Finite element analysis of mechanical properties of 3D five-directional braided composites

Materials Science and Engineering A 487 (2008) 499–509

Finite element analysis of mechanical properties of 3D five-directional braided composites K. Xu ∗ , X.W. Xu Research Institute of Structures and Strength, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing 210016, People’s Republic of China Received 13 June 2007; received in revised form 14 October 2007; accepted 15 October 2007

Abstract As for 3D five-directional rectangular braided composites, a three-dimensional (3D) finite element model (FEM) based on a representative volume element (RVE) is established under the periodical displacement boundary conditions, which truly simulates the spatial configuration of the braiding yarns and the axial yarns. The software ABAQUS is adopted to study the mechanical properties and the meso-scale mechanical response of the composites. The effects of the braiding angle and the fiber-volume fraction on the engineering elastic constants are investigated in detail. The predicted effective elastic properties are in good agreement with the available experimental data, demonstrating the applicability of the FEM in the case of tension in the primary loading direction z. By analyzing the stress distribution and deformation of the model, it is proved that the RVE-based FEM can successfully predict the meso-scale mechanical response of 3D five-directional braided composites containing periodical structures. © 2007 Elsevier B.V. All rights reserved. Keywords: Braided composites; Five-directional; Finite element model; Effective elastic properties; Stress distribution

1. Introduction Three-dimensional (3D) four-directional braided composites have a number of advantages over the conventional laminated composites, including better out-of-plane stiffness, strength, high damage tolerance, etc. Due to the above merits, they have been widely used in many industries and received a great attention [1–15]. While the mechanical performance in the thickness direction is improved, the in-plane mechanical properties of 3D four-directional braided composites have been relatively weakened because of their special yarn configuration. In order to strengthen the in-plane mechanical performance in the predetermined loading direction and maintain the integrity of the preforms as the reinforcements, 3D five-directional braided composites have been developed by adding the uniaxial reinforced yarns in the braiding direction based on the four-step braiding patterns. Since 3D five-directional braided composites have better integrated mechanical properties, they have great potential applications in the aeronautics and astronautics

∗

Corresponding author. Tel.: +86 25 8489 1781; fax: +86 25 8489 1422. E-mail address: [email protected] (K. Xu).

0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.10.030

industries and are especially suited for application in primary loading-bearing structures. Up to now, many models have been developed to analyze the microstructure [1–6] and the mechanical properties [7–15] of 3D four-directional rectangular braided composites, which, for simplicity, are often called as 3D braided composites. Ma et al. studied the effective elastic properties of 3D braided composites by using the ‘Fiber interlock model’ based on the maximum strain energy principle [7] and the ‘Fiber inclination model’ based on the modified laminated theory [8], respectively. Sun and Sun [9] reported a volume-average-compliance method to calculate the elastic constants. Recently, two numerical prediction models based on finite element methods proposed by Chen et al. [10] and Sun et al. [11], respectively, were developed to evaluate the elastic performance of 3D braided composites. Gu [12] presented an analytical model to predict the tensile strength based on the strain energy conservation law. Tang and Postle [13] analyzed the nonlinear deformation of 3D braided composites by means of the finite element method. Zeng et al. [14] established a simple multiphase element model for analyzing the tensile strength. Yu and Cui [15] developed a two-scale method to predict the mechanics parameters of 3D braided composites.

