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A random waveness model for the stiffness and strength evaluation of 3D woven composites
Composite Structures xxx (2016) xxx–xxx
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
A random waveness model for the stiffness and strength evaluation of 3D woven composites Suyang Zhong a,b, Licheng Guo a,⇑, Gang Liu a, Li Zhang a, Shidong Pan c a
Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, PR China Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621999, PR China c Center for Composite and Structure, Harbin Institute of Technology, Harbin 150080, PR China b
a r t i c l e
i n f o
Article history: Received 19 December 2015 Revised 12 June 2016 Accepted 23 June 2016 Available online xxxx Keywords: 3D woven composites Stiffness and strength Random waviness model (RW model) Finite element analysis
a b s t r a c t Three-dimensional (3D) woven composites are typical anisotropic textile composites and their mechanical properties mainly depend on the distribution and the mechanical properties of the fiber yarns. An important phenomenon should be emphasized that the fiber yarns usually deviate from their designed positions during the manufacturing process, which results in the random waviness of the fiber yarns. In some cases, these waviness phenomena may affect the mechanical responses significantly. In this paper, a random waviness model (RW model) is proposed which consider the fiber yarns’ random waviness to investigate the stiffness and strength of 3D woven composites. Based on the actual statistics of the fiber yarns’ random waviness, the stiffness and strength of a type of 3D woven composites are predicted and the corresponding experiments are also carried out. The predicted results agree well with the experimental ones. The variations of the modulus and the strength with the random characteristics of the fiber yarns’ waviness are discussed. Results show that the random waviness of fiber yarns should be taken into consideration for actual 3D woven composites and the present method is very useful for the prediction of the mechanical properties of 3D woven composites. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In the preparation process of the fiber reinforced composites including 3D woven composites, the fibers or the fiber yarns deviate from their designed positions and become wavelike. In 1970s, Argon [1] observed that practical fiber composites have regions of fiber waviness and proposed an infinite band model to derive the compressive strength of unnotched fibrous material. Based on Argon’s model, Fleck et al. [2,3] proposed an infinite band analytical method, which included fiber bending, to estimate the compressive strength of the fiber composites. Chan and Chou [4] developed an analytical method capable of quantifying the axial and bending stiffness of laminates and the wavy plies are considered as the shape of sine wave. Hsiao and Daniel [5] developed an analytical model for predicting the elastic properties and compressive strength of the unidirectional composites with fiber waviness. These researches for the unidirectional fiber or laminated composites are the precedent of that of 3D woven composites. Karami and Garnich [6] used a FE micromechanical model to determine the
⇑ Corresponding author. E-mail address: [email protected] (L. Guo).
coefficients of thermal expansion for fibrous materials and the fibers were assumed to be sinusoidal. For 3D woven composites, Cox et al. [7–9] observed the waviness of the fiber yarns and proposed a binary model which can predict the influence on the stiffness by the waviness. Callus et al. [10] and Karahan et al. [11] observed and measured the waviness of the fiber yarns and found that all the shape of the waviness are similar. Scida et al. [12] proposed a MESOTEX (MEchanical Simulation Of TEXtile) model to predict the moduli, Poisson’s rations and shear moduli of the 3D woven composites considering the waviness of the fiber yarns. Siron and Lamon [13] investigated the mechanical behavior of a 3D carbon/carbon composites considering the factor of crimped fiber yarns. Mahadik et al. [14,15] observed and measured the waviness of the fiber yarns in 3D woven composites and predicted the compressive modulus and strength considering the waviness. Chun et al. [16] studied the effects of fiber waviness on tensile and compressive behaviors of composite materials and all fibers are assumed to have sinusoidal curvature along one spatial coordinate. Stig and Hallström [17] proposed the zig–zag, trapezoid and helix models to build the waviness of the fiber yarns, respectively. Mahadik et al. [14,15] conducted a thorough experimental analysis of the effect of yarn waviness on the mechanical properties and
http://dx.doi.org/10.1016/j.compstruct.2016.06.051 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
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compressed failure mechanisms of an angel-interlock fabric. Jiang et al. [18] presented a theoretical model considering the helix waviness of the fiber yarns in the 3D braided composites to predict the effective elastic constants and the failure strength. A continuum damage model [19] is proposed for evaluating the damage behavior of fibers and matrix in 3D woven materials. In the above review, the waviness form of the fiber yarns was frequently assumed to be definite sine curves. The random characteristics of the waviness were usually not considered in the published papers. Moreover, the published models usually only considered the waveness in very local scale (e.g., representative volume element). In the current study, we aim to build a model to predict the stiffness and strength of 3D woven composites, considering the random waviness of the fiber yarns in the specimen scale. The corresponding experiments are conducted. 2. The random waviness model (RW model) for the 3D woven composites In the binary model [7,9], the line element (a kind of user-defined element) was used to represent the fiber yarns and all other properties of the fiber yarns and the matrix are represented by the solid elements, respectively. Different from the binary model [7,9], beam elements, random waveness and damage laws are introduced in the RW model so that the strength and damage behavior can be simulated practically and efficiently. The RW model is proposed in this paper in which the fiber yarns and the matrix are represented by the Timoshenko beam element and solid element. In the RW model, the solid elements are not only used to fill the space of the resin matrix but also the space of the fiber yarns, which will make it convenient to build and mesh the model but result in duplicate calculation of the material stiffness of the matrix at the positions of the fiber yarns (Fig. 1). In order to solve this problem, the elastic constants of the Timoshenko beam elements are related to those of the fiber yarns and the matrix. Thus, the elastic constants for the beam element can be given by
8 Eb1 ¼ Ef 1 Em > > > < E ¼E ¼E E 3 m b2 f2 > G ¼ E =2ð1 m b23 b2 f 23 Þ > > : Gb12 ¼ Gb13 ¼ Gf 12 Gm
ð1Þ
8 < eb11 SEf 1 ¼ 1;
eb11 P 0
: e
eb11 < 0
f 1t
Ef 1 b11 S f 1c
ð2Þ
where the subscript t and c represent tension and compression,
eb11 is the longitudinal strain of the beam element, Sf 1c and Sf 1c
are the longitudinal tensile and compressive strengths of the fiber yarns. A continuum damage model is used for the matrix [19]. The stress–strain relationship for the matrix can be expressed as:
2 6 6 6 6 1 6 6 Sm ðdÞ ¼ 6 Em 6 6 6 6 4
1 1dm
mm
mm
1 1dm
mm
3 0
1 1dm 2ð1mm Þ 1dm 2ð1mm Þ 1dm
sym:
2ð1mm Þ 1dm
8 l mt < dmt ¼ 1 1d exp Amt ð1 r em Þ if I1 P 0 r em dm ¼ l : mc exp Amc ð1 r em Þ if I1 < 0 dmc ¼ 1 1d re
Fig. 1. Beam and solid element in random waviness model (RW model).
