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The effect of yarn distortion on the mechanical properties of 3D four-directional braided composites
Composites: Part A 40 (2009) 343–350
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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
The effect of yarn distortion on the mechanical properties of 3D four-directional braided composites Fang Guo-dong *, Liang Jun, Wang Yu, Wang Bao-lai Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 1 September 2008 Received in revised form 9 December 2008 Accepted 13 December 2008
Keywords: A. Polymer (textile) fiber B. Mechanical properties C. Stochastic function theory
a b s t r a c t A representative volume cell (RVC) is chosen to analyze the mechanical properties of 3D (3 dimensions) four-directional braided composites. Owing to braid yarns (an assembly of fibers) squeezing against each other in the braided composites, the braid yarns are distorted. Based on geometrical characteristics of the braided composites, cross-section of each braid yarn is supposed to be an octagon and divided into seven regions in the RVC. The distortion characteristics of yarns are considered in each region. Elastic properties of each region obtained by stochastic function theory are introduced into finite element model to calculate the mechanical properties of the RVC. The influences of yarn distortion on the stiffness and strength of the braided composites are obtained and discussed. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The mechanical properties of fibrous composites are determined by the physical and mechanical behavior, as well as the geometrical configurations of the material constituents. The in situ geometrical flaws produced in manufacturing process of 3D braided composites, such as voids, micro-crack and fiber misalignment, etc., have significant effect on the mechanical properties as well. Therefore, it is significant to study the effect of microstructures and flaws of the braided structure on the mechanical properties of braided composites. For braided composites, the geometrical shapes of cross-sections of braid yarns (an assembly of fibers) exhibit high mount of irregularities due to the variations in the manufacturing process. Before yarns are braided to form braided preform, the cross-section of braid yarn is usually considered a circle. But the cross-section of braid yarn becomes more irregular along longitudinal direction of yarn due to braid yarns compressing against each other when they are subjected to a jamming action. Chen [1] has investigated the microstructures of 3D four-directional braided composites by cutting longitudinally at a 45° angle with respect to composites surface as shown in Fig. 1a. The interior braid yarns are peeled from the interior structure of the braided composites as shown in Fig. 1b. It can be demonstrated from the SEM micrograph of the braided composites that the braid yarns remain linear in the interior but distort at the surfaces. And the cross-sections of twisted yarns become irregular. Therefore, the twisted yarn direction can * Corresponding author. Tel./fax: +86 451 86412613. E-mail address: [email protected] (F. Guo-dong). 1359-835X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2008.12.007
be characterized statistically. In recent years, some scholars [2,3] have considered the squeezing deformation of cross-section for braid yarn, and simplified the distorted cross-section as a hexagon or an octagon. However, to date, no studies have focused on the effect of yarn distortion on the mechanical properties of the braided composites. As generally recognized, the mechanical behaviors of braided composites can be derived from the mechanical properties of yarn and matrix by using microscopic mechanical method. Up to now, the effect of different inner microstructures and material behaviors on the yarn properties has been considered by many scholars, such as the constituent material nonlinearity [4], the wave structure of yarn [5] and the slack of different length of fiber in yarn [6,7], etc. Meanwhile, the strength of yarn has been studied by means of several models, such as randomly critical-core probability model [8], simplified micromechanical model based on shear model [9], Monte Carlo technique based on lattice model [10] and multilayer crack model [11], etc. In this paper, based on the geometrical characteristics of 3D four-directional braided composites, each braid yarn in the inner braided composites is divided into several regions (Section 2). In Section 3, random variables are introduced to describe the path of fiber in the twisted yarn. And the engineering elastic constants for each region are obtained by stochastic function theory. The effect of yarn distortion on the yarn strength is considered in Section 4. The finite element model to calculate the stiffness and strength of 3D four-directional braided composites is given in Section 5. Some numerical results are discussed in Section 6. Finally, several conclusions are obtained in Section 7.
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Fig. 1. The SEM micrographs of 3D four-directional braided composites. (a) The interior of a four-step braided preform. (b) A braiding yarn in the preform interior.
2. Geometrical model of 3D four-directional braided composites 3D four-directional braided preforms are composed of fourdirectional yarns which are braided with the same braid angles in the interior of the material. One braiding loop is one period. The braided preforms are solidified with epoxy matrix to form 3D four-directional braided composites. Because of the complexity of braid structures, a representative volume cell (RVC) including four yarns is chosen from the interior braid structures of the 3D braided composites to analyze their mechanical properties. Due to different directional yarns compressing against each other, the yarn shape becomes rather complex. In this paper, it is assumed that the yarn in the interior of the RVC has an octagon cross-section and has surface contact to neighboring yarns. The layout of the RVC is shown in Fig. 2. The braid angle c and the height h of the RVC can be measured by microscopic image analysis. The relations of geometrical parameters of the RVC can be expressed as follows:
h ¼ 8b= tan c
ð1Þ
La ¼ La0 sin c ¼ 2b cos c Lb ¼ 2b ð2a La Þ cos c Lb 2a Lm
ð2Þ ð3Þ
Lm ¼ 4b cos c
ð4Þ
where the height of cross-section 2a can be determined by K-number (a thousand fiber is counted as 1 K) of yarn and cross-section area A = 6b2cos c (Lm 2a)2/(2cos c). All geometrical parameters of 3D four-directional braided composites are given in Table 1.
One-directional yarns of an RVC are arranged as shown in Fig. 3a. After the several yarns are pieced together, it appears an intact yarn in the RVC as shown in Fig. 3b. It can be found that the appearance of squeezing regions among yarns in different directions is periodic. The length of one period is denoted by L with the following geometrical relation:
L ¼ 4b= sin c
ð5Þ
In order to evaluate the mechanical properties of yarn properly, each yarn in the RVC is divided into seven regions. The regions and local coordinates in cross-section of the divided yarn are shown in Fig 4a. The path of yarn in each region, r(n), can be described by the functions F(n) in x1–x2 plane and G(n) in x1–x3 plane as shown in Fig. 4b. It can be expressed as follows:
rðnÞ ¼ FðnÞx2 þ GðnÞx3
ð6Þ
where n is a variable in x1 direction. Because the squeezing regions are periodic, F(n) and G(n) are the periodical functions as well. They can be expressed as
Table 1 Braid parameters and RVC geometrical parameters. Interior braid angle c (°)
Height for RVC h (mm)
K
Fiber volume fraction vf (%)
b (mm)
45 30
3.04 5.265
12 12
51.7 50.9
0.38 0.38
Fig. 2. Geometrical model of 3D four-directional braided composites.
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Fig. 3. The braid yarns in the interior of RVC.
