SIMPLE MODELS FOR MOLD FILLING STAGE IN LIQUID COMPOSITE MOLDING AND THEIR APPLICATIONS TO STRUCTURE-PROCESS COUPLED OPTIMIZATION

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1 Douai, FRANCE July 006 SIMPLE MODELS FOR MOLD FILLING STAGE IN LIQUID COMPOSITE MOLDING AND THEIR APPLICATIONS TO STRUCTURE-PROCESS COUPLED OPTIMIZATION Chung Hae Park 1, Abdelghani Saouab 1, Joel Bréard 1, and Rodolphe Le Riche 1 Laboratory o Mechanics, Physics and Geosciences, University o Le Havre 5 rue Philippe Lebon, BP540, F-76058, LE HAVRE, FRANCE: [email protected], [email protected], [email protected] CNRS UMR 5146 and 3MI, Ecole des Mines de Saint-Etienne, 158 cours Fauriel, F-4100, SAINT-ETIENNE, FRANCE: [email protected] ABSTRACT: The current paper is composed o two parts. In the irst part, we present the analytical and semi-analytical models to estimate mold illing time in Liquid Composite Molding processes. Their accuracy and eiciency are examined through a comparison with Control Volume / Finite Element simulations. In the second part, we propose an integrated optimization o structural perormance and manuacturing cost. The simple models are incorporated into the optimization procedure to investigate the couplings between structural perormance and manuacturing costs. By considering manuacturing at the early stage o product design, excessive manuacturing costs which sometimes arise or the best structural perormance can be avoided. In order to be cost eective, dierent manuacturing routes need to be selected depending on part dimensions, loading conditions and design criteria. KEYWORDS: LCM (Liquid Composite Molding), Analytical model, Mold illing, Integrated optimization, Cost eective manuacturing INTRODUCTION LCM (Liquid Composite Molding) processes reers to composites manuacturing processes which employ liquid resin iniltration into a preorm i.e. a dry abric reinorcement put in the closed mold. LCM processes have been widely used in aeronautic industries, because o their advantages in terms o cost reduction, part integration and control o volatile problem. From the manuacturing point o view, the main issues are reducing cost (part o which is process cycle time) and eliminating deects such as dry spots and micro/macro voids in the inished article. Thus, predicting resin low kinetics and pressure distribution in the mold is essential to optimize the process. There have been numerous studies on numerical simulation o mold illing [1]. On the one hand, numerical simulations accurately predict the resin low kinetics at the expense o a heavy computational cost, 89

2 Douai, FRANCE July 006 and even more so when the simulations are repeated to optimize the process. For example, computational cost is a burden when simulating resin inusion with high permeability layer because the resin lows through the thickness as well as in planar directions which require three dimensional meshes. On the other hand, closed orm models, which typically make more assumptions but are computationally more eicient than numerical simulations, may be preerred or design i their inaccuracy does not invalidate the inal solutions. This is the case or the optimization o injection gates and vents. From the viewpoint o design procedure, it has been a common practice to optimize process parameters only ater the structural design is decided. In LCM processes, however, there exist strong couplings between mechanical perormance and manuacturing. Fiber volume raction and orientation are key parameters to structural perormance such as stiness and strength. On the other hand, they are also major actors inluencing the preorm permeability, a key parameter to productivity and manuacturability in LCM processes. Hence, this procedure, where the manuacturing is considered ater the structural design is inalized, may require excessively high manuacturing cost or labor even i it may lead to the best structural perormance. For example, it is acknowledged that up to 80% o the manuacturing cost o the structure is ixed once the preliminary structural coniguration has been decided []. This dilemma calls or an optimization method that simultaneously considers structural perormance and manuacturing cost. To investigate the couplings between mechanical perormance and manuacturing and to consider many design solutions in the preliminary design stage, it is more eicient to use closed orm models rather than numerical simulations. Besides, this optimization approach can provide a good guideline or optimal selection o manuacturing route. In act, the criteria or process selection are numerous: the complexity o part geometry, the environmental regulation, industrial strategies, level o part quality (quantity o residual void) etc. Arguably, cost eectiveness and manuacturability are the primary criteria among them. The cost o composite structures is composed o material cost, labor cost and tooling cost. Generally, it is not an easy task to accurately predict the total manuacturing cost, since it is aected by many actors such as labor rate, machine rate, actory lay-out, batch size, etc. However, it is evident that the mold illing time plays a major role in process cycle time since the polymer curing time is usually ixed or a speciic resin. Hence, mold iling time can be a metric or the cost-eectiveness o manuacturing process. In addition, it can be a guide to estimate the manuacturability to prevent premature gelation o resin. In the irst part o this article, we present analytical and semi-analytical models or RTM, CRTM and LRI process. In the second part, using these models, a preliminary conceptual design is perormed through simultaneous optimization o the structure and the process. Resin Transer Molding SIMPLE MODELS FOR LCM PROCESSES In RTM processes, analytical solutions are easily derived or the speciic mold geometries such as linear channel-like injection and radial injection. To deal with general shaped mold, a simple model was developed or resin transer molds containing thin lat preorms with isotropic permeabilities [3]. The time required to ill the mold can be calculated by treating the resin low inside the mold as partly radial and partly channel-like low. This simple model or mold ill time o two dimensional resin 90

