SIMULATION OF INFUSION PROCESS FOR COMPOSITES MATERIALS COUPLING STOKES, DARCY AND SOLID FINITE DEFORMATION

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1 SIMULATION OF INFUSION PROCESS FOR COMPOSITES MATERIALS COUPLING STOKES, DARCY AND SOLID FINITE DEFORMATION ARNAUD DEREIMS ESI GROUP (LYON) & CLAUDE GOUX LABORATORY UMR CNRS ECOLE NATIONALE SUPÉRIEURE DES MINES DE SAINT-ETIENNE SYLVAIN DRAPIER CLAUDE GOUX LABORATORY UMR CNRS ECOLE NATIONALE SUPÉRIEURE DES MINES DE SAINT-ETIENNE JEAN-MICHEL BERGHEAU LABORATORY OF TRIBOLOGY AND SYSTEMS DYNAMIC - ECOLE NATIONALE D INGENIEUR DE SAINT-ETIENNE PATRICK DE LUCA ESI GROUP (BORDEAUX) ABSTRACT Infusion processes are a cheaper alternative to the usual injection process to realize big composite parts. However the bad control of the final properties of the molded part is a disincentive to their democratization at industrial level. We are presenting an advanced modeling method, coupling fluid flows with finite deformation solid mechanics, in order to anticipate as best as possible thickness variation of the part and infusion time. INTRODUCTION Composite manufacturing processes by resin infusion have been developed for years to bring a cheaper solution to big parts production. Those processes allow a significant cost reduction in raw materials storage and mold fabrication, a shorter infusion time and limit void formation. However, the lack of control on the final properties of the part, implying long and expensive process settings, significantly reduce the above-mentioned advantages. So, it seems necessary to develop simulation methods to anticipate those properties. LRI PROCESS DESCRIPTION Liquid Resin Infusion (LRI) is among the most used infusion process in industrial application. It consists in creating a liquid film of resin, on the top of the dry preforms, thanks to a very permeable material, the distribution medium. Then, it is the pressure differential between the vent (0 bar) and the injection line (atmospheric pressure) that provokes the infusion of the resin in the thickness direction (fig. 1a). Finally, a pressure and temperature cycle is applied to the system to bring the resin in a solid state. The flexibility of the vacuum bag does not allow maintaining a constant thickness during the whole process, while its good control is mandatory in the geometrical and mechanical properties of the final part. (a) (b) Fig. 1. (a) Principle of the Liquid Resin Infusion process (LRI). (b) Proposed model [1]

2 THE MODEL The chosen model is the one proposed by P. Celle in his PhD thesis [1-2]. This innovative model, which particularity is the distribution medium modeling, based on a three zones division (fig. 1b): - Stokes zone: fast flow zone constituted of the distribution medium and the resin, - Darcy zone: incompressible flow of the resin in the preforms submitted to finite deformation, - Dry preforms zone: zone constituted of non-impregnated preforms submitted to finite strains. Moreover the process can be split in four time periods: - Pre-filling: initial compaction of the preforms due to the vacuuming of the system, - Filling, - Post-filling: re-compaction or rest period ending by the mechanical equilibrium mandatory to the dimensional quality of the final part, - Curing (not studied here). PRE-FILLING SIMULATION During the pre-filling stage, there is no resin in the system, so only the dry preforms zone need to be considered. At this stage, since vacuum is established in the mold preforms are submitted to atmospheric pressure (1 bar) resulting in deformations. Neglecting volume forces we can write the momentum conservation equation as follow: (1) with the overall constrain and the contact forces. In addition, preforms are represented by an equivalent homogenous medium composed of rigid fibers, so that each macroscopic strain is reflected by fiber rearrangements at microscopic scale. This approach allows computing directly porosity from deformations with no need to use semi-empiric laws, expressing porosity or thickness variation as a function of pressure, as it is usually done in literature [3-4]. FILLING SIMULATION During the filling stage, the resin is first absorbed by the distribution media and then infuses the preforms in the thickness direction. So the three zones described above (fig. 1b) have to be considered, implying a complex coupling of two flows (Stokes and Darcy) with solid mechanics. Resin Flow Due to the high permeability ratio of the distribution media with respect to fibers, in first approximation, we can consider an incompressible stokes flow (2): ( ( )) with the resin viscosity, the resin velocity, the resin pressure and ( ) ( ) the eurlerian strain rate associated to the velocity field. While in Darcy s zone, the incompressible flow across a low permeability (less than 10-9 m²) porous medium is governed by Darcy s law (3): (2)

3 (3) with Darcy s velocity or the mean macroscopic resin velocity with respect to the fibers, permeability tensor, the resin viscosity and the pore pressure. The solid mechanics is the same than for the pre-filling phase, except that now it needs to be coupled to fluid mechanics, to take into account the mutual influence of fluid on structure and fibers on resin. The coupling methods are described in the following. Coupling method Stokes/Darcy coupling: The coupling between the two domains (Darcy and Stokes) is realized by considering the normal stress and normal velocity continuities across the interface. So, normal velocity on the interface taken into Darcy s domain is applied as boundary condition to stokes equations, and the hydrostatic pressure taken into stokes domain is applied to Darcy s law (fig 3). Then the problem is solved by an iterative method and the convergence checked by a relative error computation on velocity and pressure fields between two iterations. Fig. 3: Stokes/Darcy coupling Fluid/structure coupling: The fluid/structure strong coupling is treated iteratively in a quasi-static context. Indeed, the Stokes/Darcy coupled problem is solved on fix grid (i.e. rigid preforms), the resin influence on the fibers being taken into account, through pore pressure by Terzaghi s law (4): (4) with overall stress applied to the system, the effective stress in the fibers and the unity tensor. The non-linear mechanical problem in finite strain is, then, solved for a given pore pressure (i.e. for a Terzaghi equivalent behavior), finally, the strain influence is reflected by the porosity variation modifying the permeability ( ) of the medium, determined with Karman-Cozeny law (5), ( ) (5)