2p FðnÞ ¼ A2 cos n w2 L 2p GðnÞ ¼ A3 cos n w3 L
ð7Þ ð8Þ
where w2 and w3 are phase angles. The variables A2 and A3 are the amplitude values of F(n) and G(n), respectively. They shall be considered random in the following study. 3. Elastic properties for each regions of twisted yarn Because the path of fiber in the surface of braid yarn has some random characteristics, it can be expressed by stochastic function theory [12,13]. The mechanical properties of surface region in twisted yarn can be obtained as follows. The path vector of fiber can be expressed as
rðnÞ ¼ ðn; x02 ðnÞ; x03 ðnÞÞ
ð9Þ
where x0i ¼ xi hxi iði ¼ 2; 3Þ is stochastic centered function. hxii is the mean value of random variable xi. The stochastic tangent vector to the fiber path can be expressed as
r_ ðnÞ ¼ ð1; x_ 02 ðnÞ; x_ 03 ðnÞÞ
ð10Þ
The mean value and covariance function of the tangent vector are
2
hr_ i ¼ f1; 0; 0g;
0 0 ^ r_ r_ ¼ 6 K 4 0 K x_ 2 x_ 2 0 0
0
3
7 0 5
in the variation of their amplitude values and phase angles with some prescribed probabilistic distributions. In order to reduce the complexity of question, the variation of phase angle is ignored in this paper. Therefore, the mean values of x2(n) and x3(n) are zero. The covariance functions of x2(n) and x3(n) are
* 0 dx2 ðnÞ dx1 * 0 dx3 ðnÞ K x_ 3 x_ 3 ðn; nÞ ¼ dx1
K x_ 2 x_ 2 ðn; nÞ ¼
0
dx2 ðnÞ dx1 0
dx3 ðnÞ dx1
+ ¼ htan2 h2 i ¼ + ¼ htan2 h3 i ¼
2p2 r2A2 L2 2p2 r2A3 L2
ð12Þ ð13Þ
where rA2 and rA3 are the variances of random variables A2 and A3, respectively. h2 and h3 are, respectively, the deflection angles of fiber in the x1–x2 and x1–x3 planes. The twisted angle h can be obtained as
tan2 h ¼ tan2 h2 þ tan2 h3
ð14Þ
3.1. Mean and covariance of stochastic local basis vector A local coordinate system e01 e02 e03 is established on the fiber path as shown in Fig. 5. e01 is the tangent vector of the fiber path. The orientation of other two unit vectors can be chosen voluntarily to form a stochastic local orthonormal basis. The stochastic local basis vectors can be expanded into Taylor’s series about the point r_ ¼ hr_ i as follows:
ð11Þ
K x_ 3 x_ 3
where K x_ i x_ i ¼ hx_ 0i ; x_ 0i iði ¼ 2; 3Þ. hxi denotes the mean value of x. The comparison of Eq. (6) and (9) reveals that x2(n) and x3(n) can be substituted by F(n) and G(n), respectively. Because the length of period in the RVC has the geometrical relation as Eq. (5), the stochastic characteristics of F(n) and G(n) can be embodied
e0i
@e0 @ 2 e0i ¼ e0i r_ ¼hr_ i þ i ðr_ hr_ iÞ þ @ r_ r_ ¼hr_ i 2@ r_ @ r_ r_ ¼hr_ i
: ðr_ hr_ iÞ ðr_ hr_ iÞ þ ð15Þ
In the above formulation, the first item of the expansion is the directional cosines of the fiber path. The second item represents covariance properties of the local coordinate system. The third item gives the random fluctuations around the mean fiber path.
Fig. 4. The section and function of yarn surface fiber path.
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hS13 i ¼ S013 he011 i2 he033 i2 þ K x_ 3 x_ 3 ðhe011 i2 S011 þ he033 i2 S033 he011 ihe033 iS055 Þ hS22 i ¼ hS23 i ¼ hS33 i ¼
þ K x_ 2 x_ 2 he033 i2 S023 0 S22 he022 i4 þ K x_ 2 x_ 2 he022 i2 ð2S012 þ S066 Þ S023 he022 i2 he033 i2 þ K x_ 2 x_ 2 he033 i2 S013 þ K x_ 3 x_ 3 he022 i2 S012 S033 he033 i4 þ K x_ 3 x_ 3 he033 i2 ð2S013 þ S055 Þ
hS44 i ¼ S044 he022 i2 he033 i2 þ K x_ 2 x_ 2 he033 i2 S055 þ K x_ 3 x_ 3 he022 i2 S066 hS55 i ¼ S055 he011 i2 he033 i2 þ K x_ 2 x_ 2 he033 i2 S044 þ K x_ 3 x_ 3 ðhe011 i2 S011 Fig. 5. The stochastic local coordinates system and global coordinates system.
Therefore, the mean value of the second item is zero. Then the second-order approximation for the mean value of local basis vector can be provided as follows:
@ 2 e0i 0 0 hei i ¼ ei r_ ¼hr_ i þ 2@ r_ @ r_
^ rr_ ðn; nÞ :K
ð16Þ
r_ ¼hr_ i
b _ ðn; nÞ ¼ hr_ 0 r_ 0 i is the covariance of the tangent vector where K rr_ along the fiber path r_ r_ 0 ¼ ðr_ hr_ iÞ is the stochastic centered function of r_ . The covariance matrix of stochastic local basis vector is 2 00 3 00 00 00 00 he1 e00 1 i he1 e2 i he1 e3 i 00 00 00 00 00 7 ^ e0 e0 ðn;nÞ ¼ hðe0 he0 iÞðe0 he0 iÞi ¼ 6 K 4 he00 i i i i 2 e1 i he2 e2 i he2 e3 i 5 00 00 00 00 00 he00 3 e1 i he3 e2 i he3 e3 i
ð17Þ 3.2. Mean value for compliance tensors of twisted yarn The compliance tensors for yarn in the global system S and in the local system S0 are linked by the following relation: 0
0
S ¼ SðS ; e Þ; that is Sijlk ¼
S0mnop e0mi e0nj e0ok e0pk
ð18Þ
where e0ij ði; j ¼ 1; 2; 3Þ are the direction cosines between the global and local system. In order to obtain the mean value and covariance of compliance tensor in the global system, the expansion of compliance tensor into Taylor’s series about e0 = he0 i is applied as follows:
@S @ 2 S 0 0 S ¼ Sje0 ¼he0 i þ 0 ðe he iÞ þ @e e0 ¼he0 i 2@e0 @e0
e0 ¼he0 i
: ðe0 he0 iÞ ðe0 he0 iÞ þ
ð19Þ
The second-order approximation for the mean value of compliance tensor can be provided as
@ S hSi ¼ Sje0 ¼he0 i þ 2@e0 @e0 2
b e0 e0 :K
ð20Þ
e0 ¼he0 i
The mean value of stochastic local basis vector can be obtained by Eqs. (11)–(13), (16) in the form
1 1 he01 i ¼ 1 K x_ 2 x_ 2 K x_ 3 x_ 3 ; 0; 0 ; 2 2 1 1 he02 i ¼ 0; 1 K x_ 2 x_ 2 ; 0 ; he01 i ¼ 0; 0; 1 K x_ 3 x_ 3 2 2
ð21Þ
By substituting Eq. (21) into Eq. (17), Eq. (17) into Eq. (20), the mean value of each term in compliance matrix for yarn in the global coordinate system is obtained as follows:
hS11 i ¼ S011 he011 i4 þ K x_ 2 x_ 2 he011 i2 ð2S012 þ S066 Þ þ K x_ 3 x_ 3 he011 i2 ð2S013 þ S055 Þ hS12 i ¼ S012 he011 i2 he022 i2 þ K x_ 2 x_ 2 he011 i2 S011 þ he022 i2 S022 he011 ihe022 iS066 þ
K x_ 3 x_ 3 he022 i2 S023
hS66 i ¼
þ he033 i2 S033 2he011 ihe033 iS013 Þ 0 S66 he011 i2 he022 i2 þ K x_ 3 x_ 3 he022 i2 S044 2he011 ihe022 iS012 Þ
þ K x_ 2 x_ 2 ðhe011 i2 S011 þ he022 i2 S022 ð22Þ
The values of the matrix terms unlisted above are zero. compliance matrix for straight and aligned yarn:
2 6 6 6 6 6 0 ½Sij ¼ 6 6 6 6 6 4
S011
S012
S012
0
0
S022
S023
0
0
0
0
2ðS022 S023 Þ
0
S022 sym
S066
0
S0ij
is the
3
7 0 7 7 7 0 7 7 7 0 7 7 0 7 5 S066
ð23Þ
Because the mechanical properties of yarn can be regarded as transversally isotropic, there are five independent constants: E011 ; E022 ; G012 ; m012 ; m023 Which can be obtained as follows [14]:
E011 ¼ V f Ef 11 þ ð1 V f ÞEm qffiffiffiffiffiffi E022 ¼ E033 ¼ Em = 1 V f ð1 Em =Ef 22 Þ qffiffiffiffiffiffi G012 ¼ G013 ¼ Gm = 1 V f ð1 Gm =Gf 12 Þ qffiffiffiffiffiffi G023 ¼ Gm = 1 V f ð1 Gm =Gf 23 Þ
ð24Þ
m012 ¼ m013 ¼ V f mf 12 þ ð1 V f Þmm m023 ¼ E022 =ð2G023 Þ 1 As a result, the terms in Eq. (23) can be expressed by: S0ii ¼ 1=E0ii ; S0ij ¼ ðm0ij =E0ij Þði; j ¼ 1; 2; 3Þ; S044 ¼ 1=G023 ; S055 ¼ 1=G013 ; S066 ¼ 1=G012 . Finally, by substituting Eq. (12), (13), (23) into Eq. (20), the explicit formulation of the compliance matrix for the twisted yarn is obtained. 4. Strength of a twisted yarn Zhu et al. [15] has studied the axial tensile statistical strength of unidirectional fiber reinforced composites. The statistical theory can be summarized as follows. It is assumed that the probability distribution of fiber strength is Weibull function. That is
FðrÞ ¼ 1 expðadrb Þ
ð25Þ
where a, b and d are the Weibull scale parameter, the Weibull shape parameter and the ineffective length, respectively. a can be deter[16]. According to Ref. [17], it can be assumed mined by a ¼ rb f that b equals 6 in this paper.The yarn is composed of N fibers. When the tensile load P is applied on the longitudinal direction of yarn, Nf fibers are broken. When the applied load increases by dP, the average stress and the number of broken fibers will increase by dr and dNf, respectively. The broken fibers Nf should be satisfied with the following differential equation and boundary conditions [15]:
dNf dFðrÞ dFðK s rÞ dFðrÞ þ ða þ 1Þ a Nf ¼ N dr dr dr dr
Nf ð0Þ ¼ 0
ð26Þ
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where F(Ksr) is the probability of fiber breakage in which the concentration is considered. a is the number of fibers adjacent to a broken fiber. If there are 4 or 6 fibers adjacent to a broken fiber, Ks = 1.146 or 1.104, respectively. The solution of the above differential equation is
Nf ¼ N exp½ða þ 1Þ expðadrb Þ a expðadmrb ÞT
ð27Þ
R1
where T ¼ x exp½ða þ 1ÞX þ aX m dX; X ¼ expðarrb Þ; m ¼ K bs . The expectation of average fiber stress at yarn failure is expressed as follows:
r B ¼ 1 Nf ðr m Þ=N r m
ð28Þ
m is the maximum stress of unbroken fiber, which can be where r obtained by
d ð1 Nf =NÞr jr¼rm ¼ 0 dr
ð29Þ
When N is large enough, the strength distribution of yarn approaches a normal distribution obtained by Daniels [18]. The density distribution function is
" # BÞ 1 ðrB r gðrB Þ ¼ pffiffiffiffiffiffiffi exp 2w2B 2pwB
ð30Þ
where wB is standard deviation of normal distribution, which can be expressed by
wB ¼ rm fFðrm Þ½1 Fðrm Þg1=2 M 1=2 " #
1=2 2 df Ef 11 1 1 þ ð1 uÞ 1=2 d¼ 1Þ Ch ðV f 2 2Gm 2ð1 uÞ
ð32Þ
where df, Vf and Ef11 are the diameter, the volume fraction of fiber and the longitudinal elastic modulus of fiber, respectively. Gm is the matrix shear modulus. / is the fraction of the undisturbed stress value. Considering the yarn as n links in a chain, the weakest link theory [19] is applied to determinate the failure probability of yarn. The probability density distribution function is defined by
In addition, the strength properties of yarn except for longitudinal tensile strength can be obtained by
X yc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ef 11 Em V f Em ¼ 2 V f þ ð1 V f Þ Ef 11 3ð1 V f Þ
Y yt ¼
1 þ Vf
1
gy
1
K my
Y yc ¼ 4Y yt Sy ¼
1 þ Vf
1
gs
K ms
ð37Þ
1
Xm
ð38Þ ð39Þ
Sm
ð40Þ
In the above equations, Xyc, Yyt, Yyc and Sy are the longitudinal compressive, transverse tensile, transverse compressive and shear strength of the yarn, respectively. Xm is the tensile strength of matrix. gy and gs are Cai experience coefficients. Kmy and Kms are the tensile and shear stress concentration coefficients of matrix, respectively. Sm is the shear strength of matrix. gy and gs equal 0.5 [16]. 5. Finite element model of 3D four-directional braided composites Four-node tetrahedron elements are adopted to mesh the representative volume cell of 3D four-directional braided composites as shown in Fig. 6a. The regions of braid yarn have different local coordinate systems as shown in Fig. 6b. The braid yarns and matrix are T300 carbon 12 K and epoxy resin, respectively. Their properties are listed in Table 2.
Table 2 Constituent properties of composites [21]. 3D four-directional braided carbon/epoxy composites
vðrÞ ¼ ng 0 ðrÞ½1 gðrÞn1
ð33Þ
f , is obtained by The mean value of yarn strength, r
ð34Þ
Then the tensile strength of the unidirectional composites with straight yarn is expressed as
f þ ð1 V f ÞX m X tu ¼ V f r
ð36Þ
ð31Þ
The ineffective length d can be obtained by
dvðrÞ=dr ¼ 0
X yt ¼ X tu cos2 h
ð35Þ
Considering the influence of yarn distortion, the tensile strength of twisted yarn can be expressed as a function of the twisted angle h [20]:
T300 carbon fiber
Epoxy resin
Longitudinal tensile modulus Ef11 231 GPa Transverse tensile modulus Ef22 40 GPa Longitudinal Poisson’s ratio mf12 0.26 Longitudinal shear modulus Gf12 24 GPa
Tensile modulus Em 3.5 GPa Poisson’s ratio mm 0.35 Tensile strength Xm 80 MPa Compressive strength Xmc 241 MPa Shear strength Sm 60 MPa
Transverse shear modulus Gf23 14.3 GPa Longitudinal tensile strength rf 3528 MPa Longitudinal compressive strength rfc 2470 MPa Fiber volume fraction of yarn Vf 0.8 Fiber diameter df 7l
Fig. 6. The finite element model for 3D four-directional braided composites.
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Table 3 The displacement boundary conditions to obtain the different coefficients of stiffness matrix. Stiffness matrix coefficients The The The The
first column of [C]b second column of [C] third column of [C] forth column of [C]
The fifth column of [C] The sixth column of [C]
Nonzero BCs u1(L1, x2, x3) u1(0, x2, x3) = L1a u2(x1, L2, x3) u2(x1, 0, x3) = L2 u3(x1, x2, L3) u3(x1, x2, 0) = L3 u2(x1, x2, L3) u2(x1, x2, 0) = L3/2 u3(x1, L2, x3) u3(x1, 0, x3) = L2/2 u1(x1, x2, L3) u1(x1, x2, 0) = L3/2 u3(L1, x2, x3) u3(0, x2, x3) = L1/2 u1(x1, L2, x3) u1(x1, 0, x3) = L2/2 u2(L1, x2, x3) u2(0, x2, x3) = L1/2
a
ui (i = 1, 2, 3), xi (i = 1, 2, 3) and Li (i = 1, 2, 3) are the displacements, variables and edge lengths of RVC in 123 directions, respectively. b [C] is the stiffness matrix of 3D four-directional braided composites.