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There also are a few of works dealing with 3D five-directional rectangular braided composites. For example, Wu [16] proposed a simple geometrical model to predict the mechanical properties based on the laminated theory. Li et al. [17] analyzed the process of 3D five-directional braiding and established a topological model. Lu and Liu [18] presented an analytical model to calculate the elastic constants by the stiffness volume-averaging method. Zheng [19] and Chen et al. [20] experimentally studied the mechanical performance by conducting the tensile test, respectively. However, to the authors’ knowledge, the mechanical analyses of 3D five-directional braided composites are very limited. As the analytical models based on the laminated theory [16,18] have inherent limitation in modeling the microstructure, it is difficult to obtain an accurate stress field for the prediction of the mechanical performance of 3D five-directional braided composites. In this case, 3D mesomechanical finite element methods can be used to truly model the microstructure of 3D five-directional braided composites. The main objective of the present work is to develop a new 3D FEM for predicting the mechanical properties and the meso-scale mechanical response of 3D five-directional braided composites. The new model based on a representative volume element (RVE) has taken into account the periodical structure of the composites and the cross-section shape deformation of the yarns due to their mutual squeezing in the process. In order to fully exploit the potential of 3D five-directional braided composites, the effects of the braiding angle and the fiber-volume fraction on the mechanical properties are discussed. It is shown that the predicted effective elastic properties are in good agreement with the available experimental data, demonstrating the applicability of the mesomechanical FEM in the case of tension in the strong anisotropic direction z. Finally, by analyzing the stress distribution and the deformation of the model, some conclusions are drawn herein. 2. Microstructure analysis and unit cell model The whole braiding process of 3D five-directional braided preforms is similar with that of 3D four-directional braided preforms produced by the four-step 1 × 1 rectangular braiding procedures. The main difference is that some axial yarn carriers are needed to be placed between the braiding yarn carriers in rows on the machine bed in order to add the axial yarns, as shown in Fig. 1. During one machine cycle of 3D five-directional braiding process, the track movements of all the braiding yarn carriers are the same as those of the braiding yarn carriers in 3D four-directional braiding process. There are four carrier movement steps in one machine cycle and each braiding yarn carrier moves one position at each step [5]. More attention should be paid to the track movements of the axial yarn carriers during one machine cycle. At the first braiding step, the axial yarn carriers in rows are moved horizontally one position in an alternating manner. At the second step, all the axial yarn carriers are not moved and stay in the current positions. The third step reverses the axial yarn carrier movements in the first step. The fourth step is the same as the second step in which all the axial yarn carriers are not moved. After these four

Fig. 1. Schematic illustration of carrier position on the machine bed.

steps, all the yarn carriers on the machine bed return to the original pattern, completing one machine cycle [17]. Then a certain “jamming” action is imposed on all the yarns, which makes the yarns stabilized and straightened in space. The unit length of the resultant preforms is defined as the braiding pitch, denoted by h. As these steps of motion continue, the braiding yarns move throughout the cross-section and interlace with the axial yarns to form the braided preforms. To ensure consistent and uniform fabric structure, suppose the braiding procedure keep relatively steady, at least in a specified length of braiding. According to the movements of carriers, 3D five-directional braided composites can be regarded to comprise an infinite of two kinds of repeated sub-cells, A and B. Fig. 2 schematically shows the distribution of sub-cell A and sub-cell B in the cross-section of rectangular specimen and the configuration of the yarn axes in the sub-cells. As shown in Fig. 2, sub-cell A and sub-cell B are constructed, respectively, based on two braiding yarns in the interlaced directions and four half of axial yarns in the braiding direction of the z axis. The axis of each braiding yarn is oriented γ with the z axis. The main difference between sub-cell A and sub-cell B is the spatial directions of the braiding yarns. It is noteworthy that sub-cell A and subcell B marked with the dash lines distribute alternately every half of a pitch length h in the braiding direction of the z axis, as shown in Fig. 3.

Fig. 2. Distribution of unit cells on the cross-section of rectangular specimen.

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Fig. 4. Yarn configuration of the solid unit cell.

Fig. 3. Topological relationship of the main yarns in unit cell.

and Due to the complicated microstructure of 3D five-directional braided materials, it makes unfeasible to undertake a full micromechanical simulation aiming at a whole structure. Instead, RVE-based approach can be used to perform the mechanical analysis in the macro-meso-scales. Considering the periodical distribution of sub-cells, A and B, a unit cell that is the smallest RVE is selected as shown in Fig. 2. According to the unit cell partition scheme, all the unit cells are oriented in the same reference frame as the specimen, which is quite favorable for the analysis of the mechanical performance. Fig. 3 shows the topological relationship of the main yarns in the parallelepiped unit cell, which represents the central axes of the yarns in space. Two angles (γ, α) are used to describe the orientation of the braiding yarn axis, where γ is the interior angle between the central axis of the braiding yarn and the z axis and α is called the braiding angle. The relationship between the angle γ and the braiding angle α, is defined as √ tanγ = 2 tanα.

