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Unidirectional compression of fibre reinforcements. Part 2: A continuous permeability tensor measurement
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 67 (2007) 638–645 www.elsevier.com/locate/compscitech
Unidirectional compression of fibre reinforcements. Part 2: A continuous permeability tensor measurement S. Comas-Cardona *, C. Binetruy, P. Krawczak Polymers and Composites Technology and Mechanical Engineering Department, Ecole des Mines de Douai, 941 rue Charles Bourseul, BP 10838, 59508 Douai, France Received 4 January 2006; received in revised form 19 July 2006; accepted 25 July 2006 Available online 2 October 2006
Abstract Modelling LCM processes, where resin flow and compression of the fibre reinforcements are involved, requires a proper modelling of the fibre reinforcements behaviour in compression and to define how the permeability tensor evolves. The permeability measurement techniques based on fluid injection lead to data scattering. Using a material testing machine, unidirectional compression tests are performed on impregnated fibre reinforcements to induce in-plane and through-thickness fluid flows. With proper viscosities and compression speeds, the expelled-fluid pressure becomes sufficiently high to be measured. Then, using that fluid pressure and following a methodology based on numerical optimizations, the in-plane equivalent permeability and through-thickness permeabilities can be extracted. An additional measurement is needed to know the anisotropy ratio and calculate the two in-plane principal permeabilities. Due to the nature of the compression test, a major advantage of the present method is that it is continuous with respect to fibre volume fraction and that it limits errors involved with injection techniques. Samples can previously be sheared before compression and permeability measurements. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: A. Fabrics/textiles; Permeability; Unidirectional compression
1. Introduction Modelling the liquid composite moulding (LCM) manufacturing processes requires to input accurate material data such as fibre reinforcement permeability. Two types of permeability, saturated and unsaturated, can be considered. The saturated permeability is measured using an exiting flow rate measurement technique whereas the unsaturated one is usually measured recording the position of a moving flow front. This study will focus on the saturated permeability because it is the most used, measured and the most reliable. A great amount of effort has been spent on measuring such material property. Most of the techniques employed rely on fluid injection experiments with:
*
Corresponding author. Tel.: +33 3 27 71 21 87; fax: +33 3 27 71 29 81. E-mail address: [email protected] (S. Comas-Cardona).
0266-3538/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.07.020
– unidirectional flow and constant injection pressure [1,2] or constant flow rate [3,4], – biaxial flow and constant injection pressure [5,6] or constant flow rate [7], – through-thickness flow with constant pressure [1,8] or constant flow rate [9,10], – three dimensional flows [11–13]. Based on fluid injections, those techniques require a pressure pot, a vacuum pump, or an injection piston, a mould and a data acquisition system. Pressures, flow rates, pressure sensors and scales also have to be monitored during the injection. A limitation of the injection techniques is the use of all the equipment previously cited. Also, during the injection, common errors arise when proceeding to the measurement [14,15] such as the deflection of the mould, edge effects (racetracking), capillary effects and error in inlet hole dimension.
S. Comas-Cardona et al. / Composites Science and Technology 67 (2007) 638–645
639
Nomenclature b Co Cs h h_ Ki p r R
Biot’s coefficient compression modulus of the impregnated porous medium in drained conditions bulk modulus of the material constituting the porous medium (fibres) current sample thickness compression speed hydraulic permeability in the i-direction fluid (resin) pressure radial position sample radius
Because of those limitations and the scattering of the experimental data, authors also worked on numerical predictions of fibre reinforcement permeability. In that case, a periodic unit cell of the fabric has to be defined. Then fluid flow dynamics are solved in the unit cell domain (inter and intra yarns) to extract permeabilities. Fluid flow dynamics solutions can be obtained from numerical methods such as finite volume [16], finite element [17], lattice Boltzmann [18], meshless Lagrangian smoothed particle hydrodynamics [19] or the homogenization [20] method. One limitation of such numerical solution is the difficulty to define a periodic unit cell that would be representative of the fibre reinforcement. Moreover, the unit cell has to be computerized in terms of geometry and mesh, which is not an easy task for fibre reinforcements. Finally, even when the geometry and mesh are given for a fibre volume fraction, the work (especially the meshing, which is the most time consuming stage) has to be reiterated for another fibre volume fraction target. In order to overcome that issue, another type of continuous permeability measurement based on the compression of saturated fibre reinforcements has been introduced [21,22]. Impregnated fibre textiles are placed between two compression platens. The compression of the sample produced by the mechanical test machine forces the fluid to flow. Buntain and Bickerton measured the fluid pressure with a pressure transducer during the compression [22] and calculated the in-plane permeability using the equation proposed by Gutowski for composites consolidation [23]. Ait si Ahmad et al. proposed an even simpler experimental setup. The fluid pressure is measured via two compression tests using the force cell of the mechanical test machine [21]. These two methods have the advantage to limit the amount of equipment required and to continuously measure permeability in term of fibre volume fraction. Limitations however remain, the methods are limited to transversely isotropic materials and do not allow to measure the through thickness permeability. A simple compression setup based on a material testing machine has been previously proposed to test fibre rein-
