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Meso-macro numerical modeling of noncircular braided composite parts based on braiding process parameters
Composite Structures 224 (2019) 111065
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Meso-macro numerical modeling of noncircular braided composite parts based on braiding process parameters
T
Jalil Hajrasoulihaa, Reza Jafari Nedoushanb, , Mohammad Sheikhzadeha, Tohid Dastana ⁎
a b
Department of Textile Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran
ARTICLE INFO
ABSTRACT
Keywords: Noncircular braided composites Meso scale relationships Finite element modeling Variable stiffness Bending test
In the past few decades, the applications of composite structures reinforced with braided preforms have been growing increasingly in various industries due to their intrinsic properties. Therefore, their mechanical properties are worth studying experimentally, analytically and numerically. In this respect, the goal of the current study is to perform a finite element modelling on biaxially and triaxially braided composites with constant arbitrary cross-section. Thus, the braid angles formed by clockwisely and counter-clockwisely moving of carriers on noncircular parts, have been initially demonstrated to be unequal, unlike in circular parts. As a result, considering a parallelogram shape for the representative unit cell (RUC), the geometrical relationships for a RUC of 2D biaxially and triaxially braided composite are derived and implemented in VUMAT subroutine. Then, after determination of 3D effective stiffness matrix for each RUC, the finite element modelling is performed to predict bending behavior of final braided composite part. In these simulations, the stiffness variation all over the part is considered and mechanical behavior of composite parts is predicted based on the initial parameters of the braiding process. The obtained results from both experimental and simulations showed an acceptable agreement.
1. Introduction Composites reinforced with braided preforms have found vast application area in various industries due to their intrinsic properties such as high load bearing capacities, superior toughness and fatigue strength than conventional composite structures, the ability of being fabricated near-net-shape preform and cost savings because of rapid manufacturing process [1–3]. Among available braiding techniques, circular braiding process, which involves depositing bias and optional axial yarns on a mandrel, is the most common technique in manufacturing circular-shaped braided composite parts [4]. The mechanical properties of braided composites are affected by various variables (braid angle, fabric cover factor and yarn volume fraction); hence, many researchers put a lot of effort into investigating the effect of mentioned variables on the performance of braided composites experimentally and analytically. In this regard, various tests like tension, bending, compression and shear tests were performed to study the braided composite properties [5,6]. The mechanical properties of triaxially braided composites were studied experimentally and analytically by Masters et al. [7]. They focused on different reinforcing fiber architectures in braided composite parts, which govern the composite behavior under tensile loading. Potluri et al. performed flexural and ⁎
torsional tests on biaxially and triaxially braided composites [8]. The effect of number of layers and braid angles was considered in their work. The braid angle, along with the cover factor, is the key parameter in circular braiding process which governs the mechanical properties of braided composites [9]. Thus, knowing and controlling the braid angle in regard to the braiding process parameters is crucial in achieving control on mechanical properties of final product. In this respect, Du and Popper [10] suggested a model based on kinematic analysis for the geometry of fiber structures deposited on an axisymmetric mandrel in braiding process. The proposed model is based on braid angle, fabric cover factor and yarn volume fraction. Zhang et al. [11] determined the braid angle by considering the interlacing forces through a mechanical model. They concluded that the mechanical model is superior in predicting the final braid structure than kinematic analysis. A 3D geometrical modelling of tubular braids was proposed by Tuba [12] which is applicable for different braid structures with varying parameters like braid angle, yarn and mandrel diameter. The elastic properties of triaxially braided composites could be determined using a simple analytical model proposed by Redman and Douglas [13]; however, the braided preform was considered to be composed of three separate plies and the undulating fibers were neglected. The effect of braid angle,
Corresponding author. E-mail address: [email protected] (R. Jafari Nedoushan).
https://doi.org/10.1016/j.compstruct.2019.111065 Received 12 December 2018; Received in revised form 8 April 2019; Accepted 27 May 2019 Available online 29 May 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.
Composite Structures 224 (2019) 111065
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Fig. 1. The plot of obtained braid angles for mandrels with elliptical and egg-shaped cross-sections; the red and blue line corresponds to clockwisely and counterclockwisely braider yarns, respectively.
yarn size and axial yarn content was investigated analytically and experimentally on 2D triaxially braided composites [14]. A volume averaging approach, considering the iso-strain assumption, was employed to determine the three-dimensional effective stiffness of the representative unit cell (RUC). It was found that the elastic properties were only affected by braid angle and axial yarn content. Another analytical study based on curved unit-cell geometry was proposed by Ayranci and Carey to predict the elastic constants of braided composites [15]. In addition to proposed models based on analytical methods, other accurate modelling techniques such as numerical modelling were used to obtain the elastic properties of braided composite. Ji et al. performed
a multi-scale modelling, which involved three steps, to determine the elastic properties of braided composites [16]. Xu et al. analyzed biaxially and triaxially braided composites with different weave patterns using finite element (FE) method through applying periodical boundary conditions on RUCs [17]. Pickett et al. compared the analytical and FE modelling of 2D braiding. They demonstrated that analytical approach is computationally fast; however, the FE method is capable of accurately predicting yarn paths and interactions [18,19]. The effect of yarn shape, lenticular and flattened shapes, on mechanical behavior of braided composites was investigated by Goyal et al. [20]. To date, few of studies dealt with predicting braid angle and mechanical properties of braided composite with constant arbitrary cross2
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braiding machine. In their study, they focused on the position of fell point on the surface of mandrel and the yarn length between fell point and the carrier, which considerably affect the braid angle. Followed by our research on the theoretical study of braid pattern in mandrels with arbitrary cross-section as published in [22] the aim of the current research is to propose a FE modelling through developing the meso scale linear elastic stress-strain relation for both biaxially and triaxially braided composites on noncircular mandrels. In this regard, it is initially shown that the braid angles formed by clockwisely and counter-clockwisely moving of carriers on an arbitrary cross-section are not equal. Consequently, the shape of the RUC of braided preforms changes from diamond shape into a parallelogram shape. Furthermore, geometrical relationships for a RUC of 2D biaxially and triaxially noncircular braided composite will be derived. Using these relations, the stiffness variation induced by variable braid angles on each point of a noncircular braided composite can be calculated and be used in FE modeling. The stiffness is calculated based on the braiding process parameters and in this manner the final part behavior is directly obtained from braiding process. 2. Numerical modeling Fig. 2. The geometry of RUC for mandrels with arbitrary cross-sections.
Predicting properties of composite parts, through numerical modeling helps the corresponding industry in saving cost and time. Since several parameters are in braiding process that have influence on the braided composite sample properties, in this study both of meso and macro scale aspect of the parts are considered. The mechanical behavior of a material point in the braided part was estimated by the meso scale approach that was implemented in ABAQUS software by user material subroutine (VUMAT). Using this subroutine, FE modeling of
sections. On this point, Hans et al. proposed a simulation approach based on FE modelling which is suitable for any mandrel geometry [21]. They aimed at analyzing braided structures with respect to yarn alignment. Hajrasouliha et al. [22] developed a theoretical model to predict the braid angle in braided composites with constant non-circular cross-section by considering the kinematic parameters of circular
Fig. 3. Pictures taken from braided mandrels with a) an elliptical cross-section with a = 3.9, b = 2.15, b) an egg-shaped cross-section. 3
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Fig. 4. (a) RUC and (b) cross-section view of yarn path of 1 × 1 2D biaxially braided composites.
both biaxially and triaxially braided composite samples with various cross-section profiles was carried out. Using the VUMAT subroutine the variable mechanical properties of the braided composite can be included in the modeling. In the following at first meso scale relationships which were used in VUMAT subroutine are discussed and then FE simulation in macro scale is explained.
et al. [22] on noncircular mandrels, the braid angles at any arbitrary position on mandrel for each sets of yarn (clockwise and counterclockwise yarns) are not equal. Therefore, a Matlab code was written on the basis of this theory [22] for tracing both braid angles formed by sets of yarn over a mandrel. Fig. 1 shows the obtained results for braid angles formed on the upper half of the elliptical and egg-shaped crosssections. Thus, the geometry of RUC, in the case of mandrel with constant arbitrary cross-section, has the shape of parallelogram, as schematically shown in Fig. 2. Furthermore, in order to verify the geometrical shape of RUC, pictures were taken using a digital camera normal to the surface of a braided mandrels. Two pictures are shown in Fig. 3 as examples, which were taken from the surface of an elliptical crosssection with a = 3.9, b = 2.15 and from an egg-shaped cross-section. Since braid angles formed by clockwisely and counter-clockwisely moving of carriers on a constant arbitrary cross-section are not equal anymore, available derived geometrical relations for 2D biaxially and triaxially braided composites with circular cross-section by other studies [10–15] are not appropriate to be used as a basis of numerical
2.1. Calculation of stiffness matrix using meso scale approach 2.1.1. Determining the RUC of a braided composite with constant arbitrary cross-section It is known that the geometry of RUC in braided composite with circular cross section is diamond-shaped, since both braid angles, which are formed by clockwisely and counter-clockwisely moving of carriers, are equal and are marked respectively as + b and b [22]; however, this is not true anymore in the case of braiding a mandrel with constant arbitrary cross-section based on the findings of previous study [22]. That is, according to the theoretical model proposed by Hajrasouliha 4
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determined is Lu' , which could be obtained using the following equation:
L'st 2
(L'u)i = S'i
i = 1, 2
(4)
where:
S'i =
sin(
L'st =
modeling of this type of braided composites. Therefore, as is described in detail in the following, the governing equations in geometrical modelling of RUC of 2D biaxially and triaxially noncircular braided composites derived.