3 Douai, FRANCE July 006 transer molds with isotropic permeability can be applied to the preorm with anisotropic permeability through the coordinate transormation [4]. We consider complex mold geometry (0.3m 0.14m) with a circular insert in Fig. 1. Using the simple model, we estimate the mold illing times or 4 dierent injection gates: B single gate at A, B, C and simultaneous injection at three gates. The injection pressure is A 0.1MPa, and the injection gate radius is 1.5mm. The preorm permeability is m and the resin viscosity is 0.1Pa s. The results are compared with those by numerical simulation by CV/FEM (Control Volume / Finite Element Method) [5]. Good agreements are observed even or the complex mold geometry with inserts (Fig.5). Compression Resin Transer Molding Mold illing time [s] Fig. 1 Mold geometry or the validation o simple model or RTM CV/FEM Saouab et al. proposed closed 150 orm solutions or CRTM processes [6]. In the present study, we consider separate injection and compression process: injection at constant A B C A,B,C pressure and compression at Injection gate constant mold closing speed. For a linear low condition (Fig. Fig. Comparison o mold illing time in RTM 3), we can derive the closed orm solutions or total mold illing time as a sum o injection time (t inj ) and compression time (t comp ). t inj μ = ( 1 V ) KP inj L 1 V V 4 V o 1 V o 4, Model H V t = 1 comp (1) U c V o C P inj y x H W L Fig. 3 Mold geometry and dimensions We can see that the total mold illing time can be decided rom U c, mold closing speed, and the V o, iber volume raction at the moment when the injection ends and the compression begins. Mold closing speed is decided by the constraints o maximum 91

4 Douai, FRANCE July 006 pressure in the mold and the total mold clamping orce which is the sum o resin pressure and compaction pressure by preorm deormation. Then, we can obtain the V o to minimize the mold illing time. 3 U V V cμl W / o 1 Fmold = Fre sin + F iber = + As () 4 3KH V / V 1 Liquid Resin Inusion ( ) LW A striking dierence o LRI rom conventional RTM process is the adoption o High Permeability Layer (HPL or High Permeability Medium, HPM) to acilities the resin low and to reduce the inusion time. The resin low in HPM leads much aster than in reinorcement, due to the big dierence in permeability. This preerential low in HPM induces the through thickness resin low. Hence, a new approach is required considering cross low. The mold illing process in LRI can be divided into 3 steps (Fig. 4). K > K h HPM 1x x 1 < h V U 1 V U 1 max φ 1, h 1, K 1x V Preorm U U φ, h, K x ( ), K y ( ) t=0 t=t o t=t t=t p Development o preerential low in HPM U 1 >U Resin low in each layer reaches the steady state Preorm illed Flow in each layer advances at the identical speed completely maintaining the constant low ront proile U1=U U 1 =0 Fig. 4 Mold illing process in LRI L 1 P inj K 1 ( ), K T ( ) Q 1 φ 1, h 1 K ( ) Q Q T φ, h L Fig. 5 Flow kinetics in LRI (0<t<t ) Until the low reaches the end o HPM (t=t ), it is assumed that the pressure distribution is linear at each layer (Fig. 5). From these pressure distributions, we can obtain average Pinj L1 L pressure o transverse low rom HPM to iber preorm Pm =. L1 From Darcy s law we can relate the resin pressure with low rate. Considering mass conservation o each layer, we can describe the next governing equations. 9

