ON AN ESTIMATION OF THE DAMPING PROPERTIES OF WOVEN FABRIC COMPOSITES

Masaru Zako1, Tetsusei Kurashiki1, Yasumasa Nakanishi2

and Kin’ya Matsumoto2

1 Department of Management of Industry and Technology, Osaka University, 2-1 Yamada-oka, Suita, Osaka,

565-0871, Japan 2 Department of Technology, Faculty of Education, Mie University, 1515 Kamihama-cho, Tsu, Mie,

514-8507, Japan

ABSTRACT In this paper, we have proposed an estimation method that the damping properties of woven fabric composites

are obtained by a simple excitation test of unidirectional coupon specimen and finite element method. The proposed method has been applied to the plain and tri-axial woven fabric composites. Experiments have been conducted to evaluate the validity of the proposed method. It is recognized that the computational results have agreed well with the experimental ones. Since an effect of orientation of fiber bundle on material damping of plain-woven fabric composites can be simulated, the damping properties for various woven fabric composites will be estimated. It is revealed that the proposed method is very useful for the estimation of material damping characteristics.

1. INTRODUCTION Material damping represents the cumulative contributions of the visco-elastic response of

the constituents, cyclic heat flow and the friction at the fiber/matrix interface. Recent work on the material damping of FRP has shown that it depends on an array of micro-mechanics and laminate parameters, including constituent material properties, fiber volume fraction, stacking sequence [1][2]. These studies, however, are mostly limited to unidirectional composites.

Woven fabric (WF) composites are applied to many fields such as the space structures, sports items and so on. Though many researchers have reported the static characteristics of WF composites, the material damping of those materials have not been investigated [3][4].

The purpose of this study is to establish an estimation method of damping properties for WF composites. The proposed procedure has been applied to the numerical study of the material damping of 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 of energy in the material bared or stress components under vibrating for unidirectional FRP [5]. Damping ratioζ is expressed as

UU∆

⋅=π

ζ41 (1)

where U∆ is the total damping energy a cycle of vibration and U is the maximum strain energy.

We have employed a three-dimensional heterogeneous finite element model for WF composites, which consists of fiber bundle and matrix. Fiber bundle and matrix have been treated macroscopically as anisotropic and isotropic homogeneous bodies, respectively. The fibers have been arranged unidirectional within lamina of the composite laminates, but they reside in textile composites as the form of bundles. The matrix is the resin part in textile composites or the interlayer without fibers in the case of laminates.

U∆ is the total damping energy a cycle under vibrating and is defined as

⎭⎬⎫

⎩⎨⎧

+= dVdVUVV rrrfbfbfb2

1εσψεσψ ∫∫∆ (2)

where σ is the stress matrix, ε is the strain matrix and the suffix “fb” and “r” represent the parts of fiber bundle and matrix resin, respectively. fbψ and rψ are the specific damping capacities (SDC) and are expressed as follows:

, (3)

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

LT

ZL

TZ

Z

T

L

ψψ

ψψ

ψψ

000000000000000000000000000000

fbψ

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

xy

zx

yz

z

y

x

ψψ

ψψ

ψψ

000000000000000000000000000000

rψ

On the other hand, U is the maximum strain energy and defined as:

⎭⎬⎫

⎩⎨⎧

+= dVdVUVV rrfbfb2

1εσεσ ∫∫ (4)

2.2 Identification method of damping capacity

The damping matrix components ψ cannot be found easily from the experiment. Even if the components obtain from experiment, they will be generally lower than the measured ones, because the fiber-matrix interactions occur. The preferred method for measuring the damping properties is to investigate unidirectional FRP material as a whole in the shape of 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 for identifying the damping parameters in the material principle direction. We define the error function as the difference between the n-th damping ratios

)(xng

Enζ measured by experiment and the calculated one )(xnζ as follows: )()( xx nEnng ζζ −= (5)

The identification is considered as a non-linear optimization problem to find a solution x that minimizes the error norm )(xΦ .

∑=

=NTM

nng

1

2)(21)( xxΦ (6)

where NTM is the total number of referring modes.

