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An analytical method for calculating stiffness of two-dimensional tri-axial braided composites
Composite Structures 92 (2010) 2901–2905
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
An analytical method for calculating stiffness of two-dimensional tri-axial braided composites Mahmood M. Shokrieh *, Mohammad S. Mazloomi Composites Research Laboratory, Center of Excellence for Experimental Solid Mechanics and Dynamics, Department of Mechanical Engineering, Iran University of Science and Technology, 16846-13114 Tehran, Iran
a r t i c l e
i n f o
Article history: Available online 11 May 2010 Keywords: Braided composites Stiffness calculation Analytical method Unit cell Tows undulation
a b s t r a c t This paper presents a new analytical method for calculation of the stiffness of two-dimensional tri-axial braided composites. A unit cell has been introduced as a representative cell of a braided composite and its components. The braided composite is considered as consisting of three layers. The first two layers represent braided tows and the third layer is the axial tow. Then, using rule of mixtures, mechanical properties of each layer are calculated. Next, using analytical relations, the undulation of representative layers of braided tows is calculated. Finally, using a volume averaging method, the total stiffness of the braided composite is calculated. The results are compared with those obtained from experimental methods and the effect of braided tows crimp on the stiffness of braided composites is examined. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Braided composites have received increasing attention from scientists during the past two decades. Due to their exceptional mechanical properties, braided composites are utilized in aerospace, automobile and marine industries [1]. The braiding process competes well with filament winding, pultrusion, and tape lay-up. Braiding compares favorably in terms of structural integrity of components, design flexibility, damage tolerance, repair ability, and low manufacturing cost. Braiding advantages are high rate of strand deposition on the mandrel, ability to produce complex shapes, low capital investment cost, and minimal labor cost. The most important braiding process disadvantage is the difficulty in producing low braid angle preforms [2]. Braided structures may be divided into two categories, namely two-dimensional and three-dimensional. Due to the complexity of braid structures, various parameters including tow and matrix mechanical properties, braid angle, tow crimp level and tow volume fraction can affect their mechanical properties [3]. Extensive studies have been conducted to investigate mechanical properties of textile composites [4]. Among textile composites, woven composites are mostly used for structural applications. Hence, most studies carried out in this area are related to this type of composites. Ishikawa and Chou [5–7] have developed three analytical models for two-dimensional woven composites based on Classical Lamination Theory (CLT). These three models are the mosaic model, fiber undulating model and the fiber bridging model. * Corresponding author. Tel./fax: +98 21 7720 8127. E-mail address: [email protected] (M.M. Shokrieh). 0263-8223/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2010.04.016
When compared with the experimental data, they show that the mosaic model provides an appropriate estimate of elastic properties of composites, the tow undulating model is useful for modeling of plain weave fabrics and the bridging model is desirable for satin weave fabrics. Extending the one-dimensional model of Ishikawa and Chou [7] into a two-dimensional model, Naik and Shembeker [8] presented a model for the analysis of plain woven fabric composites. A micromechanical model was introduced by Huang [9] for two-dimensional examination of mechanical properties of composites reinforced by braided tows or woven tows. Tsai et al. [10] developed a parallelogram spring model for analysis of twodimensional tri-axial braided composites. In this model the effect of changes in various parameters of braid such as braid angle, tow volume fraction, etc. on mechanical properties of braid were investigated. Potluri et al. [11] presented an investigation of flexural and torsional properties of two-dimensional biaxial and tri-axial braided composites with one or more layers, at different braid angles. Recent studies have shown that crimp angle and braid angle affect the strength and stiffness of the braided composites. Through experimental data, Phoenix [12] has shown that increase in crimp angle or braid angle causes decrease in the strength of braided composite. Chen et al. [13] have presented finite multiphase element method in their research. In this method, each element represents more than one material, i.e., braided composites consist of three types of unit cells in different regions which are: interior, surface and corner regions. Masters et al. [14], presented an analytical model for predicting mechanical properties of two-dimensional tri-axial braided composites, using a simple rule-of-mixtures idea. Byun [15] developed
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an analytical model based on the unit cell for the prediction of the geometric characteristics and three-dimensional engineering constants of two-dimensional tri-axial braided composites. In this model the elastic model utilizes coordinate transformation and the averaging of stiffness and compliance constants on the basis of the volume fraction of each reinforcement and matrix material. On the other hand, Quek et al. [16] presented an analytical method to examine mechanical properties of two-dimensional tri-axial braided composites. In their model, a unit cell is considered as representative of material properties and its components are examined; each layer is considered as a transversely isotropic material. The average stiffness of unit cell is calculated by locating fibers beside each other in the unit cell and applying the volume averaging formula. The method presented by Quek et al. [16] has some shortcomings in calculation of braid mechanical properties such as shear modulus and transverse modulus. The present paper introduces a new model for calculation of these properties. The results are compared with those obtained from Quek et al. [16] model and existing experimental results. 2. Explanation of the presented analytical model This paper introduces a new model for calculation of the stiffness of two-dimensional tri-axial braided composites. This model studies a unit cell which is the representative of composite’s properties. The composite is obtained repeating of this unit cell. Fig. 1 illustrates a two-dimensional tri-axial braided composite and the representative unit cell. As depicted in Fig. 1, each unit cell consists of two groups of braided tows which are aligned in diagonal direction; a group of axial tows laid into longitudinal direction, and a matrix. To calculate the total stiffness of the unit cell, this is supposed to be composed of three separate composite layers. By using the rule of mixture, mechanical properties of each composite layer are calculated:
Vm ¼ 1 Vf
ð1Þ
E11 ¼ E11f V f þ Em V m
ð2Þ
E22 ¼ 1=ððV f =E22f Þ þ ðV m =Em ÞÞ
ð3Þ
t12 ¼ t12f V f þ tm V m
ð4Þ
G12 ¼ G12f Gm =ðG12f ð1 V f Þ þ Gm ðV f ÞÞ
ð5Þ
t23 ¼ t12 ð1 t12 ðE22 =E11 ÞÞ=ð1 t12 Þ G23 ¼ E22 =2 ð1 þ t23 Þ
ð7Þ
ð6Þ
In above formulas, E11 indicates longitudinal modulus, E22 transverse modulus, 12 longitudinal Poisson’s ratio, 23 transverse Poisson’s ratio, G12 axial shear modulus and G23 transverse shear modulus. Indices f and m represent matrix and fiber, respectively.