(1)

h=

4(2 + r)b . tanγ

According to the tangent relationship of the elliptical-cylinders of the braiding yarns, the relationship between the major and minor radii of the inscribed ellipse, a and b, can be obtained:  a = b cosγ r 2 + 4r + 3 (4) The lengths of L1 and L2 , are given by: ⎡ ⎤ 2 a L1 = 2 ⎣ b2 cot2 γ + − b cotγ ⎦ sinγ, sin2 γ

(2)

(5)

⎡

⎤ 2 a ⎦ sinγ. L2 = 2 ⎣ b2 cot2 γ + sin2 γ

(6)

As the idealized braided composites considered herein are assumed to be made of the repeated unit cells, the fiber-volume

The solid yarn configuration of 3D five-directional braided composites is shown in Fig. 4. All the yarns used in the preforms are assumed to have the same component material, size and flexibility. Considering the mutual squeezing of the yarns, as shown in Fig. 5, the cross-section shape of the braiding yarn perpendicular to the central axis is assumed to be hexagonal and the hexagon contains an inscribed ellipse with major and minor radii, a and b, respectively; the cross-section shape of the axial yarn is square with the side length rb, where r is the size factor of the axial yarn and its value equals 3 in the model. The fiber-volume fraction of the axial yarn may be different from that of the braiding yarn due to their mutual squeezing. The width and the pitch length of the unit cell are, respectively √ Wx = Wy = (2 + r) 2b

(3)

Fig. 5. Cross-section shapes of yarns. (a) Braiding yarn and (b) axial yarn.

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fraction of 3D five-directional braided composites can be determined by the following expression: Vf =

Vy ϕ Vu

(7)

where Vy is the volume of all the yarns in the unit cell, Vu is the volume of the whole unit cell and ϕ is the fiber-volume fraction of the corresponding yarn. The 3D parametrical solid unit cell can be established by using the CAD/CAM software CATIA P3 V5R14. 3. Finite element model The RVE-based mesomechanical FEM mainly consists of three parts: the periodical boundary conditions and finite element meshing, the constitutive relationships of components and the definition of the effective elastic properties. The details of the finite element model are presented in the subsections. 3.1. Periodical displacement boundary conditions and finite element meshing Since the analysis is based on the RVE, the periodical boundary conditions should be applied in the model to obtain a reasonable stress distribution. Two continuities must be satisfied at the boundary surfaces of the neighboring cubic RVEs. The first is that the displacements must be continuous, and the second is that the traction distribution at the opposite parallel boundaries of the RVE must be uniform. The unified periodical displacement boundary conditions suitable for the RVE proposed by Xia et al. [21] were employed in the model. These general formulas of the boundary conditions are given as follows: ui = ε¯ ik xk + u∗i

(8)

j+

= ε¯ ik xk + u∗i

j−

= ε¯ ik xk + u∗i

ui ui

j+

j+

(9)

j−

j−

ui − ui

j+

(10) j−

j

= ε¯ ik (xk − xk ) = ε¯ ik xk

(11)

In Eq. (8), ε¯ ik is the global average strain tensor of the periodical structure and u∗i is the periodic part of the displacement components on the boundary surfaces and it is generally unknown. For a cubic RVE as shown in Fig. 4, the displacements on a pair of opposite boundary surfaces (with their normals along the Xj axis) are expressed as in Eqs. (9) and (10), in which the index “j+” means along the positive Xj direction and “j−” means along the negative Xj direction. The difference between Eqs. (9) and j (10) is given in Eq. (11). Since xk are constants for each pair of the parallel boundary surfaces, with specified ε¯ ik , the right side of Eq. (11) become constants [21]. It can be seen that Eq. (11) does not contain the periodic part of the displacement. It becomes easier to apply the nodal displacement constraint equations in the finite element procedure, instead of giving Eq. (8) directly as the boundary conditions. In order to apply the constraint Eq. (11) in the FEM, the same

Fig. 6. Finite element mesh of the unit cell model. (a) Yarns and (b) resin matrix pocket.

meshing at each two paired boundary surfaces of the RVE should be produced. As shown in Fig. 6, the model is composed of the yarns and the resin matrix pocket. The microstructure of the model is very complicated. In order to satisfy the continuities of stress and displacement on the interfaces between the yarns and the

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resin matrix pocket, it is assumed that the perfect bonding exists between the yarns and the resin matrix pocket, which means that the merged coincident meshes have been made on the interfaces between the yarns and the resin matrix pocket. Due to the complexity of the microstructure, 3D solid tetrahedron elements were applied to mesh the whole model.

erties of the yarn have been assigned to the yarns, the command of “Assign Material Orientation” in Module/Property can be used to define the local material orientations for the transversely isotropic yarns.