(1)
(L 'u)i 2
x
(L 'u) i 2
i = 1, 2
i = 1, 2
S1 =
2B v sin( N
b2 )
S2 =
2B v sin( N
b1)
i = 1, 2
(6) (7) (8)
(L'u)1 =
2B v sin( N
b2 )
(L'u)2 =
2B v sin( N
b1 )
(L pb)i = 4(S'b) i
L st 2 sin(
b1
1 +
L st 2 sin(
b1
(9)
b2 )
1 +
(10)
b2 )
(11)
i = 1, 2
Furthermore, vertical distance between axial yarns (da ) can be obtained based on the presented schematic in Fig. 6(a):
da = (s'b)2sin b2 + (s'b)1sin b1 2B v = (sin b1sin Nsin( 1 + 2 )
da =
(2)
The spatial orientation of yarn could be defined by the in-plane angle (which is the braid angle b ) and the out-of-plane angle (which is between the tangential line to the sinusoidal path and XY plane), as shown in Fig. 5. Thus, differentiating the undulating function with respect to x gives the angle :
dz(x)i t x = |z' (x)i| = |tan i| = cos( ' ) dx 2(L'u)i (L u)i
b2 )
2.1.3. Geometrical modelling of noncircular 2 × 2 2D triaxially braided composites The RUC and cross-section view of yarn path in the weave (BB section) corresponded to2 × 2 2D triaxially braided composites is schematically shown in Fig. 6. In order to implement the same approach on the RUC 2D triaxially braided composite, some general parameters such as vertical distance between axial yarns (da), number of filaments in braider yarn (nb), number of filaments in braider yarn (na), the diameter of braider yarn filaments (dfb), the diameter of axial yarn filaments (dfa), the thickness of braider yarn (tb) and the thickness of axial yarn (ta) must be specified to determine their cross-section areas, the yarn geometrical path and subsequent yarn spatial orientation. In the following, it should be noted that subscripts “a” and “b” are related to axial and braider yarns, respectively. Applying the same approach, the length of the projected yarn (L pb)i along the x-axis in an RUC can be defined as following:
where the index ‘i” is attributed to braider yarns, in which i = 1 and i = 2 is corresponded to braider yarns having braided angles + b1 and b2 , respectively. The yarn path, as shown in Fig. 4(b), comprises of two straight and three undulating portions in which the undulating path follows the sinusoidal function. Therefore, the path function for the braider yarn in the undulating portion could be obtained as a function of Lu' and yarn thickness t:
t x z(x)i = ± sin 2 (L 'u) i
L st +
b1
(5)
The projection of undulating portion length of yarn path can be determined using a new approach, which is based on cross-section area of yarn, and will be discussed later.
)
i = 1, 2
sin(
i = 1, 2
(Lu' )
2.1.2. Geometrical modelling of noncircular1 × 1 2D biaxially braided composites The RUC and cross-section view of yarn path in the weave (AA section) corresponded to1 × 1 2D biaxially braided composites is schematically shown in Fig. 4. It should be noted that the RUC has the shape of parallelogram, based on the earlier findings. As could be observed, a global coordinate system (XYZ) is considered by such manners H H Z that Z-axis is along the thickness (H) direction and braid 2 2 angles are defined with respect to Y-axis (+ b1 and ), as depicted b2 in Fig. 4(a). Besides, a local coordinate system (xyz) is considered to determine yarn path in a way that x-axis is along the yarn projection in XY plane and z-axis is in Z direction, as shown in Fig. 4(b). Thus, all the geometrical parameters along the yarn path will be projected over the related axis. The length of the projected yarn (Lp) along the x-axis in an RUC can be defined as following, which is the summation of both undulating (Lu' ) and straight (Lst' ) portions of yarn path projection over x-axis:
(L p)i=2(L'u)i + L 'st
b2 )
where B v is Take-up length per a carrier revolution (cm/rev) and N is number of carriers. By substituting Eqs. (5)–(8) in Eq. (4) yields:
Fig. 5. A schematic of considered yarn spatial orientation.
(
Si +
b1
4B v sin b1sin b2 Nsin( b1 + b2 )
(d'a ) i =
da sin bi
b2
+ sin
b1sin b2 )
(12) (13)
i = 1, 2
(14)
The length of axial yarns in a RUC can also be calculated using the following equation:
La = 4da (cotg
b2
+ cotg
b1)
(15)
Considering the sinusoidal function for undulating portion, so the path function for the braider yarn in the undulating portion could be ' obtained as a function of Lub and subsequent differentiating with
(3)
According to Eq. (3), the only unknown parameter that should be 5
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Fig. 6. (a) RUC and (b) cross-section view of yarn path of 2 × 2 2D triaxially braided composites.
6
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observed, since the arc “AB” is a segment of a circle with the center of “O” and the radius of “r”, then the area of section “ABC” could be calculated from parameters “a” and “t”:
1 2 r 2
AreaABC =
1 a r 2
t 2
(20)
where according to the Fig. 7:
r=
Fig. 7. A schematic of yarn cross-section in 2D braided composites.
respect to × will yield the angle :
z(x) bi = ±
|tan
b i|
=
(t a + t b) 2(L 'ub) i
cos
= sin
(L 'ub)i 2
t a + tb x sin 2 (L 'ub)i x
x
(L 'ub)i i = 1, 2 2
i = 1, 2
(L 'ub) i
(L'ub) i = (d'a ) i
Area = 2sin
da sin bi
L st b b1 +
b2 )
(18)
i = 1, 2
1
4at + t2
4a2
4a2 + t2 4t
2
4a3 + 3at2 + wt 2t df 2nf 4pd
(23) ”, where df
is the diameter of each filament, nf is the number of filaments in the yarn cross-section and Pd is the packing density, which its value is considered in the range of 0.7–0.8 according to the image processing observation. The unknown parameter “a” and subsequently “b” can be ' calculated by iteratively solving Eq. (23). Consequently, Lu' and Lub are now a known parameter for 2D biaxially and triaxially braided composites.
'
2sin(
(22)
The yarn cross-section could be measured as “ A =
where d a can be obtained using Eq. (14) and L stb can be determined using the yarn cross-section area approach, which is discussed in the following. Therefore, Eq. (18) can be rewritten as:
(L'ub) i =
a r
(17)
i = 1, 2
'
1
(21)
Thus, the yarn cross-section can be obtained as a function of known parameters “w” (yarn width), “t” (yarn thickness) and the unknown parameter “a”:
(16)
According to Eq. (17), the only unknown parameter that should be ' determined is Lub , which could be determined using the following equation, based on Fig. 6(b):
L 'st b 2
a2 t + t 4
(19)
2.2. Determination of 3D effective stiffness matrix for each material point of noncircular braided composites
2.1.4. Determination of Lu' by using the cross-section area approach The calculation of Lu' and Lu' b , in 2D biaxially and triaxially braided composites in that order, can be done by determining the area of yarn cross-section. Based on the studies [24–34], the yarn cross-section in 2D braided composite structures is supposed to be composed of a flat and two lenticular-shaped sections, as shown in Fig. 7. As could be
In this study, a braided composite part with constant arbitrary crosssection is assumed an anisotropic linear elastic material with variable stiffness in different points. A typical stress-strain relation for this kind of material in a Cartesian coordinate system (X-Y-Z) is as follows:
Fig. 8. A typical assembly of FE modelling of braided composite sample with egg-shaped cross-section under four-point bending. 7
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Table 1 The characteristics of carbon fiber roving. Number of filaments
Density (kg/m3)
Filament diameter ( µm)
Fiber count (Tex )
Tensile strength (MPa )
Failure strain (%)
12,000
1740
7
800
1447.3
2.00
In order to obtain the stiffness matrix of an RUC of 2D biaxially braided composite, according to Eq. (26), the volume of each yarn segment must be multiplied by its corresponded stiffness matrix with respect to the segment spatial orientation (angles and b ), so:
Table 2 The mechanical properties of carbon fiber roving (Reference). Carbon fiber type
E1f (GPa)
E2f (GPa)
G12f (GPa)
G23f (GPa)
v12f
v23f
T300-12 k
126.02
23.1
27.6
14.3
0.25
0.3
V+ b1
Table 3 The mechanical properties of cured bone-shaped epoxy resin.
V b2
Epoxy resin type
Density (kg/m3)
Tensile strength (MPa )
Tensile modulus (GPa )
Poisson’s ratio
DSM EP411
1200
50
1.12
0.3
X Y Z YZ
=
ZX XY
C12 C22 C32 C42 C52 C62
C13 C23 C33 C43 C53 C63
C14 C24 C34 C44 C54 C64
C15 C25 C35 C45 C55 C65
C16 C26 C36 C46 C56 C66
X Z YZ ZX
(24)
where and are stress and strain, respectively, and C is the stiffness matrix with 21 engineering constants. The aim of the current section is to find these 21 constants based on the braiding parameters by meso scale relations. The first step to calculate stiffness matrix was to choose an appropriate RUC of the material and derive geometrical relations of yarns paths in the RUC, which is already done and discussed in detail in previous sections.
1 VRUC
V
CdV
V+ b 1
C+
+2
V b 2
C
b2 dV
+
VResin
V b2
b1
,
)dV
C(
b2
,
)dV
(27) (28)
(29)
VResin
V+ b1
C+
b1 dV
+4
V b2
C
b2 dV
+
CResin dV
Va
Ca dV (30)
where C+ b1, C b2 and CResin are determined using Eqs. (27)–(29), respectively. The stiffness matrix of axial yarns, considering the fact that axial yarns follows a straight path in RUC, is defined as: Va
Ca dV =
Va
Ca (0°, 0°)dV = Ca Va
(31)
Since, the stiffness matrices attributed to braider yarns (Eqs. (27) and (28)), which are used for both 2D biaxially/triaxially braided composites, are obtained in yarn local coordinate system (1-2-3), thus in order to determine the global stiffness matrix, a transformation must be made for these local stiffness matrices from the yarn local coordinate system (1-2-3) to global coordinate system (X-Y-Z).