To calculate the Poisson’s ratio 23, the formula introduced by Christensen et al. [17] is used. Then, each layer of the composite is considered to be made of a transversely isotropic material and the stiffness matrix is defined as follows [18]:
2
3
C 11
C 12
C 13
0
0
0
6C 6 12 6 6 C 13 C ij ¼ 6 6 0 6 6 4 0
C 22
C 23
0
0
C 23 0
C 33 0
0 C 44
0 0
0
0
0
C 55
0 7 7 7 0 7 7 0 7 7 7 0 5
0
0
0
0
0
ð8Þ
C 66
in which
E11 C 11 ¼ 1 m223 V E22 C 12 ¼ C 13 ¼ m12 ð1 þ m23 Þ V 2 E22 E22 C 23 ¼ m23 þ m12 E11 V E22 E22 C 22 ¼ C 33 ¼ 1 m212 E11 V E22 C 44 ¼ G23 ¼ 2ð1 þ m23 Þ C 55 ¼ C 66 ¼ G12
ð9Þ ð10Þ ð11Þ ð12Þ ð13Þ ð14Þ
where
E22 V ¼ ð1 þ m23 Þ 1 m23 2m212 E11
ð15Þ
Tow crimp affects the stiffness of the composite layer. Therefore, to calculate the stiffness of the composite layer which represents braided tow, the effect of the crimp should be calculated at first. For this reason, as Fig. 2 illustrates, composite crimp is considered as a sinusoidal function and 2L is the wavelength of the braided tow. The following equations are deducted:
z0 ¼ A sin
p x0
L 0 pA px tanðbÞ ¼ cos L L 1 ^ ¼ cosðbÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1 þ tan2 ðbÞ
tanðbÞ ^ ¼ sinðbÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 þ tan2 ðbÞ
ð16Þ ð17Þ ð18Þ
ð19Þ
To calculate the crimp, the rotation matrices around y-axis are multiplied by the stiffness matrix [16]. Then, the averaged transformed local stiffness is calculated over one wavelength.
Fig. 1. Schematic illustration of two-dimensional tri-axial braided composites and the representative volume cell (RUC).
Fig. 2. Side view of braided tow representative layer.
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3
2
^2 0 ^n ^ ^2 0 n 2m 0 m 7 6 6 0 1 0 0 0 0 7 7 6 6 n 2 ^ 2 0 2m ^n ^ 0 m 0 7 7 6 ^ b T1 ¼ 6 7 6 0 ^ ^7 0 0 m 0 n 7 6 6 ^^ ^2 0 7 ^n ^ 0 m ^2 n 5 4 mn 0 m ^ ^ 0 0 0 n 0 m 3 2 ^2 ^n ^ ^2 0 n 0 m 0 m 7 6 6 0 1 0 0 0 0 7 7 6 6 n 2 ^2 ^n ^ 0 m 0 m 0 7 7 6 ^ b T2 ¼ 6 7 6 0 ^ ^7 0 0 m 0 n 7 6 6 ^2 0 7 ^n ^ 0 2m ^n ^ 0 m ^2 n 5 4 2m ^ ^ 0 0 0 n 0 m Z 1 2l b 1 frg ¼ ½ T 1 ½C½ Tb 2 dxfeg 2l 0 frg ¼ ½Cfeg
ð20Þ
þa
ð21Þ
6 6 6 6 T1 ¼ 6 6 6 6 4 2 6 6 6 6 T2 ¼ 6 6 6 6 4
n2
0
0
0
n2
m2
0
0
0
0
0
1
0
0
0
0
0 m n
0
0
0
n
2mn
ð22Þ
3
2mn 7 7 7 7 0 7 7 0 7 7 5 0
m
mn mn 0 0 0 m2 n2 2 2 n 0 0 0 mn m n2 0
m2 0
0
0
0 m n
0
0
0
0 1
2mn 2mn 0
0 0 n 0
0 0
mn 0 0
m 0
0
ð24Þ
3 7 7 7 7 7 7 7 7 5
ð25Þ
2
m n2
m ¼ cosðaÞ
ð26Þ
n ¼ sinðaÞ
ð27Þ
1
frg ¼ ½T 1 ½C½T 2 feg frg ¼ ½C Global feg
0
ð30Þ
A thickness ratio is attributed to each layer and then multiplied by the corresponding stiffness matrix. In the above formula h+a, ha and h0 are thickness ratios and defined as the thicknesses of each layer to the thickness of whole composite. It must be mentioned that the thickness of all three layers are assumed to be equal.
