Complex stiffness method for damping properties of woven composites

Abstract

Damping is a significant feature of the dynamic behavior of composites to control resonant response. While some methods work effectively in forecasting the damping of laminated composites, few damping prediction approaches have been developed for woven composites. Based on the elastic-viscoelastic correspondence principle, an effective method called the complex stiffness method is presented to determine the damping properties of woven composites. The effect of damping of fiber phase in fiber tows is considered to improve the accuracy of the predictions. Experiments were performed to validate the proposed model and good agreements are achieved between test data and predictions.

Keywords

Woven composites damping properties complex stiffness free flexure vibration

Introduction

Woven composites consist of interlaced warp and weft tows.^1,2 Figure 1 illustrates the weave architecture of a typical woven composite. Weft tows (green) are stretched straight and orthogonal warp tows (yellow) go through different weft layers in undulate to interlock them as an integral reinforcement. When woven composites are used in structural applications, they are expected to experience vibration environment in which damping is an important feature of their dynamic behaviors. Effective methods for damping prediction of woven composites should be built for the improvement of their dynamic analysis,^3,4 vibration and sound control.^5–8

Damping properties of polymer matrix/glass fiber unidirectional composites depend on the viscoelastic nature of matrix and/or fiber materials, damping of fiber–matrix interface, damping due to damage including frictional damping and energy dissipation in the area of matrix cracks and broken fibers, viscoplastic damping caused by high stress level and thermoelastic damping due to cyclic heat flow from regions of compressive stress to regions of tensile stress.⁹ If undamaged polymer composites vibrate at small amplitudes, viscoelastic damping seems to be dominant contribution to the damping of composites.¹⁰ Thus, only viscoelastic damping is considered in this study.

Earlier analytical methods on damping properties of composites mainly focused the attention on laminated composites. Adams and Bacon³ and Adams et al.⁴ developed a strain energy theory to forecast the effective damping properties of laminated composites. The specific damping capacity (SDC) ψ, defined by Adams et al., is

ψ = \frac{Δ U}{U}

(1)

where

Δ U

and U are strain energy dissipated and the maximum strain energy during a stress cycle.

A simple solution of equation (1) can be obtained if the assumption is made that the damping coefficients in local coordinate system, i.e., the longitudinal (fiber direction) coefficient denoted by $ψ_{1}$ , the transverse coefficient denoted by $ψ_{2}$ and the in-plane shear coefficient denoted by $ψ_{12}$ , are independent of the stress $(σ_{1}, σ_{2}, σ_{12})$ . Ni and Adams¹¹ made an improvement on the damping prediction given by Adams and Bacon³ and Adams et al.⁴ They take the normal stress $σ_{x}$ , the normal strain $ɛ_{x}$ and the shear strain $ɛ_{xy}$ in the global coordinate system into account during the determination of the dissipated energy. Based on the elastic-viscoelastic correspondence principle, Yim¹² developed a complex stiffness model for more reliable damping prediction of laminated composite beams. The basic damping of Poisson’s ratio derived by the indirect method was proposed.

The methods mentioned above are all based on the assumption that the damping coefficients in local coordinate system are independent of stress. The advantages of Adams theory and Ni and Adams theory are that the relation between global stress $σ_{x}$ (or M_x) and local stress $(σ_{1}, σ_{2}, σ_{12})$ of each lamina is easily derived under free-flexure condition, and the damping of whole laminated composite beams can be obtained by summation of the damping of each lamina. However, this relation is quite difficult to obtain for the woven composites since they have complicated weaving structures in a repeated unit cell. It is seen that Yim’s approach can calculate the complex stiffness matrix of whole structure directly and makes the damping prediction much simpler due to circumvention of the complicated stress analysis.

From literature review, it is seen that very few damping prediction methods have been developed thus far for woven composites. Guan and Gibson¹⁰ developed a closed-form model and a finite element model for studying the viscoelastic damping in plain weave composite. Both analytical models show good agreement with the experimental data. The closed-form model was built by the formulations obtained from the mechanics of materials and elastic-viscoelastic correspondence principle and is actually a complex stiffness method. The same approach can be also found to be used for predicting the damping of 3D braided textile composites,¹³ damping of randomly oriented short-fiber composites¹⁴ and damping of aligned discontinuous fiber composites.¹⁵ Since the inclined angle of fiber bundles in plain weave composite is very small, the effect of undulate fiber bundles on the damping properties is not considered by Guan and Gibson.¹⁰ However, the undulate fiber bundles effect on damping capacities of a woven composite illustrated in Figure 1 should be taken into account, because the warp tows have larger inclined angle which cannot be neglected. In addition, Guan and Gibson¹⁰ pointed out that the predictions of the composite loss factors can be improved by using a reduced experimental fiber loss factor. It is notable that the reasonable use of fiber damping in formulations is important for the predictions of composites damping.^10,16

In this paper, the complex stiffness method is established to describe the damping properties of woven composites. The effects of micro weave parameters and damping properties of fiber phase in tows are considered. Experiments are carried out to validate the proposed model, and finally conclusions are drawn based on the results reported herein.