3.2. Constitutive relationships of component materials

To obtain the effective elastic properties of 3D fivedirectional braided composites, a homogenization approach is employed in this study by considering the heterogeneous composites in the micro-scale to be a homogeneous material in the macro-scale. Given the periodic cubic RVE, the global strain–global stress relation can be written as

Two “types” of linearly elastic materials are contained in the model, as shown in Fig. 6. They are the yarns and the resin matrix pocket, respectively. According to the orientation angles of the yarns in Fig. 2, the model comprises four groups of the braiding yarns and the axial yarns in the z axis. As all the yarn axes in the model are the straight lines, these yarns can be regarded as the unidirectional fiber-reinforced composites and are assumed to be transversely isotropic materials in each local material coordinate system. The engineering elastic constants of the yarn can be calculated by the micromechanics formulae proposed by Chamis [22]: E1 = ϕEf1 + (1 − ϕ)Em , Em E2 = E3 = , √ 1 − ϕ(1 − Em /Ef2 ) Gm G12 = G31 = , √ 1 − ϕ(1 − Gm /Gf12 ) Gm G23 = , √ 1 − ϕ(1 − Gm /Gf23 ) ν12 = ν13 = ϕνf12 + (1 − ϕ)νm , E2 ν23 = − 1. 2G23

3.3. Effective elastic properties

ε¯ i = Sij σ¯ j

where Sij is the effective compliance matrix. Assuming a set of the global strain, ε¯ ij , and applying the periodic boundary conditions in the form of Eq. (11) in the FEM analysis, we can obtain a unique stress distribution of the RVE. Then the global stress, σ¯ ij , corresponding to this set of global strain, can be obtained by 1 σ¯ ij = V

(12)

where ϕ is the fiber-volume fraction of the yarn, Efi is the Young’s elastic modulus of the fiber in the principal axis i, Gfij is the shear modulus of the fiber in the i–j plane, νf12 is the primary Poisson’s ratio of the fiber, Em , νm and Gm represent the Young’s elastic modulus, Poisson’s ratio and shear modulus of the matrix, respectively. The resin matrix pocket is assumed to be isotropic. The materials properties of the yarn and the resin matrix pocket are created by the command of the “Create Materials” in the Module/Property of ABAQUS/CAE. Based on the software ABAQUS/CAE, the command of the “Create Datum” in Tools can be adopted to define the local material coordinate systems O–123 for four groups of the braiding yarn and the axial yarn, as illustrated in Fig. 3. For the local coordinate systems of the braiding yarn, the principal direction 1 corresponds to the fiber direction and the principal direction 3 is located in the upright 1–3 plane perpendicular to the x–y plane. The local coordinate systems of the axial yarn are similar to the global coordinate systems and the principal direction 1 corresponds to the fiber direction. For other yarns in the model, the local material directions are defined similarly as shown in Fig. 3. It is emphasized that the tetrahedron elements included in one yarn, with the same local material coordinate system, should been organized and put into a same element set by the command of the “Create Set” in Tools. Once the material prop-

(13)

 V

σij dV

(14)

In the 3D case applying this set of ε¯ ij (six components), six equations thus can be obtained. For a general case where there is no orthotropic axis of symmetry of the material, the application of four linearly independent sets of the global strains ε¯ ij will have sufficient equations to determine 21 independent material constants of the compliance matrix Sij [21]. The effective compliance matrix Sij is the inherent properties for 3D five-directional braided materials with the decided microstructure and component materials. Once the boundary restrictions of the FEM are unchanged, its value should be constant without relationship with the load applied in the model. To avoid the trouble of solving the equations, six linearly independent sets of global strains were applied in the finite element analysis of RVE. The six sets of loading case of periodical displacement boundary conditions are shown in Table 1. By prescribing the six sets of the global strains, ε¯ kij (k = 1, 2, . . ., 6), the corresponding global stress σ¯ ijk can be calculated by Eq. (14). Then the following equations can be obtained {¯ε1i , ε¯ 2i , . . . , ε¯ 5i , ε¯ 6i } = Sij {σ¯ j1 , σ¯ j2 , . . . , σ¯ j5 , σ¯ j6 }.