3. Evaluation of results
ð23Þ
m2
a
½C RUC ¼ h ½C Global þa þ h ½C Global a þ h ½C Global 0
Another important step for the calculation of the layer’s stiffness is the tow rotation about the z-axis. The two rotation matrices around z direction must be considered [18] (see Fig. 3)
2
tows, no rotation is considered since these tows do not have any crimp and are aligned in direction of the axis. After calculating the stiffness matrix of each layer, the stiffness matrix of the unit cell can be deducted by the following equation:
ð28Þ ð29Þ
The stiffness matrix for each layer is thus determined. For layers representing braided tows, a rotation around y-axis and then a rotation around z-axis are applied. For the representative layer of axial
The material considered is E-glass/polyvinyl ester composites. Mechanical properties of the fiber and resin are listed in Tables 1 and 2. A comparison is carried out between experimental results [16], results obtained from the present analytical method and results proposed by Quek et al. [16], presented in Table 3. The following table demonstrates results for -30/0/+30 braided composites with fiber volume fraction of %45. Results presented in the third and fourth columns of Table 3 are obtained by Quek method and the present model, respectively. Also, the numbers in the parentheses illustrate the error percent between these models and experimental results. As it can be from Table 3, the results obtained by the present model are fairly close to experimental results. In Quek method, results achieved for the transverse and shear modulus have remarkable differences with experimental results. However, as it is shown in Table 3, this difference is dramatically decreased by applying the present method and results are closer to those obtained by experiment. It seems that Quek method is not able to calculate Gxy correctly. The main reason for this shortcoming is explained as follows. In Quek’s model, the composite is divided into four separate layers, i.e., two layers representing braided tows, one layer representing axial tow and one part representing the matrix. Then, the volume averaging of the stiffness of these four layers is conducted by Quek, i.e., the stiffness of each layer is multiplied by their volume fraction and the total stiffness is obtained
½C RUC ¼ V þa ½C Global þa þ V a ½C Global a þ V 0 ½C Global 0 þ V m ½Cm
ð31Þ
The structure of Eq. (31) is very similar to that of equation G = GfVf + GmVm. In the rule of mixture approach, the latter should not be used since it leads to wrong results; this explains why Quek’s results are inaccurate when calculating Gxy. In the present model, G G
the correct formula G ¼ G V mfþGmm V is used thus obtaining results that f
f
are more comparable with experimental results.
Table 1 Mechanical properties of E-glass. Axial modulus (E11) Transverse modulus (E22) Axial shear modulus (G12) Axial Poisson’s ratio (t12)
73 (GPa) 73 (GPa) 30 (GPa) 0.23
Table 2 Mechanical properties of polyvinyl ester.
Fig. 3. Rotation of the coordinate system around z-axis.
Young’s modulus Poisson’s ratio
3.45 (GPa) 0.35
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Table 3 Comparison between analytical and experimental results. [30/0/+30] braided composite consists of E-glass and polyvinyl ester resin with fiber volume fraction of 45% Mechanical properties
Experimental results [16]
Results obtained from Quek method [16]
Results of the present model
Ex (GPa) Ey (GPa) Gxy (GPa)
27.18 12.89 6.70 0.33
26.68 14.52 8.55 0.34
26.44 12.43 6.92 0.34
txy
(1.84%) (12.64%) (27.61%) (3.03%)
(2.72%) (3.57%) (3.28%) (3.03%)
4. Behavior of braided composite by changing of physical properties
Fig. 6. Transverse stiffness variation for different aspect ratios.
After comparison of the results obtained from the present model with experimental results, the behavior of braided composites resulting from changes of physical properties is predicted. Effects of different aspect ratios on the longitudinal, transverse and shear modulus are studied. The definition of aspect ratio (AR) is shown in Fig. 4. For each case the results of Quek model, the present method, CLT method and experimental results are compared.