In order to ensure continuity of forces and displacements compatibility of the opposite faces of the RVC, the periodic boundary conditions (BCs) are imposed in the simulation [21]. To calculate the stiffness matrix, the BCs are listed in Table 3. The macroscopic strain is displacement gradient in the load direction. And the macroscopically homogeneous stress is calculated from the following form:
r ¼
P
Pi S
ð41Þ
where Pi and S are, respectively, the reaction force and area of the RVC’s surface whose normal direction parallels the load direction.
The compliance matrix coefficients of 3D four-directional braided composites are obtained by macroscopically homogeneous stress dividing macroscopic strain. The unidirectional tensile strength which is the peak value of the macroscopically homogeneous stress and strain curve of 3D four-directional braided composites can be obtained by using progressive damage method reported in Ref. [21].
6. Results and discussion Numerical results for the elastic moduli, shear moduli and the Poisson’s rate of each region of twisted yarn with material properties listed in Table 2 are shown in Fig. 7. Because the relation between deflection angles h2 in Eq. (12) and h3 in Eq. (13) is difficult to be determined, it is assumed that h2 equals h3 to simplify the discussion on the effect of twisted angle h in Eq. (14) of twisted yarn. The local coordinates of twisted yarn are shown in Fig. 4b. As apparent from Fig. 7a, the mean value of the longitudinal elastic modulus for twisted yarn, hE11i, is greatly influenced by the twisted angle. Particularly, it decreases to 40% at h = 17°. But the mean value of the transverse elastic moduli, hE22i and hE33i (hE22i = hE33i), decrease slightly with the increasing twisted angles. Fig. 7b shows the effect of twisted angle h on the Poisson’s ratios. hm12i increases by 20% at h = 17°, while hm23i decreases to 90% at h = 17°. And the effect of twisted angle on the shear moduli is negligible as shown in Fig. 7c.
Fig. 7. The effect of fiber distortion on the mechanical behavior of twisted yarn surface regions.
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Fig. 8 shows the effect of fiber distortion on the mechanical behavior of 3D four-directional braided composites. The superscript ‘o’ denotes that the braid yarns are straight and aligned. The coordinates for the braided composites are shown in Fig. 6a. As shown in Fig. 8a, most severe effect is obtained for the longitudinal elastic modulus hE22i of the braided composites. It drops by 15% at h = 17°. The transverse elastic moduli, hE11i and hE33i, increase slightly with the increase of the twisted angle h. As shown in Fig. 8b, the Poisson’s ratios, hm12i and hm23i, decline with the increase of twisted angle, while hm31i slightly increases. Fig. 8c shows that the effect of yarn distortion on the shear moduli of the 3D braided composites is severe. With the increase of twisted angle, the shear moduli all decrease. Especially hG12i and hG23i decrease to 85% at h = 17°. Comparing with Figs. 7 and 8, the tendencies of longitudinal elastic modulus of yarn and braided composites are uniform. However, the tendencies of other elastic moduli, shear moduli and Poisson’s ratios are reverse. The phenomena can be attributed to the great decline of longitudinal elastic modulus of twisted yarn with increase of twisted angle. The longitudinal strength of twisted yarn decreases with increase of twisted angle according to Eq. (36). The strength of the braided composites can be obtained when the stiffness and strength of twisted yarn are introduced to the progressive damage model of the braided composites elaborated in Ref. [21]. Fig. 9 represents the variation curves of strengths with twisted angles in two-direction tension as shown in Fig. 6a. Again, the superscript
349
Fig. 9. The effect of fiber distortion on the unidirectional tensile strength of the braided composites.
‘o’ denotes the braid composites with straight and aligned braid yarns. The ordinate represents the strength ratios of braid composites with twisted braid yarns to aligned braid yarns. As shown in
Fig. 8. The effect of fiber distortion on the mechanical behavior of the braided composites.
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Fig. 9, two cases of braid parameters as listed in Table 1 are chosen. In both cases, the strength ratios of the braided composites decline with the increase of the interior braid angles. The strength of the braided composites with 45° interior braid angle decreases more quickly than that with 30° interior braid angle when twisted angle of braid yarn is under 9.5°. But the tendencies become reverse when twisted angle of braid yarn is above 9.5°. The strength of the braided composites with interior braid angle 45°and 30° decrease to 72.2% and 66.6% at h = 17°, respectively. 7. Conclusions In order to consider the effect of yarn distortion on mechanical properties of braided composites, each braid yarn is divided into seven regions. The stiffness of each region of twisted yarns is obtained by stochastic function theory. The statistical strength of twisted yarn is considered by statistical theory as well. By introducing the stiffness and strength properties of twisted yarn into finite element model of the representative volume cell (RVC), the elastic properties of the braided composites are obtained. The effect of twisted yarn on the unidirectional tensile strength of 3D four-directional braided composites is considered as well. The significant effect is obtained for the longitudinal elastic modulus and longitudinal Poisson’s ratio of the twisted yarn. The engineering elastic constants except for transverse elastic modulus and transverse Poisson’s rate for 3D four-direction braided composites are highly sensitive to the twisted angle of yarns. The unidirectional tensile strengths of 3D four-directional braided composites with interior braid angle 45°and 30° both decrease with increase of twisted angle of braid yarn. The tensile strength of the braided composites with interior braid angle 45° decreases more quickly than that with interior braid angle 30° when twisted angle is under 9.5°, while it is reverse when twisted angle is above 9.5°. As generally recognized, the magnitude of twisted angle of braid yarns is controlled by the manufacturing process. Therefore, it is imperative to study the relation between the twisted angle and pre-tensile loading in the manufacturing process further. Acknowledgments This work is supported by the National Natural Science Foundation of China (10772060), Hei Longjiang Province Outstanding
Youth Foundation (JC 2006-13) and the Program for New Century Excellent Talents in University (NCET-05-0346). References [1] Chen L, Tao XM, Choy CL. On the microstructure of three-dimensional braided preforms. Compos Sci Technol 1999;59:391–404. [2] Yang ZY, Lu ZX, Liu ZG, Li ZP. Finite element analysis of the mechanical properties of 3D braided composites. Acta Mater Comp Sin 2005;22(5): 155–61. [3] Xu K, Xu XW. On the microstructure model of four-step 3D rectangular braided composites. Acta Mater Comp Sin 2005;23(5):154–60. [4] Krajcinovic D, Silva MAG. Statistical aspects of the continuous damage theory. Int J Solids Struct 1982;18(7):551–62. [5] Phoenix SL. Statistical theory for the strength of twisted fiber bundles with applications to yarns and cables. Text Res J 1979:407–23. [6] Chudoba R, Vorechovsky M, Konrad M. Stochastic modeling of multi-filament yarns. I. Random properties within the cross-section and size effect. Int J Solids Struct 2006;43:413–34. [7] Vorechovsky M, Chudoba R. Stochastic modeling of multi-filament yarns: II. Random properties over the length and size effect. Int J Solids Struct 2006;43:435–58. [8] Zeng QD, Ma R, Fan FQ. Micro-statistical analysis of ultimate tensile strength of cross-ply laminates. J Mech 1994;26(4):451–61. [9] Beyerlein IJ, Phoenix SL. Comparison of shear-lag theory and continuum fracture mechanics for modeling fiber and matrix stresses in an elastic cracked composite lamina. Int J Solids Struct 1996;33(18):2543–74. [10] Ibnabdeljalil M, Curtin WA. Strength and reliability of fiber-reinforced composites: localized load sharing and associated size-effects. Int J Solids Struct 1997;34(21):2649–68. [11] Kun F, Zapperi S, Herrmann HJ. Damage in fiber bundles models. Eur Phys J B 2000;17:269–79. [12] Yushanov SP, Bogdanovich AE. Stochastic theory of composite materials with random waviness of the reinforcements. Int J Solids Struct 1998;35(22): 2901–30. [13] Yushanov SP, Bogdanovich Alexander E. Fiber waviness in textile composites and its stochastic modeling. Mech Comp Mater 2000;36(4):297–318. [14] Gommers B, Verpoest I, Van HP. Modeling the elastic properties of knittedfabric-reinforced composites. Compos Sci Technol 1996;56:685–94. [15] Zhu YT, Zhou BL, He GH, Zhang ZG. A statistical theory of composite materials strength. J Comp Mater 1989;23:280–7. [16] Wang XY, Tang YZ. Mechanical analysis and designing of composite materials. Changsha: National Defense Science and Technology University Press; 1999. [17] Zuo WW, Xiao LY, Liao DX. Statistical strength analysis of the 3-D braided composites. Compos Sci Technol 2007;67:2095–102. [18] Daniels HE. The statistical theory of the strength of bundles of threads I. Proc R Soc Lond 1945;A183:405. [19] Gucer DE, Gurland J. Comparison of statistics of two fracture nodes. J Mech Phys Solids 1962;10:365–73. [20] Hearle JWS, Grosberg P, Backer S. Structural mechanics of fibers, yarns and fabrics. New York: John Wiley; 1969. [21] Fang GD, Liang J, Wang BL. Progressive damage and nonlinear analysis of 3D four-directional braided composites under unidirectional tension. Comp Struct, 2008. doi:10.1016/j.compstruct.2008.07.016.