(15)

Table 1 Loading cases of periodical displacement boundary conditions k

ε¯ x

ε¯ y

ε¯ z

γ¯ yz

γ¯ zx

γ¯ xy

1 2 3 4 5 6

0.001 0 0 0 0 0

0 0.001 0 0 0 0

0 0 0.001 0 0 0

0 0 0 0.002 0 0

0 0 0 0 0.002 0

0 0 0 0 0 0.002

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Table 2 Mechanical properties of component materials Materials

Ef1 (GPa)

Ef2 (GPa)

Gf12 (GPa)

Gf23 (GPa)

νf12

Carbon fiber T300 Epoxy resin

221 4.5

13.8

9

4.8

0.2

νm 0.35

study is sufficient to guarantee the convergence of the solutions [23]. The effective compliance matrixes Sij for specimen No. 1 and No. 2 are given, respectively, as follows: ⎡ ⎤

It is easy to obtain the effective compliance matrix Sij directly through −1

Sij = {¯ε1i , ε¯ 2i , . . . , ε¯ 5i , ε¯ 6i }{σ¯ j1 , σ¯ j2 , . . . , σ¯ j5 , σ¯ j6 }

(16)

0.1293

4. Numerical results and discussion

−0.0579

−0.0049

0

0

0

⎢ −0.0579 0.1293 −0.0049 0 0 ⎢ ⎢ −0.0049 −0.0049 0.0112 0 0 Sij = ⎢ ⎢ 0 0 0 0.1651 0 ⎢ ⎣ 0 0 0 0 0.1651

4.1. Comparison of effective elastic properties with experimental results

0

As for 3D five-directional braided composites with special yarn configuration, since they have the advantage of strengthening the in-plane mechanical performance in the predetermined loading direction, it is necessary to pay more attention to the in-plane rigidity property in the primary loading direction z. In order to verify the applicability of the FEM based on the software ABAQUS, two tensile test examples with typical braiding angles (one is 13.2◦ , the other is 39.74◦ ) were selected from the available experiments studied by Zheng [19]. All the analyses reported herein were done for the 3D five-directional braided composites by the four-step 1 × 1 rectangular braiding procedures. The elastic properties of the component materials, including 12K T300 carbon fiber and TDE-85 epoxy resin, are listed in Table 2. The fiber-volume fraction of yarn, ϕ, is assumed to be 78% on the average in the model. Table 3 gives the braiding process parameters of the two specimens and the microstructure parameters of unit cell models used in the calculation. As the main structural parameters of the model were obtained by measuring the specimens directly, the model in the study had truly described the microstructure of 3D five-directional braiding composites. According to the meshing scheme of the FEM, the free biased mesh was used to keep element size small in the edges of the resin matrix pocket. In the study, the FEM for specimen No. 1 consists of 25,413 nodes and 127,552 linear tetrahedron elements. The FEM for specimen No. 2 consists of 27,812 nodes and 147,767 linear tetrahedron elements. It is noted that relatively fine meshing size is required in order to obtain more accurate stress distribution, especially near the boundaries of the RVE. However, if only the global stiffness is concerned, relative coarse meshing size can still provide satisfactory results. The meshing size of the models in this

0 0 0 0

0

0

0

0

0.3539

−0.0625

−0.0135

0

0

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

×10−3 MPa−1

and ⎡

0.1136

⎢ −0.0625 0.1136 −0.0135 0 0 ⎢ ⎢ −0.0135 −0.0135 0.0284 0 0 Sij = ⎢ ⎢ 0 0 0 0.0972 0 ⎢ ⎣ 0 0 0 0 0.0972 0 ×10

−3

0 MPa

0

0

0

0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0.1292

−1

It is found that 3D five-directional braided composites can be considered to be transversely isotropic materials in the macroscale under small deformation assumption. According to the relationship between the engineering elastic constants and the compliance matrix Sij , nine independent elastic constants of 3D five-directional braided composites can be calculated by ⎧ 1 1 1 ⎪ ⎪ Ey = Ez = ⎪ Ex = ⎪ S11 S22 S33 ⎪ ⎪ ⎨ S12 S13 S23 Vxy = − Vxz = − Vyz = − (17) ⎪ S S S33 22 33 ⎪ ⎪ ⎪ 1 1 1 ⎪ ⎪ ⎩ Gyz = Gxz = Gxy = S44 S55 S66 Table 4 gives the predicted results and the measured elastic constants. There is a good agreement between the measured and predicted axial tensile modulus for both of the two samples studied. The errors are limited to be about 8 percent. The predicted Poisson’ ratios are basically agreed with the measured values. The results demonstrate the applicability of the mesomechanical

Table 3 Braiding parameters of specimens and structural parameters of the unit cell model No.

