Prediction of Elastic Constants on 3D Four-directional Braided
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1 Prediction of Elastic Constants on 3D Four-directional Braided Composites Prediction of Elastic Constants on 3D Four-directional Braided Composites Liang Dao Zhou 1,2,* and Zhuo Zhuang 1 1 School of Aerospace, Tsinghua University, Beijing, , P.R. China 2 Shanghai Aircraft Design and Research Institute, Shanghai, , P.R. China Received: 22 February 2013, Accepted: 7 November 2013 SUMMARY In this paper, a finite element model with a prestressed unit cell is proposed to predict the elastic constants of 3D four-directional braided composites. Firstly, a finite element unit cell model with damage constitutive for 3D four-directional braided composites is established to investigate the influence of the prestresses in the unit cell. Secondly, the elastic constants of 3D four-directional braided composites are predicted with and without the prestresses in the unit cell. Finally, the influences of the braided direction and fibre content on the elastic constants of 3D four-directional braided composites are discussed. The results will provide an important guidance for designing and evaluating the mechanical properties of 3D four-directional braided composites. Keywords: 3D four-directional braided composite, Prestressed cell, Finite element model, Mechanical properties, Damage constitutive 1. INTRODUCTION The superiority of three-dimensional (3D) braided composites is obvious over conventional laminates composites 1,2, since they have the advantages of suppressing delamination due to the through thickness reinforcement, improved damage tolerance, high impact resistance, shear stiffness and strength and unique torsional rigidity in many advanced structures. In view of the complexity of 3D braided composites, it is difficult to accurately obtain their mechanical properties by experiment. The 3D braided unit cell is the smallest representative structure of 3D braided composites. Therefore, it is necessary to study the influence of the braiding parameters in the unit cell on the mechanical properties of 3D braided composites. Smithers Information Ltd., 2014 Nowadays the theoretical analysis of 3D braided composites is based on the geometrical model. Yang et al. 1 predicted the elastic properties of braided composites using a fibre inclination model, which treated the unit cell of the braided composites as an assemblage of inclined unidirectional laminates. Wu 2 proposed a three-cell model (basis cell, face cell and rod cell) of the 5D braided composites. In fact, finite element analysis has become a powerful tool for predicting the elastic properties of the braided composites. Li et al. 3-5 investigated the influences of the braiding parameters on the mechanical properties of 3D fourdirectional and 3D five-directional rectangular braided composites with the cross-section hexagon shape of the braiding yarn using a finite element model. Fang et al. 6,7 and Xu et al. 8 used finite element method to evaluate the mechanical behaviour of 3D fourdirectional braided composites and 3D five-directional braided composites by means of a representative volume cell. Chen et al. 9 calculated the overall responses of the braided composites based on the local elastic properties in the unit cells of circular cross-section yarn. Zeng et al. 10 discretized the unit cell into a number of rectangular elements of matrix element, yarn element and mixed element to predict the effective moduli and the local stress. However, there are few people considering the prestresses in the unit cell of 3D braided composites, which are inevitably caused by the thermal mismatch between the yarns and the matrix material in fabricating the composites. In this paper, the damage constitutive model is introduced to consider the influence of the prestresses in the unit cell of 3D braid composites. And a finite element with the prestressed unit cell model is established to investigate the influences of the prestresses and the braiding Polymers & Polymer Composites, Vol. 22, No. 9,
2 Liang Dao Zhou and Zhuo Zhuang parameters of the unit cell on the mechanical properties of 3D four-directional braided composites. 2. Damage constitutive of 3D braided composites Matzenmiller et al. 11 presented a damage mechanics model, which has the capability of modeling the damage independently in the principal directions of orthotropic materials. The damage constitutive equation is given as: ε = S iσ (1) Here, s=(s 1, s 2, s 3, s 4, s 5, s 6 ) T, where s i (i=1, 2, 3) is the normal stress component. s i (i=4, 5, 6) is the shear stress component. e=(e 1, e 2, e 3, e 4, e 5, e 6 ) T, where e i (i=1, 2, 3) is the normal strain component. e i (i=4, 5, 6) is the shear strain component. The stiffness matrix (S) can be expressed as: where E 1, E 2, E 3, G 12, G 13, G 23, u 12, u 21, u 13, u 31, u 23 and u 32 are the elastic constants of orthotropic materials without damage. w i (i=1, 2, 3, 23, 13, 12) are the damage parameters, which can be expressed as: (2) ω i =1 exp( 1 me (σ i X i ) m ) (3) where X i is the strength in the i th direction of orthotropic materials. s 1 /X i represents the dimensionless stress in the material. m is a material parameter. For brittle materials, the value of m is larger than that of a plastic material. e (= ) is a constant. Then, the incremental form of the damage constitutive can be expressed as: dε = S i dσ (4) Put Eqs. (1), (2), (3) into Eq. (4): where, (5) 818 Polymers & Polymer Composites, Vol. 22, No. 9, 2014