ð4Þ
where Amt and Amc are the softening law parameters which are l
l
described in Appendix A, dmt and dmc are the auxiliary damage variables, r em is the auxiliary damage threshold parameter, I1 is the first invariant of the stress tensor. The damage threshold parameter [19] can be given by
n o 8 > r m;L ¼ max 1; maxf/sm;L g ; > > < > > > :
/m;t ¼
3J 2 þI1 ðSm;c Sm;t Þ Sm;c Sm;t
/m;c ¼
3J þI ðS Sm;t Þ 2 S1m;cm;c Sm;t
L ¼ ft; cg; s 2 ½0; t
for I1 P 0
ð5Þ
for I1 < 0
where /m;t and /m;c are the loading functions, Sm;t and Sm;c are the tensile and compressive yield strengths of the matrix, J 2 are the second invariant of the corresponding deviatoric stress tensor. I1 and J2 can be written as:
~ m;11 þ r ~ m;22 þ r ~ m;33 I1 ¼ r h i 1 ~ m;11 r ~ m;22 Þ2 þ ðr ~ m;22 r ~ m;33 Þ2 þ ðr ~ m;33 r ~ m;11 Þ2 J 2 ¼ 6 ðr ð6Þ
There are 6 and 3 degrees of freedom in the beam and solid elements. The embedded element method is adopted to constrain this two types of elements. The node coincidence is not necessary in this method. The displacement of an arbitrary point in the solid element can be expressed as:
8 9 > ui > n < = X v Ni v i ¼ > : > : > ; i¼1 > ; w wi
b
ð3Þ
m
8 9 > =
a
7 7 7 7 7 7 7 7 7 7 7 5
where, dm is the damage variable of the matrix. It can be expressed by
(
where the subscript ‘1’ denotes the longitudinal direction of the fiber yarns, the subscript ‘2’ and ‘3’ denote the transverse directions, and the subscript ‘b’ represents the beams. Ef 1 , Ef 2 , mf 23 and Gf 12 are the elastic parameters of the fiber yarns. Em and Gm are the moduli of the matrix. The fiber breakage is brittle in the longitudinal direction of the fiber yarns under both tensile and compressive loads. The following maximum strain criterion is adopted for the fiber yarns:
¼ 1;
ð7Þ
where i is the point number, N i is the shape function, ðui ; v i ; wi Þ is the coordinate of the nodes in the solid element, n is the amount of nodes. Under the condition of small deformation, the rotation angle for an arbitrary volume differential in the solid element can be expressed as:
8 8 9 9 > < @w=@y @ v =@z > < hx > = = 1> @u=@z @w=@x hy ¼ > > : : > ; ; 2> @ v =@x @u=@y hz
ð8Þ
Substituting Eq. (7) into Eq. (8) gives
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
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8 9 2 38 9 0 @Ni =@z @Ni =@y > > < hx > = 1X < ui > = n 6 7 hy ¼ 0 @Ni =@x 5 v i 4 @N i =@z > > > 2 : ; : > ; i¼1 @Ni =@y @Ni =@x hz 0 wi
ð9Þ
Assume that the displacement of the node in the beam element is consistent with that of the corresponding volume differential. The displacement of an arbitrary node in the beam element can be expressed as:
8 9 2 Ni 0 0 ua > > > > > > 6 > > > > 0 N 0 v i a > > 6 > > > n 6 = X 6 0 0 N i a 6 ¼ 6 > 0 @Ni =2@z @Ni =@2y > hax > > i¼1 6 > > > > 6 > > > 4 @Ni =2@z 0 @Ni =2@x > hay > > > > : ; haz @Ni =2@y @N i =2@x 0
3 78 9 7 ui > 7> 7< = 7 vi 7> > 7: ; 7 wi 5 ð10Þ
Thus, the relationship of the nodes of the beam and solid elements can be built. The waviness of the fiber yarns in 3D woven composites is mainly in the thickness direction. The ‘‘amplitude” and ‘‘half-wavelength” should be random. So the amplitude and half-wavelength are assumed as random variables in the RW model (Fig. 2). The waviness form of the fiber yarns is assumed approximately to be sine curve. In the local coordinate system of the fiber yarns, the relationship of the abscissa and ordinate of an arbitrary point’s coordinate ðx1 ; x2 Þ in the fiber yarns can be expressed as:
x2 ¼
8 x1 > > > Ai sin p ki þ b ; 0 < x1 < ki ; i ¼ 1 > > 0 1 Xi1 > > i1 i > X X > kn x1 > n¼1 þ bA; < Ai sin @p kn < x1 < kn ; i ¼ 3;5;7... k i
n¼1 n¼1 > > 0 1 > Xi1 > > i1 i > X X kn x > > n¼1 þ p þ bA; @ 1 > A k kn ; i ¼ 2;4;6... n < x1 < i sin p > k i : n¼1
n¼1
ð11Þ where Ai is the amplitude of the ith half wave, ki is the corresponding half-wavelength, and b is the adjustment value. The random variables——amplitude and half-wavelength are the key to build the RW model for the 3D woven composites. These two random variables can be obtained by measuring the actual
fiber yarns in the 3D woven composites. Depending on the measurement results, the RW model can be built. The process is given as follows: (1) According to the pre-designment of 3D woven composites, establishing the ideal model without considering the waviness of the fiber yarns; (2) Cutting the specimen and measuring the amplitude and half-wavelength of the fiber yarns on the cross section of the specimen; (3) Generating the amplitude and half-wavelength data according to the statistical characters of the measurement results; (4) According to the amplitude and half-wavelength data, adjusting the coordinates of the beam elements of the fiber yarns in the ideal model; (5) Repeating the steps 3–4 and obtaining the models of more samples. 3. Quasi-static experiments of the typical 3D woven composites In order to study the stiffness and strength of 3D woven composites considering the random waviness of the fiber yarns, the tensile and compressive tests are conducted. The 3D layer-interlock woven composites are used here. The material constituents comprise 12 K T700 carbon fibers in the warp and weft yarns, 3K T300 carbon fibers in the binder yarns and TDE-86 resin in the matrix. They are shown in Table 1. The tested specimens were cut by water jet cutting system and the aluminum tabs were glued to the ends of the specimens (Fig. 3a). The pre-designed size of the specimens is shown in Table 2. The quasi-static uniaxial tensile and compressive experiments in the warp direction are carried out by referencing the testing standard ASTM D3039 on the Zwick/Roell universal testing machines (Fig. 3b and c). The extensometer and strainometer were used to measure the strain of the 3D woven composites. The experimental results are used to verify the prediction from the RW model in Section 4. 4. Establishment of the RW model The typical tested cross sections of the fiber yarns are shown in Fig. 4. Axis x and y are along the longitudinal direction of the warp and weft yarns, respectively. Axis z is along the thickness direction
(a)
(b)
+1
2
+1
1 Fig. 2. Schematic of the fiber yarn: (a) the actual (solid line) and pre-designed (dash line) fiber yarn, (b) the central line of the actual and pre-designed fiber yarns. (The direction 1 and 2 are the fiber longitudinal direction and thickness direction of the 3D woven composites).
Table 1 Properties of constituents of the 3D woven composites.
12K T700 carbon fiber 3K T300 carbon fiber TDE-86 resin
Ef 1 (GPa)
Ef 2 ¼ Ef 3 (GPa)
Gf 12 ¼ Gf 13 (GPa)
mf 12 ¼ mf 13
mf 23
St (GPa)
230 221 3.45
18.2 13.8
36.6 9.0
0.27 0.20 0.33
0.30 0.25
4.9 3.53 0.24
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Length
Width
(a)
Net length
Thickness of tab
Thickness
(b)
(c)
Fig. 3. Experiments of the 3D woven composites: (a) schematics of the specimen, (b) the uniaxial tensile experiments, (c) compressive experiment.
Table 2 Size of the specimens (mm). Item
Length
Width
Net length
Thickness
Thickness of tab
Tensile specimen Compressive specimen
250 140
18 15
110 20
5 5
1.6 2
x
Binder yarn
z
z
Fig. 5. Central lines of the specimen: (a) Central lines of the warp yarns in xz plane, (b) comparison between central lines of the actual structure (solid line) and the ideal pre-designed one (dash line).
x Warp yarn
z x
y
Weft yarn
Fig. 4. Cross-section of the 3D woven composites: (a) the binder yarns in xz plane, (b) the warp yarns in xz plane, (c) the weft yarns in yz plane.
of the 3D woven composites. All the three types of fiber yarns are wavy in the thickness direction in which the waviness in the warp yarns is more significantly than that in the weft yarns. And considering that the tensile and compressive experiments are conducted in the warp direction, the waviness of the warp yarns will be built in the RW model. The weft yarns are straight and the binder yarns are curved. They are considered as pre-designed. The volume fraction of the binder yarns is much less than that of the warp yarns. Since the woven plate is usually thin in thickness direction, the binder yarns are very short in thickness direction. Under loading in the plane, the load is not mainly carried by the binder yarns. Thus the random waviness of the binder yarns is not considered in the paper. The statistical process of the amplitude and half-wavelength of the warp yarns is shown as follows:
Probability density (%)
z
16
12
8
4
0 3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Half-wavelength λ (mm) Fig. 6. Sample statistics of the half-wavelengths k in the warp yarns.
i. Drawing the central lines of the actual and pre-designed yarns (Fig. 5) on the cross sections of the fiber yarns; ii. Measuring the amplitude and half-wavelength of each wavepond and obtaining their random characteristics.