4 where is the fiber diameter, the Kozeny constant and the porosity. Finally, convergence is verified by a relative error computation on velocity and pressure fields of resin and solid displacement between two iterations. Flow front Flow front evolution is managed by the PAM-RTM filling algorithm. This algorithm is based on decomposition of the transient phenomenon in a succession of quasi-static state. New filled elements, between two states or filling stages, are identified from the resin velocity computed in the previous time step. A filling factor going from 0 to 1 is affected at each element, fluid problems (Darcy and Stokes) being solved over elements having a non-zero filling factor. POST-FILLING SIMULATION The filling phase is immediately followed by a stage, more or less long, of mechanical balancing, consisting, eventually, of a re-compaction of the system to reach the fiber fraction desired. The simulation of this stage, known as post-filling in the literature, requires a boundary condition change. Indeed, if we close the vent or the injection or both, resin cannot flow across those boundaries anymore, so the initial pressure boundary condition needs to be converted into an impervious wall (6): (5) This stage, rarely describe in literature, is mandatory to the good modeling of physical phenomenon coming out during infusion. Indeed, simulation shows that a pressure gradient, implying a fiber fraction gradient, appears at the end of filling (fig. 4). In fact, the expected mechanical properties of the final part depend on the homogeneity of the fiber fraction, so it is necessary to obtain equilibrium before the curing. This equilibrium time could be more or less long depending on part thickness, resin density and viscosity or vent position. Fig. 4 : resin volume fraction at the end of filling of a simple plate [6] EXPERIMENTAL VALIDATION: CASE OF PLATE The model, presented above, was implemented in a finite element code based on the C++ library ProFlot. Simulation results were successfully compared to experiment in the case of a plate, during a first validation, by P. Wang [6]. The case was a plate measuring 335 mm x 335 mm, composed of 24 plies [90,0] [6]. Initial thickness measured with a viernier was 9.8mm. After initial compaction, a mean thickness of 6.18mm was measured, while the simulated one was 6.17mm. After filling stage, simulated swelling was 0.63 mm when the experimental measure was 0.59mm. Figure 5 shows initial conditions and some results.

5 (a) (b) Before initial compaction After initial compaction End of filling (c) Fig. 5: Simulation example: (a) Mechanical boundary conditions, (b )fluid boundary conditions, (c) Pressure field and geometry obtain after initial compaction and after filling stage DEVELOPMENT AND PERSPECTIVE This innovative model, even though validated for the simple case of a plate by P. Wang, needs some improvements to be industrialized. DOMAIN DECOMPOSITION In previous part, we proposed a three zones decomposition (fig 1b), making a Stokes flow hypothesis in the distribution medium. This assumption could be discussed, as the distribution medium is a porous medium with a high permeability, usually between 10-4 and 10-6 m², so a Darcy flow could be more appropriate to describe this phenomenon. Stokes/Darcy coupling remains necessary to simulate flow in distribution line and preferential flow zones which do not contain fibers (racetracking). This New approach should permit to obtain a more accurate model, in order to estimate infusion time as well as possible. THERMAL-CHEMICAL COUPLING Another mandatory point to better anticipate infusion time is to consider the thermal and chemical phenomenon during the flow phase. Indeed, viscosity could change along the filling due to temperature and curing degree in non-isothermal cases. So, it would be necessary to introduce a fourth coupling by integrating energy and chemical transport equation in our model for the filling phase. We can also imagine a thermal influence on the preforms behavior law.

6 CONCLUSION We presented an innovative model, by its global approach, to simulate fabrication process of composite parts by resin infusion. Caring about delivering to the professionals a simulation tool closer as possible to reality, we tried to consider the most of known phenomenon. To do that, we implemented a coupling method between Stokes, Darcy and solid mechanics equations which was compared successfully to experiment. However, this model needs other development to give results closer to the needs of industrials. Some improvements, as new domain decomposition or thermal-chemical coupling, have been proposed and will be presented in future communications. ACKNOWLEDGEMENTS The authors would like to thanks the support of the European Union in the INFUCOMP project (European Community s Seventh Framework Program FP7/ under grant agreement n ) financing partially these researches. REFERENCES [1] P. Celle, S. Drapier, J-M. Bergheau. «Numerical modeling of liquid infusion into fibrous media undergoing compaction», European Journal of Mechanics - A/Solids, 27(4) : , [2] P. Celle, «Couplage fluide / milieu poreux en grandes déformations pour la modélisation des procédés d élaboration par infusion», PhD Thesis, École Nationale des Mines de Saint-Étienne, [3] T. Gutowsky, T. Morigaki. «The Consolidation of laminate composites», Journal of Composite Materials, 21(2), , [4] T. Ouahbi, A. Souad, J. Bréard, P. Ouagne, S. Chatel. «Modeling of hydro-mechanical coupling in infusion processes», Composites Part A : Applied Science and Manufacturing, 38(7), , [5] P. Šimáček, D. Heider, J.W. Gillespie Jr. and S.G. Advani. «Post-filling flow in vacuum assisted resin transfer molding processes : Theoretical analysis», Composites Part A : Applied Science and Manufacturing, 40(6-7) : , [6] P. Wang, S. Drapier, J. Molimard, A. Vautrin, and J-C. Minni. «Characterization of Liquid Resin Infusion (LRI) filling by fringe pattern projection and in situ thermocouples», Composites Part A : Applied Science and Manufacturing, 41(1) : 36-44, 2010.