3 Prediction of Elastic Constants on 3D Four-directional Braided Composites (1 ω i )( σ i ) m ln(1 ω i ) 1 X A i = i (i =1, 2, ) (1 ω i )E i ee i B i = (1 ω i )( σ i ) m ln(1 ω i ) 1 X i (i = 23, 13, 12) (1 ω i )G i eg i 3. Finite element analysis Figure 1 shows the yarn configuration in the unit cell of 3D four-directional braided composites. The braided axis is along the x 1 axis, and both the x 2 axis and the x 3 axis are perpendicular to the braided axis. The size of the unit cell is a b c including 19 yarns with 7 main fibres (f1-f7 shown in Figure 1). There are four braided directions in the unit cell: (i) The yarns of f1 and f2 have the same direction of (c, -a, b), which pass through the points (0, 0, 0) and (0.5c, 0.5a, -0.5b), respectively. (ii) The yarns of f3 and f4 have the same direction of (c, a, b), which pass through the points (0, 0, -0.5b) and (0.5c, -0.5a, 0), respectively. (iii) The yarns of f5 and f6 have the same direction of (c, a, -b), which pass through the points (0, -0.5a, 0) and (0.5c, 0, 0.5b), respectively. (iv) The yarns of f7 have the direction of (c, -a, -b), which passes through the point (0.5c, 0.5a, 0). The braiding angle (g) is defined as the angle between the yarn and the braided axis (Figure 1) as follows: tanγ = a2 + b 2 c (6) The yarn content (y) can be determined as: ψ = π D2 a 2 + b 2 + c 2 abc (7) where D is the diameter of the yarn. Here, ABAQUS finite element software is used to establish a unit cell and predict the mechanical properties of 3D four-directional braided composites. Some basic assumptions are made, as follows: (i) The cross-section of multifilament braiding yarns is of circular shape. (ii) The flexibility of the yarns is not considered. (iii) All yarns in the braided composites have identical constituent material and size. (v) There are no defects in the yarns and the matrix material. (iv) There are no prestresses in the unit cell at the curing temperature. Here, the fibre volume fraction of the yarns is assumed as 100%, and both T300 carbon fibre and QY8911 resin are used as the yarns and the matrix material, respectively. The damage mechanics model in Section 2 is used for both the yarns and the matrix material. In the finite element analyses, the material properties of the T300 carbon fibre and the QY8911 resin are transversely isotropic and isotropic, respectively, which are summarized in Table Based on the incremental form of the damage mechanics model (Eq. (4)) and the yarn configuration in the unit cell (Figure 1), a finite element of 3D four-directional braided composites is established. Figure 2a shows the finite element unit cell of 3D four-directional braided composites without the matrix material. There are two steps in the finite element analysis of 3D fourdirectional braided composites. The temperature cools from the curing temperature (460K) to the room temperature (298K) in the first step, which will cause high prestresses at Figure 1. Geometry model of the unit cell Table 1. Material properties of the T300 carbon fibre and the QY8911 resin 12 Parameter Constant Value T300 carbon fibre Elastic modulus (GPa) E f1 221 E f Shear modulus (GPa) G f12 9 Poisson s ratio ν f ν f Thermal expansion coefficient (με/k) CTE f1 0.9 CTE f2 5 Strength (MPa) σ f 3500 Parameter m in Eq. (3) m f 4 QY8911 resin Elastic modulus (GPa) E m 3 Poisson s ratio ν m 0.4 Thermal expansion coefficient (με/k) CTE m 58 Strength (MPa) σ m 65 Parameter m in Eq. (3) m m 1 Polymers & Polymer Composites, Vol. 22, No. 9,
