BIOT COEFFICIENTS OF UNIDIRECTIONAL FIBRE PACKED ASSEMBLIES: EXPERIMENTAL AND THEORETICAL Thang Tran, Sébastien Comas-Cardona,*, Nor-Edine Abria 2, Christophe Binétruy Polymers and Composites Technology and Mechanical Engineering Department 2 Civil and Environmental Engineering Department Ecole des Mines de Douai, 94 rue Charles Bourseul, BP 088, 59508 Douai Cedex, France * email: comas-cardona@ensm-douai.r SUMMARY The present paper aims at predicting the Biot coeicients or perectly straight unidirectional (UD) ibre assemblies using micromechanical methods: Mori Tanaa and Ponte Castañeda Willis estimates. An experimental procedure, based on both uniaxial and triaxial material testing machines, is also detailed and applied to rubber ibre assemblies. The theoretical and experimental Biot coeicient results are compared and show a good agreement. Keywords: Unidirectional ibre reinorcement; Porous media; Biot coeicients; Poromechanics; Microporomechanics; Composite manuacturing. INTRODUCTION Processing composites oten involves either the compression o pre-impregnated ibre reinorcements or the low o a viscous liquid within ibrous materials. The moulding pressures during the process can reach high levels due to the compaction o the ibres getting closer to each other and the ibrous media permeability drop because o the ibre volume raction rise. The lower mechanical perormances and microstructure o vegetal ibres compared to traditional synthetic ibres raise the question o the eect o such moulding pressures on the structural integrity o the ibres. Usually, when the compression o impregnated ibrous media is involved, Terzaghi s equation is used. This soil mechanics result relates the total stress tensor o an impregnated porous media as the unction o the stress tensor applying on the porous seleton and the interstitial liquid pressure. However, when the constituents deorm due to their own compressibility under hydrostatic pressure, it is necessary to introduce the Biot tensor to tae this eect into account.
2. POROMECHANICAL THEORY AND TESTS 2.. Impregnated Porous Media Due to analogies between granular and ibrous media [], researchers have been using Terzaghi s equation in previous composite manuacturing studies, where compression o impregnated ibrous media is involved [7, 8,]. This soil mechanics result (Eq. ()) relates the total stress tensor Σ o an impregnated porous media as the unction o the stress tensor applying on the porous seleton Σ ' and the interstitial liquid pressure P Σ = Σ' δp () where δ is the second-order unit tensor δ = i i = j; δ = 0 i i j (2) ij ij When the constituents deorm due to their own compressibility under hydrostatic pressure, it is necessary to introduce the second-order Biot tensor B in Terzaghi s relationship Σ = Σ ' B P () The Biot tensor components are positive, inerior to one and B is diagonal [2] B b 0 0 = 0 b2 0 0 0 b (4) b b2, and b are the three components o the Biot tensor in the three respective Cartesian coordinates. For most soils, the grains are supposed ininitely rigid compared to the material they orm, thus the Biot tensor reduces to the second-order unit tensor (Terzaghi s hypothesis). Using results rom poroelasticity, the Biot tensor can be expressed as unction o the ourth-order drained ogeneous elastic tensor C (stiness tensor) o the seleton and o the ourth-order elastic compliance tensor S o the constituent []
( I S : C ) B = δ (5) where I is the ourth-order unit tensor = δ δ + δ δ 2 ( ) I ijl i jl il j (6) The main wor or the calculation o the Biot tensor consists in the calculation o the C tensor. For ibre assemblies o interest in this study, the Biot tensor s components can be considered within two particular cases o symmetry: both ibre assembly and individual ibre are isotropic (the case o glass mat, or example). In this case, b = b2 = b = b, and the well-nown relation (see, e.g. [2]) applies b = (7) and are the bul moduli o the ogeneous (ibrous) material and the where individual ibre, respectively; the ibre assembly is transversely isotropic with an axis o revolution Ox and the individual ibre is isotropic. The transverse isotropy is only due to the presence o inter-ibre aligned pores. In this case [5] 2 2 E + + E E b ; b 2 = b 2 = + = + (8) 2 2 ( ) 2 ( ) 2 2 E E E E where E, E, and 2 are the elastic moduli and the Poisson ratios o the drained ogeneous material. The relations o the Biot coeicients (Eq. (8)) are calculated as a unction o ogeneous moduli and Poisson ratios which are unnown, thereore speciic tests will be perormed so as to measure them.