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
The effect of yarn distortion on the mechanical properties of 3D four-directional braided composites Fang Guo-dong *, Liang Jun, Wang Yu, Wang Bao-lai Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 1 September 2008 Received in revised form 9 December 2008 Accepted 13 December 2008
Keywords: A. Polymer (textile) fiber B. Mechanical properties C. Stochastic function theory
a b s t r a c t A representative volume cell (RVC) is chosen to analyze the mechanical properties of 3D (3 dimensions) four-directional braided composites. Owing to braid yarns (an assembly of fibers) squeezing against each other in the braided composites, the braid yarns are distorted. Based on geometrical characteristics of the braided composites, cross-section of each braid yarn is supposed to be an octagon and divided into seven regions in the RVC. The distortion characteristics of yarns are considered in each region. Elastic properties of each region obtained by stochastic function theory are introduced into finite element model to calculate the mechanical properties of the RVC. The influences of yarn distortion on the stiffness and strength of the braided composites are obtained and discussed. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The mechanical properties of fibrous composites are determined by the physical and mechanical behavior, as well as the geometrical configurations of the material constituents. The in situ geometrical flaws produced in manufacturing process of 3D braided composites, such as voids, micro-crack and fiber misalignment, etc., have significant effect on the mechanical properties as well. Therefore, it is significant to study the effect of microstructures and flaws of the braided structure on the mechanical properties of braided composites. For braided composites, the geometrical shapes of cross-sections of braid yarns (an assembly of fibers) exhibit high mount of irregularities due to the variations in the manufacturing process. Before yarns are braided to form braided preform, the cross-section of braid yarn is usually considered a circle. But the cross-section of braid yarn becomes more irregular along longitudinal direction of yarn due to braid yarns compressing against each other when they are subjected to a jamming action. Chen [1] has investigated the microstructures of 3D four-directional braided composites by cutting longitudinally at a 45° angle with respect to composites surface as shown in Fig. 1a. The interior braid yarns are peeled from the interior structure of the braided composites as shown in Fig. 1b. It can be demonstrated from the SEM micrograph of the braided composites that the braid yarns remain linear in the interior but distort at the surfaces. And the cross-sections of twisted yarns become irregular. Therefore, the twisted yarn direction can * Corresponding author. Tel./fax: +86 451 86412613. E-mail address: [email protected] (F. Guo-dong). 1359-835X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2008.12.007
be characterized statistically. In recent years, some scholars [2,3] have considered the squeezing deformation of cross-section for braid yarn, and simplified the distorted cross-section as a hexagon or an octagon. However, to date, no studies have focused on the effect of yarn distortion on the mechanical properties of the braided composites. As generally recognized, the mechanical behaviors of braided composites can be derived from the mechanical properties of yarn and matrix by using microscopic mechanical method. Up to now, the effect of different inner microstructures and material behaviors on the yarn properties has been considered by many scholars, such as the constituent material nonlinearity [4], the wave structure of yarn [5] and the slack of different length of fiber in yarn [6,7], etc. Meanwhile, the strength of yarn has been studied by means of several models, such as randomly critical-core probability model [8], simplified micromechanical model based on shear model [9], Monte Carlo technique based on lattice model [10] and multilayer crack model [11], etc. In this paper, based on the geometrical characteristics of 3D four-directional braided composites, each braid yarn in the inner braided composites is divided into several regions (Section 2). In Section 3, random variables are introduced to describe the path of fiber in the twisted yarn. And the engineering elastic constants for each region are obtained by stochastic function theory. The effect of yarn distortion on the yarn strength is considered in Section 4. The finite element model to calculate the stiffness and strength of 3D four-directional braided composites is given in Section 5. Some numerical results are discussed in Section 6. Finally, several conclusions are obtained in Section 7.
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Fig. 1. The SEM micrographs of 3D four-directional braided composites. (a) The interior of a four-step braided preform. (b) A braiding yarn in the preform interior.
2. Geometrical model of 3D four-directional braided composites 3D four-directional braided preforms are composed of fourdirectional yarns which are braided with the same braid angles in the interior of the material. One braiding loop is one period. The braided preforms are solidified with epoxy matrix to form 3D four-directional braided composites. Because of the complexity of braid structures, a representative volume cell (RVC) including four yarns is chosen from the interior braid structures of the 3D braided composites to analyze their mechanical properties. Due to different directional yarns compressing against each other, the yarn shape becomes rather complex. In this paper, it is assumed that the yarn in the interior of the RVC has an octagon cross-section and has surface contact to neighboring yarns. The layout of the RVC is shown in Fig. 2. The braid angle c and the height h of the RVC can be measured by microscopic image analysis. The relations of geometrical parameters of the RVC can be expressed as follows:
h ¼ 8b= tan c
ð1Þ
La ¼ La0 sin c ¼ 2b cos c Lb ¼ 2b ð2a La Þ cos c Lb 2a Lm
ð2Þ ð3Þ
Lm ¼ 4b cos c
ð4Þ
where the height of cross-section 2a can be determined by K-number (a thousand fiber is counted as 1 K) of yarn and cross-section area A = 6b2cos c (Lm 2a)2/(2cos c). All geometrical parameters of 3D four-directional braided composites are given in Table 1.