t ur x,y,z / l rzz rozz
time interstitial fluid velocity in the r-direction Cartesian coordinates porosity fluid dynamic viscosity total stress applied to the impregnated porous medium in the z-direction stress applied to the impregnated porous medium in the z-direction
forcements in either dry or impregnated state [24]. That compression setup is further exploited in this paper to propose a reliable and generic measurement method that will provide the in-plane and through-thickness permeabilities for a wide range of fibre volume fraction and materials. 2. Methodology 2.1. Total stress constitutive equation When impregnated fibre reinforcements are compressed, the total stress response rzz results from contributions of the compaction of the fibres (effective stress rozz ) and the expelled fluid pressure p. Biot demonstrated that for porous materials in soil mechanics, the fluid pressure is multiplied by a factor b (Biot’s coefficient) representing the stress relief ratio due to pore pressure [25] rzz ¼ rozz þ bp
ð1Þ
with b¼1
Co Cs
ð2Þ
Most studies in the field of composites manufacturing model the hydro-mechanical coupling using Terzaghi’s assumption (Eq. (3)), i.e. b = 1 in Eq. (1). That coefficient b may be rather close to 1 for glass and carbon fibre reinforcements since the compressibility of the reinforcements is smaller than the bulk modulus of the constituting material. rzz ¼ rozz þ p
ð3Þ
The purpose of the methodology is to measure hydraulic permeabilities using unidirectional compression tests. Therefore, the basis of that methodology relies on measuring, with different types of experiments detailed thereafter, the total stress rzz and the effective stress rozz during unidirectional compression test. Then the fluid pressure p can be extracted and exploited to back-calculate the fibre rein-
640
S. Comas-Cardona et al. / Composites Science and Technology 67 (2007) 638–645
forcement permeability. Generating appropriate flows will allow to extract permeability data in specific directions.
2.4. In-plane permeability of a transversely isotropic fibre reinforcement
2.2. Total stress response
For such material, the two in-plane permeabilities are equal and noted Kr. Also, because of geometrical considerations and symmetries, an analytical solution for consolidation exists. For a unidirectional compression test between cylindrical platens (Fig. 3), the mass conservation equation for the fluid is: dVol dðpr2 hÞ ¼ ¼ 2prh/ur ðrÞ ð4Þ dt dt
The goal of such experiments is to measure the fibre and fluid pressure response. The in-plane and through-thickness flows are of interest. First, the in-plane fluid flow is generated using a common compression test setup constituting of two flat platens mounted on a material testing machine (Fig. 1). Fibre reinforcement plies previously impregnated with a fluid are laid up on top of a receptacle. The latter will collect the excess fluid expelled from the sample during the compression. Second, the generation of a purely transverse flow (i.e., in the through-thickness direction) using a compression test is more difficult to achieve. In order to keep the setup as simple as possible, fabric compression is performed using a perforated bottom platen that will allow the fluid to flow in the three directions of the sample (Fig. 2). The fibre reinforcements previously impregnated are placed on top of the perforated platen. During compression, the fluid will flow in both in-plane and through-thickness directions. These two types of compression tests combine (on purpose) fluid pressure and fibre effective stress response. In order to extract the fluid pressure out of those experiments, the fibre effective stress has to be measured.
Darcy’s law for the fluid confined in the sample is K r dpðrÞ ð5Þ ur ðrÞ ¼ /l dr Substituting Eq. (5) into Eq. (4) and integrating over the sample (with p(R) = 0) leads to the fluid pressure field _ 2 r2 Þ lhðR pðrÞ ¼ ð6Þ 4hK r ðhÞ Note that h_ is negative due to compression. Substituting Eq. (6) into Eq. (3) and integrating over the compression platen area leads to the in-plane permeability Kr _ 2 lhR ð7Þ K r ðhÞ ¼ 8hðrzz rozz Þ 2.5. In-plane permeabilities of an anisotropic fibre reinforcement
2.3. Effective stress response The fibre reinforcement compression response rozz is measured combining a test with a low compression speed, a low fluid viscosity and the perforated platen (Fig. 2) in order to zero the fluid pressure. That experiment is called the reference compression and can be considered as a drained compression test. Note that, for more preciseness, the reference compression test is performed on an impregnated fibre reinforcement sample and not on a dry sample.
For such material, combining the conservation of mass of both fibres and fluid, and Darcy’s law, Gutowski et al. showed that the fluid pressure p during the consolidation of an impregnated fibre reinforcement is the solution, in cartesian coordinates, of [23]: 2 o2 ^p K y o ^p þ ¼ 1 ð8Þ o^x2 K x o^y 2 _
2
with ^x ¼ Rx ; ^y ¼ Ry ; ^p ¼ pp and pc ¼ lhKhRx : c
Fig. 1. Experimental setup for impregnated fibre reinforcement compression tests that generate in-plane fluid flow.