2.2. Tests on the rubber ibre samples To obtain the Biot coeicients through experimentation or the transversely isotropic material, tests have been realized on the rubber ibre samples. The ibres (5 mm in diameter) are made o nitrile butadiene rubber whose Young modulus is 5 MPa and Poisson ratio is 0.48. In the present wor, the Biot coeicients are measured or the two arrangements o unidirectional rubber ibres: square and hexagonal (Fig. ). The unidirectional rubber ibres are assembled and lightly glued together to maintain the arrangement during the test and to assure the straightness o the ibres. The Biot coeicients o the square and hexagonal ibre arrangement materials are calculated with Eq. (8). Figure. Square (let) and hexagonal (right) arrangements. The determination o the Biot coeicients (square and hexagonal ibre arrangements) require to measure ive moduli: the bul modulus o the ibre ; the elastic moduli and m the Poisson ratios o the drained ogeneous material E, ho E, and 2. The direction Ox is oriented along the axis o the ibre. The various moduli (Tab. ) will be measured through several tests (Tab. 2). When the experiment requires one, the coninement liquid used is water. Table. Test types and corresponding moduli. UD compression Shear Drained isotropic compression Undrained isotropic compression E, E 2
Table 2. Samples geometry and test conditions. Tests UD compression Isotropic compression Shear Diameter 50 mm 50 mm 50 mm Height 50 mm 00 mm 50 mm Compression speed 0.2 mm/min 6 Pa/min 0. mm/min 2.2.. Unidirectional compression tests The square and hexagonal arrangement samples are tested with a unidirectional compression machine to measure the ogeneous elastic moduli E and E. The slope o the quasi linear parts o the stress vs. strain curves gives the tangent moduli and E. E 2.2.2. Shear tests The shear test and the isotropic test are realized with a triaxial machine (originally used or soils). The impregnated sample is conined in a membrane which isolate it rom the coninement water. The shear test allows to measure two parameters: the elastic modulus ( E and E ) and the Poisson ratio. During the test, the displacement o the piston (strain ε or ε ), the axial eort (stress σ or σ ) and the water volume ejected or brought in the cell (volumetric strainε V ) are measured. The elastic moduli E and E are determined rom those data. The ininitesimal volumetric strain writes: ε = ε + ε + ε (9) V 2 Moreover, or the shear test in the Ox direction, ε = ε 2, which leads to ε ε ε ε ε = = = 2 V V 2 ε 2ε (0) where the values o the volumetric strain ε V and the strain given by the shear test. The Poisson ratios ε (in the Ox are determined with Eq. (0). direction) are The Poisson ratio 2 cannot be determined directly by shear test. It is determined using the drained bul modulus test., which is measured by an isotropic compression
2.2.. Isotropic compression tests The isotropic compression test is also realized with the triaxial machine. The sample is prepared and conined as in the shear test, the amount o coninement water which enters or leaves the cell as unction o the isotropic coninement pressure is considered as the deormation o the rubber sample (or drained test) or the deormation o the ibre rubber itsel (undrained test). The drained ogeneous bul modulus is calculated as the ratio o the isotropic coninement compression rise to the volumetric strain ε V o the drained test, whereas the undrained bul modulus (or the ibre bul modulus) is calculated rom the undrained test. E E Thans to the moduli,, and the Poisson ratio, the Poisson ratio 2 is determined by the relation or the transversely isotropic material E 2 = ( 4 ) () 2 E 2.2.4. Results The curve o the Biot coeicients and their mean values o the square and hexagonal arrangements are presented in Fig. 2 and in Tab., respectively. Firstly, the Biot coeicients are clearly lower than one. This shows that the Terzaghi s hypothesis (Biot coeicient is equal to one) may not be applied or all cases o saturated ibrous materials. Figure 2. Biot coeicients o the square and hexagonal arrangements. Then, or the same arrangement, the Biot coeicients (in the weaest modulus direction Ox ) are always superior to b (in the highest modulus direction Ox ). This shows the inluence o the material structure on the Biot coeicients. Moreover, or the same direction o two dierent structures, the Biot coeicients o the square b