(25)
b1 dV
=
C(+
CResin dV = CResin VResin
+
2.2.3. Transformation matrix from yarn local coordinate system (1-2-3) to global coordinate system (X-Y-Z) In order to calculate the transformation matrix, considering a segment of braider yarn which is shown in Fig. 5, two consecutive transformations must be done. First, a transformation must be made from local yarn coordinate system (1-2-3) to coordinate system (x-y-z) by rotating around the y axis using the angle , which its calculation is discussed earlier. Second, a transformation must be done from the coordinate system (x-y-z) to global coordinate system (X-Y-Z) by rotating
(CRUC ) 2axial 1 2 VRUC
b2 dV
1 4 VRUC
=
The above relationship is driven under the iso-strain assumption. Since 1 × 1 2D biaxially braided composites with arbitrary crosssection includes braider yarns with braid angles of + b1 and b2 , and resin, therefore the RUC stiffness matrix for 1 × 1 2D biaxially braided composite is obtained from the summation of each stiffness matrix attributed to braider yarns and resin:
=
C
V+ b1
(CRUC )3axial
2.2.1. Calculating the stiffness matrix of noncircular 1 × 1 2D biaxially braided composites As previously shown in Fig. 4(a), the RUC of 1 × 1 2D biaxially braided composites is composed of three linear elastic phases including both braider yarn (with the angle of + b1and b2 ) and matrix. The stiffness matrix for an RUC (CRUC ) of a textile composite can be estimated as:
CRUC =
=
2.2.2. Calculating the stiffness matrix of noncircular 2 × 2 2D triaxially braided composites The RUC stiffness matrix of 2 × 2 2D triaxially braided composites is derived similar to what have been discussed in previous section. The volume of 2D triaxially braided composite’s RUC consists of four braider yarns (for each braid angles + b1and b2 ), axial yarns and resin. Therefore, the stiffness matrix of RUC is defined as:
Y
XY
b1 dV
It should be noted that the value of angle is determined through Eq. (3). The stiffness matrix of resin, as a homogenous and isotropic material, is defined as: VResin
C11 C21 C31 C41 C51 C61
C+
CResin dV (26)
Table 4 General specifications of circular braiding machine. Number of carriers
Number of braiding carbon yarns
Number of axial carbon yarns
Radius of main braiding plate (R g ) (cm )
Mean revolutionary velocity of carriers ( c ) (rad/s)
Take-up velocity (V ) (cm/s)
Take-up length per a carrier revolution (B v ) (cm/rev)
48
48
24
115.25
0.14
0.19
8.5
8
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Fig. 9. Four mandrels with egg-shaped cross-sections a) before and b) after the tape covering process.
Fig. 10. A typical pictures of braiding process on mandrels with circular (a) elliptical (b and c) and egg-shaped (d) cross-sections.
[T ]
=
a112 a12 2 a132 a11a12 a11a13 a12a13 a21a22 a21a23 a22a23 a212 a222 a232 a31a32 a31a33 a32a33 a312 a322 a332 2a11a21 2a12a22 2a13a23 a11a22 +a12 a21 a13a21 + a11a23 a12a23 + a13a22 2a11a31 2a12a32 2a13a33 a31a12 +a32a11 a11a33 + a13a31 a32a13 + a33a12 2a21a31 2a22a32 2a23a33 a21a32 +a22a31 a23a31 + a21a33 a22a33 + a23a32
Fig. 11. Prepared biaxially/triaxially braided composite samples with various cross-section shapes.
(34)
around the Z axis using the angle b , which is the braid angle. The transformation matrix ([a ij ]3 × 3 ) which transforms form the yarn local system (1-2-3) to global system (X-Y-Z) is defined as:
[aij]3× 3 =
cos cos b sin b sin cos b
cos sin b sin cos b 0 sin sin b cos
C1 2 3 in Eq. (33) is the stiffness matrix of impregnated yarn that is discussed in the next section. 2.2.4. Stiffness matrix of impregnated yarn in local coordinate system The impregnated carbon fiber with resin is considered as unidirectional reinforced composite material, which its properties are assumed to be transversely isotropic. In the other words, five independent engineering constants are required to define the mechanical behavior of an impregnated carbon fiber with resin. In this regard, the Chamis micromechanical model [23] was used to obtain these five independent engineering constants, which are as follows:
(32)
From tensor transformations relations it could be shown that [35]:
CX
Y Z
= TT C1
2 3T
(33)
where [35]:
(35)
E1 = Ef Vf + Em Vm Em
E2 = 1 9
(
Vf 1
Em Ef
)
= E3 (36)
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Table 5 Characteristics of prepared biaxially braided composite samples. Sample Coding
Type of cross-section
Mean thickness (mm )
Mean width of axial rovings (mm )
Mean width of braiding rovings (mm )
Mean thickness of axial rovings (mm )
Mean thickness of braiding rovings (mm )
2AxCirc 3AxCirc
Biaxially Triaxially
0.7 1.35
– 3.2
4.6 2.84
– 0.22
0.15 0.27
2AxElip35 3AxElip35
Biaxially Triaxially
0.72 1.2
– 3.05
4.4 2.97
– 0.25
0.16 0.26
2AxElip39 3AxElip39
Biaxially Triaxially
0.78 1.3
– 2.96
4.22 2.86
– 0.26
0.17 0.27
2AxEgg 3AxEgg
Biaxially Triaxially
0.7 1.15
– 3.3
4.1 3.1
– 0.22
0.18 0.25
Fig. 12. A typical braided composite sample under four-point bending test. Table 6 Comparing the stiffness values obtained from experimental and FE modelling. Sample Coding
2AxCirc 3AxCirc 2AxElip35 3AxElip35 2AxElip39 3AxElip39 2AxEgg 3AxEgg
Maximum Force (N )
Experimental Stiffness (N.mm−1/mn)
Numerical Stiffness (N.mm−1/mn)
Stiffness Difference (% )
471.75 1211.40 481.75 967.90 390 1075.75 185 527.25
106.43 242.89 87.15 190.92 62.82 191.73 33.12 109.56
93.08 218.86 78.64 175.49 66.97 167.11 22.25 88.83
12.54 9.89 9.77 8.07 6.60 12.84 32.83 18.93
Gm
G12 = 1
(
Vf 1
Gm Gf
)
1
Vf 1
Gm G 23f
=
23
=
f
Vf +
E2 2G23
m Vm
=
13
1
(39) (40)
where Vf and Vm are the fiber volume fraction and matrix volume fraction, respectively. It should be noted that the fiber volume fraction is equal to the fiber packing density (Pd). The stiffness matrix of resin is constructed by E and v based on stiffness matrix of an isotropic material. 2.3. Macro scale finite element modeling The main difference between mechanical properties of a circular braided composite part and a part with arbitrary cross section is due to their braid angles. While braid angles are equal for both clockwise and counterclockwise yarns and is constant around the section of a circular part, in braiding of an arbitrary cross section, braid angles vary from point to point and also each material point contain two different braid angles for clockwise and counterclockwise yarns. To use the above presented meso relationships, at first step, braid angles should be estimated at each material points. Our previously published relationships
= G13 (37)
Gm
G23 =
12
(38) 10
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Fig. 13. Force-displacement curves obtained from experimental and FE modeling for biaxially braided composite samples (a: 2AxCirc, b: 2AxElip35, c: 2AxElip39, d: 2AxEgg, see Table 5).
[22] were used to estimate the braid angles in each material points of each cross sections. Then using the presented meso scale relationship and braid angles of a point, stiffness matrix of each point is calculated using the VUMAT subroutine. This subroutine was written using FOTRAN programming language in Microsoft Visual Studio (version 2008) for both biaxially and triaxially braided composites according to the earlier discussed geometrical equations to determine the effective stiffness matrix of each material point. After calculation of the braid angles of each material point based on braiding process parameters these angles are saved in a text file in order of the coordinates of the point and this file is used by VUMAT subroutine to read braid angles of each point. Using this subroutine, stiffness of a braided composite can be calculated in each point of the braided part based on the estimated braid angles. In this section a macro scale FE simulation was carried out using ABAQUS software to analyze braided parts. To this aim four-point bending tests on biaxially and triaxially braided composite samples with circular, elliptical and egg-shaped cross-sections are simulated. At first the loading support, which includes two loading cylinders and two support cylinders, was designed as rigid half cylinders with the diameter and the length of 4.5 cm and 130 cm, respectively. Then, biaxially and triaxially braided composite samples were designed based on the geometrical parameters of cross-section shape, which were measured from experimental samples. A C3D8R element type, which is an 8-node linear brick with reduced integration element, was used to discretize composite parts.
After assigning the properties to the defined sections on composite parts, all parts were assembled in their proper position in accordance with the four-point bending loading, as shown in Fig. 8. Then, a surface to surface contact, based on penalty method, was defined between four cylinders and composite specimen. All the degree of freedom of supporting cylinders, which were defined as analytical rigid materials, was fixed. The magnitude of displacement applied on loading cylinders was 40 mm in Y-direction. Finally, using prepared VUMAT subroutine the FE modeling was done by ABAQUS explicit solver considering composite parts as variable stiffness parts. 3. Experimental procedure To validate the implemented FE modelling and all the derived geometrical and mechanical equation to determine the stiffness in each point of 2D biaxially and triaxially braided composite, an attempt was made to fabricate 2D biaxially and triaxially braided composite samples. Therefore, in this section materials and experimental procedure which are used to manufacture both biaxially and triaxially braided parts are described. Then the method of carrying out the four-point bending test is also explained. 3.1. Materials T300-12K carbon fiber roving and two-part EP411 epoxy resin were 11
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Fig. 14. Force-displacement curves obtained from experimental and FE modeling for triaxially braided composite samples (a: 3AxCirc, b: 3AxElip35, c: 3AxElip39, d: 3AxEgg, see Table 5).
used as reinforcement part (braiding and axial yarns) and matrix part, in that order, in both biaxially and triaxially braided composites. The width and thickness of carbon fiber roving, in as received state, was measured 6.15 mm and 0.13 mm, respectively. In order to obtain tensile properties of carbon fiber roving, a unidirectional tensile test was performed on it based on the ASTM D4018-17 standard test method instructions. Some of the carbon fiber roving characteristics are tabulated in Table 1. The mechanical properties (engineering constants) of the carbon fiber roving are also presented in Table 2. The characteristics of two-part EP411 epoxy resin (cured with the curing agent H3 by mixing ratio of 10:1) are presented in Table 3. The presented values are obtained through performing the uniaxial tensile test on cured bone-shaped epoxy resin according to the ASTM D638-14 [36] standard test method instructions.