Theory

Complex stiffness analysis

From the classic thin laminated plate theory, the relationship between the in-plane stress resultants N_i, moment resultants M_i, in-plane strains $ɛ_{j}$ and curvatures $κ_{j}$ is given by¹⁷

{N_{i} M_{i}} = [A_{ij} B_{ij} B_{ij} D_{ij}] {ɛ_{j} κ_{j}} (i, j = 1, 2, 6)

(2)

where

[A_{ij}]

[B_{ij}]

and

[D_{ij}]

are the extensional, stretching-bending coupling and flexural stiffness matrices defined by

A_{ij}, B_{ij}, D_{ij} = \int_{- h / 2}^{h / 2} (1, z, z^{2}) {\bar{Q}}_{ij}^{k} d z

(3)

where h is the thickness of the composite,

[{\bar{Q}}_{ij}^{k}]

can be calculated by

{\bar{Q}}_{ij}^{k} = (T^{k})^{- 1} Q_{ij} (T^{k})^{- T}

(4)

In equation (4), $[T^{k}]$ is transformation matrix of the $k th$ lamina defined by

T^{k} = [c^{2} s^{2} 2 cs c^{2} c^{2} - 2 cs cs - cs c^{2} - s^{2}]

(5)

where

c = \cos θ^{k}

s = \sin θ^{k}

θ^{k}

is the ply angle of the

k th

lamina and

[Q_{ij}]

is given by

Q_{11} = \frac{E_{1}}{1 - μ_{12} μ_{21}} Q_{12} = \frac{μ_{12} E_{2}}{1 - μ_{12} μ_{21}} Q_{22} = \frac{E_{2}}{1 - μ_{12} μ_{21}} Q_{66} = G_{12}

(6)

According to the elastic-viscoelastic correspondence principle, the complex form of the basic engineering constants is described as

E_{1}^{*} = E_{1} (1 + i η_{1}) E_{2}^{*} = E_{2} (1 + i η_{2}) G_{12}^{*} = G_{12} (1 + i η_{12})

(7)

where

η_{1}

η_{2}

, and

η_{12}

represent longitudinal, transverse, and in-plane shear damping loss factor, respectively, and the loss factor η is related to specific damping capacity ψ by the relation

ψ = 2 π η

In the case of laminated composite beams with different layers, the effective flexural modulus under free-flexure condition is given as¹⁸

E_{lf} = \frac{12}{h^{3} D_{11}^{- 1}}

(8)

where

D_{11}^{- 1}

is the ﬂexural compliance along the beam (longitudinal direction), derived as the element of the inverse matrix of

[D_{ij}]

Hence, the complex bending modulus can be obtained as

E_{lf}^{*} = \frac{12}{h^{3} D_{11}^{* - 1}} = E'_{lf} + iE ″_{lf}

(9)

where

D_{11}^{* - 1}

represents the complex ﬂexural compliance along the beam.

E'_{lf}

and

E ″_{lf}

denote storage and loss modulus of laminated composite beam. The flexural loss factor of laminated composite is then calculated as the ratio of the imaginary to real part value, namely

η_{lf} = \frac{E ″_{lf}}{E'_{lf}}

(10)

Complex stiffness matrix of woven composite

Several analytical models are available to predict the stiffness of woven composite, such as the models given by Ishikawa and Chou and¹⁹, Chou and Ishikawa,²⁰ Naik and Ganesh,^21,22 and Scida et al.^17,23 Among them, the model given by Scida et al.,^17,23 called MEchanical Simulation of TEXtile (MESOTEX), is established based on the classic thin laminate theory and used to forecast the elastic properties of woven composites with non-hybrid weave and hybrid weave fabrics. MESOTEX is applied to different weaving structures for woven composite and has good correlation with experimental results^17,23 In this paper, a simplified MESOTEX model is presented to calculate the stiffness matrix of woven composite for obtaining the correct damping by using the complex stiffness method.

The stiffness prediction of woven composite requires a characteristic repeated unit cell, the definition of woven architecture and properties of different constituents. The periodicity of woven composite enables us to isolate a repeated unit cell as illustrated in Figure 2.