arrangement are always slightly superior to that o the hexagonal arrangement. Finally, the Biot coeicients decrease when the applied stress is raised. Table. Experimental mean values o the Biot coeicients. b b porosity Square 0.87 0.79 0.25 Hexagonal 0.746 0.682 0.09. MICROPOROMECHANICAL APPROACH.. Bacground The estimated Mori Manaa [] and Ponte Castañeda Willis [0] schemes will be studied. They are calculated based on the Eshelby tensor SE or Hill tensor P which are determined analytically [4,9,2] with inogeneities o the ellipsoidal shape (three principal semi-axis a, a2 and a ). The SE and P tensors can be also calculated numerically or the general case o the ellipsoidal inogeneity with an arbitrary orientation [5,6]. The solutions o two above estimated schemes will be compared to choose the adapted schemes. The irst scheme studied is the one derived by Mori Tanaa. It accounts or the mechanical interaction between pores. The porosity in the Mori Tanaa scheme can reach up to 20% [], the individual ibre is considered as the reerence medium. The ogeneous stiness tensor or the Mori Tanaa scheme is (see, e.g. []) ( ( ) ) C = ( c) C : ( c) I + c I P : C (2) MT where c is the porosity; C is the ourth-order stiness tensor o the ibre; P is the Hill tensor, P = SE : S, SE is the Eshelby tensor which depends on both the geometry o the inogeneity and the reerence medium. The second scheme studied is the Ponte Castañeda Willis estimate which, urthermore, taes into account the spatial distribution o the pores through a distribution unction. The development o this scheme is based on the variational ormulation o Hashin and Shtriman, and the reerence material is always taen identically to the individual ibre. For the case where all the pores have the same orientation, the ogeneous stiness tensor o the Ponte Castañeda Willis scheme writes (see, or more details [0])
CPCW = C + ( Pi S ) P d c () Here P i pores) while is equivalent to P in the previous scheme (it characterizes the shape o the P d characterizes the spatial distribution o the pores in the material..2. Application to unidirectional ibre assemblies The ibre assembly has been modelled with the ollowing assumptions: it is linear elastic; it consists o aligned, parallel, linear elastic and isotropic ibres; it is modelled with cylindrical pores. To acilitate calculation with ourth-order tensors, the Walpole basis is used [,4]. The ellipsoidal pores are considered cylindrical or both schemes. The pores distribution in the Ponte Castañeda Willis scheme is circular (readers may reer to [0] or more details). The ogeneous stiness tensors C MT and C o the two schemes are determined using Eqs. (2) (). The results obtained are then substituted in Eq. (5) to calculate the Biot tensor. The calculations lead to the ollowing results or the Biot coeicients or the two schemes. In the case o the Mori Tanaa scheme PCW ( ) + ( 2 ) 2c 2 ; c c b MT = b MT = b MT = 2 + c 2 + c (4) and or the Ponte Castañeda Willis scheme, the Biot coeicients are b PCW PCW 2 2 2 b PCW 2 2 [ c ] 6( c ) 2(4 5) + 5( ) = b = 4 c ( + )(4 5 ) + c(60 99 + 57) + 45( )( 2 ) [ c ] ( c )4(4 5) + 5 = 4 c ( + )(4 5 ) + c(60 99 + 57) + 45( )( 2 ) (5) The Biot coeicients o the transversely isotropic material, which has isotropic constituents, only depend on the porosity c and the Poisson ratio o the constitutive ibre. The modulus o the ibre cancels out during derivations.
4. THEORETICAL AND EXPERIMENTAL COMPARISON The results o the Biot coeicients o the two schemes are presented in Fig. with a Poisson ratio equal to 0.48. The curve o the Mori Tanaa scheme tends towards one when the porosity tends towards one, the Ponte Castañeda Willis scheme or the circular distribution o the ibre is not much dierent to the Mori Tanaa, except the act that the Biot coeicients is only approximated to one when the porosity tends towards one. For the very low values o porosity, the three schemes give almost identical Biot coeicients. The value o is always greater than that o b. b The experimental mean values o the Biot coeicients or the square and hexagonal arrangements (Tab. ) are also presented in Fig. as unction o the initial porosity with the theoretical curves o the Mori Tanaa and Ponte Castañeda Willis schemes. A rather good agreement between the theoretical and experimental results has been ound. Figure. Theoretical and experimental results o the Biot coeicients, with = 0.48. 5. CONCLUSION Experimental Biot coeicients or unidirectional ibre assemblies have been obtained ollowing a procedure that involved uniaxial and triaxial testing machines. Micromechanical schemes have also been used and combined to the poromechanical theory in order to obtain Biot coeicients o unidirectional ibre assemblies. Both micromechanical and experimental results are in good agreement. The theoretical results are based on the assumptions o isotropy and perect straightness o the ibres. That irst result is somehow limited to some applications, but it is a starting point or ormer evolutions towards the reality o composites manuacturing. It suggests that the Biot coeicients o a UD assembly o isotropic ibres are lower than one and that Terzaghi s assumption is not valid or this speciic case. Improvements should tae into account speciicities such as the transverse isotropy o ibres, and localized contacts, inite and non-linear deormations o ibre assemblies.
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