S. Comas-Cardona et al. / Composites Science and Technology 67 (2007) 638–645
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Fig. 2. Experimental setup for impregnated fibre reinforcement compression tests that generate in-plane and through-thickness fluid flows.
Fig. 3. Domain and boundary conditions applied to solve for the equivalent in-plane permeabilities.
A finite element method solver (Comsol Multiphysics) is used to solve for liquid pressure (Table 1 and Fig. 3). With an inverse method algorithm the in-plane equivalent permeability is calculated in order to match the numerical liquid pressure to the experimental one. An additional measurement is needed to know the anisotropy ratio and calculate the two in-plane principal permeabilities (Fig. 4). A compression test with a little amount of catalysed resin inserted in a central hole can be done. Once the resin has cured, the elliptic axes developed during the compression can be measured to calculate the permeability ratio.
Fig. 4. Inverse method algorithm used to extract the in-plane permeability.
2.6. Through-thickness permeability of an anisotropic fibre reinforcement Once the in-plane permeabilities are known, the compression tests inducing through-thickness flow can be analysed using a similar methodology as previously detailed
Table 1 Methods to extract permeabilities depending on the material permeability tensor properties
Analytic FDM (2D cylindrical) FEM
Transverse isotropic fibre reinforcement
Anisotropic fibre reinforcement
In-plane permeability (Kx = Ky)
In-plane permeabilities (Kx 5 Ky)
Through-thickness permeability Kz
X
X
X X X
Through-thickness permeability Kz
X X
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Fig. 5. Domain and boundary conditions to apply to solve for the through-thickness permeability measurements.
sample transversely isotropic and the carbon NCF sample anisotropic in terms of permeability (Table 2). Prior to each compression test, the zero-thickness reference point has to be verified, especially for thin samples (few millimetres). Also, for the setup used in this study, in order to measure sufficient fluid pressure from the compression, it has been observed that the fluid viscosity and the compression _ speed has to be properly chosen so that hl=K >4 5 1 10 Pa m . It should also be noted that the use of compression speeds greater than 5 mm min1 induces incoherent results for these materials. It may be due to visco-elastic effects becoming significant. 3.1. Glass twill-weave fabric The reference compression test is performed at a compression speed of 0.5 mm min1 and with a silicone oil fluid viscosity of 0.1 Pa s. That test gives the mechanical Table 2 Material used for compression tests Material
Areal weight (g/m2)
Stacking sequence
Initial height (mm)
Glass twill-weave Carbon NCF
1500 230
[0°,90°,90°,0°] [+45°,0°,0°, 45°,90°, 90°, 45°,0°,0°,+45°]
6.8 4.5
Table 3 Silicone oil viscosities and compression speeds applied for the compression of the glass twill-weave fibre reinforcement
Fig. 6. Inverse method algorithm used to extract the through-thickness permeability.
In-plane permeability
Through-thickness permeability
Viscosity (Pa s)
Compression speed (mm min1)
Viscosity (Pa s)
Compression speed (mm min1)
0.27 1.04 1.04
2 0.5 2
5 10 10
5 2 5
with the proper boundary conditions (Fig. 5) and algorithm (Fig. 6). The fluid pressure is governed by the full 3D Gutowski’s equation: 2 o2 ^p Ky o ^ K z R2 h2o o2 ^ p p þ þ ¼ 1 ð9Þ 4 2 2 o^x o^z2 K x o^y Kxh _
2
with ^x ¼ Rx ; ^y ¼ Ry ; ^z ¼ hz ; ^ p ¼ pp and pc ¼ lhKhRx : c
3. Compression test results Compressions are performed on two types of fibre reinforcements: a glass twill-weave fabric and a carbon NCF having very different textile morphology and fibre type. Because of the fibre volume fractions of interest and the level of stress needed to perform the compressions, force cells of 10 kN and 100 kN will be respectively used for the glass and carbon fibre reinforcements. The stacking sequence of the two materials makes the glass twill-weave
Fig. 7. Experimental response of impregnated glass fibre twill-weave in compression using the setup in Fig. 1 that generates in-plane fluid flow.
S. Comas-Cardona et al. / Composites Science and Technology 67 (2007) 638–645
response of the impregnated glass fibre twill-weave while zeroing the pressure response of the expelled-fluid flow. When compression tests are run with higher compression speeds and higher viscosities (Table 3), the results show, as expected, an increase of stress on the sample for compression with in-plane flow (Fig. 7) and through-thickness flow (Fig. 8) conditions. 3.2. Carbon NCF Compression tests generating in-plane flow display similar trends as the ones observed for the glass twill-weave fabric (Fig. 9). The reference compression test is performed with a fluid having a viscosity of 0.1 Pa s and a compres-
643
sion speed of 0.5 mm min1, experimental conditions that zero fluid pressure. For the other tests, the stress response increases when compression speeds and fluid viscosities are also increased. 4. Permeability results In-plane permeability measurements obtained with the compression tests methodology detailed in this paper are compared to in-plane permeabilities measured with conventional unidirectional or radial injections. Throughthickness permeability measurements using injection techniques have not been realized because of the numerous experimental difficulties and the poor reliability involved with the injection method. 4.1. Glass twill-weave fabric
Fig. 8. Experimental response of impregnated glass fibre twill-weave in compression using the setup in Fig. 2 that generates both in-plane and through-thickness fluid flows.