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
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approached by the normal distribution. Namely, the probability density function of the amplitude A can be expressed as:
Probability density (%)
25
1 ðA lA Þ2 f ðAÞ ¼ pffiffiffiffiffiffiffi exp 2r2A 2prA
20
15
10
5
0 0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Amplitude A (mm) Fig. 7. Sample statistics of the amplitude A in the warp yarns.
ð13Þ
where lA is the mean and rA is the standard deviation of the amplitude A. The finite element model of the pre-designed 3D women composite without considering the waviness of the fiber yarns is shown in Fig. 8. The amplitude and half-wavelength are generated using Eqs. (12) and (13). Then, for the model considering the waveness of the yarns, the element nodes of the beam elements are adjusted according Eq. (11) and thus, the RW mode for the 3D woven composites can be established. The beam elements of the warp yarns in a typical RW mode are shown in Fig. 9. Consequently, the actual waviness of the warp yarns can be simulated by the RW model. 5. Result and discussion
The sample statistics results of the half-wavelengths k in the warp yarns are shown in Fig. 6. The mean of the half-wavelengths is 3.39 mm and the standard deviation is 0.056 mm. On the whole, the statistical results are approximate to the normal distribution. So we assume that the probability density function of the half-wavelength k can be expressed as:
1 ðk lk Þ2 f ðkÞ ¼ pffiffiffiffiffiffiffi exp 2r2k 2prk
!
!
ð12Þ
where lk is the mean and rk is the standard deviation of the half-wavelength k. The sample statistics results of the amplitude A in the warp yarns is shown in Fig. 7. The mean of the amplitude is 0.0899 mm and the standard deviation is 0.0065 mm. The statistical results can be
In order to verify the accuracy of the combined beam elements and solid elements in the RW model, the traditional model using only solid elements is utilized for comparison. It is assumed that a cylindrical T700 fiber yarn with radius 1 mm is packed in a cuboid TDE-86 resin (2 mm * 2 mm * 10 mm). The solid elements model and the present model with combined beam elements and solid elements are built and shown in Fig. 10. Three kinds of boundary conditions are exerted in both models. The boundary conditions can be described as: (1) Tension in the longitudinal direction of the yarn: wjz¼0 ¼ 0, wjz¼10 mm ¼ 0:1 mm; (2) Compression in the longutidinal direction of the yarn: wjz¼0 ¼ 0, wjz¼10 mm ¼ 0:1 mm;
a
b
c
d
Fig. 8. Finite element model of the pre-designed 3D women composite without considering the waviness of the fiber yarns: (a) the whole model, (b) the warp yarns, (c) the weft yarns, (d) the binder yarns.
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
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y = y1
y = y5
y = y2
y = y6
z
y = y3
y = y7
x
y = y8
y = y4
Fig. 9. Warp yarns in different cross sections (plane xz) generated by the RW model (y ¼ yi ; i ¼ 1; 2; . . . 8 denotes the different the cross section).
(a)
(b)
Fig. 10. Comparison of the solid elements model (a) and the present model with combined beam elements and solid elements (b).
Table 3 Reaction forces of the two models under three types of boundary conditions.
Tension Compression Shear load
Reaction force (N)
Error (%)
Traditional model
Model in this paper
7429 7429 65.66
7582 7582 66.59
(3) Shear load of the ujz¼10 mm ¼ 0:1 mm.
yarn:
Caculated by RW model 800
2.06 2.06 1.42
ujz¼0 ¼ v jz¼0 ¼ wjz¼0 ¼ 0,
Stress (MPa)
Boundary conditions
1000
600
400
Experimental data 200
Under the three displacement boundary conditions, the reaction forces on the loading surface are calculated and the results are shown in Table 3. The relative errors are 2.06%, 2.06% and 1.42% under tension, compression and shear load, respectively, which are completely acceptable for the 3D woven composites. On the basis of the statistical results of the amplitude and halfwavelength, five samples of the RW model are generated. Under tensile quasi-static load in the longitudinal direction of the warp yarn, the mean stress–strain curves of the five models are obtained and shown in Fig. 11. The experimental stress–strain curves are given in Fig. 11, too. The mean modulus and tensile strength calculated by the RW model is 60.2 GPa and 1015 MPa, respectively. The experimental moduli and strengths under tension are provided in
0 0.0
0.5
1.0
1.5
2.0
Strain (%) Fig. 11. Stress–strain curves of the 3D woven composites under tensile load compared with the experimental data. Table 4 Statistics of the tensile experimental results.
Modulus (GPa) Tensile strength (MPa)
No. t1
No. t2
No. t3
No. t4
No. t5
Mean
56.9 910.3
53.28 871.4
55.91 890.9
54.44 903.2
55.19 885.8
55.1 892
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
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2000
Reaction force (N)
1500
Warp yarns
1000
500
0
Binder yarns
-500
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Strain (%) Fig. 12. Reaction force-strain curves of the warp and binder yarns on the tensile loading surface of the 3D woven composites.
7
Table 4. Comparing the experimental and the predicted results, the error of the moduli is +9.3% and that of the tensile strength is +13.8%. It should be mentioned that for 3D woven composites, the dispersion range of the strength and the modulus may be big, which is limited by the level of the processing procedures, especially for the strength. So the errors of the predicted moduli and strength are reasonable. The reaction forces vs strain curves of the warp and binder yarns on the loading surface are shown in Fig. 12. From Fig. 12, it can be found that the reaction forces of the warp yarns are much larger than that of the binder yarns. It is due to the reason that the cross section of the warp yarns is larger than that of the binder yarns and the binder yarns are S-shaped curve. The reaction forces of the fiber yarns drop at the breaking strain between 1.65% and 1.75%, which implies that the fiber yarns do not break simultaneously. The MISES stress distribution patterns of the warp yarns on two typical samples’ models generated by the RW model using the same experimental random characteristics are shown in Fig. 13. It can be found that the stress distribution patterns of two models
Fig. 13. MISES stress distribution of the warp yarns in two typical samples.
Fig. 14. Damage distribution in the matrix of the 3D woven composites in the typical examples.
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
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1200 600
1000
Caculated by RW model 800
400
Stress (MPa)
Stress (MPa)
500
A=0 A=0.06 mm A=0.09 mm A=0.12 mm
300
200
600
400
Experimental data 100
200
0 0.0
0.2
0.4
0.6
0.8
0
1.0
0.0
Strain (%)
0.5
1.0
1.5
2.0
Strain (%) Fig. 15. Stress–strain curves of the 3D woven composites under compressive load compared with the experimental data.
Table 5 Statistics of the compressive experimental results.
Modulus (GPa) Compressive strength (MPa)
No. c1
No. c2
No. c3
No. c4
Mean
56.05 464
54.11 422.7
55.38 540
55.77 538.6
55.3 491
Fig. 17. Influence on the stress–strain curves by the amplitude of the warp yarns.
Table 6 Influence on the mechanical properties of the 3D woven composites by the amplitude of the warp yarns. (mm) Mean of the amplitude A
0.12
0.0899
0.06
0
Modulus E (GPa) Tensile strength S (MPa)
59.8 949
60.3 1000
60.5 1055
60.8 1115
is equal to 0 that means there is no waviness in the fiber Note: The amplitude A yarns.
