Deep drawing simulation of thermoplastic reinforced composite sheet

Transcription

1 CIFMA1 - IFCAM1, 2-4 mai/may 26 Deep drawing simulation of thermoplastic reinforced composite sheet S.A.sadough,F.R.Biglari,M.Shirani,A.Agahi Mechanical Engineering Department, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran Center of Excellence in Solid Mechanics mehdishirani@cic.aut.ac.ir Abstract: This paper covers numerical investigations of the deep drawing of thermoplastic reinforced composite sheet into a hat shape, combining a hemispherical cup with a wide flat rim. A non-orthogonal constitutive model to characterize the anisotropic material behavior of woven composite fabrics under large deformation is proposed. Also the thermoplastic resin assumed to be a Newtonian fluid and the constitutive model for Newtonian fluid used to model the thermoplastic resin.by using the developed constitutive models, user defined material subroutine was coded for commercial FEM package, ABAQUS/Explicit (VUMAT) code. By using this method the fiber orientation, logarithmic strain and the boundary profile of final part predicted. Keywords: Composite, Finite element, Forming, Reinforce, Simulation. 1. Introduction Woven fabric (as shown in Figure 1) reinforced composites have been widely used in the aerospace, automotive, and sporting goods industries due to a number of advantages such as high strength to weight ratio, ease of handling, and well developed weaving technology. As the use of these materials become more prevalent, automated manufacturing processes will be needed to replace the hand shaping of woven preforms. For the design of such automated manufacturing processes, the use of accurate numerical tools for simulating the deformation behavior of preforms becomes indispensable. In order to accurately reflect the material behavior in the simulation, constitutive equations need to be defined that take into account the mechanical behavior of the material. There have been several studies conducted for the constitutive modeling of woven preforms, with the majority of these focusing on the prediction of fiber angle change (angle between warp and weft fibers) due to the shear deformation [1 5]. Severe fiber angle changes in woven preforms have been shown to cause defects such as wrinkling and buckling during forming. When the part geometry possesses a double curvature, the forming process usually results in significant in-plane shear deformation, which consequently leads to complex redistribution and reorientation of fibers in the woven fabric reinforced composite. The change of fiber orientation during large deformation induces additional anisotropy into the composite. However, existing material models involving this non-orthogonality are very limited. There are two different approaches for the simulation of the draping of fabrics, the geometrical and the mechanical approach. The geometrical approach involves a type of mapping mostly based on a kinematics model. The pin-joint model, or so-called kinematics approach, was one kind of approach applied to simulate fabric forming [6 8]. This method is fast and fairly efficient but this method does not account for the mechanical behavior of the fabric. The model was extended in Ref. [9]. The effects of boundary conditions, such as friction and binder force, on fabric deformation and loads generated during the forming process, however, were not considered. The fishnet model [1 16] is commonly used for the geometrical analysis of the draping of fabrics. Other geometrical technique is the optical projection method [11, 18]. The effect of material properties is not generally included in the geometrical approach. The solid mechanics analysis of the draping of fabrics by using finite element methodology [19 22] takes into account the mechanical properties of the fabric and, hence, describes the physical process of draping. The fabric is considered as a continuum and the analysis of the draping of fabrics is essentially an extension of the deep drawing [23, 24] or diaphragm forming simulation of metal forming processes [25]. For woven composites, there are two different approaches to predict the material properties and simulate the forming processes, micromechanical and macroscopic approaches. The micromechanical approach is represented by the work of Hsiao and Kikuchi [26], which can take into account all geometrical and mechanical characteristics associated with the woven fabric and the resin, and is capable of calculating stresses in the microstructure. Nevertheless, it is time-consuming and computational costly for complex fiber architectures and forming processes. The micromechanical approach for fabric can be found in Ref. [27]. The macroscopic approach has its own merits because it is efficient and able to consider various bounding conditions and material behaviors provided that a good constitutive law is used. A lot of effort has been made in this area, such as the mosaic model and the fiber undulation model [28], which are orthogonal models, as well as the viscoelastic model [29], which considered the initial non-orthogonality of fibers, however, not its evolution. Compared to extensive literatures on the orthogonal material models for woven composites and fabrics [3 37], little efforts have been given to the non-orthogonal constitutive relations.