To simplify the calculation, the weft tows which slightly undulate are supposed to be straight, and their cross sections are hexagon. Warp tows are simplified as several horizontal and inclined cubes with same rectangular sections. Hence, the parameters of the architecture of woven composite can be determined as illustrated in Figure 3. The length and width of repeated unit cell are $L_{t}$ and $L_{p}$ . The height of tows can be derived by

h_{y} = H / (N_{p} + N_{t} + 2)

(11)

where H is the thickness of woven composite,

N_{p}

and

N_{t}

are the ply number of warp and weft tows. Here, it is assumed that the heights of warp and weft tows are equal.

The relations of weaving parameters are given as

θ_{p} = \tan^{- 1} \frac{h_{y}}{2 w}

(12)

W_{t} - w - A_{f} / (V_{f} h_{y}) = 0

(13)

W_{p} - A_{f} / (V_{f} h_{y}) = 0

(14)

where

W_{p}

and

W_{t}

are the widths of cross section of warp and weft tows, w is the width of triangle part in the cross section of warp tow,

θ_{p}

is the inclined angle of warp tow,

A_{f}

is the cross-sectional area of fiber phase in tows, and

V_{f}

is the fiber volume fraction in tows. The widths of cross section of weft tows are larger than warp tows, since the straight weft tows are encountered more compression by undulate warp tows. However, the cross section area of weft tows and warp tows should be equal in the fact that warp tows and weft tows are made of same fiber bundles. Thus, equations (13) and (14) are given to ensure this condition.

The technique proposed in Scida et al.²³ is a point-wise lamination approach considering that the unit cell of woven composite is treated as a composite composed of many slices with fiber tows and resin. The stiffness matrix $[Q_{ij}^{s}]$ of each slice is given as a summation of the stiffness matrices of resin, weft tow, and warp tow

Q_{ij}^{s} (x, y) = \sum_{I} V_{I} (x, y) {\bar{Q}}_{ij}^{I} (x, y) (i, j = 1, 2, 6)

(15)

where superscript ‘I’ denotes resin (R), weft tow (T) or warp tow (P),

{\bar{Q}}_{ij}^{I} (x, y)

is the stiffness matrix of the ‘I’ phase in slice in global coordinate system, and

V_{I} (x, y)

is the volume fraction of the ‘I’ phase in a repeated unit cell. If the thickness effect is not considered based on the classic laminated plate theory, the stiffness matrix of each constituent in global coordinate system (

x, y, z

) is given as

{\bar{Q}}_{ij}^{I} = [\frac{E_{x}^{I}}{d} μ_{xy}^{I} \frac{E_{y}^{I}}{d} 0 μ_{yx}^{I} \frac{E_{x}^{I}}{d} \frac{E_{y}^{I}}{d} 000 G_{xy}^{I}] where d = 1 - μ_{xy}^{I} μ_{yx}^{I}

(16)

For the resin

E_{x}^{R} = E_{r} E_{y}^{R} = E_{r} μ_{xy}^{R} = μ_{r} G_{xy}^{R} = G_{r}

(17)

where

E_{r}

μ_{r}

and

G_{r}

are engineering elastic constant of resin.

For the weft tows

E_{x}^{T} = E_{2} E_{y}^{T} = E_{1} μ_{xy}^{T} = μ_{21} G_{xy}^{T} = G_{12}

(18)

For the warp tows

E_{x}^{P} = [\frac{1}{E_{1}} c_{p}^{4} + (\frac{1}{G_{13}} - 2 \frac{μ_{13}}{E_{1}}) c_{p}^{2} s_{p}^{2} + \frac{1}{E_{2}} s_{p}^{4}]^{- 1} E_{y}^{P} = E_{2} μ_{xy}^{P} = E_{x}^{P} [\frac{μ_{12}}{E_{1}} c_{p}^{2} + \frac{μ_{23}}{E_{2}} s_{p}^{2}] G_{xy}^{P} = [\frac{s_{p}^{2}}{G_{23}} + \frac{c_{p}^{2}}{G_{12}}]^{- 1}

(19)

where

c_{p} = \cos (θ_{p})

and

s_{p} = \sin (θ_{p})

Since in a repeated unit cell shown in Figure 2, the warp tows have three different local off-axis angles in x direction, i.e. $0^{o}$ , $θ_{p}$ and $- θ_{p}$ , hence the average stiffness matrix of warp tow is a volume average of the stiffness matrix of the warp tow with three different local off-axis angles.