1 2

Braiding parameters of specimens

Structural parameters of unit cell model

Dimensions (mm)

α

(◦ )

Vf (%)

Wx (mm)

γ (◦ )

a (mm)

b (mm)

Vf (%)

Wx = Wy (mm)

h (mm)

250 × 25 × 5 250 × 25 × 5

13.20 39.74

44.44 44.84

2.78 2.91

18.35 49.62

0.914 0.653

0.197 0.206

42.12 41.40

2.78 2.91

11.85 3.50

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Table 4 Comparison of the effective elastic constants predicted by the model and experimental data Elastic constants

No. 1 Measured

Ex (GPa) Ey (GPa) Ez (GPa) Gxz (GPa) Gyz (GPa) Gxy (GPa) νxz νyz νxy

96.70

0.40 0.40

No. 2 Predicted 7.73 7.73 89.29 6.06 6.06 2.83 0.44 0.44 0.45

Measured

33.07

0.56 0.56

Predicted 8.80 8.80 35.21 10.28 10.28 7.74 0.48 0.48 0.55

FEM in the case of tension in the primary loading direction z. It is noted that though it is convenient to simultaneously predict all the global elastic constants by using the FEM, the applicability of the FEM in other directions and for shear modes has not been validated. 4.2. Deformation of unit cell and stress distribution For 3D five-directional braided composites with periodical structures, the RVE-based FEM can also be used to calculate the mechanical quantities in the meso-scale, such as the deformation of the model, distribution of stress, stress concentration, etc. To demonstrate the application, the FEM of specimen No. 2 subjected to typical loading case is chosen to show the mesomechanical response of 3D five-directional braided composites. Fig. 7 shows the deformation of two parallel boundary surfaces vertical to the z axis of the model subjected to loading case 3, k = 3. Under such loading case, ε¯ 33 equals 0.001 and all the other five effective average strains equal zero. From Fig. 7, the two opposite boundary surfaces do not remain plane any more and are warped after the deformation (the deformation scale factor is 600). The warped deformation occurs simultaneously at the other two sets of opposite boundary surfaces of unit cell, but the warped extent is relatively weak. The reason resulting in the phenomena is that the unit cell model of 3D five-directional braided composites does not have the symmetries of geometrical structure and physical properties. Fig. 8 shows the deformation of the model subjected to loading case 6, k = 6. Under such loading case, γ¯ xy equals 0.002 and all the other five effective average strains equal zero. From Fig. 8, two sets of opposite boundary surfaces vertical to the x axis and y axis before deformation, respectively, do not remain plane and the warped deformation has occurred (the deformation scale factor is 200). The warped deformation occurs simultaneously at the opposite boundary surfaces vertical to the z axis. However, the warped extent is still weak. From Figs. 7 and 8, the FEM based on the periodical displacement boundary conditions guarantees the displacement continuity at the opposite surfaces between the neighboring RVEs. Based on the software ABAQUS, banded-type stress contour has been selected to show the calculated stress results and field output instead of discontinuities is chosen as the quantity to be

Fig. 7. Surface deformation of the model under loading case 3. (a) Deformation of positive surface vertical to z axis. (b) Deformation of negative surface vertical to z axis.

plot. For the banded-type stress contour, ABAQUS/CAE extrapolates the stress components at the integration points to the nodes and makes the extrapolated values at nodes common to two or more elements averaged by selecting the averaging criteria to be 75%, which is the default threshold value. If the stress invariants or the stress components are requested, ABAQUS/CAE then operates on the extrapolated data to calculate the desired quantity using standard invariant or component formulas.

Fig. 8. Deformation of the model under loading case 6.

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By analyzing the stress results, whether the stress components or the stress invariants, the uniform stress distribution has been obtained at the opposite boundary surfaces of the cubic model. In order to demonstrate that the FEM based on the periodical displacement boundary conditions has guaranteed the traction continuity at the opposite surfaces between the neighboring RVEs, the maximal principal stress distributions

of the model under two kinds of loading case were selected to be shown in Figs. 9 and 10 as examples. For 3D fivedirectional braided composites with special yarn configuration, the mechanical response characteristic of the model subjected to the different loading cases varies, which mainly means that the yarns participating in bearing the primary load are different.

