ON AN ESTIMATION OF THE DAMPING PROPERTIES OF WOVEN FABRIC COMPOSITES Masaru Zao 1, Tetsusei Kurashii 1, Yasumasa Naanishi 2 and Kin ya Matsumoto 2 1 Department o Management o Industry and Technology, Osaa University, 2-1 Yamada-oa, Suita, Osaa, 565-871, Japan 2 Department o Technology, Faculty o Education, Mie University, 1515 Kamihama-cho, Tsu, Mie, 514-857, Japan ABSTRACT In this paper, we have proposed an estimation method that the damping properties o woven abric composites are obtained by a simple excitation test o unidirectional coupon specimen and inite element method. The proposed method has been applied to the plain and tri-axial woven abric composites. Experiments have been conducted to evaluate the validity o the proposed method. It is recognized that the computational results have agreed well with the experimental ones. Since an eect o orientation o iber bundle on material damping o plain-woven abric composites can be simulated, the damping properties or various woven abric composites will be estimated. It is revealed that the proposed method is very useul or the estimation o material damping characteristics. 1. INTRODUCTION Material damping represents the cumulative contributions o the visco-elastic response o the constituents, cyclic heat low and the riction at the iber/matrix interace. Recent wor on the material damping o FRP has shown that it depends on an array o micro-mechanics and laminate parameters, including constituent material properties, iber volume raction, stacing sequence [1][2]. These studies, however, are mostly limited to unidirectional composites. Woven abric (WF) composites are applied to many ields such as the space structures, sports items and so on. Though many researchers have reported the static characteristics o WF composites, the material damping o those materials have not been investigated [3][4]. The purpose o this study is to establish an estimation method o damping properties or WF composites. The proposed procedure has been applied to the numerical study o the material damping o a plain and a triaxial WF composite materials. 2. PROCEDURE 2.1 Simulation method Adams and Bacon have reported that the material damping energy can obtain the sum o energy in the material bared or stress components under vibrating or unidirectional FRP [5]. Damping ratioζ is expressed as = 1 U ζ 4 π U (1) where U is the total damping energy a cycle o vibration and U is the maximum strain energy. We have employed a three-dimensional heterogeneous inite element model or WF composites, which consists o iber bundle and matrix. Fiber bundle and matrix have been treated macroscopically as anisotropic and isotropic homogeneous bodies, respectively. The ibers have been arranged unidirectional within lamina o the composite laminates, but they reside in textile composites as the orm o bundles. The matrix is the resin part in textile composites or the interlayer without ibers in the case o laminates.
U is the total damping energy a cycle under vibrating and is deined as 1 U = ψ dv + dv V bσ bε b ψ V rσ rε r 2 (2) where σ is the stress matrix, ε is the strain matrix and the suix b and r represent the parts o iber bundle and matrix resin, respectively. ψ b and ψ r are the speciic damping capacities (SDC) and are expressed as ollows: ψ b ψ L ψ T ψ Z = ψ TZ ψ ZL ψ LT ψ x ψ y ψ z, ψ r = (3) ψ yz ψ zx ψ xy On the other hand, U is the maximum strain energy and deined as: U 1 = dv + dv V b b V r r 2 σ ε σ ε (4) 2.2 Identiication method o damping capacity The damping matrix components ψ cannot be ound easily rom the experiment. Even i the components obtain rom experiment, they will be generally lower than the measured ones, because the iber-matrix interactions occur. The preerred method or measuring the damping properties is to investigate unidirectional FRP material as a whole in the shape o beam specimens [6]. Considering the relationship between the SDC matrix and the modal damping ratios as a nonlinear system, the quasi-newton method is applied or identiying the damping parameters in the material principle direction. We deine the error unction g n (x) as the dierence between the n-th damping ratios ζ En measured by experiment and the calculated one ζ n (x ) as ollows: gn ( x) = ζ ζ ( x) (5) En The identiication is considered as a non-linear optimization problem to ind a solution x that minimizes the error norm Φ (x). 1 = NTM g n 2 n= 1 n 2 Φ ( x) ( x) (6) where NTM is the total number o reerring modes. By quasi-newton method taes an initial value x, and the value is improved by the iteration ormula using the direction vector d and the step size parameter λ. x +1 = x + λ d (7)
A step size parameter λ is chosen by the line searcher algorithm, and the vector d can be given as a solution o the equation by H d = Φ ( x ) = J ( x ) g( x ) T (8) where H and J are the Hessian and Jacobian matrices o error unction g( x ), respectively. On the other hand, iber volume raction has a great inluence on the SDC matrix components. In this study, SDC components o iber bundle are calculated by Eqs. (9), (1) and (11). LT Em ψ L =ψ m ( 1 V ) (9) E 2 2 ψ m ( 1 V ){( G + 1) + V ( G 1) } { G(1 + V ) + 1 V }{ G(1 V ) + 1+ V } ψ = (1) m G G = (11) G where ψ is the SDC o the matrix, E is the modulus o elasticity o iber, E is the m modulus o elasticity o the matrix, V is the iber volume raction, G is the lateral modulus o elasticity o the iber and G m is the lateral modulus o elasticity o the matrix, respectively. 2.3 An estimation method o damping properties or WF composites The estimation procedures o damping properties or WF composites are as ollows: (i) Vibration tests o unidirectional FRP and resin (ii) Identiication o SDC matrix or unidirectional FRP and resin (iii) Generation o inite element mesh o WF composites (iv) Finite element analysis o WF composites (v) Estimation o damping and natural requency At irst, the vibration test is carried out in the low-pressure condition in order to obtain the material damping o unidirectional FRP and resin. And, the SDC matrix o unidirectional FRP and resin are identiied rom experimental data. And then, inite element mesh model o WF composites is generated by the digital image processing as shown in Fig.1. Ater that, the inite element analysis is carried out. m (a) Cross section image o WF composites (b) Cross section image o inite element mesh model Fig. 1. Numerical model o WF composites. 3. PLAIN WF COMPOSITES 3.1 Specimen Test specimens have been abricated by the hand-lay up method. The vinylester resin (supplied rom Showa polymer Co. LTD.: R-86) is used E-glass woven cloth abric with 3 bundles (supplied rom Asahi iber glass Co. LTD.: WR57B) is used as the reinorcement.