AR ¼ A=L
ð32Þ
As it is shown in Fig. 5, as the aspect ratio increases, the stiffness in longitudinal direction decreases. This may be explained considering that by increasing the aspect ratio, the load bearing of the tow in longitudinal direction decreases. In addition, CLT method does not consider crimp variations. Therefore, the longitudinal stiffness, calculated by this method, is constant for different magnitudes of crimps. As it can be observed in Fig. 6, by increasing the aspect ratio the transverse stiffness increases. As shown in Fig. 6, the present model is in a good agreement with the experimental result. The shear modulus, as shown in Fig. 7, decreases by increasing the aspect ratio. Also in this case, the present model is in good agreement with the experimental results. A key parameter for braided composite stiffness is the braid angle. The effect of braid angle on mechanical properties of braided composites is explained below. As shown in Fig. 8, as the braid angle increases, longitudinal stiffness decreases. For tri-axial braided
Fig. 7. Shear stiffness variation for different aspect ratios.
Fig. 4. Side view of a composite representing braided tows. Fig. 8. Longitudinal stiffness variation with change of braid angle.
Fig. 5. Longitudinal stiffness variation for different aspect ratios.
composites, longitudinal stiffness degradation due to the change of braid angle is much less than that of bi-axial braided composites. Since bi-axial braided composites do not have tows in the longitudinal direction, they show substantial stiffness degradation with the change of the braid angle. It is shown in Fig. 9 that by increasing the braid angle, transverse stiffness increases. This is due to increasing of the angle between tows and the y-axis, as the braid angle increases. Thus, tows load bearing in this direction increases which results in increasing of the transverse stiffness. As Fig. 10 illustrates, shear stiffness increases as braid angle increases, reaching a peak between 40° and 50°, but as the braid angle increases, shear stiffness starts to decline. In case of absolute symmetry, the peak would occur at 45°; however, since tri-axial
M.M. Shokrieh, M.S. Mazloomi / Composite Structures 92 (2010) 2901–2905
2905
The results obtained by applying this method are compared with those achieved from Quek et al. and experimental methods. Finally, after validation of the new model, predictions are made for longitudinal, transverse and shear stiffnesses according to different tow aspect ratios and braid angles. It is shown that longitudinal, transverse and shear stiffnesses are sensitive to variations of braid parameters such as tow aspect ratio and braid angle. In comparison with Quek method, results obtained from the present method show better agreement with experimental results. References
Fig. 9. Transverse stiffness variation with change of braid angle.
Fig. 10. Shear stiffness variation for different braid angles.
braided composites have tows in longitudinal direction, the peak stands at an angle less than the 45°. For a bi-axial braided composite, the peak will exactly occur at 45°. 5. Conclusion This research presents an analytical model to calculate mechanical properties of tri-axial two-dimensional braided composites.
[1] Chawla KK. Composite materials: science and engineering. 2nd ed. Science; 1998. [2] Cagri Ayranci, Jason Carey. 2D braided composites: a review for stiffness critical applications. Compos Struct 2008;85(1):43–58. [3] Shunjun S, Anthony W, Khaled S, Xinran X, Omar F. Braided textile composites under compressive loads: modeling the response, strength and degradation. Compos Sci Technol 2007;67:3059–70. [4] Miravette A. 3-D textile reinforcements in composite materials. Cambridge: Woodhead Publishing Limited; 1999. [5] Ishikawa T, Chou TW. Elastic behavior of woven hybrid composites. J Compos Mater 1982;16:2–19. [6] Ishikawa T, Chou TW. Stiffness and strength behavior of woven fabric composites. J Mater Sci 1982;17:3211–20. [7] Ishikawa T, Chou TW. Onedimensional analysis of woven fabric composites. AIAA J 1983;21:1714–21. [8] Naik NK, Shembekar PS. Elastic behaviour of woven fabric composites: I – lamina analysis. J Compos Mater 1992;26:2196–225. [9] Huang ZM. The mechanical properties of composites reinforced with woven and braided fabrics. Compos Sci Technol 2000;60:479–98. [10] Tsai KH, Chung LH, Wen LC, Chang HCA. Parallelogram spring model for predicting the effective elastic properties of 2D braided composites. Compos Struct 2008;83:273–83. [11] Potluri P, Manan A, Francke M, Day RJ. Flexural and torsional behaviour of biaxial and triaxial braided composite structures. Compos Struct 2006;75:377–86. [12] Phoenix SL. Mechanical response of a tubular braided cable with an elastic core. Text Res J 1978;48(2):81–91. [13] Chen L, Tao XM, Choy CL. Mechanical analysis of 3-D braided composites by the finite multiphase element method. Compos Sci Technol 1999;59:2383–91. [14] Masters J, Foye R, Pastore C, Gowayed Y. Mechanical properties of triaxial braided composites: experimental and analytical results. J Compos Technol Res 1993;15(2):112–22. [15] Joon-Hyung Byun. The analytical characterization of 2-D braided textile composites. Compos Sci Technol 2000;60(5):705–16. [16] Quek SC, Waas AM, Shahwan KW, Agaram V. Analysis of 2-D triaxial flat braided textile composites. Int J Mech Sci 2003;45:1077–96. [17] Christensen RM. Tensor transformations and failure criteria for the analysis of fiber composite materials. J Compos Mater 1988;22:874. [18] Whitney JM. Structural analysis of laminated anisotropic plates. Technomic 1987.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
An analytical method for calculating stiffness of two-dimensional tri-axial braided composites Mahmood M. Shokrieh *, Mohammad S. Mazloomi Composites Research Laboratory, Center of Excellence for Experimental Solid Mechanics and Dynamics, Department of Mechanical Engineering, Iran University of Science and Technology, 16846-13114 Tehran, Iran
a r t i c l e
i n f o
Article history: Available online 11 May 2010 Keywords: Braided composites Stiffness calculation Analytical method Unit cell Tows undulation