1200
1200
1000
Warp yarns 600
λ= λ =2.55mm λ =3.39mm λ =4.25mm
800
Stress (MPa)
Reaction force (N)
900
300
0
-300 0.0
Binder yarns
0.2
0.4
0.6
0.8
1.0
600
400
200
Strain (%) Fig. 16. Reaction force–strain curves of the warp and binder yarns on the compressive loading surface of the 3D woven composites.
are different. The stress distribution of some yarns has reduced to a low level losing the bearing capacity. The damage accumulation of the two model in the matrix is shown in Fig. 14. By comparing Fig. 13 with Fig.14, it can be found that the regions where the damage has accumulated seriously correspond to those where the stress distribution of the yarns has reduced to a low level. Under compressive quasi-static load in the longitudinal direction of the warp yarns, the mean stress–strain curves of the five models generated by the RW model are obtained and compared with the experimental results, as shown in Fig. 15. The mean modulus and compressive strength, which is calculated by the RW model, is 60.3 GPa and 553 MPa, respectively. The experimental moduli and strengths under compression are shown in Table 5, the relative errors are +9.0% and +12.6%, respectively. As mentioned above, the errors are reasonable for 3D woven composites. The reaction forces vs strain curves of the warp and binder
0 0.0
0.5
1.0
1.5
2.0
Strain (%) Fig. 18. Influence on the stress–strain curves of the 3D woven composites by the half-wavelength k of the warp yarns.
yarns on the loading surface are shown in Fig. 16. The fiber yarns do not break simultaneously, which is similar to that under tensile load. The breaking strain is between 0.9% and 1.0%. In order to study the influences of the waviness degree of the fiber yarns on the mechanical properties, the means of the random variables (amplitude and half-wavelength) are changed and then the modulus and the tensile strength are calculated. The influence of the amplitude of the warp yarns on the stress–strain curves under tensile loading is shown in Fig. 17 and Table 6. It is found that the reduction of the tensile strength is obvious and that of the moduli is unobvious. The influence of the half-wavelength of the warp yarns on the stress–strain curves is shown in the Fig. 18 and Table 7. It is found that the reduction of the tensile
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S. Zhong et al. / Composite Structures xxx (2016) xxx–xxx Table 7 Influence on the mechanical properties of the 3D woven composites by the half-wavelength k of the warp yarns. Mean of the half-wavelength k (mm)
2.55
3.39
4.25
1
Modulus E (GPa) Tensile strength S (MPa)
60.1 964
60.3 1000
60.4 1041
60.8 1115
Note: The half-wavelength k is equal to 1 that means there is no waviness in the fiber yarns.
strength is obvious and that of the moduli is unobvious, too. In general, the influence on the tensile strength is greater than that on the modulus by the degree of the fiber yarns’ waviness. 6. Conclusion i. A RW model for 3D woven composites is proposed, in which the strength and damage of the 3D woven composites considering the fiber yarns random waviness can be studied with high efficiency. ii. The amplitude and half-wavelength of the fiber yarns’ waviness are assumed to be random variables in the RW model. The experimental statistical data of the two random variables are tested for building the RW model. iii. The moduli, tensile and compressive strengths of a typical 3D woven composites with fiber yarns’ random waviness are investigated via the RW model and they agree well with the experimental results. The influence of the waviness degree of the fiber yarns on the strength is greater than that on the modulus. It indicates that the random waviness of fiber yarns should be taken into consideration for actual 3D woven composites and the present method is very useful for the prediction of the mechanical properties of 3D woven composites.
Acknowledgement This work is sponsored by NSFC (11322217, 11432005). Appendix A The softening law parameters Amt and Amc are obtained by regularizing the softening branch of the stress–strain curve [19]. On the basis on the Bazant’s crack band theory [20], the energy dissipated per unit volume g mt and g mc can be determined by the fracture toughness Gmt and Gmc :
g mt ¼
Gmt ; l
g mc ¼
Gmc l
ðA1Þ
where l is the characteristic length of the finite element. The energy dissipated per unit volume g mt and g mc can be obtained from the integration of the damage energy dissipation during the process of damage development:
Z 1 Z 1 8 e @Gm _ @Gm @dmt > > d g ¼ dt ¼ drm mt < mt @dmt @dmt @r em 0 1 Z 1 Z 1 > e @Gm _ @Gm @dmc > : g mc ¼ d dt ¼ dr m mc @dmc @dmc @r e 0
1
m
where the Helmholtz free energy Gm can be written as:
ðA2Þ
Gm ¼
9
r2m;11 þ r2m;22 þ r2m;33 tm
ðrm;11 rm;22 þ rm;22 rm;33 þ rm;33 rm;11 Þ 2ð1 dm ÞEm Em 2 2 s sm;23 þ sm;31 þ ð1 dm ÞEm =ð1 tm Þ ðA3Þ 2 m;12 þ
where Em and tm are the modulus and Poisson’s ratio, rm;11 , rm;22 , rm;33 , sm;12 , sm;23 and sm;31 are the stresses of the matrix. The softening law parameters Amt and Amc can be determined by Eqs. (A1)–(A3) and Eq. (4). Finally, the variables Amt and Amc are in the integrals which can be calculated by the corresponding iterative algorithm. For the present problem, the parameters Amt and Amc will only have effects on the softening process of matrix, and the woven material is almost brittle fracture under both compression and tension. Therefore, the strength of the matrix has much more effects than the softening parameters Amt and Amc on the woven materials in the present model. If the experiments are not performed for fracture toughness Gmt and Gmc , the parameter Amt and Amc may be approximately obtained from the experimental tensile and compressive strain–stress curves. References [1] Argon AS. Fracture of composites. Amsterdam: Elsevier; 1972. [2] Fleck NA, Deng L, Budiansky B. Prediction of kink width in compressed fiber composites. J Appl Mech Trans ASME 1995;62:329–37. [3] Fleck NA, Shu JY. Microbuckle initiation in fibre composites: a finite element study. J Mech Phys Solids 1995;43:1887–918. [4] Chan WS, Chou CJ. Effects of delamination and ply fiber waviness on effective axial and bending stiffnesses in composite laminates. Compos Struct 1995;30:299–306. [5] Hsiao HM, Daniel IM. Effect of fiber waviness on stiffness and strength reduction of unidirectional composites under compressive loading. Compos Sci Technol 1996;56:581–93. [6] Karami G, Garnich M. Micromechanical study of thermoelastic behavior of composites with periodic fiber waviness. Compos B Eng 2005;36:241–8. [7] Xu J, Cox BN, McGlockton MA, Carter WC. A binary model of textile composites. 2. The elastic regime. Acta Metall Mater 1995;43:3511–24. [8] Cox BN, Dadkhah MS. The macroscopic elasticity of 3D woven composites. J Compos Mater 1995;29:785–819. [9] Cox BN, Carter WC, Fleck NA. A binary model of textile composites. 1. Formulation. Acta Metall Mater 1994;42:3463–79. [10] Callus PJ, Mouritz AP, Bannister MK, Leong KH. Tensile properties and failure mechanisms of 3D woven GRP composites. Compos A Appl Sci Manuf 1999;30:1277–87. [11] Karahan M, Lomov SV, Bogdanovich AE, Mungalov D, Verpoest I. Internal geometry evaluation of non-crimp 3D orthogonal woven carbon fabric composite. Compos A Appl Sci Manuf 2010;41:1301–11. [12] Scida D, Aboura Z, Benzeggagh ML, Bocherens E. A micromechanics model for 3D elasticity and failure of woven-fibre composite materials. Compos Sci Technol 1999;59:505–17. [13] Siron O, Lamon J. Damage and failure mechanisms of a3-directional carbon/carbon composite under uniaxial tensile and shear loads. Acta Mater 1998;46:6631–43. [14] Mahadik Y, Brown KAR, Hallett SR. Characterisation of 3D woven composite internal architecture and effect of compaction. Compos A Appl Sci Manuf 2010;41:872–80. [15] Mahadik Y, Hallett SR. Effect of fabric compaction and yarn waviness on 3D woven composite compressive properties. Compos Part A Appl Sci Manuf 2011;42:1592–600. [16] Chun H-J, Shin J-Y, Daniel IM. Effects of material and geometric nonlinearities on the tensile and compressive behavior of composite materials with fiber waviness. Compos Sci Technol 2001;61:125–34. [17] Stig F, Hallström S. Influence of crimp on 3D-woven fibre reinforced composites. Compos Struct 2013;95:114–22. [18] Jiang L, Zeng T, Yan S, Fang D. Theoretical prediction on the mechanical properties of 3D braided composites using a helix geometry model. Compos Struct 2013;100:511–6. [19] Zhong S, Guo L, Liu G, Lu H, Zeng T. A continuum damage model for threedimensional woven composites and finite element implementation. Compos Struct 2015;128:1–9. [20] Bazˇant Z, Oh BH. Crack band theory for fracture of concrete. Matér Constr 1983;16:155–77.