4 Liang Dao Zhou and Zhuo Zhuang the interface between the T300 yarns and the QY8911 resin. In the second step, the loads are applied on the cell to predict the material parameters of 3D four-directional braided composites, which is transversely isotropic with five material parameters: the elastic modulus in the braided axis (E c1 ), the elastic modulus perpendicular to the braided axis (E c2 ), the shear modulus (G c12 ), Poisson s ratio (u c12 ) and Poisson s ratio (u c23 ), where u cij characterizes the transverse strain in the j-direction, when the material is loaded in the i-direction. Here, three kinds of loading cases are applied to the cell, respectively: (i) Uniaxial tensile loads in the x 1 -direction are applied (Figure 2b). Due to the displacements of the faces in the x 1 -direction and the x 2 -direction, the strain in the x 1 - direction and the x 2 -direction can be obtained. Then E c1 can be predicted by the ratio between the tensile loads and the strain in the x 1 -direction. u c12 can be predicted by the ratio between the strains in the x 1 -direction and the x 2 -direction. (ii) Uniaxial tensile loads in the x 2 -direction are applied (Figure 2c). Due to the displacements of the faces in the x 2 -direction and the x 3 -direction, the strain in the x 1 - direction and the x 2 -direction can be obtained. Then E c2 can be predicted by the ratio of the tensile loads and the strain in the x 2 -direction. u c23 can be predicted by the ratio of the strain in the x 2 -direction and the x 3 -direction. (iii) Shear loads on the two x 1 -x 2 faces are applied (Figure 2d). Due to the displacements on two x 1 -x 2 faces, the shear strain can be obtained. G c12 can be predicted by the ratio of the shear loads and the shear strain. Figure 2. Finite element model and loading form of the unit cell 4. Results and Discussion 4.1. Influence of prestresses A unit cell with braiding angle (g) 20 and yarn content (y) 40% (Figure 2) is selected to analyze the influence of prestresses on the material properties of the unit cell. Here, the diameter of the yarn (D) is assumed to be 1 mm. Based on Eqs. (6) and (7), the sizes of the unit cell can be determined as a=b=2.89 mm, c=11.23 mm. Then two finite element models with and without prestresses are performed on the unit cell. For the grid independency, the models with prestresses are taken as an example. If the seed of the grid is doubled, there will be elements for the unit cell. The error of E c1 between elements and elements is less than 1%, which indicates that the meshes of the finite element model are credible. Table 2 shows the predicted mechanical properties of the unit cell with and without prestresses, respectively. It is shown that the elastic constants of E c1, E c2 and G c12 without prestresses are smaller than those with prestresses. The reason is that the thermal mismatch during the curing process causes high prestresses in the interface between the T300 yarn and the QY8911 resin. The prestresses introduce damage into the unit cell, which causes the stiffness of the unit cell to decrease. The results indicate that the prestresses caused by thermal mismatch cannot be ignored, which has an important influence on predicting the mechanical properties of 3D fourdirectional braided composites. Table 2. Predicted mechanical properties of the unit cell with and without prestresses Parameter Constant Without With prestress prestress Elastic modulus (GPa) E c E c Poisson s ratio ν c ν c Shear modulus (GPa) G c Influence of Braided Direction Here, it is assumed that the yarn content (y) of 3D four-directional braided composites is 40%. And the braiding angles (g) of the unit cell are 20, 30 and 40, respectively. The size of the unit cell is listed in Table 3. Figure 3 shows the geometrical configuration of the unit cell for different braiding angles with and without resin, respectively. 820 Polymers & Polymer Composites, Vol. 22, No. 9, 2014
5 Prediction of Elastic Constants on 3D Four-directional Braided Composites Based on the finite element analysis with prestressed unit cell, the mechanical properties of 3D fourdirectional braided composites with the T300 fibre and the QY8911 resin are predicted. Figure 4 shows the predicted mechanical properties (E c1, E c2, G c12, u c12 and u c23 ) with and without prestress of the unit cell for different braiding angles. The elastic modulus in the braided axis (E c1 ) decreases rapidly with increasing braiding angle (Figure 4a). The elastic modulus perpendicular to the braided axis (E c2 ) decreases slowly and the shear modulus (G c12 ) first increases and then decreases with increasing braiding angle (Figure 4b and c). However, the influence of the braiding angle on E c2 and G c12 is relatively little. For the Poisson s ratio, u c23 decreases and u c12 has little variation with the braiding angle. The result indicates the influence of the braiding angle is mainly on the mechanical properties in the braided axis. However, it has little effect on the mechanical properties perpendicular to the braided axis. Table 3. Size of the unit cell with different braiding angles and different yarn contents Braiding angle ( ) Yarn content (%) a=b (mm) c (mm) Figure 3. Unit cell for different braiding angles: (a) 20 without resin. (b) 30 without resin. (c) 40 without resin. (d) 20 with resin. (e) 30 with resin. (f) 40 with resin 4.3. Influence of Yarn Content The unit cells with braiding angle (g) 20 and yarn contents (y) 30%, 40% and 50% are analyzed by finite element method. Table 3 lists the size of the cell with braiding angle (g) of 20 and yarn content (y) of 30%, 40% and 50%. Figure 5 shows the unit cell for different yarn contents with and without resin. Figure 4. Predicted mechanical properties of the unit cell for different braiding angles: (a) E c1. (b) E c2. (c) G c12. (d) u c12 and u c23 Then the mechanical properties (E c1, E c2, G c1, u c12 and u c23 ) of the unit cell for different yarn contents were predicted by the finite element method with a prestressed unit cell. Figure 6 shows the relationship between the predicted mechanical properties and the yarn content. The elastic moduli E c1, E c2 and G c12 increase with the increasing of the yarn content (Figure 6a). For the Poisson s ratio, u c23 decreases rapidly and u c12 has little variation with increasing yarn content (Figure 6b). It is shown that the mechanical Polymers & Polymer Composites, Vol. 22, No. 9,
6 Liang Dao Zhou and Zhuo Zhuang Figure 5. Unit cell for different yarn contents: (a) 30% without resin. (b) 40% without resin. (c) 50% without resin. (d) 30% with resin. (e) 40% with resin. (f) 50% with resin Figure 6. Predicted mechanical properties of the unit cell for different yarn contents: (a) E c1. (b) E c2. (c) G c12. (d) u c12 and u c23 By comparing the predicted mechanical properties of 3D four-directional braided composites with and without prestresses, it was shown that the influence of the prestresses cannot be ignored, which will introduce the damage into the unit cell and decreases the stiffness of the unit cell. With decreasing braiding angle, the mechanical properties in the braided axis of 3D four-directional braided composites will be increased. However, the influence of the braiding angle on the mechanical properties perpendicular to the braided axis is much smaller. For the yarn content, the mechanical properties of 3D fourdirectional braided composites will be improved with the increasing of the yarn content. References properties of 3D four-directional braided composites will be improved with increasing yarn content. 5. Conclusions In this paper, the mechanical properties of 3D four-directional braided composites were predicted, based on the prestressed unit cell model. Some important conclusions are summarized. A finite element model with damage constitutive relations was established, based on a prestressed unit cell for analyzing the mechanical properties of 3D four-directional braided composites. 1. Yang J.M., Ma C.L. and Chou T.W., Fiber inclination model of 3-dimensional textile structural composites J. Compos. Mater., 20 (1986) Wu D.L., Three-cell model and 5D braided structural composites, Compos. Sci. Technol., 56 (1996) Li D.S., Li J.L., Chen L., Lu Z.X. and Fang D.N., Finite element analysis of mechanical properties of 3d four-directional rectangular braided composites part 1: microgeometry and 3d finite element model, Appl. Compos. Mater., 17 (2010) Li D.S., Fang D.N., Lu Z.X., Yang Z.Y. and Jiang N., Finite element analysis of mechanical properties of 3d four-directional rectangular braided composites-part 2: validation of the 3d finite element model, Appl. Compos. Mater., 17 (2010) Li D.S., Fang D.N., Jiang N. and Yao X.F., Finite element modeling of mechanical properties of 3D five-directional rectangular braided composites Composites Part B., 42 (2011) Fang G.D., Liang J., Wang Y. and Wang B.L., The effect of yarn distortion on the mechanical 822 Polymers & Polymer Composites, Vol. 22, No. 9, 2014
7 Prediction of Elastic Constants on 3D Four-directional Braided Composites properties of 3D four-directional braided composites, Composites Part A., 40 (2009) Fang G.D., Liang J., Wang B.L. and Wang Y., Effect of interface properties on mechanical behavior of 3d four-directional braided composites with large braid angle subjected to uniaxial tension. Appl. Compos. Mater., 18 (2011) Xu K. and Xu X.W., Finite element analysis of mechanical properties of 3D five-directional braided composites, Mater. Sci. Eng. A., 487 (2008) Chen L., Tao X.M. and Choy C.L., Mechanical analysis of 3-D braided composites by the finite multiphase element method, Compos. Sci. Technol., 59 (1999) Zeng T., Wu L.Z. and Guo L.C., Mechanical analysis of 3D braided composites: a finite element model, Compos. Struct., 64 (2004) Matzenmiller A., Lubliner J. and Taylor R.L., A constitutive model for anisotropic damage in fibercomposites, Mech. Mater., 20 (1995) Li J., Yao X.F., Liu Y.H., Chen S.S., Kou Z.J. and Dai D., Curing deformation analysis for the composite T-shaped integrated structures, Appl. Compos. Mater., 15 (2008) Polymers & Polymer Composites, Vol. 22, No. 9,
8 Liang Dao Zhou and Zhuo Zhuang 824 Polymers & Polymer Composites, Vol. 22, No. 9, 2014
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