A two-scale method for predicting the mechanical properties of 3D braided composites with internal defects

Composite Structures 152 (2016) 1–10

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A two-scale method for predicting the mechanical properties of 3D braided composites with internal defects Jiwei Dong a,⇑, Ningfei Huo b a b

School of Mechanics & Civil Engineering, China University of Mining and Technology, Jiangsu 221116, China School of Mechanical Engineering, Tianjin University, Tianjin 300072, China

a r t i c l e

i n f o

Article history: Received 6 January 2016 Revised 4 April 2016 Accepted 4 May 2016 Available online 7 May 2016 Keywords: Braided composites Two-scale Periodical boundary condition Effective elastic constant Defect

a b s t r a c t Pore defects have significant effect on the mechanical properties of 3D braided composites. Two-scale finite element models of the fiber tows and the braided composites are developed to predict the elastic properties and the micro stress of 3D braided composites. Two basic types of pore defects, the voids in resin matrix and the dry patches in fiber tows, have been taken into account. Periodical boundary conditions are applied on the two-scale FE models by the coupling and constraint equation defining commands in ANSYS. The predicted effective elastic constants agree well with the available experimental data, demonstrating the validity of the two-scale FE models. The effects of braiding angle and fiber volume fraction on the engineering elastic constants are discussed. Furthermore, the effects of two types of defects on the effective elastic properties of 3D braided composites are also discussed in detail. Some useful conclusions are drawn herein. From simulation, dry patches in fiber tows have more significant effect on the elastic properties than voids in matrix. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Having excellent mechanical properties such as better out-ofplane stiffness and strength, high damage tolerance and good impact resistance, 3D braided composites have great potentialities in aerospace, automobile, marine and other industries, especially suited for application for primary loading-bearing structures. Therefore, the microstructure and mechanical properties of these materials have received more and more attention in recent years. Although 3D braided composites have complicated spatial structures, their heterogeneity is mainly presented in the meso and micro scale. Generally, representative volume element (RVE) models were proposed to predict the homogenous properties of heterogeneous materials. For example, on the micro scale, a fiber tow of resin matrix composites contains thousands of fibers and bonded resin matrix. In some articles, micro RVE models of the fiber tows were proposed to predict its stiffness and strength. Zhou et al. [1] proposed a RVE model of fiber tows by cutting a rectangular area from the bulk materials. Lu et al. [2] created a RVE of fiber tows based on the collision algorithm. Meanwhile, on the meso scale, 3D braided composites are made up of fiber tows and resin

⇑ Corresponding author. E-mail address: [email protected] (J. Dong). http://dx.doi.org/10.1016/j.compstruct.2016.05.025 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.

matrix. According to the periodicity of the braided structure, kinds of meso RVE models of 3D braided composites called unit cells were proposed, such as ‘‘four straight yarns model” [3], ‘‘four bending yarns model” [4] and ‘‘twelve straight yarns model” [5]. Considering different yarn paths in the interior, surface and corner regions, Chen et al. [6] created three kinds of unit cells: body cell, face cell and corner cell, the microstructure has high accuracy, and is widely used in many articles [7–10]. Based on the RVE models, the mechanical properties of 3D braided composites have been extensively studied. Up to now, great achievements have been made on the elastic properties of these materials [11–14]. With the rapid development of computer technology, the elastic properties and the micro stress of 3D braided composites were simulated by using finite element (FE) method. Chen et al. [15] proposed a finite multiphase element method to predict the elastic properties of 3D braided composites by applying uniform strain boundary conditions on three types of unit cells. Xu and Zhang [16,17] proposed a strain energy-based method to predict the elastic properties of 3D braided composites by using the strain energy of RVE under specific displacement boundaries. Yu and Cui [18] applied a two-scale finite element method to predict the tensile properties of 3D four-directional braided composites. In recent years, the micro failure mechanism and the strength have become two hot and challenging topics for