a b s t r a c t This paper presents a new analytical method for calculation of the stiffness of two-dimensional tri-axial braided composites. A unit cell has been introduced as a representative cell of a braided composite and its components. The braided composite is considered as consisting of three layers. The first two layers represent braided tows and the third layer is the axial tow. Then, using rule of mixtures, mechanical properties of each layer are calculated. Next, using analytical relations, the undulation of representative layers of braided tows is calculated. Finally, using a volume averaging method, the total stiffness of the braided composite is calculated. The results are compared with those obtained from experimental methods and the effect of braided tows crimp on the stiffness of braided composites is examined. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Braided composites have received increasing attention from scientists during the past two decades. Due to their exceptional mechanical properties, braided composites are utilized in aerospace, automobile and marine industries [1]. The braiding process competes well with filament winding, pultrusion, and tape lay-up. Braiding compares favorably in terms of structural integrity of components, design flexibility, damage tolerance, repair ability, and low manufacturing cost. Braiding advantages are high rate of strand deposition on the mandrel, ability to produce complex shapes, low capital investment cost, and minimal labor cost. The most important braiding process disadvantage is the difficulty in producing low braid angle preforms [2]. Braided structures may be divided into two categories, namely two-dimensional and three-dimensional. Due to the complexity of braid structures, various parameters including tow and matrix mechanical properties, braid angle, tow crimp level and tow volume fraction can affect their mechanical properties [3]. Extensive studies have been conducted to investigate mechanical properties of textile composites [4]. Among textile composites, woven composites are mostly used for structural applications. Hence, most studies carried out in this area are related to this type of composites. Ishikawa and Chou [5–7] have developed three analytical models for two-dimensional woven composites based on Classical Lamination Theory (CLT). These three models are the mosaic model, fiber undulating model and the fiber bridging model. * Corresponding author. Tel./fax: +98 21 7720 8127. E-mail address: [email protected] (M.M. Shokrieh). 0263-8223/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2010.04.016
When compared with the experimental data, they show that the mosaic model provides an appropriate estimate of elastic properties of composites, the tow undulating model is useful for modeling of plain weave fabrics and the bridging model is desirable for satin weave fabrics. Extending the one-dimensional model of Ishikawa and Chou [7] into a two-dimensional model, Naik and Shembeker [8] presented a model for the analysis of plain woven fabric composites. A micromechanical model was introduced by Huang [9] for two-dimensional examination of mechanical properties of composites reinforced by braided tows or woven tows. Tsai et al. [10] developed a parallelogram spring model for analysis of twodimensional tri-axial braided composites. In this model the effect of changes in various parameters of braid such as braid angle, tow volume fraction, etc. on mechanical properties of braid were investigated. Potluri et al. [11] presented an investigation of flexural and torsional properties of two-dimensional biaxial and tri-axial braided composites with one or more layers, at different braid angles. Recent studies have shown that crimp angle and braid angle affect the strength and stiffness of the braided composites. Through experimental data, Phoenix [12] has shown that increase in crimp angle or braid angle causes decrease in the strength of braided composite. Chen et al. [13] have presented finite multiphase element method in their research. In this method, each element represents more than one material, i.e., braided composites consist of three types of unit cells in different regions which are: interior, surface and corner regions. Masters et al. [14], presented an analytical model for predicting mechanical properties of two-dimensional tri-axial braided composites, using a simple rule-of-mixtures idea. Byun [15] developed
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an analytical model based on the unit cell for the prediction of the geometric characteristics and three-dimensional engineering constants of two-dimensional tri-axial braided composites. In this model the elastic model utilizes coordinate transformation and the averaging of stiffness and compliance constants on the basis of the volume fraction of each reinforcement and matrix material. On the other hand, Quek et al. [16] presented an analytical method to examine mechanical properties of two-dimensional tri-axial braided composites. In their model, a unit cell is considered as representative of material properties and its components are examined; each layer is considered as a transversely isotropic material. The average stiffness of unit cell is calculated by locating fibers beside each other in the unit cell and applying the volume averaging formula. The method presented by Quek et al. [16] has some shortcomings in calculation of braid mechanical properties such as shear modulus and transverse modulus. The present paper introduces a new model for calculation of these properties. The results are compared with those obtained from Quek et al. [16] model and existing experimental results. 2. Explanation of the presented analytical model This paper introduces a new model for calculation of the stiffness of two-dimensional tri-axial braided composites. This model studies a unit cell which is the representative of composite’s properties. The composite is obtained repeating of this unit cell. Fig. 1 illustrates a two-dimensional tri-axial braided composite and the representative unit cell. As depicted in Fig. 1, each unit cell consists of two groups of braided tows which are aligned in diagonal direction; a group of axial tows laid into longitudinal direction, and a matrix. To calculate the total stiffness of the unit cell, this is supposed to be composed of three separate composite layers. By using the rule of mixture, mechanical properties of each composite layer are calculated:
Vm ¼ 1 Vf
ð1Þ
E11 ¼ E11f V f þ Em V m
ð2Þ
E22 ¼ 1=ððV f =E22f Þ þ ðV m =Em ÞÞ
ð3Þ
t12 ¼ t12f V f þ tm V m
ð4Þ
G12 ¼ G12f Gm =ðG12f ð1 V f Þ þ Gm ðV f ÞÞ
ð5Þ
t23 ¼ t12 ð1 t12 ðE22 =E11 ÞÞ=ð1 t12 Þ G23 ¼ E22 =2 ð1 þ t23 Þ
ð7Þ
ð6Þ
In above formulas, E11 indicates longitudinal modulus, E22 transverse modulus, 12 longitudinal Poisson’s ratio, 23 transverse Poisson’s ratio, G12 axial shear modulus and G23 transverse shear modulus. Indices f and m represent matrix and fiber, respectively.