The global stiffness matrix of the unit cell is calculated by $Q_{ij}^{s}$ for each slice with an average in both x and y direction.

Q_{ij}^{wc} = \frac{1}{L_{t} L_{p}} \int_{0}^{L_{t}} \int_{0}^{L_{p}} Q_{ij}^{s} (x, y) d x d y = \sum_{I} V_{I} {\bar{Q}}_{ij}^{I} (i, j = 1, 2, 6)

(20)

where

{\bar{Q}}_{ij}^{I}

is the stiffness matrix of the ‘I’ phase and V_I is the volume fraction of the ‘I’ phase in a repeated unit cell.

Then, the stiffness matrix of repeated unit cell is given as

A_{ij}^{wc}, B_{ij}^{wc}, D_{ij}^{wc} = \int_{- H / 2}^{H / 2} (1, z, z^{2}) Q_{ij}^{wc} d z

(21)

Finally, the complex stiffness matrix of repeated unit cell can be obtained by

A_{ij}^{* wc}, B_{ij}^{* wc}, D_{ij}^{* wc} = \int_{- H / 2}^{H / 2} (1, z, z^{2}) Q_{ij}^{* wc} d z

(22)

Damping of woven composite beams

Under free-flexure condition, the effective bending modulus of woven composite similar to laminated composite in equation (8) is given as

E_{bending}^{wc} = \frac{12}{H^{3} D_{11}^{- 1 wc}}

(23)

where

D_{11}^{- 1 wc}

represents the ﬂexural compliance of woven composite along the beam (longitudinal direction) and is obtained by the element of the inverse matrix of

D_{ij}^{wc}

derived in equation (21).

Using elastic-viscoelastic correspondence principle, the effective complex bending modulus $E_{bending}^{* wc}$ of woven composite is

E_{bending}^{* wc} = \frac{12}{h^{3} D_{11}^{* - 1 wc}} = E_{' wc}^{bending} + iE {″ wc}_{bending}

(24)

where

D_{11}^{* - 1 wc}

represents the complex ﬂexural compliance of woven composite along the beam and is derived as the element of inverse matrix of

D_{ij}^{* wc}

in equation (22).

E {' wc}_{bending}

and

E {″ wc}_{bending}

denote the storage and loss bending modulus of woven composite.

According to elastic-viscoelastic correspondence principle and assuming that all basic damping of Poisson’s ratios are constants, the complex engineering constants of unidirectional composite used in $D_{ij}^{* wc}$ are defined as

E_{r}^{*} = E_{r} (1 + i η_{r}) E_{1}^{*} = E_{1} (1 + i η_{1}) E_{2}^{*} = E_{2} (1 + i η_{2}) E_{3}^{*} = E_{3} (1 + i η_{3}) G_{12}^{*} = G_{12} (1 + i η_{12}) G_{13}^{*} = G_{13} (1 + i η_{13}) G_{23}^{*} = G_{23} (1 + i η_{23})

(25)

where

η_{r}

is the loss factor of resin,

η_{3}

η_{13}

and

η_{23}

are the transverse loss factor, in-plane shear loss factor and transverse shear loss factor of unidirectional composite, respectively. Since unidirectional composites are treated as transverse isotropic material and

G_{23}^{*} = E_{2}^{*} / 2 (1 + μ_{23})

, hence

η_{3} = η_{2}

η_{13} = η_{12}

and

η_{23} = η_{2}

Finally, the flexural loss factor $η_{bending}^{wc}$ (damping) of woven composite under free-flexure condition is obtained by complex bending modulus in equation (24) as the ratio of the imaginary to real part value

η_{bending}^{wc} = \frac{E {″ wc}_{bending}}{E {' wc}_{bending}}

(26)

Experiment

Materials and specimen

Epoxy resin (E51) and glass fiber/epoxy matrix composite beams with different fiber orientations ( $0^{o}$ , $45^{o}$ , and $90^{o}$ ) were prepared to determine their basic static elastic constants and damping properties. The shape of the cross-section of beams is rectangular. Pure resin and unidirectional composites plates were first cured at $25^{o} C$ for 24 h under a pressure of 0.6 MPa on a mold plate. Four groups of slender beam specimen, i.e., pure resin, $0^{o}$ , $45^{o}$ , and $90^{o}$ unidirectional composite specimen, were cut from the rectangular plates. Each group has three specimens. The dimensions of the specimen are 280 × 25 × 2 mm.

Woven composites which are constructed with glass fiber tows and epoxy matrix were fabricated by using hot molding press method. Two types of woven composites were supplied by Nanjing Fiberglass Research & Design Institute. The major physical properties of the two woven composites, namely, woven-a and woven-b, are listed in Table 1. Figure 4 illustrates two cross sections of a woven composite along different direction, captured by optical microscope.