Fig. 9. Experimental response of impregnated carbon NCF in compression using the setup in Fig. 1 that generates in-plane fluid flow.
Once compression tests have been realized (Figs. 7 and 8), they are compared to the reference test. Using the analytical or numerical methodology presented previously, permeabilities are calculated. In-plane permeability measurements obtained using unidirectional injection with constant pressure are compared to the ones obtained from the compression tests in Fig. 10. The error bars represent the standard deviation of the permeability calculated from compressions realized at three different speeds and fluid velocities (Table 3). Considering that Terzaghi’s assumption (b = 1) applies, the in-plane permeability results obtained by compression show a similar trend with the ones obtained by fluid injection in a mould having a PMMA window and using constant injection pressure. Even though the data differ by a factor 2, the measurement could be considered as satisfying regarding the impreciseness that can occur during the injection techniques.
Fig. 10. In-plane and through-thickness permeabilities of the glass fibre twill-weave extracted from the compression tests using Terzaghi’s assumption (b = 1). The black circles (d) are in-plane permeabilities obtained using the unidirectional injection measurement method.
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References
Fig. 11. In-plane (x-axis) and through-thickness permeabilities of the carbon fibre NCF extracted from the compression tests using Terzaghi’s assumption. The black circles (d) are in-plane permeabilities obtained using the central injection measurement method with constant injection flow rate.
4.2. Carbon NCF Similarly, the permeabilities of the carbon NCF have been calculated using Terzaghi’s assumption. Due to the stacking sequence, the in-plane anisotropy ratio is Kx/Ky = 1.1. The in-plane permeabilities (x-axis) are in excellent agreement with the ones measured with central injection in an aluminium mould mounted onto a press using constant flow rate (Fig. 11). The error bars represent the standard deviation calculated from compressions realized with three different fluid viscosities and compression speeds. For the through-thickness permeability, the results are in a range that is satisfying. 5. Conclusion A methodology to measure fibre reinforcement permeabilities using a compression test setup has been described. The results match very well the permeability measured with injection techniques. The method offers the main advantage of being continuous over a wide range of fibre volume fractions. Moreover, once the anisotropy ratio is known, the method allows for the determination of the in-plane and through-thickness permeabilities. That method also reduces lots of drawbacks present in the injection methods such as edge effects, mould deflection or the use of expensive data acquisition. Finally, permeabilities can also be measured on fibre reinforcements that have been sheared prior to compression tests. Acknowledgement The authors would like to thank the Advanced Technologies Development Centre, Dassault Aviation (France) for providing the carbon fibre reinforcements and funding a part of the study.
[1] Kardos JL, Lam RC. The permeability and compressibility of aligned and cross-plied carbon fiber beds during processing of composites. Polym Eng Sci 1991;31(14):1064–70. [2] Kenny JM, Molina G, Trivisano A. Mathematical modelling of resin transfer molding of composites for automotive applications. In: ICCM proceedings, Madrid, Spain, July 12–16th, 1993; 3: p. 513–20. [3] Chibani M, Gauvin R. The modeling of mold filling in resin transfer molding. Int Polym Proces 1986:42–6. [4] Bruno A, Mazzola M, Molina G. Mode´lisation de l’e´coulement et validation expe´rimentale du proce´de´ RTM. Compos 1992;3: 397–404. [5] Adams KL, Miller B, Rebenfeld L. Forced in-plane flow of an epoxy resin in fibrous networks. Polym Eng Sci 1986;26(20):1434–41. [6] Advani SG, Bruschke MV. A finite element/control volume approach to mold filling in anisotropic porous media. Polym Compos 1990;11(6):398–405. [7] Lee LJ, Perry MJ, Wang TJ. Analysis of permeability and void formation in resin transfer moding. In: ANTEC proceedings, Detroit, USA, May 3–7th, 1992; p. 756–60. [8] Endruweit A, Luthy T, Ermanni P. Investigation of the influence of textile compression on the out of plane permeability of a bidirectional glass fiber fabric. Polym Compos 2002;23(4):538–54. [9] Bauer S, Boucher D, Gutowski TG. Consolidation experiments for laminate composites. J Compos Mater 1987;21:650–69. [10] Han KK, Lee CW, Rice BP. Measurements of the permeability of fibre preforms and applications. Compos Sci Technol 2000;60(12–13): 2435–41. [11] Woerdeman DL, Phelan FR, Parnas RS. Interpretation of 3D permeability measurements for RTM modelling. Polym Compos 1995;16(6):470–80. [12] Ahn SH, Lee WI, Springer GS. Measurement of the three-dimensional permeability of fiber preforms using embedded fiber optic sensors. J Compos Mater 1995;29(6):714–33. [13] Weitzenbock JR, Shenoi RA, Wilson PA. Measurements of three-dimensional permeability. Compos Part A 1998;29(2): 159–69. [14] Lundstrom TS, Stenberg R, Bergstrom R, Partanen H, Birkeland PA. In-plane permeability measurements: a nordic round-robin study. Compos Part A 2000;31(1):29–43. [15] Parnas