To calculate the Poisson’s ratio 23, the formula introduced by Christensen et al. [17] is used. Then, each layer of the composite is considered to be made of a transversely isotropic material and the stiffness matrix is defined as follows [18]:
2
3
C 11
C 12
C 13
0
0
0
6C 6 12 6 6 C 13 C ij ¼ 6 6 0 6 6 4 0
C 22
C 23
0
0
C 23 0
C 33 0
0 C 44
0 0
0
0
0
C 55
0 7 7 7 0 7 7 0 7 7 7 0 5
0
0
0
0
0
ð8Þ
C 66
in which
E11 C 11 ¼ 1 m223 V E22 C 12 ¼ C 13 ¼ m12 ð1 þ m23 Þ V 2 E22 E22 C 23 ¼ m23 þ m12 E11 V E22 E22 C 22 ¼ C 33 ¼ 1 m212 E11 V E22 C 44 ¼ G23 ¼ 2ð1 þ m23 Þ C 55 ¼ C 66 ¼ G12
ð9Þ ð10Þ ð11Þ ð12Þ ð13Þ ð14Þ
where
E22 V ¼ ð1 þ m23 Þ 1 m23 2m212 E11
ð15Þ
Tow crimp affects the stiffness of the composite layer. Therefore, to calculate the stiffness of the composite layer which represents braided tow, the effect of the crimp should be calculated at first. For this reason, as Fig. 2 illustrates, composite crimp is considered as a sinusoidal function and 2L is the wavelength of the braided tow. The following equations are deducted:
z0 ¼ A sin
p x0
L 0 pA px tanðbÞ ¼ cos L L 1 ^ ¼ cosðbÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1 þ tan2 ðbÞ
tanðbÞ ^ ¼ sinðbÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 þ tan2 ðbÞ
ð16Þ ð17Þ ð18Þ
ð19Þ
To calculate the crimp, the rotation matrices around y-axis are multiplied by the stiffness matrix [16]. Then, the averaged transformed local stiffness is calculated over one wavelength.
Fig. 1. Schematic illustration of two-dimensional tri-axial braided composites and the representative volume cell (RUC).
Fig. 2. Side view of braided tow representative layer.
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3
2
^2 0 ^n ^ ^2 0 n 2m 0 m 7 6 6 0 1 0 0 0 0 7 7 6 6 n 2 ^ 2 0 2m ^n ^ 0 m 0 7 7 6 ^ b T1 ¼ 6 7 6 0 ^ ^7 0 0 m 0 n 7 6 6 ^^ ^2 0 7 ^n ^ 0 m ^2 n 5 4 mn 0 m ^ ^ 0 0 0 n 0 m 3 2 ^2 ^n ^ ^2 0 n 0 m 0 m 7 6 6 0 1 0 0 0 0 7 7 6 6 n 2 ^2 ^n ^ 0 m 0 m 0 7 7 6 ^ b T2 ¼ 6 7 6 0 ^ ^7 0 0 m 0 n 7 6 6 ^2 0 7 ^n ^ 0 2m ^n ^ 0 m ^2 n 5 4 2m ^ ^ 0 0 0 n 0 m Z 1 2l b 1 frg ¼ ½ T 1 ½C½ Tb 2 dxfeg 2l 0 frg ¼ ½Cfeg
ð20Þ
þa
ð21Þ
6 6 6 6 T1 ¼ 6 6 6 6 4 2 6 6 6 6 T2 ¼ 6 6 6 6 4
n2
0
0
0
n2
m2
0
0
0
0
0
1
0
0
0
0
0 m n
0
0
0
n
2mn
ð22Þ
3
2mn 7 7 7 7 0 7 7 0 7 7 5 0
m
mn mn 0 0 0 m2 n2 2 2 n 0 0 0 mn m n2 0
m2 0
0
0
0 m n
0
0
0
0 1
2mn 2mn 0
0 0 n 0
0 0
mn 0 0
m 0
0
ð24Þ
3 7 7 7 7 7 7 7 7 5
ð25Þ
2
m n2
m ¼ cosðaÞ
ð26Þ
n ¼ sinðaÞ
ð27Þ
1
frg ¼ ½T 1 ½C½T 2 feg frg ¼ ½C Global feg
0
ð30Þ
A thickness ratio is attributed to each layer and then multiplied by the corresponding stiffness matrix. In the above formula h+a, ha and h0 are thickness ratios and defined as the thicknesses of each layer to the thickness of whole composite. It must be mentioned that the thickness of all three layers are assumed to be equal.