process using a circular braiding machine. Table 4 shows general specifications of circular braiding machine. The surface of each prepared mandrel was covered with tape and subsequent releasing agent was applied to facilitate the removal of braided composite sample from the mandrel and to avoid any significant damages to the specimen during removal process. Fig. 9 shows four mentioned mandrels before and after the tape covering process. It should be noted that, for each cross-section, four identical mandrels were prepared so that two of them were used for biaxially braiding process and two other mandrels were used for triaxially braiding process. Further, braiding process was performed on prepared mandrels to produce braided composite samples. Fig. 10 shows the mandrels during the braiding process. 3.2.2. Biaxially and triaxially braided composite samples fabrication After mandrels were braided biaxially and triaxially, the matrix part was evenly applied on braided carbon fiber roving. An attempt was taken to keep the fiber volume fraction of all braided composite samples at 0.5. Soon afterwards a uniform pressure was applied on the impregnated braided samples by evenly tightly wrapping the heat shrink plastic film around them and subsequent heating using a heat blower. The braided composite samples were cured based on matrix technical data sheet instructions. Finally, after unwrapping the plastic film, fully cured samples were easily removed from their mandrels. Fig. 11 shows the typical biaxially and triaxially braided composites with various cross-sections and characteristics of prepared biaxially and
3.2. Biaxially and triaxially braided composite sample manufacturing To produce braided composite samples with various cross-sections, different mandrels were braided with carbon rovings and then were impregnated with resin, as it is discussed in the followings. 3.2.1. Braiding process Four wooden mandrels with different cross-section shapes, including one circular, two elliptical and one egg-shaped (non-circular cross-sections are shown in Fig. 1), were prepared prior to braiding 12
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Fig. 15. Variation of braid angle around the cross-section of braided composite samples (a: 2AxCirc, b: 2AxElip35, c: 2AxElip39, d: 2AxEgg, see Table 5).
triaxially braided composite samples are given in Table 5.
composite samples from experimental and FE modelling results were compared and presented in Table 6. Since fabricated braided composite samples were different in thickness, the obtained stiffness values were normalized by thickness. One can see that the difference between stiffness values predicted by FE modelling, for each braided composite samples, and experimental results are acceptable. Figs. 13 and 14 show the obtained results from both experimental and FE modelling for biaxially and triaxially braided composites with different cross-sections, respectively. As could be observed, there is a good agreement between the experimental and FE modelling results in the elastic region, considering the fact that only the linear elastic behavior of braided composites was modeled. From the obtained results it could be inferred that all the mechanical relations and equations that were derived based on the current geometrical modeling of RUC of both biaxially and triaxially braided composites with constant arbitrary cross-section are of good accuracy. Therefore, these derived equations and subsequent FE modeling can be used to analyze braided structures with constant arbitrary cross-sections. It is worth mentioning that one source of the discrepancy of the predicted and measured stiffness can be the inaccuracy of the predicted braid angles due to ignoring the real conditions including: the effect of the yarn interaction and friction with mandrel [22]. As it was mentioned earlier in current simulations, braid angles and consequently material stiffness matrix vary around the part. The predicted braid angles variation along the length of braided composite samples, which were used in VUMAT subroutine, is presented in Fig. 15. The variation is the same as variation presented in Fig. 1 and is the results of relations presented by Hajrasouliha et al. [22]. It is worth mentioning that braid angle variation can be more remarkable in
3.3. Four-point bending test Four-point bending test was performed on biaxially and triaxially braided composite samples in accordance with ASTM D6272-17 [37] standard test method instructions. This test was carried out on Hounsfield testing machine with a 50 kN load cell and the rate of displacement was 10 mm/min for all bending tests. The span length was also kept 24 cm for all biaxially and triaxially braided composite samples. Fig. 12 shows a typical composite sample under four-point bending. As previously noted, four-point bending test was performed on two specimens for each biaxially and triaxially braided composite samples. 4. Results and discussion Table 6 presents the measured experimental results, maximum force and stiffness, for both biaxially and triaxially braided composite samples. As can be seen, in the case of comparing the biaxially and triaxially braided composite samples for each cross-section shape, introducing carbon rovings in longitudinal direction of tubular braided parts contribute a considerable enhancement in their flexural properties. Considering the egg-shaped braided composite as an example, its maximum flexural load and stiffness increased approximately 185% and 230%, respectively, when carbon rovings are added in its longitudinal direction. This shows the flexibility of braiding process in designing braided structures with tunable mechanical properties, as required. In addition, the obtained stiffness value for each braided 13
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Fig. 16. Predicted stress for biaxially braided composites for different cross-sections at displacement of 24 mm (a: 2AxCirc, b: 2AxElip35, c: 2AxElip39, d: 2AxEgg, see Table 5).
special cases like asymmetrical braiding process. Details about the effect of the braiding condition on the braid angle variation can be find in [38]. In analyzing braided parts with large braid angle variation, the effect of the current calculation on the result will be more significant. Figs. 16 and 17 show the stress distribution for biaxially and triaxially braided composite samples, respectively. And, as an example, Fig. 18 shows the deformation of egg-shaped biaxially braided composite sample under four-point loading obtained experimentally and numerically. Based on the findings of the current research, it was demonstrated that considering the shape of RUC of biaxially and triaxially braided composite with arbitrary constant cross-sections as parallelogram, due to the braid angles inequality, was correct. And also the whole derived geometrical equations and subsequent mechanical relations based on this assumption, which were derived for the first time to predict the mechanical performance of braided composite parts with various crosssection shape, are of great importance. Therefore, the current FE modelling is capable of being implemented on other biaxially and triaxially braided composite with arbitrary constant cross-sections. In this type of modeling, the final part mechanical behavior is directly linked to braiding process parameters. The presented meso scale model should be extended to capture the strength of a braided composite parts as a function of braiding parameters to predict failure in braided composite parts. This will be covered in author future works.
5. Conclusion The present research initially proves the inequality of braid angles formed by clockwisely and counter-clockwisely moving of carriers on a constant arbitrary cross-section. Since the braid angles are not equal, the rectangular shape of RUC alters to parallelogram. Therefore, geometrical relations are derived based on this assumption for the RUC of biaxially and triaxially braided composites. Furthermore, after determining the 3D effective stiffness matrix of each RUC, the FE modelling is performed using VUMAT subroutine. To validate the obtained results from FE modelling, 16 mandrels with different cross-sections were biaxially and triaxially braided using circular braiding machine and then composite samples were fabricated. By comparing the obtained results from both experimental and FE modelling, an acceptable agreement was found between the obtained data. Therefore, it can be concluded that all the derived geometrical equations and mechanical relations based on parallelogramical shape of RUC on mandrels with non-circular cross-section shape are correct. Also these derived relations can be used for other cross-sections with different geometries. Presented meso scale relationships in conjunction with FE modeling not only can be used to model braided composite parts with variable stiffness but also construct a tool to model final composite parts based on the initial parameters of the braiding process.
14
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Fig. 17. Predicted stress for triaxially braided composites for different cross-sections (a: 3AxCirc, b: 3AxElip35, c: 3AxElip39, d: 3AxEgg, see Table 5).
Fig. 18. Deformation of egg-shaped biaxially braided composite sample under four-point bending loading.
References
[9] Braiding Ko F. Engineering materials handbook. Netherlands: ASM International; 1987. [10] Du G, Popper P, Chou T. Process model of circular braiding. Am Soc Mech Eng Mater Div (Publ) MD 1990;19:119–33. [11] Zhang Q, Beale D, Adanur S, Broughton RM, Walker RP. Structural analysis of a two-dimensional braided fabric. J Text Inst 1997;88:41–52. [12] Tuba A. 3D geometrical modelling of tubular braids. Text Res J 2012;82:443–53. [13] Redman CJ, Douglas CD. Theoretical prediction of the tensile elastic properties of braided composites (1). International SAMPE symposium and exhibition (proceedings)1993. p. 719–27. [14] Naik RA, Ifju PG, Masters JE. Effect of fiber architecture parameters on deformation fields and elastic moduli of 2-D braided composites. J Compos Mater 1994;28(7):656–81. [15] Ayranci C, Carey J. 2D braided composites: a review for stiffness critical applications. Compos Struct 2008;85(1):43–58. [16] Ji Xianbai, Khatri Aditya M, Chia Elvin SM, Cha Ryan KH, Yeo Bern TB, Joshi Sunil C, Chen Zhong. Multi-scale simulation and finite element-assisted computation of elastic properties of braided textile reinforced composites. J Compos Mater 2014;48(8):931–9. [17] Lei Xu, Kim Seong Jong, Ong Cheng-Huat, Ha Sung Kyu. Prediction of material
[1] Braiding Popper P. Handbook of Composite Reinforcements. Sons: John Wiley and; 1992. p. 24–40. [2] Yuksekkaya ME, Adanur S. Analysis of polymeric braided tubular structures intended for medical applications. Text Res J 2009;79(2):99–109. [3] Sturm R, Heieck F. Energy absorption capacity of braided frames under bending loads. Compos Struct 2015;134:957–65. [4] Du GW, Popper P. Analysis of a circular braiding process for complex shapes, The. J Textile Inst 1994;85(3). [5] Falzon PJ, Herszberg I. Mechanical performance of 2-D braided carbon/epoxy composites. Compos Sci Technol 1998;58:253–65. [6] Dauda B, Oyadiji SO, Potluri P. Characterising mechanical properties of braided and woven textile composite beams. Appl Compos Mater 2009;16:15–31. [7] Masters J, Foye R, Pastore C, Gowayed Y. Mechanical properties of triaxially braided composites: experimental and analytical results. J Compos Technol Res 1993;15:112–22. [8] Potluri P, Manan A, Francke M, Day RJ. Flexural and torsional behaviour of biaxial and triaxial braided composite structures. Compos Struct 2006;75(1–4):377–86.
