent wrinkling. The total punch stroke is set to be 60 mm. To improve convergence of numerical simulation, a soft-contact formulation with a friction coefficient of 0.3 is assigned on contacts between the fabric and tooling. Fig. 12 shows the simulation result of deformed fabric with shear angle distribution for the 0° and 45° samples, respectively, under a punch stroke of 60 mm. The large shear deformation areas appear in the flange of the larger curvature surface area for both samples. The maximum shear angle is 54.4°and 55.7°for the 0° and 45° samples respectively. Although the maximum shear angle of the 45° sample is larger, its shear angle variation is obviously smoother than that of the 0° woven fabric. This indicates that the formability of the 0° woven fabric is worse than that of the 45° fabric when forming the wingtip part. Fig. 13 presents the comparison between numerical and experimental shear angles of marked points in Fig. 9. As shown in Fig. 13, overall, for both cases, the predicted shear angles are in a good agreement with experimental ones. Experimental results show that wrinkles appear for both cases at points 12 and 14, which have shear angles smaller than 34° However, the picture-frame test shows that the critical shear angle of the woven carbon fabric is about 35°. Hence, the critical shear angle cannot be used as an solely indicator for wrinkling criterion. The mechanism of wrinkling needs to be further investigated. Fig. 14 shows the top views of deformed boundary profiles of both fabrics draping over the double curvature mold. The experimental deformed boundary profiles for both cases have a good repeatability. Large discrepancy on the deformed boundary profile is found between the 0° and 45° fabrics. As demonstrated in Figs. 10 and 12–14, fiber orientation plays an important role in forming double-curvature parts. As shown in Fig. 14, numerically predicted boundary profiles have good agreement with the experimental ones. The companions on shear angle and deformed boundary profile between numerical results and experimental data indicate that the simple anisotropic hyper-elastic model is effective in characterizing the nonlinear material behavior of the woven carbon fabric in forming over a double curvature mold. 5. Conclusions Draping of a plain woven carbon fabric over a double curvature mold is investigated numerically and experimentally in this paper. A previously developed simple anisotropic hyper-elastic model is used to characterize the material behavior of the woven carbon fabric under complex deformation modes. Material parameters for the constitutive model are obtained by using experimental uni-axial tensile and bias-extension testing results of the fabric. Very good agreements on shear angle and deformed boundary profile between experimental data and numerical results are obtained. It is shown that fiber orientation plays an important role in forming double-curvature parts. Although the present work has demonstrated the validity of the material model on predicting the complex fabric redistribution and reorientation during draping over double curvature mold, further investigations on the influence of fabric lay-up and bending stiffness are necessary to make a design tool for practical application of the model. Acknowledgements The supports from the National Nature Science Foundation of China (11172171 and 11272362), Ph.D. Programs Foundation of
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