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Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
A random waveness model for the stiffness and strength evaluation of 3D woven composites Suyang Zhong a,b, Licheng Guo a,⇑, Gang Liu a, Li Zhang a, Shidong Pan c a
Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, PR China Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621999, PR China c Center for Composite and Structure, Harbin Institute of Technology, Harbin 150080, PR China b
a r t i c l e
i n f o
Article history: Received 19 December 2015 Revised 12 June 2016 Accepted 23 June 2016 Available online xxxx Keywords: 3D woven composites Stiffness and strength Random waviness model (RW model) Finite element analysis
a b s t r a c t Three-dimensional (3D) woven composites are typical anisotropic textile composites and their mechanical properties mainly depend on the distribution and the mechanical properties of the fiber yarns. An important phenomenon should be emphasized that the fiber yarns usually deviate from their designed positions during the manufacturing process, which results in the random waviness of the fiber yarns. In some cases, these waviness phenomena may affect the mechanical responses significantly. In this paper, a random waviness model (RW model) is proposed which consider the fiber yarns’ random waviness to investigate the stiffness and strength of 3D woven composites. Based on the actual statistics of the fiber yarns’ random waviness, the stiffness and strength of a type of 3D woven composites are predicted and the corresponding experiments are also carried out. The predicted results agree well with the experimental ones. The variations of the modulus and the strength with the random characteristics of the fiber yarns’ waviness are discussed. Results show that the random waviness of fiber yarns should be taken into consideration for actual 3D woven composites and the present method is very useful for the prediction of the mechanical properties of 3D woven composites. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In the preparation process of the fiber reinforced composites including 3D woven composites, the fibers or the fiber yarns deviate from their designed positions and become wavelike. In 1970s, Argon [1] observed that practical fiber composites have regions of fiber waviness and proposed an infinite band model to derive the compressive strength of unnotched fibrous material. Based on Argon’s model, Fleck et al. [2,3] proposed an infinite band analytical method, which included fiber bending, to estimate the compressive strength of the fiber composites. Chan and Chou [4] developed an analytical method capable of quantifying the axial and bending stiffness of laminates and the wavy plies are considered as the shape of sine wave. Hsiao and Daniel [5] developed an analytical model for predicting the elastic properties and compressive strength of the unidirectional composites with fiber waviness. These researches for the unidirectional fiber or laminated composites are the precedent of that of 3D woven composites. Karami and Garnich [6] used a FE micromechanical model to determine the
⇑ Corresponding author. E-mail address: [email protected] (L. Guo).
coefficients of thermal expansion for fibrous materials and the fibers were assumed to be sinusoidal. For 3D woven composites, Cox et al. [7–9] observed the waviness of the fiber yarns and proposed a binary model which can predict the influence on the stiffness by the waviness. Callus et al. [10] and Karahan et al. [11] observed and measured the waviness of the fiber yarns and found that all the shape of the waviness are similar. Scida et al. [12] proposed a MESOTEX (MEchanical Simulation Of TEXtile) model to predict the moduli, Poisson’s rations and shear moduli of the 3D woven composites considering the waviness of the fiber yarns. Siron and Lamon [13] investigated the mechanical behavior of a 3D carbon/carbon composites considering the factor of crimped fiber yarns. Mahadik et al. [14,15] observed and measured the waviness of the fiber yarns in 3D woven composites and predicted the compressive modulus and strength considering the waviness. Chun et al. [16] studied the effects of fiber waviness on tensile and compressive behaviors of composite materials and all fibers are assumed to have sinusoidal curvature along one spatial coordinate. Stig and Hallström [17] proposed the zig–zag, trapezoid and helix models to build the waviness of the fiber yarns, respectively. Mahadik et al. [14,15] conducted a thorough experimental analysis of the effect of yarn waviness on the mechanical properties and
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S. Zhong et al. / Composite Structures xxx (2016) xxx–xxx
compressed failure mechanisms of an angel-interlock fabric. Jiang et al. [18] presented a theoretical model considering the helix waviness of the fiber yarns in the 3D braided composites to predict the effective elastic constants and the failure strength. A continuum damage model [19] is proposed for evaluating the damage behavior of fibers and matrix in 3D woven materials. In the above review, the waviness form of the fiber yarns was frequently assumed to be definite sine curves. The random characteristics of the waviness were usually not considered in the published papers. Moreover, the published models usually only considered the waveness in very local scale (e.g., representative volume element). In the current study, we aim to build a model to predict the stiffness and strength of 3D woven composites, considering the random waviness of the fiber yarns in the specimen scale. The corresponding experiments are conducted. 2. The random waviness model (RW model) for the 3D woven composites In the binary model [7,9], the line element (a kind of user-defined element) was used to represent the fiber yarns and all other properties of the fiber yarns and the matrix are represented by the solid elements, respectively. Different from the binary model [7,9], beam elements, random waveness and damage laws are introduced in the RW model so that the strength and damage behavior can be simulated practically and efficiently. The RW model is proposed in this paper in which the fiber yarns and the matrix are represented by the Timoshenko beam element and solid element. In the RW model, the solid elements are not only used to fill the space of the resin matrix but also the space of the fiber yarns, which will make it convenient to build and mesh the model but result in duplicate calculation of the material stiffness of the matrix at the positions of the fiber yarns (Fig. 1). In order to solve this problem, the elastic constants of the Timoshenko beam elements are related to those of the fiber yarns and the matrix. Thus, the elastic constants for the beam element can be given by
8 Eb1 ¼ Ef 1 Em > > > < E ¼E ¼E E 3 m b2 f2 > G ¼ E =2ð1 m b23 b2 f 23 Þ > > : Gb12 ¼ Gb13 ¼ Gf 12 Gm
ð1Þ
8 < eb11 SEf 1 ¼ 1;
eb11 P 0
: e
eb11 < 0
f 1t
Ef 1 b11 S f 1c
ð2Þ
where the subscript t and c represent tension and compression,
eb11 is the longitudinal strain of the beam element, Sf 1c and Sf 1c
are the longitudinal tensile and compressive strengths of the fiber yarns. A continuum damage model is used for the matrix [19]. The stress–strain relationship for the matrix can be expressed as:
2 6 6 6 6 1 6 6 Sm ðdÞ ¼ 6 Em 6 6 6 6 4
1 1dm
mm
mm
1 1dm
mm
3 0
1 1dm 2ð1mm Þ 1dm 2ð1mm Þ 1dm
sym:
2ð1mm Þ 1dm
8 l mt < dmt ¼ 1 1d exp Amt ð1 r em Þ if I1 P 0 r em dm ¼ l : mc exp Amc ð1 r em Þ if I1 < 0 dmc ¼ 1 1d re
Fig. 1. Beam and solid element in random waviness model (RW model).