Fig. 9. Maximal principal stress nephogram of the model No. 2 under loading case 3. (a) The whole model. (b) Yarns and principal directions. (c) Resin matrix pocket.

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Fig. 10. Maximal principal stress nephogram of the model No. 2 under loading case 6. (a) The whole model. (b) Yarns and principal directions. (c) Resin matrix pocket.

Fig. 9 shows the maximal principal stress nephogram of the whole FEM subjected to loading case 3. As shown in Fig. 9a, the traction continuity at the opposite boundary surfaces has been guaranteed and satisfied the periodic condition. From Fig. 9b and c, the stresses in yarns are more 10 times than that in the matrix pocket region on the average. In particularly, the stresses in the axial yarns are more three times than that in the braiding yarns under such loading case. This indicates that the axial yarns of 3D five-directional braided composites bear the primary load. The principal directions of the yarns are plotted in

Fig. 9b with a zoom. As shown in Fig. 9c, the stress concentration is produced in the contacting region between the yarns and the matrix pocket. The closer to this region, the greater stress is produced. Fig. 10 shows the maximal principal stress nephogram of the whole FEM subjected to loading case 6. The stress distribution is different from that of the model subjected to loading case 3. Similarly, the traction continuity at the corresponding parallel boundary surfaces has still been guaranteed. From Fig. 10b, the stresses of the braiding yarns, with the projection directions ori-

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ented 45◦ with the positive x axis and the positive y axis, are more eight times than that in the other yarns. This indicates that these braiding yarns of 3D five-directional braided composites bear the primary shear load. The principal directions of the yarns are plotted in Fig. 10b with a zoom. As shown in Fig. 10c, the stresses in the matrix pocket are relatively smaller and the produced stress concentration is not obvious as that subjected to loading case 3.

4.3. Discussion on the effective properties of 3D five-directional braided composites The unit cell of 3D five-directional braided composites produced by the four-step 1 × 1 rectangular braiding procedures, shown in Figs. 3 and 4, can be characterized by two micro-structural parameters, including the braiding angle and the fiber-volume fraction. In this section, the effects

Fig. 11. Variation of engineering elastic constants with structural parameters.

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of the two parameters on the effective elastic properties of 3D five-directional braided composites are studied with the mesomechanical FEM. The models were established as shown in Section 2 and Section 3. The models with different fiber-volume fractions under a same braiding angle are obtained by defining the fiber-volume fraction of the yarn from Eq. (7). The elastic properties of fibers and matrix are shown in Table 2. Fig. 11 shows the variation of the predicted elastic constants with the increasing braiding angle, including three samples with different fiber-volume fractions. Fig. 11a describes that the elastic modulus Ez decreases sharply as the braiding angle increases. With the fiber-volume fraction increasing, the elastic modulus Ez increases as a whole. However, when the braiding angle is small, the change of the elastic modulus Ez caused by the increment of fiber-volume fraction becomes comparatively significant. Fig. 11b gives that the elastic moduli, Ex and Ey (Ex = Ey ), vary with the braiding angle. The elastic modulus Ex increases steadily as the braiding angle increases. With the fibervolume fraction increasing, the elastic modulus Ex increases in a similar tendency. Fig. 11c depicts that the shear modulus Gxy increases monotonically with the increasing braiding angle. As the fiber-volume fraction increases, the shear modulus Gxy increases. When the braiding angle is large, the change of the shear modulus Gxy caused by the increment of the fiber-volume fraction is comparatively larger with increasing the fiber-volume fraction. Fig. 11d presents that the shear moduli, Gxz and Gyz (Gxz = Gyz ), increase steadily as the braiding angle increases. With the increase of the fiber-volume fraction, the shear moduli, Gxz and Gyz , increase. When the braiding angle is large, the increments of the shear moduli, Gxz and Gyz , are relatively larger with increasing the fiber-volume fraction. Fig. 11e and f shows that the variation of the Poisson’s ratios, νxy and νxz (νxz = νyz ), with the braiding angle. With increasing the braiding angle, νxy firstly decreases and then increases; νxz and νyz firstly increase and then decrease. As the fiber-volume fraction increases, the Poisson’s ratio νxy decreases a little. The decrement of the Poisson’s ratio νxy is reduced when the braiding angle is large. With increasing the fiber-volume fraction, the Poisson’s ratios, νxz and νyz , increase. The changes of the Poisson’s ratios caused by the increment of fiber-volume fraction, νxz and νyz , are relatively obvious when the braiding angle is about 25◦ . On the whole, the effect of the fiber-volume fraction on the Poisson’s ratios is relatively weak. As shown in Fig. 11, the effective elastic properties of the composites have been influenced by the two structural parameters. Therefore, optimization of the structural parameters can help to reduce the design time and save the manufacture costs. 5. Conclusions A new 3D mesomechanical FEM is proposed to predict the effective elastic properties and the meso-scale mechanical response of 3D five-directional braided composites. The RVEbased model has taken into amount the periodical structure