One-directional yarns of an RVC are arranged as shown in Fig. 3a. After the several yarns are pieced together, it appears an intact yarn in the RVC as shown in Fig. 3b. It can be found that the appearance of squeezing regions among yarns in different directions is periodic. The length of one period is denoted by L with the following geometrical relation:
L ¼ 4b= sin c
ð5Þ
In order to evaluate the mechanical properties of yarn properly, each yarn in the RVC is divided into seven regions. The regions and local coordinates in cross-section of the divided yarn are shown in Fig 4a. The path of yarn in each region, r(n), can be described by the functions F(n) in x1–x2 plane and G(n) in x1–x3 plane as shown in Fig. 4b. It can be expressed as follows:
rðnÞ ¼ FðnÞx2 þ GðnÞx3
ð6Þ
where n is a variable in x1 direction. Because the squeezing regions are periodic, F(n) and G(n) are the periodical functions as well. They can be expressed as
Table 1 Braid parameters and RVC geometrical parameters. Interior braid angle c (°)
Height for RVC h (mm)
K
Fiber volume fraction vf (%)
b (mm)
45 30
3.04 5.265
12 12
51.7 50.9
0.38 0.38
Fig. 2. Geometrical model of 3D four-directional braided composites.
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Fig. 3. The braid yarns in the interior of RVC.
2p FðnÞ ¼ A2 cos n w2 L 2p GðnÞ ¼ A3 cos n w3 L
ð7Þ ð8Þ
where w2 and w3 are phase angles. The variables A2 and A3 are the amplitude values of F(n) and G(n), respectively. They shall be considered random in the following study. 3. Elastic properties for each regions of twisted yarn Because the path of fiber in the surface of braid yarn has some random characteristics, it can be expressed by stochastic function theory [12,13]. The mechanical properties of surface region in twisted yarn can be obtained as follows. The path vector of fiber can be expressed as
rðnÞ ¼ ðn; x02 ðnÞ; x03 ðnÞÞ
ð9Þ
where x0i ¼ xi hxi iði ¼ 2; 3Þ is stochastic centered function. hxii is the mean value of random variable xi. The stochastic tangent vector to the fiber path can be expressed as
r_ ðnÞ ¼ ð1; x_ 02 ðnÞ; x_ 03 ðnÞÞ
ð10Þ
The mean value and covariance function of the tangent vector are
2
hr_ i ¼ f1; 0; 0g;
0 0 ^ r_ r_ ¼ 6 K 4 0 K x_ 2 x_ 2 0 0
0
3
7 0 5
in the variation of their amplitude values and phase angles with some prescribed probabilistic distributions. In order to reduce the complexity of question, the variation of phase angle is ignored in this paper. Therefore, the mean values of x2(n) and x3(n) are zero. The covariance functions of x2(n) and x3(n) are
* 0 dx2 ðnÞ dx1 * 0 dx3 ðnÞ K x_ 3 x_ 3 ðn; nÞ ¼ dx1
K x_ 2 x_ 2 ðn; nÞ ¼
0
dx2 ðnÞ dx1 0
dx3 ðnÞ dx1
+ ¼ htan2 h2 i ¼ + ¼ htan2 h3 i ¼
2p2 r2A2 L2 2p2 r2A3 L2
ð12Þ ð13Þ
where rA2 and rA3 are the variances of random variables A2 and A3, respectively. h2 and h3 are, respectively, the deflection angles of fiber in the x1–x2 and x1–x3 planes. The twisted angle h can be obtained as
tan2 h ¼ tan2 h2 þ tan2 h3
ð14Þ
3.1. Mean and covariance of stochastic local basis vector A local coordinate system e01 e02 e03 is established on the fiber path as shown in Fig. 5. e01 is the tangent vector of the fiber path. The orientation of other two unit vectors can be chosen voluntarily to form a stochastic local orthonormal basis. The stochastic local basis vectors can be expanded into Taylor’s series about the point r_ ¼ hr_ i as follows:
ð11Þ
K x_ 3 x_ 3
where K x_ i x_ i ¼ hx_ 0i ; x_ 0i iði ¼ 2; 3Þ. hxi denotes the mean value of x. The comparison of Eq. (6) and (9) reveals that x2(n) and x3(n) can be substituted by F(n) and G(n), respectively. Because the length of period in the RVC has the geometrical relation as Eq. (5), the stochastic characteristics of F(n) and G(n) can be embodied
e0i
@e0 @ 2 e0i ¼ e0i r_ ¼hr_ i þ i ðr_ hr_ iÞ þ @ r_ r_ ¼hr_ i 2@ r_ @ r_ r_ ¼hr_ i
: ðr_ hr_ iÞ ðr_ hr_ iÞ þ ð15Þ
In the above formulation, the first item of the expansion is the directional cosines of the fiber path. The second item represents covariance properties of the local coordinate system. The third item gives the random fluctuations around the mean fiber path.
Fig. 4. The section and function of yarn surface fiber path.
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hS13 i ¼ S013 he011 i2 he033 i2 þ K x_ 3 x_ 3 ðhe011 i2 S011 þ he033 i2 S033 he011 ihe033 iS055 Þ hS22 i ¼ hS23 i ¼ hS33 i ¼
þ K x_ 2 x_ 2 he033 i2 S023 0 S22 he022 i4 þ K x_ 2 x_ 2 he022 i2 ð2S012 þ S066 Þ S023 he022 i2 he033 i2 þ K x_ 2 x_ 2 he033 i2 S013 þ K x_ 3 x_ 3 he022 i2 S012 S033 he033 i4 þ K x_ 3 x_ 3 he033 i2 ð2S013 þ S055 Þ
hS44 i ¼ S044 he022 i2 he033 i2 þ K x_ 2 x_ 2 he033 i2 S055 þ K x_ 3 x_ 3 he022 i2 S066 hS55 i ¼ S055 he011 i2 he033 i2 þ K x_ 2 x_ 2 he033 i2 S044 þ K x_ 3 x_ 3 ðhe011 i2 S011 Fig. 5. The stochastic local coordinates system and global coordinates system.