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properties of biaxial and triaxial braided textile composites. J Compos Mater 2012;46(18):2255–70. Pickett AK, Erber A, von Reden T, Drechsler K. Comparison of analytical and finite element simulation of 2d braiding Plast Rubber. Compos 2009;38:387–95. Pickett AK, Sirtautas J, Erber A. Braiding simulation and prediction of mechanical properties. Appl Compos Mater 2009;16:345–64. Goyal D, Tang X, Whitcomb JD, Kelkar AD. Effect of various parameters on effective engineering properties of 22 braided composites. Mech Adv Mater Struct 2005;12. 113e128. Hans T, Cichosz J, Brand M, Hinterhölzl R. Finite element simulation of the braiding process for arbitrary mandrel shapes. Compos A Appl Sci Manuf 2015;77:124–32. Hajrasouliha J, Nedoushan RJ, Sheikhzadeh M, Na W, Yu W-R. Theoretical and experimental study of braid pattern in mandrels with arbitrary cross-sections. J Compos Mater 2018;52(29):4009–22. Chamis CC. Simplified composite micromechanics equations for hygral, thermal and mechanical properties. 38th Annual Conference of Society of the Plastics Industry, Houston, TX, USA. 1983. Raju IS, Foye RL, Awa VS. A Review of Analytical Methods for Fabric and Textile Composites. Proceedings of Indo-US Workshop on Composite Materials for Aerospace Applications, Bangalore, India, July 23-27, 1990, available as Composite Structures, Testing, Analysis and Design. New Delhi, India: Narosa Publishing House; 1992. p. 274–93. Chou TW, Ishikawa T. Analysis and Modeling of Two-Dimensional Fabric Composites. Composite Materials Series 3, Textile Structural Composites. Elsevier Science Publishers B. V.; 1989. p. 210–64. Ko FK, Pastore CM, Lei C, Whyte DW. A fabric geometry model for 3D braid reinforced FP/AI-Li composites. International SAMPE Metals Conference: Competitive
Advances in Metals/Metal Processing, Cherry Hill, NJ. 1987. [27] Yang JM, Ma CL, Chou TW. Fiber inclination model of three-dimensional textile structural composites. J Compos Mater 1986;20:472–84. [28] Xu L, Kim SJ, Ong CH, Ha SK. Prediction of material properties of biaxial and triaxial braided textile composites. J Compos Mater 2012;46(18):2255–70. [29] Lomov SV, Parnas RS, Ghosh SB, Verpoest I, Nakai A. Experimental and theoretical characterization of the geometry of two-dimensional braided fabrics. Text Res J 2002;72:706–12. [30] Lomov SV, Huysmans G, Verpoest I. Hierarchy of textile structures and architecture of fabric geometric models. Textile Res J 2001;71(6):534–43. [31] Wang Y, Sun X. Digital-element simulation of textile processes. Compos Sci Technol 2001;61(2):311–9. [32] Miao Y, Zhou E, Wang Y, Cheeseman B. Mechanics of textile composites: microgeometry. Compos Sci Technol 2008;68(7–8):1671–8. [33] Potluri P, Parlak I, Ramgulam R, Sagar T. Analysis of tow deformations in textile preforms subjected to forming forces. Compos Sci Technol 2006;66(2):297–305. [34] Fang G, Liang J. A review of numerical modeling of three-dimensional braided textile composites. J Compos Mater 2011;45(23):2415–36. [35] Chretien N, Numerical constitutive models of woven and braided textile structural composites; 2002. [36] ASTM D 638-14, “Standard Test Method for Tensile Properties of Plastics”, ASTM International, West Conshohocken, PA; 2014. [37] ASTM D 6272-17, Standard Test Method for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials by Four-Point Bending, ASTM International, West Conshohocken, PA; 2017. [38] Fouladi A, Jafari Nedoushan R. Prediction and optimization of yarn path in braiding of mandrels with flat faces. J Compos Mater 2017;52(5):581–92.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Meso-macro numerical modeling of noncircular braided composite parts based on braiding process parameters
T
Jalil Hajrasoulihaa, Reza Jafari Nedoushanb, , Mohammad Sheikhzadeha, Tohid Dastana ⁎
a b
Department of Textile Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran
ARTICLE INFO
ABSTRACT
Keywords: Noncircular braided composites Meso scale relationships Finite element modeling Variable stiffness Bending test
In the past few decades, the applications of composite structures reinforced with braided preforms have been growing increasingly in various industries due to their intrinsic properties. Therefore, their mechanical properties are worth studying experimentally, analytically and numerically. In this respect, the goal of the current study is to perform a finite element modelling on biaxially and triaxially braided composites with constant arbitrary cross-section. Thus, the braid angles formed by clockwisely and counter-clockwisely moving of carriers on noncircular parts, have been initially demonstrated to be unequal, unlike in circular parts. As a result, considering a parallelogram shape for the representative unit cell (RUC), the geometrical relationships for a RUC of 2D biaxially and triaxially braided composite are derived and implemented in VUMAT subroutine. Then, after determination of 3D effective stiffness matrix for each RUC, the finite element modelling is performed to predict bending behavior of final braided composite part. In these simulations, the stiffness variation all over the part is considered and mechanical behavior of composite parts is predicted based on the initial parameters of the braiding process. The obtained results from both experimental and simulations showed an acceptable agreement.
1. Introduction Composites reinforced with braided preforms have found vast application area in various industries due to their intrinsic properties such as high load bearing capacities, superior toughness and fatigue strength than conventional composite structures, the ability of being fabricated near-net-shape preform and cost savings because of rapid manufacturing process [1–3]. Among available braiding techniques, circular braiding process, which involves depositing bias and optional axial yarns on a mandrel, is the most common technique in manufacturing circular-shaped braided composite parts [4]. The mechanical properties of braided composites are affected by various variables (braid angle, fabric cover factor and yarn volume fraction); hence, many researchers put a lot of effort into investigating the effect of mentioned variables on the performance of braided composites experimentally and analytically. In this regard, various tests like tension, bending, compression and shear tests were performed to study the braided composite properties [5,6]. The mechanical properties of triaxially braided composites were studied experimentally and analytically by Masters et al. [7]. They focused on different reinforcing fiber architectures in braided composite parts, which govern the composite behavior under tensile loading. Potluri et al. performed flexural and ⁎
torsional tests on biaxially and triaxially braided composites [8]. The effect of number of layers and braid angles was considered in their work. The braid angle, along with the cover factor, is the key parameter in circular braiding process which governs the mechanical properties of braided composites [9]. Thus, knowing and controlling the braid angle in regard to the braiding process parameters is crucial in achieving control on mechanical properties of final product. In this respect, Du and Popper [10] suggested a model based on kinematic analysis for the geometry of fiber structures deposited on an axisymmetric mandrel in braiding process. The proposed model is based on braid angle, fabric cover factor and yarn volume fraction. Zhang et al. [11] determined the braid angle by considering the interlacing forces through a mechanical model. They concluded that the mechanical model is superior in predicting the final braid structure than kinematic analysis. A 3D geometrical modelling of tubular braids was proposed by Tuba [12] which is applicable for different braid structures with varying parameters like braid angle, yarn and mandrel diameter. The elastic properties of triaxially braided composites could be determined using a simple analytical model proposed by Redman and Douglas [13]; however, the braided preform was considered to be composed of three separate plies and the undulating fibers were neglected. The effect of braid angle,
Corresponding author. E-mail address: [email protected] (R. Jafari Nedoushan).
https://doi.org/10.1016/j.compstruct.2019.111065 Received 12 December 2018; Received in revised form 8 April 2019; Accepted 27 May 2019 Available online 29 May 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. The plot of obtained braid angles for mandrels with elliptical and egg-shaped cross-sections; the red and blue line corresponds to clockwisely and counterclockwisely braider yarns, respectively.
yarn size and axial yarn content was investigated analytically and experimentally on 2D triaxially braided composites [14]. A volume averaging approach, considering the iso-strain assumption, was employed to determine the three-dimensional effective stiffness of the representative unit cell (RUC). It was found that the elastic properties were only affected by braid angle and axial yarn content. Another analytical study based on curved unit-cell geometry was proposed by Ayranci and Carey to predict the elastic constants of braided composites [15]. In addition to proposed models based on analytical methods, other accurate modelling techniques such as numerical modelling were used to obtain the elastic properties of braided composite. Ji et al. performed
a multi-scale modelling, which involved three steps, to determine the elastic properties of braided composites [16]. Xu et al. analyzed biaxially and triaxially braided composites with different weave patterns using finite element (FE) method through applying periodical boundary conditions on RUCs [17]. Pickett et al. compared the analytical and FE modelling of 2D braiding. They demonstrated that analytical approach is computationally fast; however, the FE method is capable of accurately predicting yarn paths and interactions [18,19]. The effect of yarn shape, lenticular and flattened shapes, on mechanical behavior of braided composites was investigated by Goyal et al. [20]. To date, few of studies dealt with predicting braid angle and mechanical properties of braided composite with constant arbitrary cross2
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braiding machine. In their study, they focused on the position of fell point on the surface of mandrel and the yarn length between fell point and the carrier, which considerably affect the braid angle. Followed by our research on the theoretical study of braid pattern in mandrels with arbitrary cross-section as published in [22] the aim of the current research is to propose a FE modelling through developing the meso scale linear elastic stress-strain relation for both biaxially and triaxially braided composites on noncircular mandrels. In this regard, it is initially shown that the braid angles formed by clockwisely and counter-clockwisely moving of carriers on an arbitrary cross-section are not equal. Consequently, the shape of the RUC of braided preforms changes from diamond shape into a parallelogram shape. Furthermore, geometrical relationships for a RUC of 2D biaxially and triaxially noncircular braided composite will be derived. Using these relations, the stiffness variation induced by variable braid angles on each point of a noncircular braided composite can be calculated and be used in FE modeling. The stiffness is calculated based on the braiding process parameters and in this manner the final part behavior is directly obtained from braiding process. 2. Numerical modeling Fig. 2. The geometry of RUC for mandrels with arbitrary cross-sections.