2 CIFMA1 - IFCAM1, 2-4 mai/may 26 Taking into account the effect of fiber orientation on material anisotropy, Yu et al [1] developed a non-orthogonal constitutive model for woven composites based on force analysis along weft and warp fiber yarns under a continuum mechanics framework. Based on the stress and strain transformations in the orthogonal and non-orthogonal coordinate Figure 1: Woven fabric. systems and the rigid body rotation matrix, Xue et al [3] developed a constitutive model for characterizing the nonorthogonal material behavior of woven composite fabrics under large deformation. The model was validated by comparing numerical shearing results of a plain weave composite sheet with experimental data. Peng and Cao [38] presented a systematic framework for predicting the effective nonlinear orthotropic elastic moduli of textile composites with the combination of the homogenization method and finite element analysis. This pure numerical approach provides a generic method for the material characterization of textile composites. No experimental test is needed to obtain the effective material properties of textile composites. However, the accuracy of this approach mainly depends on the geometric description of the unit cell and the homogenized material properties imposed. Gaps between fiber yarns and fiber threads usually result in a much lower tensile modulus in the transverse direction than that modeled in the unit cell. No feasible approach is available to accurately evaluate the effect of gaps. In addition, the stochastic distribution of geometric parameters and material properties in the textile composites makes it very hard to build a unit cell that can accurately represent the material behavior of the composite. To circumvent the difficulties involved in the purely numerical approach and to capture the anisotropic material behavior of woven composite fabrics during forming, a nonorthogonal constitutive model is developed by Peng and Cao [39] under the same material characterization framework as in Peng and Cao [38].They use the real material properties in their model..in this paper we use discrete layers of woven and viscous materials to model the thermoplastic reinforced composite sheet. Each ply of the viscous material is modeled as a transversely isotropic incompressible Newtonian fluid, also we use the experimental results and nonorthogonal constitutive model that developed by Peng and Cao [39] to model each ply of the woven material. The developed constitutive equations were implemented into the ABAQUS/Explicit code using the user material subroutine VUMAT [4]. By using this code the fiber orientation, logarithmic strain and the boundary profile of final part predicted. Assumptions used are: (1) the tensile shear behavior are assumed to be decoupled in the non-orthogonal coordinate system. This tensile shear decoupling was well verified in the experimental studies on the biaxial tensile tests of woven composite fabrics [41, 42]. This observation makes the modeling work much easier; (2) the tensile behavior of fiber yarns is assumed not to be affected by gap closing due to shearing. The first two assumptions are related and both are reasonable when the shear angle is low. However, when the shear angle is close to the locking angle, the first two assumptions might be invalid; (3) yarn slippage is negligible; (4) the thickness of the composite plate should be small enough, so that the plane stress approach can be used. 2. Constitutive model for woven fabric To trace the rotation of fiber yarns in woven fabrics during deformation, we choose the two vectors, g 1 and g 2, to coincide with the current weft and warp directions of the fabrics, as shown in Figure 2. Figure 2: Schematic of a deformed plain weave structure with shear deformation [39].

3 CIFMA1 - IFCAM1, 2-4 mai/may 26 For simplicity, g 1 and 2 g i construct a non-orthogonal coordinate system which reflects the fiber reorientation during deformation. Experimental studies on the biaxial tensile tests of woven composite fabrics have verified that the shear stress and the direct stresses can be treated as uncoupled. Hence, it should be reasonable to assume that in the non-orthogonal coordinate system { g i } the elastic matrix, which relates the stresses to the strains, has an orthotropic format. Consequently, we can rewrite the constitutive relation in a rate form as g are chosen to be unit vectors. The vectors { } d σ dσ d σ D ( ε) 12 = D ( ε) D D ( ε) ( ε) d ε11 d ε D ( ε) dγ 12 (1) D in non- g. The balanced plain weave fabric used is a fiber reinforced thermoplastic composite (glass fiber and polypropylene resin). As a natural way, we are expected to express these elastic moduli as functions of the strain components in the non-orthogonal coordinate system. The real tensile moduli D 11 and D 22 were determined by Peng and Cao [39] has the following formulation 3. Determination of the elastic matrix [ D ] In this section we will using experimental force displacement curves [39] to obtain the elastic matrix [ ] orthogonal coordinate system { i }.1 ii 2 D ( MPa) = 1+ exp( 6( εi.1)) 12 i< εi <.22 ε ε i.22 (2) where i=1 or 2 and no summation for i. Considering the weak interaction between the weft and warp yarns under tension, the term D 12 is assumed to have a very small value ( MPa).2 min( D D 22 ) D = (3) The real shear modulus were determined by Peng and Cao [39] has the following formulation D ( MPa) = 5.27γ γ γ (4) 4. Constitutive model for thermoplastic resin Constitutive approaches have considered the composite sheet either as a continuum, or as layers of discrete materials. In this paper we use discrete layers of woven and viscous materials to model the composite sheet. Each ply of the viscous material is modeled as a transversely isotropic incompressible Newtonian fluid. Such a material is characterized by a single fluid viscosity η with the following constitutive equation: = (5) ij σ Pδij + 2η εij Assumption of plane stress condition leads to the following equation set: 11 σ 4η 22 = σ 2η 12 σ 2η 4η ε11 ε 22 η 2ε 12 (6) Where η is the resin shear viscosity. The resin viscosity is taken to be η=5 Pa s. 5. Implementation of user material subroutine VUMAT In user subroutine VUMAT, tensor quantities are defined in the non-orthogonal coordinate system that rotates with the material point so that we directly use the components of the stress and strain tensors in our non-orthogonal constitutive equation and we don t have to convert the components of stress and strain tensors from non-orthogonal coordinate