Therefore, the mean value of the second item is zero. Then the second-order approximation for the mean value of local basis vector can be provided as follows:
@ 2 e0i 0 0 hei i ¼ ei r_ ¼hr_ i þ 2@ r_ @ r_
^ rr_ ðn; nÞ :K
ð16Þ
r_ ¼hr_ i
b _ ðn; nÞ ¼ hr_ 0 r_ 0 i is the covariance of the tangent vector where K rr_ along the fiber path r_ r_ 0 ¼ ðr_ hr_ iÞ is the stochastic centered function of r_ . The covariance matrix of stochastic local basis vector is 2 00 3 00 00 00 00 he1 e00 1 i he1 e2 i he1 e3 i 00 00 00 00 00 7 ^ e0 e0 ðn;nÞ ¼ hðe0 he0 iÞðe0 he0 iÞi ¼ 6 K 4 he00 i i i i 2 e1 i he2 e2 i he2 e3 i 5 00 00 00 00 00 he00 3 e1 i he3 e2 i he3 e3 i
ð17Þ 3.2. Mean value for compliance tensors of twisted yarn The compliance tensors for yarn in the global system S and in the local system S0 are linked by the following relation: 0
0
S ¼ SðS ; e Þ; that is Sijlk ¼
S0mnop e0mi e0nj e0ok e0pk
ð18Þ
where e0ij ði; j ¼ 1; 2; 3Þ are the direction cosines between the global and local system. In order to obtain the mean value and covariance of compliance tensor in the global system, the expansion of compliance tensor into Taylor’s series about e0 = he0 i is applied as follows:
@S @ 2 S 0 0 S ¼ Sje0 ¼he0 i þ 0 ðe he iÞ þ @e e0 ¼he0 i 2@e0 @e0
e0 ¼he0 i
: ðe0 he0 iÞ ðe0 he0 iÞ þ
ð19Þ
The second-order approximation for the mean value of compliance tensor can be provided as
@ S hSi ¼ Sje0 ¼he0 i þ 2@e0 @e0 2
b e0 e0 :K
ð20Þ
e0 ¼he0 i
The mean value of stochastic local basis vector can be obtained by Eqs. (11)–(13), (16) in the form
1 1 he01 i ¼ 1 K x_ 2 x_ 2 K x_ 3 x_ 3 ; 0; 0 ; 2 2 1 1 he02 i ¼ 0; 1 K x_ 2 x_ 2 ; 0 ; he01 i ¼ 0; 0; 1 K x_ 3 x_ 3 2 2
ð21Þ
By substituting Eq. (21) into Eq. (17), Eq. (17) into Eq. (20), the mean value of each term in compliance matrix for yarn in the global coordinate system is obtained as follows:
hS11 i ¼ S011 he011 i4 þ K x_ 2 x_ 2 he011 i2 ð2S012 þ S066 Þ þ K x_ 3 x_ 3 he011 i2 ð2S013 þ S055 Þ hS12 i ¼ S012 he011 i2 he022 i2 þ K x_ 2 x_ 2 he011 i2 S011 þ he022 i2 S022 he011 ihe022 iS066 þ
K x_ 3 x_ 3 he022 i2 S023
hS66 i ¼
þ he033 i2 S033 2he011 ihe033 iS013 Þ 0 S66 he011 i2 he022 i2 þ K x_ 3 x_ 3 he022 i2 S044 2he011 ihe022 iS012 Þ
þ K x_ 2 x_ 2 ðhe011 i2 S011 þ he022 i2 S022 ð22Þ
The values of the matrix terms unlisted above are zero. compliance matrix for straight and aligned yarn:
2 6 6 6 6 6 0 ½Sij ¼ 6 6 6 6 6 4
S011
S012
S012
0
0
S022
S023
0
0
0
0
2ðS022 S023 Þ
0
S022 sym
S066
0
S0ij
is the
3
7 0 7 7 7 0 7 7 7 0 7 7 0 7 5 S066
ð23Þ
Because the mechanical properties of yarn can be regarded as transversally isotropic, there are five independent constants: E011 ; E022 ; G012 ; m012 ; m023 Which can be obtained as follows [14]:
E011 ¼ V f Ef 11 þ ð1 V f ÞEm qffiffiffiffiffiffi E022 ¼ E033 ¼ Em = 1 V f ð1 Em =Ef 22 Þ qffiffiffiffiffiffi G012 ¼ G013 ¼ Gm = 1 V f ð1 Gm =Gf 12 Þ qffiffiffiffiffiffi G023 ¼ Gm = 1 V f ð1 Gm =Gf 23 Þ
ð24Þ
m012 ¼ m013 ¼ V f mf 12 þ ð1 V f Þmm m023 ¼ E022 =ð2G023 Þ 1 As a result, the terms in Eq. (23) can be expressed by: S0ii ¼ 1=E0ii ; S0ij ¼ ðm0ij =E0ij Þði; j ¼ 1; 2; 3Þ; S044 ¼ 1=G023 ; S055 ¼ 1=G013 ; S066 ¼ 1=G012 . Finally, by substituting Eq. (12), (13), (23) into Eq. (20), the explicit formulation of the compliance matrix for the twisted yarn is obtained. 4. Strength of a twisted yarn Zhu et al. [15] has studied the axial tensile statistical strength of unidirectional fiber reinforced composites. The statistical theory can be summarized as follows. It is assumed that the probability distribution of fiber strength is Weibull function. That is
FðrÞ ¼ 1 expðadrb Þ
ð25Þ
where a, b and d are the Weibull scale parameter, the Weibull shape parameter and the ineffective length, respectively. a can be deter[16]. According to Ref. [17], it can be assumed mined by a ¼ rb f that b equals 6 in this paper.The yarn is composed of N fibers. When the tensile load P is applied on the longitudinal direction of yarn, Nf fibers are broken. When the applied load increases by dP, the average stress and the number of broken fibers will increase by dr and dNf, respectively. The broken fibers Nf should be satisfied with the following differential equation and boundary conditions [15]:
dNf dFðrÞ dFðK s rÞ dFðrÞ þ ða þ 1Þ a Nf ¼ N dr dr dr dr
Nf ð0Þ ¼ 0
ð26Þ
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where F(Ksr) is the probability of fiber breakage in which the concentration is considered. a is the number of fibers adjacent to a broken fiber. If there are 4 or 6 fibers adjacent to a broken fiber, Ks = 1.146 or 1.104, respectively. The solution of the above differential equation is
Nf ¼ N exp½ða þ 1Þ expðadrb Þ a expðadmrb ÞT
ð27Þ
R1
where T ¼ x exp½ða þ 1ÞX þ aX m dX; X ¼ expðarrb Þ; m ¼ K bs . The expectation of average fiber stress at yarn failure is expressed as follows:
r B ¼ 1 Nf ðr m Þ=N r m
ð28Þ
m is the maximum stress of unbroken fiber, which can be where r obtained by
d ð1 Nf =NÞr jr¼rm ¼ 0 dr
ð29Þ
When N is large enough, the strength distribution of yarn approaches a normal distribution obtained by Daniels [18]. The density distribution function is
" # BÞ 1 ðrB r gðrB Þ ¼ pffiffiffiffiffiffiffi exp 2w2B 2pwB
ð30Þ
where wB is standard deviation of normal distribution, which can be expressed by
wB ¼ rm fFðrm Þ½1 Fðrm Þg1=2 M 1=2 " #
1=2 2 df Ef 11 1 1 þ ð1 uÞ 1=2 d¼ 1Þ Ch ðV f 2 2Gm 2ð1 uÞ
ð32Þ
where df, Vf and Ef11 are the diameter, the volume fraction of fiber and the longitudinal elastic modulus of fiber, respectively. Gm is the matrix shear modulus. / is the fraction of the undisturbed stress value. Considering the yarn as n links in a chain, the weakest link theory [19] is applied to determinate the failure probability of yarn. The probability density distribution function is defined by
In addition, the strength properties of yarn except for longitudinal tensile strength can be obtained by
X yc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ef 11 Em V f Em ¼ 2 V f þ ð1 V f Þ Ef 11 3ð1 V f Þ
Y yt ¼
1 þ Vf
1
gy
1
K my
Y yc ¼ 4Y yt Sy ¼
1 þ Vf
1
gs
K ms
ð37Þ
1
Xm
ð38Þ ð39Þ
Sm
ð40Þ
In the above equations, Xyc, Yyt, Yyc and Sy are the longitudinal compressive, transverse tensile, transverse compressive and shear strength of the yarn, respectively. Xm is the tensile strength of matrix. gy and gs are Cai experience coefficients. Kmy and Kms are the tensile and shear stress concentration coefficients of matrix, respectively. Sm is the shear strength of matrix. gy and gs equal 0.5 [16]. 5. Finite element model of 3D four-directional braided composites Four-node tetrahedron elements are adopted to mesh the representative volume cell of 3D four-directional braided composites as shown in Fig. 6a. The regions of braid yarn have different local coordinate systems as shown in Fig. 6b. The braid yarns and matrix are T300 carbon 12 K and epoxy resin, respectively. Their properties are listed in Table 2.