Predicting properties of composite parts, through numerical modeling helps the corresponding industry in saving cost and time. Since several parameters are in braiding process that have influence on the braided composite sample properties, in this study both of meso and macro scale aspect of the parts are considered. The mechanical behavior of a material point in the braided part was estimated by the meso scale approach that was implemented in ABAQUS software by user material subroutine (VUMAT). Using this subroutine, FE modeling of
sections. On this point, Hans et al. proposed a simulation approach based on FE modelling which is suitable for any mandrel geometry [21]. They aimed at analyzing braided structures with respect to yarn alignment. Hajrasouliha et al. [22] developed a theoretical model to predict the braid angle in braided composites with constant non-circular cross-section by considering the kinematic parameters of circular
Fig. 3. Pictures taken from braided mandrels with a) an elliptical cross-section with a = 3.9, b = 2.15, b) an egg-shaped cross-section. 3
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Fig. 4. (a) RUC and (b) cross-section view of yarn path of 1 × 1 2D biaxially braided composites.
both biaxially and triaxially braided composite samples with various cross-section profiles was carried out. Using the VUMAT subroutine the variable mechanical properties of the braided composite can be included in the modeling. In the following at first meso scale relationships which were used in VUMAT subroutine are discussed and then FE simulation in macro scale is explained.
et al. [22] on noncircular mandrels, the braid angles at any arbitrary position on mandrel for each sets of yarn (clockwise and counterclockwise yarns) are not equal. Therefore, a Matlab code was written on the basis of this theory [22] for tracing both braid angles formed by sets of yarn over a mandrel. Fig. 1 shows the obtained results for braid angles formed on the upper half of the elliptical and egg-shaped crosssections. Thus, the geometry of RUC, in the case of mandrel with constant arbitrary cross-section, has the shape of parallelogram, as schematically shown in Fig. 2. Furthermore, in order to verify the geometrical shape of RUC, pictures were taken using a digital camera normal to the surface of a braided mandrels. Two pictures are shown in Fig. 3 as examples, which were taken from the surface of an elliptical crosssection with a = 3.9, b = 2.15 and from an egg-shaped cross-section. Since braid angles formed by clockwisely and counter-clockwisely moving of carriers on a constant arbitrary cross-section are not equal anymore, available derived geometrical relations for 2D biaxially and triaxially braided composites with circular cross-section by other studies [10–15] are not appropriate to be used as a basis of numerical
2.1. Calculation of stiffness matrix using meso scale approach 2.1.1. Determining the RUC of a braided composite with constant arbitrary cross-section It is known that the geometry of RUC in braided composite with circular cross section is diamond-shaped, since both braid angles, which are formed by clockwisely and counter-clockwisely moving of carriers, are equal and are marked respectively as + b and b [22]; however, this is not true anymore in the case of braiding a mandrel with constant arbitrary cross-section based on the findings of previous study [22]. That is, according to the theoretical model proposed by Hajrasouliha 4
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determined is Lu' , which could be obtained using the following equation:
L'st 2
(L'u)i = S'i
i = 1, 2
(4)
where:
S'i =
sin(
L'st =
modeling of this type of braided composites. Therefore, as is described in detail in the following, the governing equations in geometrical modelling of RUC of 2D biaxially and triaxially noncircular braided composites derived.
(1)
(L 'u)i 2
x
(L 'u) i 2
i = 1, 2
i = 1, 2
S1 =
2B v sin( N
b2 )
S2 =
2B v sin( N
b1)
i = 1, 2
(6) (7) (8)
(L'u)1 =
2B v sin( N
b2 )
(L'u)2 =
2B v sin( N
b1 )
(L pb)i = 4(S'b) i
L st 2 sin(
b1
1 +
L st 2 sin(
b1
(9)
b2 )
1 +
(10)
b2 )
(11)
i = 1, 2
Furthermore, vertical distance between axial yarns (da ) can be obtained based on the presented schematic in Fig. 6(a):
da = (s'b)2sin b2 + (s'b)1sin b1 2B v = (sin b1sin Nsin( 1 + 2 )
da =
(2)
The spatial orientation of yarn could be defined by the in-plane angle (which is the braid angle b ) and the out-of-plane angle (which is between the tangential line to the sinusoidal path and XY plane), as shown in Fig. 5. Thus, differentiating the undulating function with respect to x gives the angle :
dz(x)i t x = |z' (x)i| = |tan i| = cos( ' ) dx 2(L'u)i (L u)i
b2 )
2.1.3. Geometrical modelling of noncircular 2 × 2 2D triaxially braided composites The RUC and cross-section view of yarn path in the weave (BB section) corresponded to2 × 2 2D triaxially braided composites is schematically shown in Fig. 6. In order to implement the same approach on the RUC 2D triaxially braided composite, some general parameters such as vertical distance between axial yarns (da), number of filaments in braider yarn (nb), number of filaments in braider yarn (na), the diameter of braider yarn filaments (dfb), the diameter of axial yarn filaments (dfa), the thickness of braider yarn (tb) and the thickness of axial yarn (ta) must be specified to determine their cross-section areas, the yarn geometrical path and subsequent yarn spatial orientation. In the following, it should be noted that subscripts “a” and “b” are related to axial and braider yarns, respectively. Applying the same approach, the length of the projected yarn (L pb)i along the x-axis in an RUC can be defined as following:
where the index ‘i” is attributed to braider yarns, in which i = 1 and i = 2 is corresponded to braider yarns having braided angles + b1 and b2 , respectively. The yarn path, as shown in Fig. 4(b), comprises of two straight and three undulating portions in which the undulating path follows the sinusoidal function. Therefore, the path function for the braider yarn in the undulating portion could be obtained as a function of Lu' and yarn thickness t:
t x z(x)i = ± sin 2 (L 'u) i
L st +
b1
(5)
The projection of undulating portion length of yarn path can be determined using a new approach, which is based on cross-section area of yarn, and will be discussed later.
)
i = 1, 2
sin(
i = 1, 2
(Lu' )
2.1.2. Geometrical modelling of noncircular1 × 1 2D biaxially braided composites The RUC and cross-section view of yarn path in the weave (AA section) corresponded to1 × 1 2D biaxially braided composites is schematically shown in Fig. 4. It should be noted that the RUC has the shape of parallelogram, based on the earlier findings. As could be observed, a global coordinate system (XYZ) is considered by such manners H H Z that Z-axis is along the thickness (H) direction and braid 2 2 angles are defined with respect to Y-axis (+ b1 and ), as depicted b2 in Fig. 4(a). Besides, a local coordinate system (xyz) is considered to determine yarn path in a way that x-axis is along the yarn projection in XY plane and z-axis is in Z direction, as shown in Fig. 4(b). Thus, all the geometrical parameters along the yarn path will be projected over the related axis. The length of the projected yarn (Lp) along the x-axis in an RUC can be defined as following, which is the summation of both undulating (Lu' ) and straight (Lst' ) portions of yarn path projection over x-axis:
(L p)i=2(L'u)i + L 'st
b2 )
where B v is Take-up length per a carrier revolution (cm/rev) and N is number of carriers. By substituting Eqs. (5)–(8) in Eq. (4) yields:
Fig. 5. A schematic of considered yarn spatial orientation.
(
Si +
b1
4B v sin b1sin b2 Nsin( b1 + b2 )
(d'a ) i =
da sin bi
b2
+ sin
b1sin b2 )
(12) (13)
i = 1, 2
(14)
The length of axial yarns in a RUC can also be calculated using the following equation:
La = 4da (cotg
b2
+ cotg
b1)
(15)
Considering the sinusoidal function for undulating portion, so the path function for the braider yarn in the undulating portion could be ' obtained as a function of Lub and subsequent differentiating with
(3)
According to Eq. (3), the only unknown parameter that should be 5
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Fig. 6. (a) RUC and (b) cross-section view of yarn path of 2 × 2 2D triaxially braided composites.
6
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observed, since the arc “AB” is a segment of a circle with the center of “O” and the radius of “r”, then the area of section “ABC” could be calculated from parameters “a” and “t”:
1 2 r 2
AreaABC =
1 a r 2
t 2
(20)
where according to the Fig. 7:
r=
Fig. 7. A schematic of yarn cross-section in 2D braided composites.
respect to × will yield the angle :
z(x) bi = ±
|tan
b i|
=
(t a + t b) 2(L 'ub) i
cos
= sin
(L 'ub)i 2
t a + tb x sin 2 (L 'ub)i x
x
(L 'ub)i i = 1, 2 2
i = 1, 2
(L 'ub) i
(L'ub) i = (d'a ) i
Area = 2sin
da sin bi
L st b b1 +
b2 )
(18)
i = 1, 2
1
4at + t2
4a2
4a2 + t2 4t
2
4a3 + 3at2 + wt 2t df 2nf 4pd
(23) ”, where df
is the diameter of each filament, nf is the number of filaments in the yarn cross-section and Pd is the packing density, which its value is considered in the range of 0.7–0.8 according to the image processing observation. The unknown parameter “a” and subsequently “b” can be ' calculated by iteratively solving Eq. (23). Consequently, Lu' and Lub are now a known parameter for 2D biaxially and triaxially braided composites.
'
2sin(
(22)
The yarn cross-section could be measured as “ A =
where d a can be obtained using Eq. (14) and L stb can be determined using the yarn cross-section area approach, which is discussed in the following. Therefore, Eq. (18) can be rewritten as:
(L'ub) i =
a r
(17)
i = 1, 2
'
1
(21)
Thus, the yarn cross-section can be obtained as a function of known parameters “w” (yarn width), “t” (yarn thickness) and the unknown parameter “a”:
(16)
According to Eq. (17), the only unknown parameter that should be ' determined is Lub , which could be determined using the following equation, based on Fig. 6(b):
L 'st b 2
a2 t + t 4
(19)
2.2. Determination of 3D effective stiffness matrix for each material point of noncircular braided composites
2.1.4. Determination of Lu' by using the cross-section area approach The calculation of Lu' and Lu' b , in 2D biaxially and triaxially braided composites in that order, can be done by determining the area of yarn cross-section. Based on the studies [24–34], the yarn cross-section in 2D braided composite structures is supposed to be composed of a flat and two lenticular-shaped sections, as shown in Fig. 7. As could be
In this study, a braided composite part with constant arbitrary crosssection is assumed an anisotropic linear elastic material with variable stiffness in different points. A typical stress-strain relation for this kind of material in a Cartesian coordinate system (X-Y-Z) is as follows:
Fig. 8. A typical assembly of FE modelling of braided composite sample with egg-shaped cross-section under four-point bending. 7
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Table 1 The characteristics of carbon fiber roving. Number of filaments
Density (kg/m3)
Filament diameter ( µm)
Fiber count (Tex )
Tensile strength (MPa )
Failure strain (%)
12,000
1740
7
800
1447.3
2.00
In order to obtain the stiffness matrix of an RUC of 2D biaxially braided composite, according to Eq. (26), the volume of each yarn segment must be multiplied by its corresponded stiffness matrix with respect to the segment spatial orientation (angles and b ), so:
Table 2 The mechanical properties of carbon fiber roving (Reference). Carbon fiber type
E1f (GPa)
E2f (GPa)
G12f (GPa)
G23f (GPa)
v12f
v23f
T300-12 k
126.02
23.1
27.6
14.3
0.25
0.3
V+ b1
Table 3 The mechanical properties of cured bone-shaped epoxy resin.