The volume raction in a iber bundle has been measured by the digital image processing and laser microscope image. Fig.2 shows the cross section o the iber bundles by laser microscope. From this image, the volume raction or a iber bundle can be estimated by an image process. In case o a plain WF composite as shown in Fig.2, the volume raction is 45.%. (a) Cross section o the plain WF composites. (b) Cross section o iber bundle. Fig. 2. Laser microscope image o the specimen. 3.2 Damping capacity To calculate SDC components o the iber bundle and matrix resin, the identiication method has been applied. On the other hand, authors have been reported that the aerodynamic orce has a great inluence on the material damping and the vibration test at below 1 3 Pa has been required to obtain the real damping ratio [8]. In order to consider the eect o the air on the damping properties, the excitation tests have been carried out in the low-pressure chamber (4Pa) at room temperature. Fig.3 is a scheme o the experimental apparatus used. The exciting point is the center o the specimen. The orces o input and the acceleration o output have been measured by an impedance head. The damping ratios have been obtained by modal analysis. The specimens have been made o unidirectional GFRP (E-glass/vinylester). The length, width and thicness o GFRP specimen are 18mm, 15.mm and.57mm, respectively. And the length, width and thicness o vinylester specimen are 186mm, 15.mm and 3.3mm, respectively. From the measured damping ratios, the SDC components o the GFRP and vinylester resin have been identiied. Table1 shows the identiied SDC matrix o each material. From these tables, it is recognized that the identiied ψ T is higher than ψ L. It is ound that the vibration energy has been diused in the matrix resin. Thereore, matrix resin plays an important part in the material damping.
RP Vacuum chamber Specimen θ Impedance head Exciter Acceleration Force Fig. 3. Experimental set up. Table 1. Identiied damping parameters, %. (a) Unidirectional GFRP (b) Vinylester ψ L ψ T ψ ψ TZ LT ψ x ψ xy 1.47 4.63 5.69 8.34 9.4 1.8 3.3 Finite element analysis The shape and dimensions o analytical model are shown in the Fig.4. To estimate the woven architecture, inite element model has been generated and the mechanical properties have been calculated. The number o total nodes and elements o inite element mesh are 2,653 and 17,424, respectively. To conirm the validity o inite element mesh, the experimental data and inite element mesh or iber bundle orientation have been compared. The result is shown in Fig.5. From this result, it can be recognized that the undulation o iber bundle by inite element mesh model has a good agreement with measured values. Fiber bundle is treated as unidirectional FRP, and the mechanical properties can be calculated by the rule o mixture based on the obtained iber volume ractions. 13 11.7 θ.5 Angle θ [deg] Unit mm Fig. 4. Dimensions o test specimen..3.2.1 Measured Finite element model -.1 -.2 -.3. 1. 2. 3. 4. 5. Unit: mm Fig. 5. Comparison o undulation o iber bundles.
Table 2. Mechanical properties o iber bundle and matrix resin. Modulus o elasticity, GPa Shear modulus, GPa Poisson s ratios SDC, % Fiber bundle E-glass/Vinylester E L 34.6 E T 8.94 E Z 8.94 G TZ 3.46 G ZL 3.33 G LT 3.33 ν TZ.291 ν ZL.67 ν LT.254 ψ L.479 ψ T 5.3 ψ TZ 4. ψ LT 8.72 Matrix resin Vinylester E 3.3 G 1.17 ν.35 ψ x 9.4 ψ xy 1.8 In order to estimate the material damping, inite element analysis has been carried out. Numerical results o plain WF composite ( θ = ) are shown in Figs.6, 7 and 8. As shown in Fig.7, it is recognized that the damping ratio does not depend on requency. The eects o the orientation θ o the iber bundle on natural requency and damping ratio are shown in Fig.8. The natural requency has maximum at θ =, 9. Because the stiness has maximum at θ =, 9. On the other hand, the damping ratio shows a maximum when θ = 45. In this case, since the shear deormation at 45 is bigger than any other angles. SDC ψ LT, ψ TZ, ψ ZL and ψ xy show maximum values. In order to examine the numerical results, the vibration test has been carried out in a lowpressure chamber (4Pa). The numerical results both natural requency and damping ratio have a good agreement with the experimental ones. From these results, it is recognized that the material damping o plain WF composites can be estimated by the proposed numerical method. (a) 1st mode (b) 2nd mode (c) 3rd mode Fig. 6. Vibration modes o plain WF composites ( θ = ).