3. Evaluation of results
ð23Þ
m2
a
½C RUC ¼ h ½C Global þa þ h ½C Global a þ h ½C Global 0
Another important step for the calculation of the layer’s stiffness is the tow rotation about the z-axis. The two rotation matrices around z direction must be considered [18] (see Fig. 3)
2
tows, no rotation is considered since these tows do not have any crimp and are aligned in direction of the axis. After calculating the stiffness matrix of each layer, the stiffness matrix of the unit cell can be deducted by the following equation:
ð28Þ ð29Þ
The stiffness matrix for each layer is thus determined. For layers representing braided tows, a rotation around y-axis and then a rotation around z-axis are applied. For the representative layer of axial
The material considered is E-glass/polyvinyl ester composites. Mechanical properties of the fiber and resin are listed in Tables 1 and 2. A comparison is carried out between experimental results [16], results obtained from the present analytical method and results proposed by Quek et al. [16], presented in Table 3. The following table demonstrates results for -30/0/+30 braided composites with fiber volume fraction of %45. Results presented in the third and fourth columns of Table 3 are obtained by Quek method and the present model, respectively. Also, the numbers in the parentheses illustrate the error percent between these models and experimental results. As it can be from Table 3, the results obtained by the present model are fairly close to experimental results. In Quek method, results achieved for the transverse and shear modulus have remarkable differences with experimental results. However, as it is shown in Table 3, this difference is dramatically decreased by applying the present method and results are closer to those obtained by experiment. It seems that Quek method is not able to calculate Gxy correctly. The main reason for this shortcoming is explained as follows. In Quek’s model, the composite is divided into four separate layers, i.e., two layers representing braided tows, one layer representing axial tow and one part representing the matrix. Then, the volume averaging of the stiffness of these four layers is conducted by Quek, i.e., the stiffness of each layer is multiplied by their volume fraction and the total stiffness is obtained
½C RUC ¼ V þa ½C Global þa þ V a ½C Global a þ V 0 ½C Global 0 þ V m ½Cm
ð31Þ
The structure of Eq. (31) is very similar to that of equation G = GfVf + GmVm. In the rule of mixture approach, the latter should not be used since it leads to wrong results; this explains why Quek’s results are inaccurate when calculating Gxy. In the present model, G G
the correct formula G ¼ G V mfþGmm V is used thus obtaining results that f
f
are more comparable with experimental results.
Table 1 Mechanical properties of E-glass. Axial modulus (E11) Transverse modulus (E22) Axial shear modulus (G12) Axial Poisson’s ratio (t12)
73 (GPa) 73 (GPa) 30 (GPa) 0.23
Table 2 Mechanical properties of polyvinyl ester.
Fig. 3. Rotation of the coordinate system around z-axis.
Young’s modulus Poisson’s ratio
3.45 (GPa) 0.35
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Table 3 Comparison between analytical and experimental results. [30/0/+30] braided composite consists of E-glass and polyvinyl ester resin with fiber volume fraction of 45% Mechanical properties
Experimental results [16]
Results obtained from Quek method [16]
Results of the present model
Ex (GPa) Ey (GPa) Gxy (GPa)
27.18 12.89 6.70 0.33
26.68 14.52 8.55 0.34
26.44 12.43 6.92 0.34
txy
(1.84%) (12.64%) (27.61%) (3.03%)
(2.72%) (3.57%) (3.28%) (3.03%)
4. Behavior of braided composite by changing of physical properties
Fig. 6. Transverse stiffness variation for different aspect ratios.
After comparison of the results obtained from the present model with experimental results, the behavior of braided composites resulting from changes of physical properties is predicted. Effects of different aspect ratios on the longitudinal, transverse and shear modulus are studied. The definition of aspect ratio (AR) is shown in Fig. 4. For each case the results of Quek model, the present method, CLT method and experimental results are compared.