V b2
Epoxy resin type
Density (kg/m3)
Tensile strength (MPa )
Tensile modulus (GPa )
Poisson’s ratio
DSM EP411
1200
50
1.12
0.3
X Y Z YZ
=
ZX XY
C12 C22 C32 C42 C52 C62
C13 C23 C33 C43 C53 C63
C14 C24 C34 C44 C54 C64
C15 C25 C35 C45 C55 C65
C16 C26 C36 C46 C56 C66
X Z YZ ZX
(24)
where and are stress and strain, respectively, and C is the stiffness matrix with 21 engineering constants. The aim of the current section is to find these 21 constants based on the braiding parameters by meso scale relations. The first step to calculate stiffness matrix was to choose an appropriate RUC of the material and derive geometrical relations of yarns paths in the RUC, which is already done and discussed in detail in previous sections.
1 VRUC
V
CdV
V+ b 1
C+
+2
V b 2
C
b2 dV
+
VResin
V b2
b1
,
)dV
C(
b2
,
)dV
(27) (28)
(29)
VResin
V+ b1
C+
b1 dV
+4
V b2
C
b2 dV
+
CResin dV
Va
Ca dV (30)
where C+ b1, C b2 and CResin are determined using Eqs. (27)–(29), respectively. The stiffness matrix of axial yarns, considering the fact that axial yarns follows a straight path in RUC, is defined as: Va
Ca dV =
Va
Ca (0°, 0°)dV = Ca Va
(31)
Since, the stiffness matrices attributed to braider yarns (Eqs. (27) and (28)), which are used for both 2D biaxially/triaxially braided composites, are obtained in yarn local coordinate system (1-2-3), thus in order to determine the global stiffness matrix, a transformation must be made for these local stiffness matrices from the yarn local coordinate system (1-2-3) to global coordinate system (X-Y-Z).
(25)
b1 dV
=
C(+
CResin dV = CResin VResin
+
2.2.3. Transformation matrix from yarn local coordinate system (1-2-3) to global coordinate system (X-Y-Z) In order to calculate the transformation matrix, considering a segment of braider yarn which is shown in Fig. 5, two consecutive transformations must be done. First, a transformation must be made from local yarn coordinate system (1-2-3) to coordinate system (x-y-z) by rotating around the y axis using the angle , which its calculation is discussed earlier. Second, a transformation must be done from the coordinate system (x-y-z) to global coordinate system (X-Y-Z) by rotating
(CRUC ) 2axial 1 2 VRUC
b2 dV
1 4 VRUC
=
The above relationship is driven under the iso-strain assumption. Since 1 × 1 2D biaxially braided composites with arbitrary crosssection includes braider yarns with braid angles of + b1 and b2 , and resin, therefore the RUC stiffness matrix for 1 × 1 2D biaxially braided composite is obtained from the summation of each stiffness matrix attributed to braider yarns and resin:
=
C
V+ b1
(CRUC )3axial
2.2.1. Calculating the stiffness matrix of noncircular 1 × 1 2D biaxially braided composites As previously shown in Fig. 4(a), the RUC of 1 × 1 2D biaxially braided composites is composed of three linear elastic phases including both braider yarn (with the angle of + b1and b2 ) and matrix. The stiffness matrix for an RUC (CRUC ) of a textile composite can be estimated as:
CRUC =
=
2.2.2. Calculating the stiffness matrix of noncircular 2 × 2 2D triaxially braided composites The RUC stiffness matrix of 2 × 2 2D triaxially braided composites is derived similar to what have been discussed in previous section. The volume of 2D triaxially braided composite’s RUC consists of four braider yarns (for each braid angles + b1and b2 ), axial yarns and resin. Therefore, the stiffness matrix of RUC is defined as:
Y
XY
b1 dV
It should be noted that the value of angle is determined through Eq. (3). The stiffness matrix of resin, as a homogenous and isotropic material, is defined as: VResin
C11 C21 C31 C41 C51 C61
C+
CResin dV (26)
Table 4 General specifications of circular braiding machine. Number of carriers
Number of braiding carbon yarns
Number of axial carbon yarns
Radius of main braiding plate (R g ) (cm )
Mean revolutionary velocity of carriers ( c ) (rad/s)
Take-up velocity (V ) (cm/s)
Take-up length per a carrier revolution (B v ) (cm/rev)
48
48
24
115.25
0.14
0.19
8.5
8
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Fig. 9. Four mandrels with egg-shaped cross-sections a) before and b) after the tape covering process.
Fig. 10. A typical pictures of braiding process on mandrels with circular (a) elliptical (b and c) and egg-shaped (d) cross-sections.
[T ]
=
a112 a12 2 a132 a11a12 a11a13 a12a13 a21a22 a21a23 a22a23 a212 a222 a232 a31a32 a31a33 a32a33 a312 a322 a332 2a11a21 2a12a22 2a13a23 a11a22 +a12 a21 a13a21 + a11a23 a12a23 + a13a22 2a11a31 2a12a32 2a13a33 a31a12 +a32a11 a11a33 + a13a31 a32a13 + a33a12 2a21a31 2a22a32 2a23a33 a21a32 +a22a31 a23a31 + a21a33 a22a33 + a23a32
Fig. 11. Prepared biaxially/triaxially braided composite samples with various cross-section shapes.
(34)
around the Z axis using the angle b , which is the braid angle. The transformation matrix ([a ij ]3 × 3 ) which transforms form the yarn local system (1-2-3) to global system (X-Y-Z) is defined as:
[aij]3× 3 =
cos cos b sin b sin cos b
cos sin b sin cos b 0 sin sin b cos
C1 2 3 in Eq. (33) is the stiffness matrix of impregnated yarn that is discussed in the next section. 2.2.4. Stiffness matrix of impregnated yarn in local coordinate system The impregnated carbon fiber with resin is considered as unidirectional reinforced composite material, which its properties are assumed to be transversely isotropic. In the other words, five independent engineering constants are required to define the mechanical behavior of an impregnated carbon fiber with resin. In this regard, the Chamis micromechanical model [23] was used to obtain these five independent engineering constants, which are as follows:
(32)
From tensor transformations relations it could be shown that [35]:
CX
Y Z
= TT C1
2 3T
(33)
where [35]:
(35)
E1 = Ef Vf + Em Vm Em
E2 = 1 9
(
Vf 1
Em Ef
)
= E3 (36)
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Table 5 Characteristics of prepared biaxially braided composite samples. Sample Coding
Type of cross-section
Mean thickness (mm )
Mean width of axial rovings (mm )
Mean width of braiding rovings (mm )
Mean thickness of axial rovings (mm )
Mean thickness of braiding rovings (mm )
2AxCirc 3AxCirc
Biaxially Triaxially
0.7 1.35
– 3.2
4.6 2.84
– 0.22
0.15 0.27
2AxElip35 3AxElip35
Biaxially Triaxially
0.72 1.2
– 3.05
4.4 2.97
– 0.25
0.16 0.26
2AxElip39 3AxElip39
Biaxially Triaxially
0.78 1.3
– 2.96
4.22 2.86
– 0.26
0.17 0.27
2AxEgg 3AxEgg
Biaxially Triaxially
0.7 1.15
– 3.3
4.1 3.1
– 0.22
0.18 0.25
Fig. 12. A typical braided composite sample under four-point bending test. Table 6 Comparing the stiffness values obtained from experimental and FE modelling. Sample Coding
2AxCirc 3AxCirc 2AxElip35 3AxElip35 2AxElip39 3AxElip39 2AxEgg 3AxEgg
Maximum Force (N )
Experimental Stiffness (N.mm−1/mn)
Numerical Stiffness (N.mm−1/mn)
Stiffness Difference (% )
471.75 1211.40 481.75 967.90 390 1075.75 185 527.25
106.43 242.89 87.15 190.92 62.82 191.73 33.12 109.56
93.08 218.86 78.64 175.49 66.97 167.11 22.25 88.83
12.54 9.89 9.77 8.07 6.60 12.84 32.83 18.93
Gm
G12 = 1
(
Vf 1
Gm Gf
)
1
Vf 1
Gm G 23f
=
23
=
f
Vf +
E2 2G23
m Vm
=
13
1
(39) (40)
where Vf and Vm are the fiber volume fraction and matrix volume fraction, respectively. It should be noted that the fiber volume fraction is equal to the fiber packing density (Pd). The stiffness matrix of resin is constructed by E and v based on stiffness matrix of an isotropic material. 2.3. Macro scale finite element modeling The main difference between mechanical properties of a circular braided composite part and a part with arbitrary cross section is due to their braid angles. While braid angles are equal for both clockwise and counterclockwise yarns and is constant around the section of a circular part, in braiding of an arbitrary cross section, braid angles vary from point to point and also each material point contain two different braid angles for clockwise and counterclockwise yarns. To use the above presented meso relationships, at first step, braid angles should be estimated at each material points. Our previously published relationships
= G13 (37)
Gm
G23 =
12
(38) 10
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Fig. 13. Force-displacement curves obtained from experimental and FE modeling for biaxially braided composite samples (a: 2AxCirc, b: 2AxElip35, c: 2AxElip39, d: 2AxEgg, see Table 5).