1 Damping ratio, % 1-1 Experiment (4Pa) Numerical analysis 1 1 1 2 1 3 Frequency, Hz Fig. 7. Comparison o experimental and numerical results ( θ = ). 3 1 Frequency, Hz 2 1 Experiment (4Pa) Numerical analysis 45 9 Angle θ, deg (a) Natural requency Damping ratio, % Experiment (4Pa) Numerical analysis 1-1 45 9 Angle θ, deg (b) Damping ratio Fig. 8. Relationship between vibration characteristics and angle θ. 4. TWF COMPOSITES 4.1 Specimen The proposed estimation method has been also applied to the tri-axial woven abric (TWF) composites in order to conirm whether the proposed estimation method is able to predict the multi-axial WF composites. The specimens have been made o a TWF (T3) and vinylester. The length, width and thicness o specimens are 126mm, 13.mm and.2mm, respectively. The longitudinal direction o specimen is shown in Fig.9. The volume raction in a iber bundle has been measured by the digital image processing and laser microscope image. The volume raction inside a iber bundle can be evaluated by an image process. In case o TWF composites used, the volume raction is 44.%. The mechanical properties o iber bundle and matrix are shown in Table3. These properties have been calculated by the rule o mixture and identiication method.
Longitudinal direction Fig. 9. CCD image o reinorcement o TWF composites. Table 3. Mechanical properties o iber bundle and matrix resin. Young s modulus, GPa Shear modulus, GPa Poisson s ratio SDC, % Fiber bundle T3/Vinylester E L 98.9 E T 5.44 E Z 5.44 G TV 1.94 G VL 2.61 G LT 2.61 ν TV.43 ν VL.144 ν LT.261 ψ L.176 ψ T 4.9 ψ TV 6.3 ψ LT 7.63 Matrix resin Vinylester E 3.3 G 1.2 ν.35 ψ x 9.4 ψ xy 1.8 4.2 Numerical analysis and Vibration test The inite element mesh model is shown in Fig.1. The number o total nodes and elements o FE mesh are 41,539 and 34,68, respectively. To estimate the material damping, inite element analysis has been carried out. A numerical result o TWF composites is shown in Fig.11. In order to examine the veriication o these numerical results, the vibration test have been carried out under low-pressure condition (4Pa). The numerical results have good agreement with the experimental ones. From these results, it is also recognized that material damping o TWF composites can be predicted by the proposed numerical method. Fig. 1. Finite element mesh o TWF composites.
1. Damping ratio, % Experiment (4Pa) Numerical analysis.1 1 1 1 2 1 3 Frequency, Hz Fig. 11. Comparison o experimental and numerical results. 5. SUMMARY An estimation method o damping properties o plain and tri-axial woven abric composites has been described. Comparing the numerical results with the experimental ones, it is veriied that the proposed method is able to estimate the material damping o woven abric composites. Thereore, even i the composites are composed by complex architecture lie tri-axial woven abric, the damping characteristics will be able to estimated by a simple excitation test o unidirectional coupon specimen and inite element method. Reerences 1. D.A.Saravanos, C.C.Chamis, Uniied micromechanics o damping or unidirectional and o-axis iber composites, Journal o Composite Technology Research, 12(1), (199), 31 4 2. R.Chandra, S.P.Singh, K.Gupta, Micromechanical damping models or iber-reinorced composites: a comparative study, 33, (22), Composites Part A, 787-796 3. V.Carvelli, C.A.Poggi, A homogenisation procedure or the numerical analysis o woven abric composites, Composites Part A, 32, (21), 1425-1432 4. J.Whitcomb, K.Sriengan, Eect o various approximations on predicted progressive ailure in plain weave composites, Composite structures, 34, (1996), 13-2 5. R.D.Adams, D.G.C.Bacon, Eect iber-orientation and laminate geometry on properties o CFRP, Journal o Composites Materials, 7, (1973), 42 428 6. K.Matsumoto, K.Hosoawa and M.Zao, Identiication o Vibration Damping Parameters or Laminated Composite Materials by Using FEM Eigenvalue Analysis, Vol.374, (1998), 319-324 7. R.D.Adams, Damping Properties Analysis o Composites, Composites (Engineered Materials Handboo Volume 1), (1984), 26 217 8. M.Zao, K.Matsumoto, T.Kurashii and Y.Naanishi, On the damping properties o triaxial woven abric composites, Proceedings o the 8th Japan international SAMPE symposium, (23), 1169-1172