5 Douai, FRANCE July 006 ( L L ) K P P 1 inj KT inj 1 dl1 h1 = φ1h1 μ L1 μ h1 L1 dt (3) K Pinj K P T inj ( L1 L ) dl h + = φh μ L μ h1 L1 dt Once the low reaches the tip o HPM, the transverse low rom HPM to preorm and the longitudinal low in preorm ill the remaining dry preorm (Fig. 6). L 1 =L K 1 ( ), K T ( ) φ 1 P inj K ( ), K T ( ) φ Q 1 Q A P m h 1 h F K ( ), K T ( ) φ Q B Q T h -h F L Fig. 6 Flow kinetics in LRI (t <t<t p ) L-L We adopt the assumption o linear pressure distribution again. P m is the average pressure in the zone o (L-L ) and (h 1 +h F ). From Darcy s law, we can describe the relations o each low rate. Taking into consideration the mass conservation o each zone, we can derive the governing equations. Q1 + Q A = QT d( ( h h )( L L )) (4) F Q B + QT = φ We introduce an assumption that the Total Filling Time (tp) unilled zone (L-L and h -h F ) o preorm maintains the constant tp-t aspect ratio. These set o coupled PDEs can be solved by simple numerical integration t scheme such as 40 Runge-Kutta method. 0 We present the 0 comparison o the results by models and CV/FEM simulations in Tables 1~ and Fig. 7. Case Fig. 7 Comparison o mold illing time in LRI Even though it is not a closed orm solution, the proposed model shows a much better numerical eiciency than numerical simulation. A CV/FEM simulation with 3507 nodes and 6000 triangular elements takes 1897 seconds o CPU time with Pentium 4 processor o.6ghz. With the same CPU, on the contrary, the simple model needs only old illing time [s] M A (CVFEM) A (Model) B (CVFEM) dt B (Model) C (CVFEM) C (Model) D (CVFEM) D (Model) 93

6 Douai, FRANCE July second or one calculation (10 seconds or 100 calculations) using the time increasing step o 10-4 second in Runge-Kutta method. Table 1 Material properties or simple LRI model φ 1 φ K T [m ] K [m ] μ [Pa s] h 1 [m] L [m] P inj [Pa] Table Sample cases or preorm permeability and thickness Case K 1 /K h /h 1 A B C D 10 5 STRUCTURE-PROCESS COUPLED OPTIMIZATION IN LCM Integrated Optimization o Structural Perormance and Manuacturing Process We suggest an integrated optimization method simultaneously considering structural perormance and manuacturing process. The design objective is the minimization o structural weight. We assign structural and process constraints at the same time. As a structural constraint, the stiness is considered to constrain the strain under the load. As process constraints, the mold illing time, the mold clamping orce and the maximal pressure are treated. To achieve these purposes, our parameters are optimized: layer number, layer stacking sequence, inal iber volume raction and inal part thickness. Fiber orientation is selected rom the pre-assigned angle set. In this work, we employ the 4 angle set composed o 0, 45, 90, -45. The layup o laminated plate is assumed to be symmetric. As the optimization scheme, we apply the genetic algorithm. To deal with the layer number variation, crossover and mutation operators are modiied as in Park et al. [6]. Elastic moduli o composites can be obtained rom the moduli o the constituents by the Halpin-Tsai equations. As a metric o the structural stiness, we deine the strain norm using classical lamination theory. ε = Max 0.5 ε + ε + 0. ε ε (5) ( 1 ) ( 1 ) The permeability according to the iber volume raction variation is obtained using Kozeny-Carman equation. The anisotropic permeability tensor in each layer can be related to the principal permeabilities by tensor transormation equation. The gapwise averaged permeability model is applied to obtain the perorm permeability composed o layers with dierent orientations, assuming that the in-plane permeabilities in principal directions are o the same order. Sample Problem top, bottom 5 As a design object, the rectangular plate under the lexural bending is regarded. The mold geometry and injection port location are illustrated in Fig. 3. We consider RTM, CRTM and LRI as a candidate or manuacturing route. The unidirectional carbon stitched mat (iber density: 1.79g/cm 3, areal weight: 15g/m ) is used as reinorcement. The in-plane permeabilities o mat are 10-9 m in iber direction and m in 94