RS, Flynn KM, Dal-Favero ME. A permeability database for composites manufacturing. Polym Compos 1997;18(5): 623–33. [16] Song YS, Chung K, Kang TJ, Youn JR. Prediction of permeability tensor for three dimensional circular braided preform by applying a finite volume method to a unit cell. Compos Sci Technol 2004; 64(10–11):1629–36. [17] Fournier R, Coupez T, Vincent M. Numerical determination of the permeability of fibre reinforcement for the RTM process. Revue Europe´enne des Ele´ments Finis 2005;14(6–7):803–18. [18] Belov EB, Lomov SV, Verpoest I, Peters T, Roose D, Parnas RS, Hoes K, Sol H. Modelling of permeability of textile reinforcements: lattice Boltzmann method. Compos Sci Technol 2004;64(7–8): 1069–80. [19] Comas-Cardona S, Groenenboom PHL, Binetruy C, Krawczak P. Simulation of liquid molding processes using a generic mixed FE-SPH method. Revue Europe´enne des Ele´ments Finis 2005;14: 867–83. [20] Nedanov PB, Advani SG. Numerical computation of the fiber preform permeability tensor by the homogenization method. Polym Compos 2002;23(5):758–70. [21] Ait Si Ahmad M, Fasi-Ferhi O, Poitou A, Bojji C. Model for determining the permeability variation of fiber mat reinforcement submitted to compression. Me´caniques et Industries 2002;3: 27–34.
S. Comas-Cardona et al. / Composites Science and Technology 67 (2007) 638–645 [22] Buntain MJ, Bickerton S. Compression flow permeability measurement: a continuous technique. Compos Part A 2003;34(5): 445–57. [23] Gutowski TG, Morigaki T, Cai Z. The consolidation of laminate composites. J Compos Mater 1987;21:172–88.
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[24] Comas-Cardona S, Binetruy C, Krawczak P. Unidirectional compression of fibre reinforcements Part 1 : A non-linear elastic–plastic behaviour. Compos Sci Technol, submitted for publication. [25] Biot MA. Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 1955;26(2):182–5.
Unidirectional compression of fibre reinforcements. Part 2: A continuous permeability tensor measurement S. Comas-Cardona *, C. Binetruy, P. Krawczak Polymers and Composites Technology and Mechanical Engineering Department, Ecole des Mines de Douai, 941 rue Charles Bourseul, BP 10838, 59508 Douai, France Received 4 January 2006; received in revised form 19 July 2006; accepted 25 July 2006 Available online 2 October 2006
Abstract Modelling LCM processes, where resin flow and compression of the fibre reinforcements are involved, requires a proper modelling of the fibre reinforcements behaviour in compression and to define how the permeability tensor evolves. The permeability measurement techniques based on fluid injection lead to data scattering. Using a material testing machine, unidirectional compression tests are performed on impregnated fibre reinforcements to induce in-plane and through-thickness fluid flows. With proper viscosities and compression speeds, the expelled-fluid pressure becomes sufficiently high to be measured. Then, using that fluid pressure and following a methodology based on numerical optimizations, the in-plane equivalent permeability and through-thickness permeabilities can be extracted. An additional measurement is needed to know the anisotropy ratio and calculate the two in-plane principal permeabilities. Due to the nature of the compression test, a major advantage of the present method is that it is continuous with respect to fibre volume fraction and that it limits errors involved with injection techniques. Samples can previously be sheared before compression and permeability measurements. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: A. Fabrics/textiles; Permeability; Unidirectional compression
1. Introduction Modelling the liquid composite moulding (LCM) manufacturing processes requires to input accurate material data such as fibre reinforcement permeability. Two types of permeability, saturated and unsaturated, can be considered. The saturated permeability is measured using an exiting flow rate measurement technique whereas the unsaturated one is usually measured recording the position of a moving flow front. This study will focus on the saturated permeability because it is the most used, measured and the most reliable. A great amount of effort has been spent on measuring such material property. Most of the techniques employed rely on fluid injection experiments with:
*
Corresponding author. Tel.: +33 3 27 71 21 87; fax: +33 3 27 71 29 81. E-mail address: [email protected] (S. Comas-Cardona).
0266-3538/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.07.020
– unidirectional flow and constant injection pressure [1,2] or constant flow rate [3,4], – biaxial flow and constant injection pressure [5,6] or constant flow rate [7], – through-thickness flow with constant pressure [1,8] or constant flow rate [9,10], – three dimensional flows [11–13]. Based on fluid injections, those techniques require a pressure pot, a vacuum pump, or an injection piston, a mould and a data acquisition system. Pressures, flow rates, pressure sensors and scales also have to be monitored during the injection. A limitation of the injection techniques is the use of all the equipment previously cited. Also, during the injection, common errors arise when proceeding to the measurement [14,15] such as the deflection of the mould, edge effects (racetracking), capillary effects and error in inlet hole dimension.