Table 1.

Major physical properties of the woven composites.

Composite	Mass (g)	Length (mm)	Width (mm)	Thickness (mm)	$V_{f}$ (%)	$N_{p}$	$N_{t}$
Woven-a	41.2	205	27.1	4.44	54.3	6	7
Woven-b	40.1	199	25.7	5.20	65.9	8	9

Experimental procedures

The impulse method is chosen to measure the damping properties of the resin and composites due to the ease of implementation and the quickness of the experiment. The experimental setup is shown in Figure 5.

The specimen is supported horizontally in a clamping block. The excitation of the flexural vibration of the beam is induced by an impulse hammer near the clamping block. Due to no added mass influence on the evaluation of the damping, a laser sensor head (LK-G30) is used to detect the displacements response of the beam near the free end. The response signals are processed by a sensor controller (LK-G3001) connected with a PC. The system allows the simultaneous acquisition of two signals with a maximum sampling frequency of 50 kHz with a repeatability of 0.05 $μ m$ .

After initial excitation, the Fast Fourier Transform (FFT) of time-domain signal gotten from the sensor leads to the frequency response functions of the beams. The peaks of the response correspond to the natural frequencies of the flexural vibration of the beams. Based on the rational fraction polynomial (RFP) method, the modal analysis is implemented via curve ﬁtting the frequency responses by using the toolbox of Matlab to get the values of the natural frequencies w_i and the modal damping coeﬃcient $ξ_{i}$ or the loss factor $η_{i}$ . The bending stiﬀness E_i for each natural frequency of the beam in the case of low damping is evaluated by²⁴

E_{i} = \frac{4 π^{2} ρ A}{I λ^{4}} w_{i}^{2}

(27)

where ρ is the density of material, A is the cross-sectional area of the beam, I is the moment of inertial of the beam, and λ is the parameter determined by boundary condition, respectively.

Resins, unidirectional composites ( $0^{o}$ , $45^{o}$ , $90^{o}$ ), and woven composites beam specimens are used in free vibrating experiment and each type of material has three specimens, respectively. Since the thickness shear effect may be a significant role for composite damping especially in higher mode,²⁵ the identification of damping was carried out in the first flexural mode with different specimen lengths to describe the evolution of the damping of each type of material at low frequency range (0–250 Hz). The free lengths of resin and unidirectional composites beam specimens are 200, 160, 120 and 80 mm, while the lengths of woven composites beam specimens are 180, 160, 140 and 120 mm for two types of woven composites, respectively.

Results and discussion

Material properties

The elastic properties of unidirectional composite can be expressed by law of mixture, but a better evaluation can be obtained according to Berthelot.²⁶ Therefore, the prediction method by Berthelot²⁶ is used to predict the elastic properties of the fiber tows treated as the unidirectional composites. The elastic properties of resin and fiber tows (fiber tows-a and fiber tows-b) in two woven composites are given in Table 2. The geometry parameters of the architecture of woven composite are listed in Table 3. The length

L_{t}

and width

L_{p}

of repeated unit cell and the widths of weft tows and warp tows (

W_{t}

and

W_{p}

) are directly measured by using optical microscope.

h_{y}

θ_{p}

, and w are derived from equations (11) to (14), respectively.

Table 2.

Elastic properties of resin, glass fiber and fiber tows.

Material	E₁ (GPa)	E₂ (GPa)	$μ_{12}$	$G_{12}$ (GPa)	$G_{23}$ (GPa)	$V_{f}$
Matrix E51	3.5	–	0.3	1.35	–	–
Glass fiber	73	–	0.2	30.42	–	–
Fiber tows-a	49.3	12.75	0.234	5.44	5.174	0.659
Fiber tows-b	41.3	9.59	0.246	4.01	3.858	0.543

Table 3.

The parameters of the architecture geometry of woven composites.

Composites	L_t (mm)	$L_{p}$ (mm)	h_y (mm)	$W_{t}$ (mm)	$W_{p}$ (mm)	w (mm)	θ (deg)
Woven-a	4.900	1.125	0.262	1.262	1.000	0.256	27.100
Woven-b	4.440	1.125	0.276	1.160	1.000	0.160	40.780

Bending stiffness

The experimental results of the bending stiffness of resin, unidirectional composites ( $0^{o}$ , $45^{o}$ , and $90^{o}$ ) and woven composites beams with the variation of the frequency are obtained by equation (27) and are shown in Figure 6. The four experimental frequencies for the first flexural mode of each material increase as the beam lengths decrease. It is seen that the bending moduli of all specimens increase very slightly (maximum about 3.2%) with the frequency within the frequency domain under consideration. Similar trend of the bending stiffness of composites is also found by Berthelot and Sefrani.²⁴

The analytical bending moduli for unidirectional composites and woven composites, calculated by equation (8) and equation (23), are listed in Table 4. Compared with the experimental results for a frequency of 100 Hz, a good agreement is observed.