ð4Þ
where Amt and Amc are the softening law parameters which are l
l
described in Appendix A, dmt and dmc are the auxiliary damage variables, r em is the auxiliary damage threshold parameter, I1 is the first invariant of the stress tensor. The damage threshold parameter [19] can be given by
n o 8 > r m;L ¼ max 1; maxf/sm;L g ; > > < > > > :
/m;t ¼
3J 2 þI1 ðSm;c Sm;t Þ Sm;c Sm;t
/m;c ¼
3J þI ðS Sm;t Þ 2 S1m;cm;c Sm;t
L ¼ ft; cg; s 2 ½0; t
for I1 P 0
ð5Þ
for I1 < 0
where /m;t and /m;c are the loading functions, Sm;t and Sm;c are the tensile and compressive yield strengths of the matrix, J 2 are the second invariant of the corresponding deviatoric stress tensor. I1 and J2 can be written as:
~ m;11 þ r ~ m;22 þ r ~ m;33 I1 ¼ r h i 1 ~ m;11 r ~ m;22 Þ2 þ ðr ~ m;22 r ~ m;33 Þ2 þ ðr ~ m;33 r ~ m;11 Þ2 J 2 ¼ 6 ðr ð6Þ
There are 6 and 3 degrees of freedom in the beam and solid elements. The embedded element method is adopted to constrain this two types of elements. The node coincidence is not necessary in this method. The displacement of an arbitrary point in the solid element can be expressed as:
8 9 > ui > n < = X v Ni v i ¼ > : > : > ; i¼1 > ; w wi
b
ð3Þ
m
8 9 > =
a
7 7 7 7 7 7 7 7 7 7 7 5
where, dm is the damage variable of the matrix. It can be expressed by
(
where the subscript ‘1’ denotes the longitudinal direction of the fiber yarns, the subscript ‘2’ and ‘3’ denote the transverse directions, and the subscript ‘b’ represents the beams. Ef 1 , Ef 2 , mf 23 and Gf 12 are the elastic parameters of the fiber yarns. Em and Gm are the moduli of the matrix. The fiber breakage is brittle in the longitudinal direction of the fiber yarns under both tensile and compressive loads. The following maximum strain criterion is adopted for the fiber yarns:
¼ 1;
ð7Þ
where i is the point number, N i is the shape function, ðui ; v i ; wi Þ is the coordinate of the nodes in the solid element, n is the amount of nodes. Under the condition of small deformation, the rotation angle for an arbitrary volume differential in the solid element can be expressed as:
8 8 9 9 > < @w=@y @ v =@z > < hx > = = 1> @u=@z @w=@x hy ¼ > > : : > ; ; 2> @ v =@x @u=@y hz
ð8Þ
Substituting Eq. (7) into Eq. (8) gives
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8 9 2 38 9 0 @Ni =@z @Ni =@y > > < hx > = 1X < ui > = n 6 7 hy ¼ 0 @Ni =@x 5 v i 4 @N i =@z > > > 2 : ; : > ; i¼1 @Ni =@y @Ni =@x hz 0 wi
ð9Þ
Assume that the displacement of the node in the beam element is consistent with that of the corresponding volume differential. The displacement of an arbitrary node in the beam element can be expressed as:
8 9 2 Ni 0 0 ua > > > > > > 6 > > > > 0 N 0 v i a > > 6 > > > n 6
3 78 9 7 ui > 7> 7< = 7 vi 7> > 7: ; 7 wi 5 ð10Þ
Thus, the relationship of the nodes of the beam and solid elements can be built. The waviness of the fiber yarns in 3D woven composites is mainly in the thickness direction. The ‘‘amplitude” and ‘‘half-wavelength” should be random. So the amplitude and half-wavelength are assumed as random variables in the RW model (Fig. 2). The waviness form of the fiber yarns is assumed approximately to be sine curve. In the local coordinate system of the fiber yarns, the relationship of the abscissa and ordinate of an arbitrary point’s coordinate ðx1 ; x2 Þ in the fiber yarns can be expressed as:
x2 ¼
8 x1 > > > Ai sin p ki þ b ; 0 < x1 < ki ; i ¼ 1 > > 0 1 Xi1 > > i1 i > X X > kn x1 > n¼1 þ bA; < Ai sin @p kn < x1 < kn ; i ¼ 3;5;7... k i
n¼1 n¼1 > > 0 1 > Xi1 > > i1 i > X X kn x > > n¼1 þ p þ bA; @ 1 > A k kn ; i ¼ 2;4;6... n < x1 < i sin p > k i : n¼1
n¼1
ð11Þ where Ai is the amplitude of the ith half wave, ki is the corresponding half-wavelength, and b is the adjustment value. The random variables——amplitude and half-wavelength are the key to build the RW model for the 3D woven composites. These two random variables can be obtained by measuring the actual
fiber yarns in the 3D woven composites. Depending on the measurement results, the RW model can be built. The process is given as follows: (1) According to the pre-designment of 3D woven composites, establishing the ideal model without considering the waviness of the fiber yarns; (2) Cutting the specimen and measuring the amplitude and half-wavelength of the fiber yarns on the cross section of the specimen; (3) Generating the amplitude and half-wavelength data according to the statistical characters of the measurement results; (4) According to the amplitude and half-wavelength data, adjusting the coordinates of the beam elements of the fiber yarns in the ideal model; (5) Repeating the steps 3–4 and obtaining the models of more samples. 3. Quasi-static experiments of the typical 3D woven composites In order to study the stiffness and strength of 3D woven composites considering the random waviness of the fiber yarns, the tensile and compressive tests are conducted. The 3D layer-interlock woven composites are used here. The material constituents comprise 12 K T700 carbon fibers in the warp and weft yarns, 3K T300 carbon fibers in the binder yarns and TDE-86 resin in the matrix. They are shown in Table 1. The tested specimens were cut by water jet cutting system and the aluminum tabs were glued to the ends of the specimens (Fig. 3a). The pre-designed size of the specimens is shown in Table 2. The quasi-static uniaxial tensile and compressive experiments in the warp direction are carried out by referencing the testing standard ASTM D3039 on the Zwick/Roell universal testing machines (Fig. 3b and c). The extensometer and strainometer were used to measure the strain of the 3D woven composites. The experimental results are used to verify the prediction from the RW model in Section 4. 4. Establishment of the RW model The typical tested cross sections of the fiber yarns are shown in Fig. 4. Axis x and y are along the longitudinal direction of the warp and weft yarns, respectively. Axis z is along the thickness direction
(a)
(b)
+1
2
+1
1 Fig. 2. Schematic of the fiber yarn: (a) the actual (solid line) and pre-designed (dash line) fiber yarn, (b) the central line of the actual and pre-designed fiber yarns. (The direction 1 and 2 are the fiber longitudinal direction and thickness direction of the 3D woven composites).
Table 1 Properties of constituents of the 3D woven composites.
12K T700 carbon fiber 3K T300 carbon fiber TDE-86 resin
Ef 1 (GPa)
Ef 2 ¼ Ef 3 (GPa)
Gf 12 ¼ Gf 13 (GPa)
mf 12 ¼ mf 13
mf 23
St (GPa)
230 221 3.45
18.2 13.8
36.6 9.0
0.27 0.20 0.33
0.30 0.25
4.9 3.53 0.24
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Length
Width
(a)
Net length
Thickness of tab
Thickness
(b)
(c)
Fig. 3. Experiments of the 3D woven composites: (a) schematics of the specimen, (b) the uniaxial tensile experiments, (c) compressive experiment.
Table 2 Size of the specimens (mm). Item
Length
Width
Net length
Thickness
Thickness of tab
Tensile specimen Compressive specimen
250 140
18 15
110 20
5 5
1.6 2
x
Binder yarn
z
z
Fig. 5. Central lines of the specimen: (a) Central lines of the warp yarns in xz plane, (b) comparison between central lines of the actual structure (solid line) and the ideal pre-designed one (dash line).
x Warp yarn
z x
y
Weft yarn
Fig. 4. Cross-section of the 3D woven composites: (a) the binder yarns in xz plane, (b) the warp yarns in xz plane, (c) the weft yarns in yz plane.
of the 3D woven composites. All the three types of fiber yarns are wavy in the thickness direction in which the waviness in the warp yarns is more significantly than that in the weft yarns. And considering that the tensile and compressive experiments are conducted in the warp direction, the waviness of the warp yarns will be built in the RW model. The weft yarns are straight and the binder yarns are curved. They are considered as pre-designed. The volume fraction of the binder yarns is much less than that of the warp yarns. Since the woven plate is usually thin in thickness direction, the binder yarns are very short in thickness direction. Under loading in the plane, the load is not mainly carried by the binder yarns. Thus the random waviness of the binder yarns is not considered in the paper. The statistical process of the amplitude and half-wavelength of the warp yarns is shown as follows:
Probability density (%)
z
16
12
8
4
0 3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Half-wavelength λ (mm) Fig. 6. Sample statistics of the half-wavelengths k in the warp yarns.
i. Drawing the central lines of the actual and pre-designed yarns (Fig. 5) on the cross sections of the fiber yarns; ii. Measuring the amplitude and half-wavelength of each wavepond and obtaining their random characteristics.
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S. Zhong et al. / Composite Structures xxx (2016) xxx–xxx
approached by the normal distribution. Namely, the probability density function of the amplitude A can be expressed as:
Probability density (%)
25
1 ðA lA Þ2 f ðAÞ ¼ pffiffiffiffiffiffiffi exp 2r2A 2prA
20
15
10
5
0 0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Amplitude A (mm) Fig. 7. Sample statistics of the amplitude A in the warp yarns.