4 CIFMA1 - IFCAM1, 2-4 mai/may 26 system to orthogonal coordinate system. To illustrate what this means in terms of stresses, consider the bar shown in Figure 3, which is stretched and rotated from its original configuration, AB, to its new position, A B. This deformation can be obtained in two stages; the bar is first stretched, as shown in Figure 4, and is then rotated by applying a rigid body rotation to it, as shown in Figure 5. Figure 3 : Stretched and rotated bar. Figure 4 : Stretching of bar. Figure 5 : Rigid body rotation of bar. The stress in the bar after it has been stretched is σ 11, and this stress does not change during the rigid body rotation. The X Y coordinate system that rotates as a result of the rigid body rotation is the non-orthogonal coordinate system. The stress tensor and state variables are, therefore, computed directly and updated in user subroutine VUMAT using the strain tensor. 6. Model validation To verify the model a multi-element model is built to simulate the bias extension test of sample size of 167 mm 334 mm. In Figure 6 the numerical load displacement curve obtained from the FEM simulation is compared with experimental results Simulation experimental Load (N) Displacement (mm) Figure 6: The numerical load displacement curve obtained from the FEM simulation is compared with experimental results. As can be seen from Figure 6, a very good agreement between the numerical bias extension load and the experimental results is obtained. 7. Geometrical finite element model and basic set up of deep drawing simulations The simulations were performed using the geometrical model, shown in Figure 7, includes a rigid punch, a die and a holder and the thermoplastic reinforced composite blank.

5 CIFMA1 - IFCAM1, 2-4 mai/may 26 Figure 7 : Finite element model To reduce computation time only one quarter of the specimen was modeled. The punch is a hemispherical mould with a radius of 97 mm joined with a cylindrical flange at the upper end. The die is a hemispherical hat shape mould with a radius of 1mm in the hemisphere and a radius of 25 mm in the flat disc part. The holder is a flat ring to hold the fabric during draping. The blank material is a composite sheet of thickness of 1 mm. Four-node shell elements SR4 are used to model the composite blank. This type of element is of bilinear finite strain and is designed to handle large deformations and large rotations. The holder, die and punch are represented by analytical rigid bodies. The initial shape of elements is chosen to be squares. 8. Simulation of composite sheet forming using VUMAT subroutine Simulation was done using the constitutive equations, geometrical model and the non-orthogonal elastic matrix [ D ( ε )] obtained in previous sections. The material directions of the undeformed woven fabric coincide with the sides of the initial mesh and we didn t refine the mesh during the simulation so that the finite element mesh was used to represent the fiber yarns in the composite part. The mesh size was The computer time involved in running the forming simulation using explicit time integration with a given mesh is directly proportional to the time period of the event. In explicit dynamic analyses, the stable time increment size is a function of the mesh size and the material stiffness. Investigations were carried out for different punch speeds, v, from.85 to 85 m/s. Figure 8 shows deformed mesh with contour of logarithmic strain with punch speed of.85 m/s. Figure 8 : Deformed mesh with contour of logarithmic strain with punch speed of.85 m/s.