Table 4.

Bending modulus of composites.

Bending modulus(GPa)	$0^{o}$	$45^{o}$	$90^{o}$	Woven-a	Woven-b
Experimental data	27.95	7.34	6.75	16.48	13.17
Analytical results	27.83	7.24	6.52	16.36	12.77

Damping of resin and unidirectional composites

The influence of frequency

The experimental loss factors of resin and unidirectional composites ( $0^{o}$ , $45^{o}$ , and $90^{o}$ ) in the first flexural mode are obtained by fitting the experimental responses of beam specimens with different lengths and are illustrated in Figure 7 as the function of frequency. The four experimental frequencies for the first flexural mode of each material increase as the beam lengths decrease. It is observed that the loss factors of resin and $0^{o}$ unidirectional composite stay almost unchanged, while the loss factors of the $45^{o}$ and $90^{o}$ unidirectional composites increase with the increase of frequency. The magnitude of the loss factor of the resin is the largest, since the fiber phase in unidirectional composites is a low damping material.

Damping of unidirectional composites

Since the dimensions of the specimens can significantly affect the damping of composites and the loss factors of unidirectional composites and resin used in this paper were all measured under free flexural condition by beam specimens, thus, all loss factors here are not material properties, but represent the flexural loss factors. Some analytical models^12,16,27 were developed to describe longitudinal damping ( $η_{1}$ ), transverse damping ( $η_{2}$ ), and in-plane shear damping ( $η_{12}$ ). If the viscoelastic matrix is only considered for the contributions to damping, the longitudinal damping ( $η_{1}$ ) can be given as¹⁶

η_{1} = \frac{E_{r}}{E_{1}} (1 - V_{f}) η_{r}

(28)

However, it is shown^12,16 that expression (28) underestimates the longitudinal damping considerably. Berthelot and Sefrani¹⁶ pointed out that the effect of the low stiffness fiber–matrix interface does not affect the longitudinal damping appreciably. Based on the experimental damping of unidirectional composites, the longitudinal damping can be calculated more accurately than equation (28) by

η_{1} = η_{1 f} \frac{E_{f}}{E_{1}} V_{f} + η_{r} \frac{E_{r}}{E_{1}} (1 - V_{f})

(29)

where

η_{1 f}

is the longitudinal damping of glass fiber in a viscoelastic matrix.

Analytical predictions by equation (29) are compared with the experimental results provided by Adams et al.⁴ and the present investigations, they are shown in Figure 8. It is seen that good agreements are observed, where $η_{1 f} = 0.0019$ .¹⁶

For transverse damping ( $η_{2}$ ), Chandra et al.²⁷ compared several analytical models based on elastic-viscoelastic correspondence principle. Experimental investigations on damping properties of unidirectional glass fiber-reinforced composite by Chandra et al.²⁸ were later carried out to measure six loss factors (three in normal and three in shear). The measured transverse damping agrees with theoretical results (Eshelby's method) well in Chandra et al.²⁷ for a given fiber volume fraction. However, experimental transverse damping with the fiber volume fraction was not further measured. Yim¹² developed a fitting model to describe the transverse damping by using curve ﬁtting with a parameter α. Since the parameter α is obtained only for ﬁber volume fraction of 0.65, Berthelot and Sefrani¹⁶ performed a better analysis to evaluate the transverse composite damping more reasonably. Following formulas are provided to calculate the transverse composite damping

η_{2} = η_{r 2} + η_{f 2}

(30)

where

η_{r 2} = η_{r} [\frac{E_{2}}{E_{r}} (1 - V_{f})^{3} + \frac{9}{4} \frac{E_{2}}{E_{f}} (1 - V_{f})^{2} V_{f} + \frac{3}{2} \frac{E_{2} E_{r}}{E_{f}^{2}} (1 - V_{f}) V_{f}^{2}]

(31)

η_{f 2} = η_{2 f} [\frac{E_{2}}{E_{f}} (V_{f})^{3} + \frac{3}{2} \frac{E_{2}}{E_{r}} (1 - V_{f}) V_{f}^{2} + \frac{3}{4} \frac{E_{2} E_{f}}{E_{r}^{2}} (1 - V_{f})^{2} V_{f}]