ð13Þ
where lA is the mean and rA is the standard deviation of the amplitude A. The finite element model of the pre-designed 3D women composite without considering the waviness of the fiber yarns is shown in Fig. 8. The amplitude and half-wavelength are generated using Eqs. (12) and (13). Then, for the model considering the waveness of the yarns, the element nodes of the beam elements are adjusted according Eq. (11) and thus, the RW mode for the 3D woven composites can be established. The beam elements of the warp yarns in a typical RW mode are shown in Fig. 9. Consequently, the actual waviness of the warp yarns can be simulated by the RW model. 5. Result and discussion
The sample statistics results of the half-wavelengths k in the warp yarns are shown in Fig. 6. The mean of the half-wavelengths is 3.39 mm and the standard deviation is 0.056 mm. On the whole, the statistical results are approximate to the normal distribution. So we assume that the probability density function of the half-wavelength k can be expressed as:
1 ðk lk Þ2 f ðkÞ ¼ pffiffiffiffiffiffiffi exp 2r2k 2prk
!
!
ð12Þ
where lk is the mean and rk is the standard deviation of the half-wavelength k. The sample statistics results of the amplitude A in the warp yarns is shown in Fig. 7. The mean of the amplitude is 0.0899 mm and the standard deviation is 0.0065 mm. The statistical results can be
In order to verify the accuracy of the combined beam elements and solid elements in the RW model, the traditional model using only solid elements is utilized for comparison. It is assumed that a cylindrical T700 fiber yarn with radius 1 mm is packed in a cuboid TDE-86 resin (2 mm * 2 mm * 10 mm). The solid elements model and the present model with combined beam elements and solid elements are built and shown in Fig. 10. Three kinds of boundary conditions are exerted in both models. The boundary conditions can be described as: (1) Tension in the longitudinal direction of the yarn: wjz¼0 ¼ 0, wjz¼10 mm ¼ 0:1 mm; (2) Compression in the longutidinal direction of the yarn: wjz¼0 ¼ 0, wjz¼10 mm ¼ 0:1 mm;
a
b
c
d
Fig. 8. Finite element model of the pre-designed 3D women composite without considering the waviness of the fiber yarns: (a) the whole model, (b) the warp yarns, (c) the weft yarns, (d) the binder yarns.
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y = y1
y = y5
y = y2
y = y6
z
y = y3
y = y7
x
y = y8
y = y4
Fig. 9. Warp yarns in different cross sections (plane xz) generated by the RW model (y ¼ yi ; i ¼ 1; 2; . . . 8 denotes the different the cross section).
(a)
(b)
Fig. 10. Comparison of the solid elements model (a) and the present model with combined beam elements and solid elements (b).
Table 3 Reaction forces of the two models under three types of boundary conditions.
Tension Compression Shear load
Reaction force (N)
Error (%)
Traditional model
Model in this paper
7429 7429 65.66
7582 7582 66.59
(3) Shear load of the ujz¼10 mm ¼ 0:1 mm.
yarn:
Caculated by RW model 800
2.06 2.06 1.42
ujz¼0 ¼ v jz¼0 ¼ wjz¼0 ¼ 0,
Stress (MPa)
Boundary conditions
1000
600
400
Experimental data 200
Under the three displacement boundary conditions, the reaction forces on the loading surface are calculated and the results are shown in Table 3. The relative errors are 2.06%, 2.06% and 1.42% under tension, compression and shear load, respectively, which are completely acceptable for the 3D woven composites. On the basis of the statistical results of the amplitude and halfwavelength, five samples of the RW model are generated. Under tensile quasi-static load in the longitudinal direction of the warp yarn, the mean stress–strain curves of the five models are obtained and shown in Fig. 11. The experimental stress–strain curves are given in Fig. 11, too. The mean modulus and tensile strength calculated by the RW model is 60.2 GPa and 1015 MPa, respectively. The experimental moduli and strengths under tension are provided in
0 0.0
0.5
1.0
1.5
2.0
Strain (%) Fig. 11. Stress–strain curves of the 3D woven composites under tensile load compared with the experimental data. Table 4 Statistics of the tensile experimental results.
Modulus (GPa) Tensile strength (MPa)
No. t1
No. t2
No. t3
No. t4
No. t5
Mean
56.9 910.3
53.28 871.4
55.91 890.9
54.44 903.2
55.19 885.8
55.1 892
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
S. Zhong et al. / Composite Structures xxx (2016) xxx–xxx
2000
Reaction force (N)
1500
Warp yarns
1000
500
0
Binder yarns
-500
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Strain (%) Fig. 12. Reaction force-strain curves of the warp and binder yarns on the tensile loading surface of the 3D woven composites.
7
Table 4. Comparing the experimental and the predicted results, the error of the moduli is +9.3% and that of the tensile strength is +13.8%. It should be mentioned that for 3D woven composites, the dispersion range of the strength and the modulus may be big, which is limited by the level of the processing procedures, especially for the strength. So the errors of the predicted moduli and strength are reasonable. The reaction forces vs strain curves of the warp and binder yarns on the loading surface are shown in Fig. 12. From Fig. 12, it can be found that the reaction forces of the warp yarns are much larger than that of the binder yarns. It is due to the reason that the cross section of the warp yarns is larger than that of the binder yarns and the binder yarns are S-shaped curve. The reaction forces of the fiber yarns drop at the breaking strain between 1.65% and 1.75%, which implies that the fiber yarns do not break simultaneously. The MISES stress distribution patterns of the warp yarns on two typical samples’ models generated by the RW model using the same experimental random characteristics are shown in Fig. 13. It can be found that the stress distribution patterns of two models
Fig. 13. MISES stress distribution of the warp yarns in two typical samples.
Fig. 14. Damage distribution in the matrix of the 3D woven composites in the typical examples.
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
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S. Zhong et al. / Composite Structures xxx (2016) xxx–xxx
1200 600
1000
Caculated by RW model 800
400
Stress (MPa)
Stress (MPa)
500
A=0 A=0.06 mm A=0.09 mm A=0.12 mm
300
200
600
400
Experimental data 100
200
0 0.0
0.2
0.4
0.6
0.8
0
1.0
0.0
Strain (%)
0.5
1.0
1.5
2.0
Strain (%) Fig. 15. Stress–strain curves of the 3D woven composites under compressive load compared with the experimental data.
Table 5 Statistics of the compressive experimental results.
Modulus (GPa) Compressive strength (MPa)
No. c1
No. c2
No. c3
No. c4
Mean
56.05 464
54.11 422.7
55.38 540
55.77 538.6
55.3 491
Fig. 17. Influence on the stress–strain curves by the amplitude of the warp yarns.
Table 6 Influence on the mechanical properties of the 3D woven composites by the amplitude of the warp yarns. (mm) Mean of the amplitude A
0.12
0.0899
0.06
0
Modulus E (GPa) Tensile strength S (MPa)
59.8 949
60.3 1000
60.5 1055
60.8 1115
is equal to 0 that means there is no waviness in the fiber Note: The amplitude A yarns.