Table 2 Constituent properties of composites [21]. 3D four-directional braided carbon/epoxy composites
vðrÞ ¼ ng 0 ðrÞ½1 gðrÞn1
ð33Þ
f , is obtained by The mean value of yarn strength, r
ð34Þ
Then the tensile strength of the unidirectional composites with straight yarn is expressed as
f þ ð1 V f ÞX m X tu ¼ V f r
ð36Þ
ð31Þ
The ineffective length d can be obtained by
dvðrÞ=dr ¼ 0
X yt ¼ X tu cos2 h
ð35Þ
Considering the influence of yarn distortion, the tensile strength of twisted yarn can be expressed as a function of the twisted angle h [20]:
T300 carbon fiber
Epoxy resin
Longitudinal tensile modulus Ef11 231 GPa Transverse tensile modulus Ef22 40 GPa Longitudinal Poisson’s ratio mf12 0.26 Longitudinal shear modulus Gf12 24 GPa
Tensile modulus Em 3.5 GPa Poisson’s ratio mm 0.35 Tensile strength Xm 80 MPa Compressive strength Xmc 241 MPa Shear strength Sm 60 MPa
Transverse shear modulus Gf23 14.3 GPa Longitudinal tensile strength rf 3528 MPa Longitudinal compressive strength rfc 2470 MPa Fiber volume fraction of yarn Vf 0.8 Fiber diameter df 7l
Fig. 6. The finite element model for 3D four-directional braided composites.
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Table 3 The displacement boundary conditions to obtain the different coefficients of stiffness matrix. Stiffness matrix coefficients The The The The
first column of [C]b second column of [C] third column of [C] forth column of [C]
The fifth column of [C] The sixth column of [C]
Nonzero BCs u1(L1, x2, x3) u1(0, x2, x3) = L1a u2(x1, L2, x3) u2(x1, 0, x3) = L2 u3(x1, x2, L3) u3(x1, x2, 0) = L3 u2(x1, x2, L3) u2(x1, x2, 0) = L3/2 u3(x1, L2, x3) u3(x1, 0, x3) = L2/2 u1(x1, x2, L3) u1(x1, x2, 0) = L3/2 u3(L1, x2, x3) u3(0, x2, x3) = L1/2 u1(x1, L2, x3) u1(x1, 0, x3) = L2/2 u2(L1, x2, x3) u2(0, x2, x3) = L1/2
a
ui (i = 1, 2, 3), xi (i = 1, 2, 3) and Li (i = 1, 2, 3) are the displacements, variables and edge lengths of RVC in 123 directions, respectively. b [C] is the stiffness matrix of 3D four-directional braided composites.
In order to ensure continuity of forces and displacements compatibility of the opposite faces of the RVC, the periodic boundary conditions (BCs) are imposed in the simulation [21]. To calculate the stiffness matrix, the BCs are listed in Table 3. The macroscopic strain is displacement gradient in the load direction. And the macroscopically homogeneous stress is calculated from the following form:
r ¼
P
Pi S
ð41Þ
where Pi and S are, respectively, the reaction force and area of the RVC’s surface whose normal direction parallels the load direction.
The compliance matrix coefficients of 3D four-directional braided composites are obtained by macroscopically homogeneous stress dividing macroscopic strain. The unidirectional tensile strength which is the peak value of the macroscopically homogeneous stress and strain curve of 3D four-directional braided composites can be obtained by using progressive damage method reported in Ref. [21].
6. Results and discussion Numerical results for the elastic moduli, shear moduli and the Poisson’s rate of each region of twisted yarn with material properties listed in Table 2 are shown in Fig. 7. Because the relation between deflection angles h2 in Eq. (12) and h3 in Eq. (13) is difficult to be determined, it is assumed that h2 equals h3 to simplify the discussion on the effect of twisted angle h in Eq. (14) of twisted yarn. The local coordinates of twisted yarn are shown in Fig. 4b. As apparent from Fig. 7a, the mean value of the longitudinal elastic modulus for twisted yarn, hE11i, is greatly influenced by the twisted angle. Particularly, it decreases to 40% at h = 17°. But the mean value of the transverse elastic moduli, hE22i and hE33i (hE22i = hE33i), decrease slightly with the increasing twisted angles. Fig. 7b shows the effect of twisted angle h on the Poisson’s ratios. hm12i increases by 20% at h = 17°, while hm23i decreases to 90% at h = 17°. And the effect of twisted angle on the shear moduli is negligible as shown in Fig. 7c.
Fig. 7. The effect of fiber distortion on the mechanical behavior of twisted yarn surface regions.
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Fig. 8 shows the effect of fiber distortion on the mechanical behavior of 3D four-directional braided composites. The superscript ‘o’ denotes that the braid yarns are straight and aligned. The coordinates for the braided composites are shown in Fig. 6a. As shown in Fig. 8a, most severe effect is obtained for the longitudinal elastic modulus hE22i of the braided composites. It drops by 15% at h = 17°. The transverse elastic moduli, hE11i and hE33i, increase slightly with the increase of the twisted angle h. As shown in Fig. 8b, the Poisson’s ratios, hm12i and hm23i, decline with the increase of twisted angle, while hm31i slightly increases. Fig. 8c shows that the effect of yarn distortion on the shear moduli of the 3D braided composites is severe. With the increase of twisted angle, the shear moduli all decrease. Especially hG12i and hG23i decrease to 85% at h = 17°. Comparing with Figs. 7 and 8, the tendencies of longitudinal elastic modulus of yarn and braided composites are uniform. However, the tendencies of other elastic moduli, shear moduli and Poisson’s ratios are reverse. The phenomena can be attributed to the great decline of longitudinal elastic modulus of twisted yarn with increase of twisted angle. The longitudinal strength of twisted yarn decreases with increase of twisted angle according to Eq. (36). The strength of the braided composites can be obtained when the stiffness and strength of twisted yarn are introduced to the progressive damage model of the braided composites elaborated in Ref. [21]. Fig. 9 represents the variation curves of strengths with twisted angles in two-direction tension as shown in Fig. 6a. Again, the superscript
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Fig. 9. The effect of fiber distortion on the unidirectional tensile strength of the braided composites.
‘o’ denotes the braid composites with straight and aligned braid yarns. The ordinate represents the strength ratios of braid composites with twisted braid yarns to aligned braid yarns. As shown in
Fig. 8. The effect of fiber distortion on the mechanical behavior of the braided composites.
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Fig. 9, two cases of braid parameters as listed in Table 1 are chosen. In both cases, the strength ratios of the braided composites decline with the increase of the interior braid angles. The strength of the braided composites with 45° interior braid angle decreases more quickly than that with 30° interior braid angle when twisted angle of braid yarn is under 9.5°. But the tendencies become reverse when twisted angle of braid yarn is above 9.5°. The strength of the braided composites with interior braid angle 45°and 30° decrease to 72.2% and 66.6% at h = 17°, respectively. 7. Conclusions In order to consider the effect of yarn distortion on mechanical properties of braided composites, each braid yarn is divided into seven regions. The stiffness of each region of twisted yarns is obtained by stochastic function theory. The statistical strength of twisted yarn is considered by statistical theory as well. By introducing the stiffness and strength properties of twisted yarn into finite element model of the representative volume cell (RVC), the elastic properties of the braided composites are obtained. The effect of twisted yarn on the unidirectional tensile strength of 3D four-directional braided composites is considered as well. The significant effect is obtained for the longitudinal elastic modulus and longitudinal Poisson’s ratio of the twisted yarn. The engineering elastic constants except for transverse elastic modulus and transverse Poisson’s rate for 3D four-direction braided composites are highly sensitive to the twisted angle of yarns. The unidirectional tensile strengths of 3D four-directional braided composites with interior braid angle 45°and 30° both decrease with increase of twisted angle of braid yarn. The tensile strength of the braided composites with interior braid angle 45° decreases more quickly than that with interior braid angle 30° when twisted angle is under 9.5°, while it is reverse when twisted angle is above 9.5°. As generally recognized, the magnitude of twisted angle of braid yarns is controlled by the manufacturing process. Therefore, it is imperative to study the relation between the twisted angle and pre-tensile loading in the manufacturing process further. Acknowledgments This work is supported by the National Natural Science Foundation of China (10772060), Hei Longjiang Province Outstanding
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