7 Douai, FRANCE July 006 transverse direction at the iber volume raction o 0.4, while the permeability in the thickness direction is m. For the sake o easy layup, our plies stacked in the same orientation make up one layer. The resin viscosity is 0.15Pa s. In RTM process, the injection pressure is maintained at 0.15MPa. In CRTM process, injection is perormed under the constant pressure o 0.1MPa. Maximum allowable mold clamping orce is 300kN and maximum allowable pressure is 0.15MPa. Mold closing speed should not exceed 1 mm/s. In LRI process, injection pressure is assumed to be 99.5kPa, the dierence between atmospheric pressure (0.1MPa) and vacuum pressure (500Pa). HPM permeability is 10-8 m and its thickness is 1 mm. Results and Discussion For the various loading conditions and plate dimensions, optimal material conigurations are obtained or each manuacturing process (Tables 3~4). Loading M x M y [N] [N] Loading M x M y [N] [N] Table 3 Results o structure-process coupled optimization (L=0.5m, W=0.5m, ε c =0.001, t c =40s) Process Weight Optimal coniguration [g] V H Stacking sequence Layer [mm] (symmetric layup, s) number RTM s 10 CRTM s 10 LRI s 10 RTM s 16 CRTM s 18 LRI s 16 *M x and M y denote the moment per unit length Table 4 Results o structure-process coupled optimization (L=1.0m, W=0.5m, ε c =0.001, t c =600s) Process Weight Optimal coniguration [g] V H Stacking sequence Layer [mm] (symmetric layup, s) number RTM s 10 CRTM s 10 LRI s 10 RTM s 16 CRTM s 16 LRI s 16 We can see that optimal material coniguration changes according to the manuacturing route as well as the loading conditions, the design constraints and the plate dimensions. Cost-eectiveness o each process can be investigated with the optimization results reerring to the material cost and the batch size. For example, CRTM process results in lighter structure than LRI process does under the same design constraints, in some case (e.g. M x =1000N, M y =1000N, L=0.5m, W=0.5m). However, CRTM process needs more iber mats (18) than LRI does (16) or each product. Furthermore, the total manuacturing cost also depends on the batch size. Since the equipment and tooling cost per product is critical, LRI may be better in terms o manuacturing cost, or the low 95

8 Douai, FRANCE July 006 production number. On the other hand, the equipment and tooling cost goes marginal, as the production number increases. Otherwise, we can assign dierent mold illing time constraints with the aid o more exact model or the total manuacturing cost evaluation. CONCLUSIONS Analytical and semi-analytical models have been proposed or Liquid Composite Moldings: Resin Transer Molding, Compression Resin Transer Molding and Liquid Resin Inusion. They are not only numerically eicient but also accurate enough to be applied in a global optimization procedure. The semi-analytical LCM models have been applied to the integrated optimization o structural perormance and manuacturing process. In the preliminary design stage, this approach provides a good guideline to predict the cost-eectiveness o each process or given design criteria, structural size and loading conditions. ACKNOWLEDGEMENTS The authors would like to acknowledge the Pro. Woo Il Lee and Dr. Moon Koo Kang at Seoul National University in Korea or the use o RTM simulation code. REFERENCES 1. J. Mackerle, Finite Element Analyses and Simulations o Manuacturing Processes o Composites and Their Mechanical Properties, A Bibliography ( ), Computational Materials Science, Vol. 31, 004, pp K. Wang, D. Kelly, and S. Dutton, Multi-Objective Optimisation o Composite Aerospace Structures, Composite Structures, Vol. 57, 00, pp A. Boccard, W. I. Lee, and G. S. Springer, Model or Determining the Vent Locations and the Fill Time o Resin Transer Molds, Journal o Composite Materials, Vol. 9, No. 3, 1995, pp C. H. Park, W. I. Lee, W. S. Han and A. Vautrin, "Multi-Constraint Optimization o Composite Structures Manuactured by Resin Transer Molding Process," Journal o Composite Materials, Vol. 39, No. 4, 005, pp M. K. Kang and W. I. Lee, A Flow Front Reinement Technique or the Numerical Simulation o the Resin Transer Molding Process, Composites Science and Technology, Vol. 59, 1999, pp A. Saouab and J. Bréard, Analytical Modeling o CRTM and RTM Processes: Part A. General Mathematical Development, International Journal o Forming Processes, Vol. 9, No. 3, 006, in press 96