[22] were used to estimate the braid angles in each material points of each cross sections. Then using the presented meso scale relationship and braid angles of a point, stiffness matrix of each point is calculated using the VUMAT subroutine. This subroutine was written using FOTRAN programming language in Microsoft Visual Studio (version 2008) for both biaxially and triaxially braided composites according to the earlier discussed geometrical equations to determine the effective stiffness matrix of each material point. After calculation of the braid angles of each material point based on braiding process parameters these angles are saved in a text file in order of the coordinates of the point and this file is used by VUMAT subroutine to read braid angles of each point. Using this subroutine, stiffness of a braided composite can be calculated in each point of the braided part based on the estimated braid angles. In this section a macro scale FE simulation was carried out using ABAQUS software to analyze braided parts. To this aim four-point bending tests on biaxially and triaxially braided composite samples with circular, elliptical and egg-shaped cross-sections are simulated. At first the loading support, which includes two loading cylinders and two support cylinders, was designed as rigid half cylinders with the diameter and the length of 4.5 cm and 130 cm, respectively. Then, biaxially and triaxially braided composite samples were designed based on the geometrical parameters of cross-section shape, which were measured from experimental samples. A C3D8R element type, which is an 8-node linear brick with reduced integration element, was used to discretize composite parts.
After assigning the properties to the defined sections on composite parts, all parts were assembled in their proper position in accordance with the four-point bending loading, as shown in Fig. 8. Then, a surface to surface contact, based on penalty method, was defined between four cylinders and composite specimen. All the degree of freedom of supporting cylinders, which were defined as analytical rigid materials, was fixed. The magnitude of displacement applied on loading cylinders was 40 mm in Y-direction. Finally, using prepared VUMAT subroutine the FE modeling was done by ABAQUS explicit solver considering composite parts as variable stiffness parts. 3. Experimental procedure To validate the implemented FE modelling and all the derived geometrical and mechanical equation to determine the stiffness in each point of 2D biaxially and triaxially braided composite, an attempt was made to fabricate 2D biaxially and triaxially braided composite samples. Therefore, in this section materials and experimental procedure which are used to manufacture both biaxially and triaxially braided parts are described. Then the method of carrying out the four-point bending test is also explained. 3.1. Materials T300-12K carbon fiber roving and two-part EP411 epoxy resin were 11
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Fig. 14. Force-displacement curves obtained from experimental and FE modeling for triaxially braided composite samples (a: 3AxCirc, b: 3AxElip35, c: 3AxElip39, d: 3AxEgg, see Table 5).
used as reinforcement part (braiding and axial yarns) and matrix part, in that order, in both biaxially and triaxially braided composites. The width and thickness of carbon fiber roving, in as received state, was measured 6.15 mm and 0.13 mm, respectively. In order to obtain tensile properties of carbon fiber roving, a unidirectional tensile test was performed on it based on the ASTM D4018-17 standard test method instructions. Some of the carbon fiber roving characteristics are tabulated in Table 1. The mechanical properties (engineering constants) of the carbon fiber roving are also presented in Table 2. The characteristics of two-part EP411 epoxy resin (cured with the curing agent H3 by mixing ratio of 10:1) are presented in Table 3. The presented values are obtained through performing the uniaxial tensile test on cured bone-shaped epoxy resin according to the ASTM D638-14 [36] standard test method instructions.
process using a circular braiding machine. Table 4 shows general specifications of circular braiding machine. The surface of each prepared mandrel was covered with tape and subsequent releasing agent was applied to facilitate the removal of braided composite sample from the mandrel and to avoid any significant damages to the specimen during removal process. Fig. 9 shows four mentioned mandrels before and after the tape covering process. It should be noted that, for each cross-section, four identical mandrels were prepared so that two of them were used for biaxially braiding process and two other mandrels were used for triaxially braiding process. Further, braiding process was performed on prepared mandrels to produce braided composite samples. Fig. 10 shows the mandrels during the braiding process. 3.2.2. Biaxially and triaxially braided composite samples fabrication After mandrels were braided biaxially and triaxially, the matrix part was evenly applied on braided carbon fiber roving. An attempt was taken to keep the fiber volume fraction of all braided composite samples at 0.5. Soon afterwards a uniform pressure was applied on the impregnated braided samples by evenly tightly wrapping the heat shrink plastic film around them and subsequent heating using a heat blower. The braided composite samples were cured based on matrix technical data sheet instructions. Finally, after unwrapping the plastic film, fully cured samples were easily removed from their mandrels. Fig. 11 shows the typical biaxially and triaxially braided composites with various cross-sections and characteristics of prepared biaxially and
3.2. Biaxially and triaxially braided composite sample manufacturing To produce braided composite samples with various cross-sections, different mandrels were braided with carbon rovings and then were impregnated with resin, as it is discussed in the followings. 3.2.1. Braiding process Four wooden mandrels with different cross-section shapes, including one circular, two elliptical and one egg-shaped (non-circular cross-sections are shown in Fig. 1), were prepared prior to braiding 12
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Fig. 15. Variation of braid angle around the cross-section of braided composite samples (a: 2AxCirc, b: 2AxElip35, c: 2AxElip39, d: 2AxEgg, see Table 5).
triaxially braided composite samples are given in Table 5.
composite samples from experimental and FE modelling results were compared and presented in Table 6. Since fabricated braided composite samples were different in thickness, the obtained stiffness values were normalized by thickness. One can see that the difference between stiffness values predicted by FE modelling, for each braided composite samples, and experimental results are acceptable. Figs. 13 and 14 show the obtained results from both experimental and FE modelling for biaxially and triaxially braided composites with different cross-sections, respectively. As could be observed, there is a good agreement between the experimental and FE modelling results in the elastic region, considering the fact that only the linear elastic behavior of braided composites was modeled. From the obtained results it could be inferred that all the mechanical relations and equations that were derived based on the current geometrical modeling of RUC of both biaxially and triaxially braided composites with constant arbitrary cross-section are of good accuracy. Therefore, these derived equations and subsequent FE modeling can be used to analyze braided structures with constant arbitrary cross-sections. It is worth mentioning that one source of the discrepancy of the predicted and measured stiffness can be the inaccuracy of the predicted braid angles due to ignoring the real conditions including: the effect of the yarn interaction and friction with mandrel [22]. As it was mentioned earlier in current simulations, braid angles and consequently material stiffness matrix vary around the part. The predicted braid angles variation along the length of braided composite samples, which were used in VUMAT subroutine, is presented in Fig. 15. The variation is the same as variation presented in Fig. 1 and is the results of relations presented by Hajrasouliha et al. [22]. It is worth mentioning that braid angle variation can be more remarkable in
3.3. Four-point bending test Four-point bending test was performed on biaxially and triaxially braided composite samples in accordance with ASTM D6272-17 [37] standard test method instructions. This test was carried out on Hounsfield testing machine with a 50 kN load cell and the rate of displacement was 10 mm/min for all bending tests. The span length was also kept 24 cm for all biaxially and triaxially braided composite samples. Fig. 12 shows a typical composite sample under four-point bending. As previously noted, four-point bending test was performed on two specimens for each biaxially and triaxially braided composite samples. 4. Results and discussion Table 6 presents the measured experimental results, maximum force and stiffness, for both biaxially and triaxially braided composite samples. As can be seen, in the case of comparing the biaxially and triaxially braided composite samples for each cross-section shape, introducing carbon rovings in longitudinal direction of tubular braided parts contribute a considerable enhancement in their flexural properties. Considering the egg-shaped braided composite as an example, its maximum flexural load and stiffness increased approximately 185% and 230%, respectively, when carbon rovings are added in its longitudinal direction. This shows the flexibility of braiding process in designing braided structures with tunable mechanical properties, as required. In addition, the obtained stiffness value for each braided 13
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Fig. 16. Predicted stress for biaxially braided composites for different cross-sections at displacement of 24 mm (a: 2AxCirc, b: 2AxElip35, c: 2AxElip39, d: 2AxEgg, see Table 5).
special cases like asymmetrical braiding process. Details about the effect of the braiding condition on the braid angle variation can be find in [38]. In analyzing braided parts with large braid angle variation, the effect of the current calculation on the result will be more significant. Figs. 16 and 17 show the stress distribution for biaxially and triaxially braided composite samples, respectively. And, as an example, Fig. 18 shows the deformation of egg-shaped biaxially braided composite sample under four-point loading obtained experimentally and numerically. Based on the findings of the current research, it was demonstrated that considering the shape of RUC of biaxially and triaxially braided composite with arbitrary constant cross-sections as parallelogram, due to the braid angles inequality, was correct. And also the whole derived geometrical equations and subsequent mechanical relations based on this assumption, which were derived for the first time to predict the mechanical performance of braided composite parts with various crosssection shape, are of great importance. Therefore, the current FE modelling is capable of being implemented on other biaxially and triaxially braided composite with arbitrary constant cross-sections. In this type of modeling, the final part mechanical behavior is directly linked to braiding process parameters. The presented meso scale model should be extended to capture the strength of a braided composite parts as a function of braiding parameters to predict failure in braided composite parts. This will be covered in author future works.
5. Conclusion The present research initially proves the inequality of braid angles formed by clockwisely and counter-clockwisely moving of carriers on a constant arbitrary cross-section. Since the braid angles are not equal, the rectangular shape of RUC alters to parallelogram. Therefore, geometrical relations are derived based on this assumption for the RUC of biaxially and triaxially braided composites. Furthermore, after determining the 3D effective stiffness matrix of each RUC, the FE modelling is performed using VUMAT subroutine. To validate the obtained results from FE modelling, 16 mandrels with different cross-sections were biaxially and triaxially braided using circular braiding machine and then composite samples were fabricated. By comparing the obtained results from both experimental and FE modelling, an acceptable agreement was found between the obtained data. Therefore, it can be concluded that all the derived geometrical equations and mechanical relations based on parallelogramical shape of RUC on mandrels with non-circular cross-section shape are correct. Also these derived relations can be used for other cross-sections with different geometries. Presented meso scale relationships in conjunction with FE modeling not only can be used to model braided composite parts with variable stiffness but also construct a tool to model final composite parts based on the initial parameters of the braiding process.
14
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Fig. 17. Predicted stress for triaxially braided composites for different cross-sections (a: 3AxCirc, b: 3AxElip35, c: 3AxElip39, d: 3AxEgg, see Table 5).
Fig. 18. Deformation of egg-shaped biaxially braided composite sample under four-point bending loading.
References
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