(32)

in which

η_{2 f}

is the transverse damping of fiber in a viscoelastic matrix. Predictions by the analytical model are compared with the experimental results given in Berthelot and Sefrani¹⁶ and this investigation, shown in Figure 9. It is seen that good agreements are obtained with

η_{2 f} = 0.00083

Few experimental results and analytical models on variations of in-plane shear damping ( $η_{12}$ ) with fiber fraction volume were reported in the literature. The loss factor $η_{12}$ given by Chandra et al.²⁷ is higher than the limited available experimental results.⁴ Here, an indirect method is employed to derive in-plane shear damping by identification of the experimental responses of unidirectional composite ( $45^{o}$ ) beams and the analytical expression of the flexural loss factor $η_{45^{o}}$ of them.²⁴ The specific equation of $η_{45^{o}}$ is shown as follow

η_{45^{o}} = E_{45^{o}} [\frac{η_{1}}{4 E_{1}} + \frac{η_{2}}{4 E_{2}} + \frac{η_{12}}{4 G_{12}} - (η_{1} + η_{2}) \frac{v_{12}}{4 E_{1}}]

(33)

where

η_{11}

and

η_{22}

are determined by equations (29) and (30).

E_{45^{o}}

represents Young’s modulus in longitudinal direction for unidirectional composite (

45^{o}

) and is given as

E_{45^{o}} = [\frac{1}{4 E_{1}} + \frac{1}{4 E_{2}} + (\frac{1}{4 G_{12}} - \frac{μ_{12}}{2 E_{1}})]^{- 1}

(34)

Then, the in-plane shear damping is written as

η_{12} = 4 G_{12} [\frac{η_{45^{o}}}{E_{45^{o}}} + (η_{1} + η_{2}) \frac{v_{12}}{4 E_{1}} - (\frac{η_{1}}{4 E_{1}} + \frac{η_{2}}{4 E_{2}})]

(35)

Damping of woven composites

The experimental damping of woven composites as function of the frequency is illustrated in Figure 10. The four experimental frequencies for the first flexural mode of woven composite increase as the beam lengths decrease. It is seen that the damping of woven composites grows slowly with the increase of frequency (about 1.95% for woven-a and 2.45% for woven-b).

To validate the proposed complex stiffness model, the calculated loss factors of resin and fiber tows in two woven composites at 100 Hz are listed in Table 5, where damping coefficient of resin (

η_{r}

) is the experimental result and damping coefficients of fiber tows are obtained by equations (29) to (35). Since the fiber volume fractions of fiber tows-a and fiber tows-b are 65.9% and 54.3% (Table 1),

45^{o}

unidirectional beam specimens for the fiber volume fractions approximately equal to 66% and 54% were tested, and the measured flexural loss factors

η_{45^{o}}

used in equation (35) are 0.0075 and 0.0097, respectively.

Table 5.

Damping coefficients of resin and fiber tows.

Material	$η_{r}$	$η_{1}$	$η_{2}$	$η_{12}$
Fiber tows-a	0.02	0.0023	0.0079	0.0078
Fiber tows-b	0.02	0.0026	0.0104	0.0099

For comparisons, the experimental damping of woven composites at 100 Hz is summarized in Table 6, it is seen that a good agreement is achieved.

Table 6.

Damping of woven composites.

Material	Experimental results	Analytical calculation
Woven-a	0.0155	0.0159
Woven-b	0.0181	0.0185

Conclusions

Based on elastic-viscoelastic correspondence principle, a complex stiffness method is established to predict the damping properties of woven composites. Comparisons to experimental data reveal that the proposed method works effectively. It is shown that the damping of two woven composites with frequency has similar trend and the increase is about 1.95% for woven-a and 2.45% for woven-b with the increase of the frequency. The bending stiffness of woven composites also increases slowly with the frequency and the tendency is similar to the one of laminated composites and resin matrix.

Although the strain energy method is successfully used in the damping analysis for laminated composites, it is difficult to be extended to woven composites due to the existence of undulate warp tows. Hence, the advantage of the complex stiffness method is the convenience of calculating the complex stiffness matrix so that the damping can be obtained directly. Although the method ignores the damping of Poisson’s ratio to simplify the calculation of complex stiffness matrix, it is demonstrated that this simplification affects very little on the damping predictions of woven composites.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper is partially supported by Fund of Jiangsu Innovation Program for Graduate Education (CXZZ13_0149) and the Fundamental Research Funds for the Central Universities, National Natural Science Foundation of China (11272147,10772078), Aviation Science Foundation (2013ZF52074), Fund of State Key Laboratory of Mechanical Structural Mechanics and Control (0214G02), and Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

References

Zhili

. Tensile behavior of 3D woven composites by using different fabric structures. Mater Des 2002; 23: 671–674.