By quasi-Newton method takes an initial value , and the value is improved by the iteration formula using the direction vector d and the step size parameter

0xλ .

kkkk dxx λ+=+1 (7)

A step size parameter λ is chosen by the line searcher algorithm, and the vector d can be given as a solution of the equation by (8) )()()( kk

Tkkk xgxJxdH −=−∇= Φ

where H and J are the Hessian and Jacobian matrices of error function , respectively. )( kxg

On the other hand, fiber volume fraction has a great influence on the SDC matrix components. In this study, SDC components of fiber bundle are calculated by Eqs. (9), (10) and (11).

f

mfmL E

EV )1( −=ψψ (9)

{ }

{ }{ }ffff

ffmLT

VVGVVG

GVGV

++−−++

−++−=

1)1(1)1(

)1()1()1( 22ψψ (10)

m

f

GG

G = (11)

where mψ is the SDC of the matrix, is the modulus of elasticity of fiber, is the modulus of elasticity of the matrix, is the fiber volume fraction, is the lateral modulus of elasticity of the fiber and is the lateral modulus of elasticity of the matrix, respectively.

fE mE

fV fG

mG 2.3 An estimation method of damping properties for WF composites

The estimation procedures of damping properties for WF composites are as follows:

(i) Vibration tests of unidirectional FRP and resin (ii) Identification of SDC matrix for unidirectional FRP and resin (iii) Generation of finite element mesh of WF composites (iv) Finite element analysis of WF composites

(v) Estimation of damping and natural frequency

At first, the vibration test is carried out in the low-pressure condition in order to obtain the material damping of unidirectional FRP and resin. And, the SDC matrix of unidirectional FRP and resin are identified from experimental data. And then, finite element mesh model of WF composites is generated by the digital image processing as shown in Fig.1. After that, the finite element analysis is carried out.

(a) Cross section image of WF composites (b) Cross section image of finite element mesh model

Fig. 1. Numerical model of WF composites. 3. PLAIN WF COMPOSITES 3.1 Specimen

Test specimens have been fabricated by the hand-lay up method. The vinylester resin (supplied from Showa polymer Co. LTD.: R-806) is used E-glass woven cloth fabric with 3 bundles (supplied from Asahi fiber glass Co. LTD.: WR570B) is used as the reinforcement.

The volume fraction in a fiber bundle has been measured by the digital image processing and laser microscope image. Fig.2 shows the cross section of the fiber bundles by laser microscope. From this image, the volume fraction for a fiber bundle can be estimated by an image process. In case of a plain WF composite as shown in Fig.2, the volume fraction is 45.0%.

(a) Cross section of the plain WF composites.

(b) Cross section of fiber bundle.

Fig. 2. Laser microscope image of the specimen.

3.2 Damping capacity

To calculate SDC components of the fiber bundle and matrix resin, the identification method has been applied. On the other hand, authors have been reported that the aerodynamic force has a great influence on the material damping and the vibration test at below 103 Pa has been required to obtain the real damping ratio [8]. In order to consider the effect of the air on the damping properties, the excitation tests have been carried out in the low-pressure chamber (40Pa) at room temperature. Fig.3 is a scheme of the experimental apparatus used. The exciting point is the center of the specimen. The forces of input and the acceleration of output have been measured by an impedance head. The damping ratios have been obtained by modal analysis.

The specimens have been made of unidirectional GFRP (E-glass/vinylester). The length, width and thickness of GFRP specimen are 180mm, 15.0mm and 0.57mm, respectively. And the length, width and thickness of vinylester specimen are 186mm, 15.0mm and 3.30mm, respectively.

From the measured damping ratios, the SDC components of the GFRP and vinylester resin have been identified. Table1 shows the identified SDC matrix of each material. From these tables, it is recognized that the identified Tψ is higher than Lψ . It is found that the vibration energy has been diffused in the matrix resin. Therefore, matrix resin plays an important part in the material damping.

Acceleration Force

ExciterImpedance head

θSpecimen

Vacuum chamber

RP

Acceleration Force

ExciterImpedance head

θSpecimen

Vacuum chamber

RPRP

Fig. 3. Experimental set up.

Table 1. Identified damping parameters, %.

(a) Unidirectional GFRP (b) Vinylester

Lψ Tψ TZψ LTψ xψ xyψ

1.47 4.63 5.69 8.34 9.40 10.8

3.3 Finite element analysis

The shape and dimensions of analytical model are shown in the Fig.4. To estimate the woven architecture, finite element model has been generated and the mechanical properties have been calculated. The number of total nodes and elements of finite element mesh are 20,653 and 17,424, respectively.

To confirm the validity of finite element mesh, the experimental data and finite element mesh for fiber bundle orientation have been compared. The result is shown in Fig.5. From this result, it can be recognized that the undulation of fiber bundle by finite 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 of mixture based on the obtained fiber volume fractions.

θ

103

11.7

0.50

θ

103

11.7 θθ

103

11.7

0.50

Angle θ [deg] Unit� mm

Fig. 4. Dimensions of test specimen.

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.0 1.0 2.0 3.0 4.0 5.0

Measured

Finite element model

Unit: mm

Fig. 5. Comparison of undulation of fiber bundles.

Table 2. Mechanical properties of fiber bundle and matrix resin.