S. Comas-Cardona et al. / Composites Science and Technology 67 (2007) 638–645
639
Nomenclature b Co Cs h h_ Ki p r R
Biot’s coefficient compression modulus of the impregnated porous medium in drained conditions bulk modulus of the material constituting the porous medium (fibres) current sample thickness compression speed hydraulic permeability in the i-direction fluid (resin) pressure radial position sample radius
Because of those limitations and the scattering of the experimental data, authors also worked on numerical predictions of fibre reinforcement permeability. In that case, a periodic unit cell of the fabric has to be defined. Then fluid flow dynamics are solved in the unit cell domain (inter and intra yarns) to extract permeabilities. Fluid flow dynamics solutions can be obtained from numerical methods such as finite volume [16], finite element [17], lattice Boltzmann [18], meshless Lagrangian smoothed particle hydrodynamics [19] or the homogenization [20] method. One limitation of such numerical solution is the difficulty to define a periodic unit cell that would be representative of the fibre reinforcement. Moreover, the unit cell has to be computerized in terms of geometry and mesh, which is not an easy task for fibre reinforcements. Finally, even when the geometry and mesh are given for a fibre volume fraction, the work (especially the meshing, which is the most time consuming stage) has to be reiterated for another fibre volume fraction target. In order to overcome that issue, another type of continuous permeability measurement based on the compression of saturated fibre reinforcements has been introduced [21,22]. Impregnated fibre textiles are placed between two compression platens. The compression of the sample produced by the mechanical test machine forces the fluid to flow. Buntain and Bickerton measured the fluid pressure with a pressure transducer during the compression [22] and calculated the in-plane permeability using the equation proposed by Gutowski for composites consolidation [23]. Ait si Ahmad et al. proposed an even simpler experimental setup. The fluid pressure is measured via two compression tests using the force cell of the mechanical test machine [21]. These two methods have the advantage to limit the amount of equipment required and to continuously measure permeability in term of fibre volume fraction. Limitations however remain, the methods are limited to transversely isotropic materials and do not allow to measure the through thickness permeability. A simple compression setup based on a material testing machine has been previously proposed to test fibre rein-
t ur x,y,z / l rzz rozz
time interstitial fluid velocity in the r-direction Cartesian coordinates porosity fluid dynamic viscosity total stress applied to the impregnated porous medium in the z-direction stress applied to the impregnated porous medium in the z-direction
forcements in either dry or impregnated state [24]. That compression setup is further exploited in this paper to propose a reliable and generic measurement method that will provide the in-plane and through-thickness permeabilities for a wide range of fibre volume fraction and materials. 2. Methodology 2.1. Total stress constitutive equation When impregnated fibre reinforcements are compressed, the total stress response rzz results from contributions of the compaction of the fibres (effective stress rozz ) and the expelled fluid pressure p. Biot demonstrated that for porous materials in soil mechanics, the fluid pressure is multiplied by a factor b (Biot’s coefficient) representing the stress relief ratio due to pore pressure [25] rzz ¼ rozz þ bp
ð1Þ
with b¼1
Co Cs
ð2Þ
Most studies in the field of composites manufacturing model the hydro-mechanical coupling using Terzaghi’s assumption (Eq. (3)), i.e. b = 1 in Eq. (1). That coefficient b may be rather close to 1 for glass and carbon fibre reinforcements since the compressibility of the reinforcements is smaller than the bulk modulus of the constituting material. rzz ¼ rozz þ p
ð3Þ
The purpose of the methodology is to measure hydraulic permeabilities using unidirectional compression tests. Therefore, the basis of that methodology relies on measuring, with different types of experiments detailed thereafter, the total stress rzz and the effective stress rozz during unidirectional compression test. Then the fluid pressure p can be extracted and exploited to back-calculate the fibre rein-
640
S. Comas-Cardona et al. / Composites Science and Technology 67 (2007) 638–645
forcement permeability. Generating appropriate flows will allow to extract permeability data in specific directions.
2.4. In-plane permeability of a transversely isotropic fibre reinforcement
2.2. Total stress response
For such material, the two in-plane permeabilities are equal and noted Kr. Also, because of geometrical considerations and symmetries, an analytical solution for consolidation exists. For a unidirectional compression test between cylindrical platens (Fig. 3), the mass conservation equation for the fluid is: dVol dðpr2 hÞ ¼ ¼ 2prh/ur ðrÞ ð4Þ dt dt
The goal of such experiments is to measure the fibre and fluid pressure response. The in-plane and through-thickness flows are of interest. First, the in-plane fluid flow is generated using a common compression test setup constituting of two flat platens mounted on a material testing machine (Fig. 1). Fibre reinforcement plies previously impregnated with a fluid are laid up on top of a receptacle. The latter will collect the excess fluid expelled from the sample during the compression. Second, the generation of a purely transverse flow (i.e., in the through-thickness direction) using a compression test is more difficult to achieve. In order to keep the setup as simple as possible, fabric compression is performed using a perforated bottom platen that will allow the fluid to flow in the three directions of the sample (Fig. 2). The fibre reinforcements previously impregnated are placed on top of the perforated platen. During compression, the fluid will flow in both in-plane and through-thickness directions. These two types of compression tests combine (on purpose) fluid pressure and fibre effective stress response. In order to extract the fluid pressure out of those experiments, the fibre effective stress has to be measured.