1200
1200
1000
Warp yarns 600
λ= λ =2.55mm λ =3.39mm λ =4.25mm
800
Stress (MPa)
Reaction force (N)
900
300
0
-300 0.0
Binder yarns
0.2
0.4
0.6
0.8
1.0
600
400
200
Strain (%) Fig. 16. Reaction force–strain curves of the warp and binder yarns on the compressive loading surface of the 3D woven composites.
are different. The stress distribution of some yarns has reduced to a low level losing the bearing capacity. The damage accumulation of the two model in the matrix is shown in Fig. 14. By comparing Fig. 13 with Fig.14, it can be found that the regions where the damage has accumulated seriously correspond to those where the stress distribution of the yarns has reduced to a low level. Under compressive quasi-static load in the longitudinal direction of the warp yarns, the mean stress–strain curves of the five models generated by the RW model are obtained and compared with the experimental results, as shown in Fig. 15. The mean modulus and compressive strength, which is calculated by the RW model, is 60.3 GPa and 553 MPa, respectively. The experimental moduli and strengths under compression are shown in Table 5, the relative errors are +9.0% and +12.6%, respectively. As mentioned above, the errors are reasonable for 3D woven composites. The reaction forces vs strain curves of the warp and binder
0 0.0
0.5
1.0
1.5
2.0
Strain (%) Fig. 18. Influence on the stress–strain curves of the 3D woven composites by the half-wavelength k of the warp yarns.
yarns on the loading surface are shown in Fig. 16. The fiber yarns do not break simultaneously, which is similar to that under tensile load. The breaking strain is between 0.9% and 1.0%. In order to study the influences of the waviness degree of the fiber yarns on the mechanical properties, the means of the random variables (amplitude and half-wavelength) are changed and then the modulus and the tensile strength are calculated. The influence of the amplitude of the warp yarns on the stress–strain curves under tensile loading is shown in Fig. 17 and Table 6. It is found that the reduction of the tensile strength is obvious and that of the moduli is unobvious. The influence of the half-wavelength of the warp yarns on the stress–strain curves is shown in the Fig. 18 and Table 7. It is found that the reduction of the tensile
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051
S. Zhong et al. / Composite Structures xxx (2016) xxx–xxx Table 7 Influence on the mechanical properties of the 3D woven composites by the half-wavelength k of the warp yarns. Mean of the half-wavelength k (mm)
2.55
3.39
4.25
1
Modulus E (GPa) Tensile strength S (MPa)
60.1 964
60.3 1000
60.4 1041
60.8 1115
Note: The half-wavelength k is equal to 1 that means there is no waviness in the fiber yarns.
strength is obvious and that of the moduli is unobvious, too. In general, the influence on the tensile strength is greater than that on the modulus by the degree of the fiber yarns’ waviness. 6. Conclusion i. A RW model for 3D woven composites is proposed, in which the strength and damage of the 3D woven composites considering the fiber yarns random waviness can be studied with high efficiency. ii. The amplitude and half-wavelength of the fiber yarns’ waviness are assumed to be random variables in the RW model. The experimental statistical data of the two random variables are tested for building the RW model. iii. The moduli, tensile and compressive strengths of a typical 3D woven composites with fiber yarns’ random waviness are investigated via the RW model and they agree well with the experimental results. The influence of the waviness degree of the fiber yarns on the strength is greater than that on the modulus. It indicates that the random waviness of fiber yarns should be taken into consideration for actual 3D woven composites and the present method is very useful for the prediction of the mechanical properties of 3D woven composites.
Acknowledgement This work is sponsored by NSFC (11322217, 11432005). Appendix A The softening law parameters Amt and Amc are obtained by regularizing the softening branch of the stress–strain curve [19]. On the basis on the Bazant’s crack band theory [20], the energy dissipated per unit volume g mt and g mc can be determined by the fracture toughness Gmt and Gmc :
g mt ¼
Gmt ; l
g mc ¼
Gmc l
ðA1Þ
where l is the characteristic length of the finite element. The energy dissipated per unit volume g mt and g mc can be obtained from the integration of the damage energy dissipation during the process of damage development:
Z 1 Z 1 8 e @Gm _ @Gm @dmt > > d g ¼ dt ¼ drm mt < mt @dmt @dmt @r em 0 1 Z 1 Z 1 > e @Gm _ @Gm @dmc > : g mc ¼ d dt ¼ dr m mc @dmc @dmc @r e 0
1
m
where the Helmholtz free energy Gm can be written as:
ðA2Þ
Gm ¼
9
r2m;11 þ r2m;22 þ r2m;33 tm
ðrm;11 rm;22 þ rm;22 rm;33 þ rm;33 rm;11 Þ 2ð1 dm ÞEm Em 2 2 s sm;23 þ sm;31 þ ð1 dm ÞEm =ð1 tm Þ ðA3Þ 2 m;12 þ
where Em and tm are the modulus and Poisson’s ratio, rm;11 , rm;22 , rm;33 , sm;12 , sm;23 and sm;31 are the stresses of the matrix. The softening law parameters Amt and Amc can be determined by Eqs. (A1)–(A3) and Eq. (4). Finally, the variables Amt and Amc are in the integrals which can be calculated by the corresponding iterative algorithm. For the present problem, the parameters Amt and Amc will only have effects on the softening process of matrix, and the woven material is almost brittle fracture under both compression and tension. Therefore, the strength of the matrix has much more effects than the softening parameters Amt and Amc on the woven materials in the present model. If the experiments are not performed for fracture toughness Gmt and Gmc , the parameter Amt and Amc may be approximately obtained from the experimental tensile and compressive strain–stress curves. References [1] Argon AS. Fracture of composites. Amsterdam: Elsevier; 1972. [2] Fleck NA, Deng L, Budiansky B. Prediction of kink width in compressed fiber composites. J Appl Mech Trans ASME 1995;62:329–37. [3] Fleck NA, Shu JY. Microbuckle initiation in fibre composites: a finite element study. J Mech Phys Solids 1995;43:1887–918. [4] Chan WS, Chou CJ. Effects of delamination and ply fiber waviness on effective axial and bending stiffnesses in composite laminates. Compos Struct 1995;30:299–306. [5] Hsiao HM, Daniel IM. Effect of fiber waviness on stiffness and strength reduction of unidirectional composites under compressive loading. Compos Sci Technol 1996;56:581–93. [6] Karami G, Garnich M. Micromechanical study of thermoelastic behavior of composites with periodic fiber waviness. Compos B Eng 2005;36:241–8. [7] Xu J, Cox BN, McGlockton MA, Carter WC. A binary model of textile composites. 2. The elastic regime. Acta Metall Mater 1995;43:3511–24. [8] Cox BN, Dadkhah MS. The macroscopic elasticity of 3D woven composites. J Compos Mater 1995;29:785–819. [9] Cox BN, Carter WC, Fleck NA. A binary model of textile composites. 1. Formulation. Acta Metall Mater 1994;42:3463–79. [10] Callus PJ, Mouritz AP, Bannister MK, Leong KH. Tensile properties and failure mechanisms of 3D woven GRP composites. Compos A Appl Sci Manuf 1999;30:1277–87. [11] Karahan M, Lomov SV, Bogdanovich AE, Mungalov D, Verpoest I. Internal geometry evaluation of non-crimp 3D orthogonal woven carbon fabric composite. Compos A Appl Sci Manuf 2010;41:1301–11. [12] Scida D, Aboura Z, Benzeggagh ML, Bocherens E. A micromechanics model for 3D elasticity and failure of woven-fibre composite materials. Compos Sci Technol 1999;59:505–17. [13] Siron O, Lamon J. Damage and failure mechanisms of a3-directional carbon/carbon composite under uniaxial tensile and shear loads. Acta Mater 1998;46:6631–43. [14] Mahadik Y, Brown KAR, Hallett SR. Characterisation of 3D woven composite internal architecture and effect of compaction. Compos A Appl Sci Manuf 2010;41:872–80. [15] Mahadik Y, Hallett SR. Effect of fabric compaction and yarn waviness on 3D woven composite compressive properties. Compos Part A Appl Sci Manuf 2011;42:1592–600. [16] Chun H-J, Shin J-Y, Daniel IM. Effects of material and geometric nonlinearities on the tensile and compressive behavior of composite materials with fiber waviness. Compos Sci Technol 2001;61:125–34. [17] Stig F, Hallström S. Influence of crimp on 3D-woven fibre reinforced composites. Compos Struct 2013;95:114–22. [18] Jiang L, Zeng T, Yan S, Fang D. Theoretical prediction on the mechanical properties of 3D braided composites using a helix geometry model. Compos Struct 2013;100:511–6. [19] Zhong S, Guo L, Liu G, Lu H, Zeng T. A continuum damage model for threedimensional woven composites and finite element implementation. Compos Struct 2015;128:1–9. [20] Bazˇant Z, Oh BH. Crack band theory for fracture of concrete. Matér Constr 1983;16:155–77.
Please cite this article in press as: Zhong S et al. A random waveness model for the stiffness and strength evaluation of 3D woven composites. Compos Struct (2016), http://dx.doi.org/10.1016/j.compstruct.2016.06.051