Fiber bundle E-glass/Vinylester

Matrix resin Vinylester

LE 34.6

TE 8.94 Modulus of elasticity, GPa

ZE 8.94

E 3.30

TZG 3.46

ZLG 3.33 Shear modulus, GPa

LTG 3.33

G 1.17

TZν 0.291

ZLν 0.067 Poisson’s ratios

LTν 0.254

ν 0.35

Lψ 0.479

Tψ 5.03 xψ 9.40

TZψ 4.00 SDC, %

LTψ 8.72 xyψ 10.8

In order to estimate the material damping, finite element analysis has been carried out. Numerical results of plain WF composite ( °= 0θ ) are shown in Figs.6, 7 and 8. As shown in Fig.7, it is recognized that the damping ratio does not depend on frequency. The effects of the orientation θ of the fiber bundle on natural frequency and damping ratio are shown in Fig.8. The natural frequency has maximum at °= 0θ , . Because the stiffness has maximum at °90

°= 0θ , . On the other hand, the damping ratio shows a maximum when °90 °= 45θ . In this case, since the shear deformation at is bigger than any other angles. SDC °45 LTψ , TZψ , ZLψ and xyψ show maximum values.

In order to examine the numerical results, the vibration test has been carried out in a low-pressure chamber (40Pa). The numerical results both natural frequency and damping ratio have a good agreement with the experimental ones.

From these results, it is recognized that the material damping of plain WF composites can be estimated by the proposed numerical method.

(a) 1st mode (b) 2nd mode (c) 3rd mode

Fig. 6. Vibration modes of plain WF composites ( °= 0θ ).

Experiment (40Pa)

Numerical analysis

101 102 103

Frequency, Hz

10-1

100

Dam

ping

ratio

, %

Experimentanalysis

Experiment (40Pa)

Numerical analysis

101 102 103

Frequency, Hz

10-1

100

Dam

ping

ratio

, %

Experimentanalysis

Fig. 7. Comparison of experimental and numerical results ( °= 0θ ).

Experiment (40Pa)Numerical analysis

Angle θ , deg

Freq

uenc

y, H

z

0 45 900

10

20

30

Experimentanalysis

Experiment (40Pa)Numerical analysis

Angle θ , deg

Freq

uenc

y, H

z

0 45 900

10

20

30

Experimentanalysis

Experiment (40Pa)Numerical analysis

Dam

ping

ratio

, %

Angle θ , deg0 45 90

10-1

100

(a) Natural frequency (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 fabric (TWF) composites in order to confirm whether the proposed estimation method is able to predict the multi-axial WF composites.

The specimens have been made of a TWF (T300) and vinylester. The length, width and thickness of specimens are 126mm, 13.0mm and 0.20mm, respectively. The longitudinal direction of specimen is shown in Fig.9.

The volume fraction in a fiber bundle has been measured by the digital image processing and laser microscope image. The volume fraction inside a fiber bundle can be evaluated by an image process. In case of TWF composites used, the volume fraction is 44.0%. The mechanical properties of fiber bundle and matrix are shown in Table3. These properties have been calculated by the rule of mixture and identification method.

Longitudinal direction

Fig. 9. CCD image of reinforcement of TWF composites.

Table 3. Mechanical properties of fiber bundle and matrix resin.

Fiber bundle T300/Vinylester

Matrix resin Vinylester

LE 98.9

TE 5.44 Young’s modulus, GPa

ZE 5.44

E 3.30

TVG 1.94

VLG 2.61 Shear modulus, GPa

LTG 2.61

G 1.20

TVν 0.403

VLν 0.0144 Poisson’s ratio

LTν 0.261

ν 0.350

Lψ 0.176

Tψ 4.90 xψ 9.40

TVψ 6.03 SDC, %

LTψ 7.63 xyψ 10.8

4.2 Numerical analysis and Vibration test

The finite element mesh model is shown in Fig.10. The number of total nodes and elements of FE mesh are 41,539 and 34,068, respectively.

To estimate the material damping, finite element analysis has been carried out. A numerical result of TWF composites is shown in Fig.11. In order to examine the verification of these numerical results, the vibration test have been carried out under low-pressure condition (40Pa). The numerical results have good agreement with the experimental ones.

From these results, it is also recognized that material damping of TWF composites can be predicted by the proposed numerical method.

Fig. 10. Finite element mesh of TWF composites.

0.1

1.0

Experiment (40Pa)Numerical analysis

101 102 103

Frequency, Hz

Dam

ping

ratio

, %

Fig. 11. Comparison of experimental and numerical results. 5. SUMMARY

An estimation method of damping properties of plain and tri-axial woven fabric composites has been described. Comparing the numerical results with the experimental ones, it is verified that the proposed method is able to estimate the material damping of woven fabric composites. Therefore, even if the composites are composed by complex architecture like tri-axial woven fabric, the damping characteristics will be able to estimated by a simple excitation test of unidirectional coupon specimen and finite element method. References

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