Darcy’s law for the fluid confined in the sample is K r dpðrÞ ð5Þ ur ðrÞ ¼ /l dr Substituting Eq. (5) into Eq. (4) and integrating over the sample (with p(R) = 0) leads to the fluid pressure field _ 2 r2 Þ lhðR pðrÞ ¼ ð6Þ 4hK r ðhÞ Note that h_ is negative due to compression. Substituting Eq. (6) into Eq. (3) and integrating over the compression platen area leads to the in-plane permeability Kr _ 2 lhR ð7Þ K r ðhÞ ¼ 8hðrzz rozz Þ 2.5. In-plane permeabilities of an anisotropic fibre reinforcement
2.3. Effective stress response The fibre reinforcement compression response rozz is measured combining a test with a low compression speed, a low fluid viscosity and the perforated platen (Fig. 2) in order to zero the fluid pressure. That experiment is called the reference compression and can be considered as a drained compression test. Note that, for more preciseness, the reference compression test is performed on an impregnated fibre reinforcement sample and not on a dry sample.
For such material, combining the conservation of mass of both fibres and fluid, and Darcy’s law, Gutowski et al. showed that the fluid pressure p during the consolidation of an impregnated fibre reinforcement is the solution, in cartesian coordinates, of [23]: 2 o2 ^p K y o ^p þ ¼ 1 ð8Þ o^x2 K x o^y 2 _
2
with ^x ¼ Rx ; ^y ¼ Ry ; ^p ¼ pp and pc ¼ lhKhRx : c
Fig. 1. Experimental setup for impregnated fibre reinforcement compression tests that generate in-plane fluid flow.
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Fig. 2. Experimental setup for impregnated fibre reinforcement compression tests that generate in-plane and through-thickness fluid flows.
Fig. 3. Domain and boundary conditions applied to solve for the equivalent in-plane permeabilities.
A finite element method solver (Comsol Multiphysics) is used to solve for liquid pressure (Table 1 and Fig. 3). With an inverse method algorithm the in-plane equivalent permeability is calculated in order to match the numerical liquid pressure to the experimental one. An additional measurement is needed to know the anisotropy ratio and calculate the two in-plane principal permeabilities (Fig. 4). A compression test with a little amount of catalysed resin inserted in a central hole can be done. Once the resin has cured, the elliptic axes developed during the compression can be measured to calculate the permeability ratio.
Fig. 4. Inverse method algorithm used to extract the in-plane permeability.
2.6. Through-thickness permeability of an anisotropic fibre reinforcement Once the in-plane permeabilities are known, the compression tests inducing through-thickness flow can be analysed using a similar methodology as previously detailed
Table 1 Methods to extract permeabilities depending on the material permeability tensor properties
Analytic FDM (2D cylindrical) FEM
Transverse isotropic fibre reinforcement
Anisotropic fibre reinforcement
In-plane permeability (Kx = Ky)
In-plane permeabilities (Kx 5 Ky)
Through-thickness permeability Kz
X
X
X X X
Through-thickness permeability Kz
X X
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Fig. 5. Domain and boundary conditions to apply to solve for the through-thickness permeability measurements.
sample transversely isotropic and the carbon NCF sample anisotropic in terms of permeability (Table 2). Prior to each compression test, the zero-thickness reference point has to be verified, especially for thin samples (few millimetres). Also, for the setup used in this study, in order to measure sufficient fluid pressure from the compression, it has been observed that the fluid viscosity and the compression _ speed has to be properly chosen so that hl=K >4 5 1 10 Pa m . It should also be noted that the use of compression speeds greater than 5 mm min1 induces incoherent results for these materials. It may be due to visco-elastic effects becoming significant. 3.1. Glass twill-weave fabric The reference compression test is performed at a compression speed of 0.5 mm min1 and with a silicone oil fluid viscosity of 0.1 Pa s. That test gives the mechanical Table 2 Material used for compression tests Material
Areal weight (g/m2)
Stacking sequence
Initial height (mm)
Glass twill-weave Carbon NCF
1500 230
[0°,90°,90°,0°] [+45°,0°,0°, 45°,90°, 90°, 45°,0°,0°,+45°]
6.8 4.5
Table 3 Silicone oil viscosities and compression speeds applied for the compression of the glass twill-weave fibre reinforcement
Fig. 6. Inverse method algorithm used to extract the through-thickness permeability.