AR ¼ A=L
ð32Þ
As it is shown in Fig. 5, as the aspect ratio increases, the stiffness in longitudinal direction decreases. This may be explained considering that by increasing the aspect ratio, the load bearing of the tow in longitudinal direction decreases. In addition, CLT method does not consider crimp variations. Therefore, the longitudinal stiffness, calculated by this method, is constant for different magnitudes of crimps. As it can be observed in Fig. 6, by increasing the aspect ratio the transverse stiffness increases. As shown in Fig. 6, the present model is in a good agreement with the experimental result. The shear modulus, as shown in Fig. 7, decreases by increasing the aspect ratio. Also in this case, the present model is in good agreement with the experimental results. A key parameter for braided composite stiffness is the braid angle. The effect of braid angle on mechanical properties of braided composites is explained below. As shown in Fig. 8, as the braid angle increases, longitudinal stiffness decreases. For tri-axial braided
Fig. 7. Shear stiffness variation for different aspect ratios.
Fig. 4. Side view of a composite representing braided tows. Fig. 8. Longitudinal stiffness variation with change of braid angle.
Fig. 5. Longitudinal stiffness variation for different aspect ratios.
composites, longitudinal stiffness degradation due to the change of braid angle is much less than that of bi-axial braided composites. Since bi-axial braided composites do not have tows in the longitudinal direction, they show substantial stiffness degradation with the change of the braid angle. It is shown in Fig. 9 that by increasing the braid angle, transverse stiffness increases. This is due to increasing of the angle between tows and the y-axis, as the braid angle increases. Thus, tows load bearing in this direction increases which results in increasing of the transverse stiffness. As Fig. 10 illustrates, shear stiffness increases as braid angle increases, reaching a peak between 40° and 50°, but as the braid angle increases, shear stiffness starts to decline. In case of absolute symmetry, the peak would occur at 45°; however, since tri-axial
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The results obtained by applying this method are compared with those achieved from Quek et al. and experimental methods. Finally, after validation of the new model, predictions are made for longitudinal, transverse and shear stiffnesses according to different tow aspect ratios and braid angles. It is shown that longitudinal, transverse and shear stiffnesses are sensitive to variations of braid parameters such as tow aspect ratio and braid angle. In comparison with Quek method, results obtained from the present method show better agreement with experimental results. References
Fig. 9. Transverse stiffness variation with change of braid angle.
Fig. 10. Shear stiffness variation for different braid angles.
braided composites have tows in longitudinal direction, the peak stands at an angle less than the 45°. For a bi-axial braided composite, the peak will exactly occur at 45°. 5. Conclusion This research presents an analytical model to calculate mechanical properties of tri-axial two-dimensional braided composites.
[1] Chawla KK. Composite materials: science and engineering. 2nd ed. Science; 1998. [2] Cagri Ayranci, Jason Carey. 2D braided composites: a review for stiffness critical applications. Compos Struct 2008;85(1):43–58. [3] Shunjun S, Anthony W, Khaled S, Xinran X, Omar F. Braided textile composites under compressive loads: modeling the response, strength and degradation. Compos Sci Technol 2007;67:3059–70. [4] Miravette A. 3-D textile reinforcements in composite materials. Cambridge: Woodhead Publishing Limited; 1999. [5] Ishikawa T, Chou TW. Elastic behavior of woven hybrid composites. J Compos Mater 1982;16:2–19. [6] Ishikawa T, Chou TW. Stiffness and strength behavior of woven fabric composites. J Mater Sci 1982;17:3211–20. [7] Ishikawa T, Chou TW. Onedimensional analysis of woven fabric composites. AIAA J 1983;21:1714–21. [8] Naik NK, Shembekar PS. Elastic behaviour of woven fabric composites: I – lamina analysis. J Compos Mater 1992;26:2196–225. [9] Huang ZM. The mechanical properties of composites reinforced with woven and braided fabrics. Compos Sci Technol 2000;60:479–98. [10] Tsai KH, Chung LH, Wen LC, Chang HCA. Parallelogram spring model for predicting the effective elastic properties of 2D braided composites. Compos Struct 2008;83:273–83. [11] Potluri P, Manan A, Francke M, Day RJ. Flexural and torsional behaviour of biaxial and triaxial braided composite structures. Compos Struct 2006;75:377–86. [12] Phoenix SL. Mechanical response of a tubular braided cable with an elastic core. Text Res J 1978;48(2):81–91. [13] Chen L, Tao XM, Choy CL. Mechanical analysis of 3-D braided composites by the finite multiphase element method. Compos Sci Technol 1999;59:2383–91. [14] Masters J, Foye R, Pastore C, Gowayed Y. Mechanical properties of triaxial braided composites: experimental and analytical results. J Compos Technol Res 1993;15(2):112–22. [15] Joon-Hyung Byun. The analytical characterization of 2-D braided textile composites. Compos Sci Technol 2000;60(5):705–16. [16] Quek SC, Waas AM, Shahwan KW, Agaram V. Analysis of 2-D triaxial flat braided textile composites. Int J Mech Sci 2003;45:1077–96. [17] Christensen RM. Tensor transformations and failure criteria for the analysis of fiber composite materials. J Compos Mater 1988;22:874. [18] Whitney JM. Structural analysis of laminated anisotropic plates. Technomic 1987.