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of the composites and truly simulated the spatial configuration of the yarns. The predicted effective elastic properties are compared favorably with the available experimental data, demonstrating the applicability of the mesomechanical FEM in the case of tension in the strong anisotropic direction z. Meanwhile, the predicted effective compliance matrix indicates that 3D five-directional braided composites can be considered to be transversely isotropic materials in the macro-scale under small deformation assumption. The effects of the braiding angle and fiber-volume fraction on the engineering elastic constants have been discussed in detail. The results show that the elastic modulus Ez is influenced significantly by the braiding angle. By analyzing the stress distribution and deformation, it is proved that the model guarantees the continuities of the displacement and the traction at the boundary surfaces of the neighboring RVEs, which is helpful to obtain reasonable stress field of the RVE in the meso-scale. Future work will focus on the strength and failure analysis of 3D five-directional braided composites subjected to tension, compression or shear load by using the mesomechanical FEM. Acknowledgements This paper is financially supported by Postgraduate Innovation Project of Jiangsu Province of China (No. 2005065). And the authors would like to acknowledge the support given by Postgraduate Innovation Fundation of Nanjing University of Aeronautics and Astronautics (BCXJ 05-03). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

W. Li, M. Hammad, S.A. El, J. Text. Inst. 81 (1990) 491–514. G.W. Du, F.K. Ko, J. Reinforced Plast. Compos. 12 (1993) 752–768. Y.Q. Wang, A.S.D. Wang, Compos. Sci. Technol. 51 (1994) 575–586. R. Pandey, H.T. Hahn, Compos. Sci. Technol. 56 (1996) 161–170. L. Chen, X.M. Tao, C.L. Choy, Compos. Sci. Tcehnol. 59 (1999) 391–404. Y. Wang, X. Sun, Compos. Sci. Technol. 61 (2001) 311–319. C.L. Ma, J.M. Yang, T.-W. Chou, Composite Materials: Testing and Design, Seven Conference, 1984, pp. 404–421. J.M. Yang, C.L. Ma, T.-W. Chou, J. Compos. Mater. 20 (1986) 472–483. X.K. Sun, C.J. Sun, Compos. Struct. 65 (2004) 485–492. L. Chen, X.M. Tao, C.L. Choy, Compos. Sci. Technol. 59 (1999) 2383–2391. H.Y. Sun, S.L. Di, N. Zhang, Comput. Struct. 81 (2003) 2021–2027. B.H. Gu, Compos. Struct. 64 (2004) 235–241. Z.X. Tang, R. Postle, Compos. Struct. 55 (2002) 307–317. T. Zeng, D.N. Fang, L. Ma, Mater. Lett. 58 (2004) 3237–3241. X.-G. Yu, J.-Z. Cui, Compos. Sci. Technol. 67 (2007) 471–480. D.-L. Wu, Compos. Sci. Technol. 56 (1996) 225–233. D.-S. Li, L. Chen, J.-L. Li, J. Tianjin Polytech. Univ. 22 (2003) 7–11. Z.-X. Lu, Z.-X. Liu, J. Beijing Univ. Aeronautics Astronautics 32 (2006) 455–460. X.-T. Zheng, Northwestern Polytechnical University, September 2003. L. Chen, Z.-Q. Liang, Z.-J. Ma, J.-Y. Liu, J.-L. Li, J. Mater. Eng. 52 (2005) 3–6. Z.-H. Xia, Y.-F. Zhang, F. Ellyin, Int. J. Solids Struct. 40 (2003) 1907–1921. C.C. Chamis, J. Compos. Technol. Res. 11 (1989) 3–14. Z.-H. Xia, C.-W. Zhou, Q.-L. Yong, Int. J. Solids Struct. 43 (2006) 266–278.