In-plane permeability
Through-thickness permeability
Viscosity (Pa s)
Compression speed (mm min1)
Viscosity (Pa s)
Compression speed (mm min1)
0.27 1.04 1.04
2 0.5 2
5 10 10
5 2 5
with the proper boundary conditions (Fig. 5) and algorithm (Fig. 6). The fluid pressure is governed by the full 3D Gutowski’s equation: 2 o2 ^p Ky o ^ K z R2 h2o o2 ^ p p þ þ ¼ 1 ð9Þ 4 2 2 o^x o^z2 K x o^y Kxh _
2
with ^x ¼ Rx ; ^y ¼ Ry ; ^z ¼ hz ; ^ p ¼ pp and pc ¼ lhKhRx : c
3. Compression test results Compressions are performed on two types of fibre reinforcements: a glass twill-weave fabric and a carbon NCF having very different textile morphology and fibre type. Because of the fibre volume fractions of interest and the level of stress needed to perform the compressions, force cells of 10 kN and 100 kN will be respectively used for the glass and carbon fibre reinforcements. The stacking sequence of the two materials makes the glass twill-weave
Fig. 7. Experimental response of impregnated glass fibre twill-weave in compression using the setup in Fig. 1 that generates in-plane fluid flow.
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response of the impregnated glass fibre twill-weave while zeroing the pressure response of the expelled-fluid flow. When compression tests are run with higher compression speeds and higher viscosities (Table 3), the results show, as expected, an increase of stress on the sample for compression with in-plane flow (Fig. 7) and through-thickness flow (Fig. 8) conditions. 3.2. Carbon NCF Compression tests generating in-plane flow display similar trends as the ones observed for the glass twill-weave fabric (Fig. 9). The reference compression test is performed with a fluid having a viscosity of 0.1 Pa s and a compres-
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sion speed of 0.5 mm min1, experimental conditions that zero fluid pressure. For the other tests, the stress response increases when compression speeds and fluid viscosities are also increased. 4. Permeability results In-plane permeability measurements obtained with the compression tests methodology detailed in this paper are compared to in-plane permeabilities measured with conventional unidirectional or radial injections. Throughthickness permeability measurements using injection techniques have not been realized because of the numerous experimental difficulties and the poor reliability involved with the injection method. 4.1. Glass twill-weave fabric
Fig. 8. Experimental response of impregnated glass fibre twill-weave in compression using the setup in Fig. 2 that generates both in-plane and through-thickness fluid flows.
Fig. 9. Experimental response of impregnated carbon NCF in compression using the setup in Fig. 1 that generates in-plane fluid flow.
Once compression tests have been realized (Figs. 7 and 8), they are compared to the reference test. Using the analytical or numerical methodology presented previously, permeabilities are calculated. In-plane permeability measurements obtained using unidirectional injection with constant pressure are compared to the ones obtained from the compression tests in Fig. 10. The error bars represent the standard deviation of the permeability calculated from compressions realized at three different speeds and fluid velocities (Table 3). Considering that Terzaghi’s assumption (b = 1) applies, the in-plane permeability results obtained by compression show a similar trend with the ones obtained by fluid injection in a mould having a PMMA window and using constant injection pressure. Even though the data differ by a factor 2, the measurement could be considered as satisfying regarding the impreciseness that can occur during the injection techniques.
Fig. 10. In-plane and through-thickness permeabilities of the glass fibre twill-weave extracted from the compression tests using Terzaghi’s assumption (b = 1). The black circles (d) are in-plane permeabilities obtained using the unidirectional injection measurement method.
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References
Fig. 11. In-plane (x-axis) and through-thickness permeabilities of the carbon fibre NCF extracted from the compression tests using Terzaghi’s assumption. The black circles (d) are in-plane permeabilities obtained using the central injection measurement method with constant injection flow rate.
4.2. Carbon NCF Similarly, the permeabilities of the carbon NCF have been calculated using Terzaghi’s assumption. Due to the stacking sequence, the in-plane anisotropy ratio is Kx/Ky = 1.1. The in-plane permeabilities (x-axis) are in excellent agreement with the ones measured with central injection in an aluminium mould mounted onto a press using constant flow rate (Fig. 11). The error bars represent the standard deviation calculated from compressions realized with three different fluid viscosities and compression speeds. For the through-thickness permeability, the results are in a range that is satisfying. 5. Conclusion A methodology to measure fibre reinforcement permeabilities using a compression test setup has been described. The results match very well the permeability measured with injection techniques. The method offers the main advantage of being continuous over a wide range of fibre volume fractions. Moreover, once the anisotropy ratio is known, the method allows for the determination of the in-plane and through-thickness permeabilities. That method also reduces lots of drawbacks present in the injection methods such as edge effects, mould deflection or the use of expensive data acquisition. Finally, permeabilities can also be measured on fibre reinforcements that have been sheared prior to compression tests. Acknowledgement The authors would like to thank the Advanced Technologies Development Centre, Dassault Aviation (France) for providing the carbon fibre reinforcements and funding a part of the study.
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