6 CIFMA1 - IFCAM1, 2-4 mai/may 26 Figure 9 displays the predicted shear angle along the diagonal line of the fabric with punch speed of.85. Grid element angle ( degrees L/S Figure 9: Plot of the values of the angle of the grid elements along the diagonal fabric direction as a function of the normalised arc distance L / S from the apex for the woven fabric draped over a hemispherical dome. Figure 1 shows edge profile of the deformed woven preform for different punch speeds. Figure.11 displays the predicted punch force for different punch speeds. The predictions for punch force and boundary profile with 85 m/s and 8.5 m/s punch speeds seem not to be accurate and result in large oscillations whereas the accuracy is much improved for speeds lower than.85 m/s. Y (mm) V= 85 m/s V= 8.5 m/s V=.85 m/s V=.85 m/s V=.85 m/s X (mm) Figure 1: Edge profile of the deformed woven preform for different punch speeds Force (N) V= 1.7 m/s V=.85 m/s V=.85 m/s V=.85 m/s Punch Displacement (mm) Figure 11 : The predicted punch force for different punch speeds.

7 CIFMA1 - IFCAM1, 2-4 mai/may 26 From the results presented in Figures 1 and 11 the precision regarding punch force and boundary profile converges with decreasing punch speed while the computation time increase. Apart from the need of more computation time due to the larger number of elements, fine meshes require smaller stable time increments, thus adding to the total number of time increments and the computation time requirements. To assess the effect of mesh size on CPU time different mesh sizes were employed for the blank, namely 8 8, 1 1 and shell elements. The punch speed was.85 m/s. Figure 12 illustrates the computational costs for different mesh sizes. Time (s) Number of meshes Figure 12: The computational costs for different mesh sizes. 9. Conclusions A non-orthogonal constitutive model to characterize the anisotropic material behavior of woven composite fabrics under large deformation was proposed. Also the thermoplastic resin assumed to be a Newtonian fluid and the constitutive model for Newtonian fluid used to model the thermoplastic resin.by using the developed constitutive models, user defined material subroutine was coded for commercial FEM package, ABAQUS/Explicit (VUMAT) code. By using this method the fiber orientation, logarithmic strain and the boundary profile of final part predicted. References [1] Yu WR, Pourboghrat F, Chung K, Zampaloni M, Kang TJ. Nonorthogonal constitutive equation for woven fabric reinforced composites. Composites Part A 22;33: [2] Yu WR, Zampaloni M, Pourboghrat F, Chung K, Kang TJ. Sheet hydroforming of woven FRT composites: nonorthogonal constitutive equation considering shear stiffness and undulation of woven structure. Compos Struct 23;61: [3] Xue P, Peng XQ, Cao J. A non-orthogonal constitutive model for characterizing woven composites. Composites Part A 23;34: [4] Dong L, Lekakou C, Bader MG. Processing of composites: simulations of the draping of fabrics with updated material behavior law. J Compos Mater 21;35(2): [5] Billoet JL, Cherouat A. New numerical model of composite fabric behavior: simulation of manufacturing of thin composite part. Adv Compos Lett 2;9(3): [6] Van West BP, Pipes RB, Keefe M. A simulation of the draping of bidirectional fabrics over arbitrary surfaces. J Textile Inst 199;81(4): [7] Chou TW, Ko FK. Textile structural composite. Composite material series, vol. 3. Amsterdam: Elsevier; [8] Laroche D, Vu-Khanh T. Modelling of the forming of complex parts from fabric composites. In: Hoa SV, editor. Development and design with advanced materials. Amsterdam: Elsevier; p [9] Long AC, Robitaille F, Souter BJ. Mechanical modeling of in-plane shear and draping for woven and non-crimp reinforcements. J Thermoplast Composite Materials 21;14(4): [1] Mack C, Taylor HM. The fitting of woven cloths to surfaces. Journal of Textile Institute 1956;47:T [11] Potter KD. The influence of accurate stretch data for reinforcements on the production of complex structural mouldings. Part 1. Deformation of aligned sheets and fabrics. Composites 1979: [12] Robertson RE, Hsiue ES, Sickafus EN, Yeh GY. Fibre rearrangements during the molding of continuous fibre composites. I. Flat cloth to a hemisphere. Polymer Composites 1981;2(3): [13] Van West BP, Meefe M, Pipes RB. The draping of bidirectional fabric over three-dimensional surfaces. Proceedings of the American Society for Composites. The Fourth Technical Conference, 3 5 October,Virginia, Lancaster: Technomic Publishing, p [14] Trochu F, Hammami A, Benoit Y. Prediction of fibre orientation and net shape definition of complex composite parts. Composites: Part A 1996;27: [15] Bickerton S, Simacek P, Guglielmi SE, Advani SG. Investigation of draping and its effects on the mold filling process during manufacturing of compound curved composite parts. Composites Part A 1997;28:81 16.

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