Baucom

Zikry

. Evolution of failure mechanisms in 2D and 3D woven composite systems under quasi-static perforation. J Compos Mater 2003; 37: 1651–1674.

Adams

Bacon

. Measurement of the flexural damping capacity and dynamic Young’s modulus of metals and reinforced plastics. J Phys D 1973; 6: 27–27.

Adams

Fox

Flood

et al.

The dynamic properties of unidirectional carbon and glass fiber reinforced plastics in torsion and flexure. J Compos Mater 1969; 3: 594–603.

Maheri

Adams

. On the flexural vibration of Timoshenko beams, and the applicability of the analysis to a sandwich configuration. J Sound Vib 1998; 209: 419–442.

Alam

Asnani

. Vibration and damping analysis of fibre reinforced composite material plates. J Compos Mater 1986; 20: 2–18.

Rao

. Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. J Sound Vib 2003; 262: 457–474.

Rao

. Dynamic analysis and design of laminated composite beams with multiple damping layers. AIAA J 1993; 31: 736–745.

Chandra

Singh

Gupta

. Damping studies in fiber-reinforced composites – a review. Compo Struct 1999; 46: 41–51.

10.

Guan

Gibson

. Micromechanical modelsfor damping in wovenfabric-reinforced polymermatrix composites. J Compos Mater 2001; 35: 1417–1434.

11.

Adams

. The damping and dynamic moduli of symmetric laminated composite beams – theoretical and experimental results. J Compos Mater 1984; 18: 104–121.

12.

Yim

. A damping analysis of composite laminates using the closed form expression for the basic damping of Poisson’s ratio. Compos Struct 1999; 46: 405–411.

13.

Yim

Jang

. Prediction and experimental verification of damping in 3-D braided textile structural composites. Polym Compos 1999; 20: 664–674.

14.

Sun

Gibson

. Prediction of Material damping in randomly oriented short fiber polymer matrix composites. J Reinf Plast Compos 1985; 4: 262–272.

15.

Gibson

Chaturvedi

Sun

. Complex moduli of aligned discontinuous fibre-reinforced polymer composites. J Mater Sci 1982; 17: 3499–3509.

16.

Berthelot

J-M

Sefrani

. Longitudinal and transverse damping of unidirectional fibre composites. Compos Struct 2007; 79: 423–431.

17.

Scida

Aboura

Benzeggagh

Bocherens

. Prediction of the elastic behaviour of hybrid and non-hybrid woven composites. ComposSci Technol 1998; 57: 1727–1740.

18.

Adams

Bacon

DGC

. Effect of Fibre orientation and laminate geometry on the dynamic properties of CFRP. J Compos Mater 1973; 7: 402–428.

19.

Ishikawa

Chou

T-W

. Stiffness and strength behaviour of woven fabric composites. J Mater Sci 1982; 17: 3211–3220.

20.

Chou

T-W

Ishikawa

. One-dimensional micromechanical analysis of woven fabric composites. AIAAJ 1983; 21: 1714–1721.

21.

Naik

Ganesh

. Prediction of on-axes elastic properties of plain weave fabric composites. Compos Sci Technol 1992; 45: 135–152.

22.

Naik

Ganesh

. Failure behavior of plain weave fabric laminates under on-axis uniaxial tensile loading: I – analytical predictions. J Compos Mater 1996; 30: 1779–1822.

23.

Scida

Aboura

Benzeggagh

Bocherens

. A micromechanics model for 3D elasticity and failure of woven-fibre composite materials. Compos Sci Technol 1999; 59: 505–517.

24.

Berthelot

J-M

Sefrani

. Damping analysis of unidirectional glass and Kevlar fibre composites. Compos Sci Technol 2004; 64: 1261–1278.

25.

Saravanos

. Analysis of passive damping in thick composite structures. AIAA J 1993; 31: 1503–1510.

26.

Berthelot

J-M

. Composite materials: Mechanical behavior and structural analysis, New York: Springer, 1999.

27.

Chandra

Singh

Gupta

. Micromechanical damping models for fiber-reinforced composites: a comparative study. Compos Part A 2002; 33: 787–796.

28.

Chandra

Singh

Gupta

. Experimental evaluation of damping of fiber-reinforced composites. J Compos Technol Res 2003; 25: 96–107.