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3D braided composites. Zeng et al. established a multiphase element model for calculating local stress [19] and predicting the nonlinear response and failure [20] of 3D braided composites. Pang et al. [21] and Fang et al. [22] simulated the microscopic damage behavior of 3D braided composites by means of Murakami’s damage theory and FE method. Dong and Feng [23] simulated the progressive damage of 3D braided composites by the asymptotic expansion homogenization method. Although lots of analytical and computational models have contributed to the stiffness and the strength analysis of 3D braided composites, most of the models have generally assumed that the materials have no internal defects and do not consider the effect of random pore defects on their mechanical properties. However, since 3D braided composites are generally compounded by RTM forming process, it is almost impossible to completely avoid the existence of defects in the composites. Now, there are few literatures focusing on the effect of the defects on the mechanical properties of 3D braided composites. Zeng et al. [24] presented a simplified model of 3D braided composites with transverse and longitudinal cracks to calculate the effective elastic properties of 3D braided composites. Lu et al. [25] simulated the progressive damage behavior of plain weave composites under uniaxial tension considering randomly distributed voids defects in the matrix. Xu and Qian [26] developed a RVE model to predict the elastic properties of 3D braided composites containing two basic types of defects, voids and dry patches. Unfortunately, the heterogeneity of the fiber tows was ignored in these articles. In fact, the dry patches caused by insufficient resin infiltration along the fiber tows are very dangerous and have great influences on the mechanical properties of the composites. Therefore, developing a FE method based on the two-scale RVEs of fiber tows and whole structures is beneficial to the precision of numerical results. In this paper, the main objective of the work is to develop twoscale finite element models of fiber tows and braided structures for predicting the mechanical properties of 3D braided composites. These FE models contain two basic types of defects, voids and dry patches. Firstly, two-scale RVE models of ideal fiber tows and ideal 3D braided composites are proposed by the periodicity of their structures. Based on the geometric RVE models, two FE models with two types of pore defects are created. Then, periodical boundary conditions are applied on the FE models to predict the effective elastic constants and micro stress of the materials. Lastly, the calculated results are compared with the experiment data to demonstrate the applicability of the two-scale models. Furthermore, the effect of braided parameters and the effect of void volume fraction on the mechanical properties of the composites are discussed in detail. 2. Double-scale RVE modeling considering internal defects 2.1. RVE modeling of ideal fiber tows For fiber reinforced composites, on the micro scale, the fiber tows are selected to be analyzed. During the process of RTM, the gaps between the fibers will be filled with resin matrix. Therefore, an ideal fiber tow is made up of thousands of fibers and bonded matrix (shown in Fig. 1(a)). It is assumed that fibers are hexagonally arranged and the cross-section of a fiber tow is a circle (in Fig. 1(b)), therefore an ideal fiber tow is assumed to be composed of hexagonal unit cells In order to apply periodic boundary conditions on a unit cell easily, a RVE model is created (in Fig. 1(c)) by cutting the rectangular area from Fig. 1(b). Defining the yarn packing factor e as the fiber volume fraction in a fiber bundle, thus the size ratio of l, w and r in Fig. 1(c) can be pffiffiffi pffiffiffiffiffiffiffiffi easily calculated as 3:1:0:93 e=p.

2.2. RVE modeling of ideal 3D braided composites Despite the complexity of macro structures, 3D braided composites can be divided into RVEs by the periodicity of their meso structures. Based on the typical structure, 3D braided structures are extensively divided into three kinds of RVEs (i.e. interior RVEs, surface RVEs and corner RVEs) [6]. Generally, if the column number and the row number of yarn carriers are large enough, surface RVEs and corner RVEs can be ignored reasonably. Therefore, interior RVEs are the only consideration for our following finite element analysis. According to the four-step procedure, the distributions of three kinds of RVEs in 3D braided composites are shown in Fig. 2. Considering the periodical distribution of two kinds of interior sub-cells, a unit cell which is the smallest RVE is selected as shown in Fig. 2. The following assumptions are made to establish RVEs of 3D braided composites: 1. Based on the experimental observation [5], and supposing that all the fiber bundles contact closely, the cross section of each fiber bundle is assumed to be a circumscribed octagon of an ellipse (shown in Fig. 3). It is assumed there are no mutual immersion and large voids between fiber bundles; 2. The braiding procedure maintains relatively steady and the braided structure is uniform; 3. The fiber is regarded as transversely isotropic in elasticity, and the matrix is isotropic; 4. Fiber bundles have adequate flexibility. According to the assumptions above, the cross section of a fiber bundle is assumed to be an octagon, and the octagon contains an inscribed ellipse with major and minor radii, a and b, respectively. When the yarns are jamming and contact each other, the relationship between a and b can be deduced as follows [6]:

pffiffiffi a ¼ b 3 cos c

ð1Þ

where c is the interior braiding angle. The total cross-sectional area of fibers in a fiber bundle is given by:

Af ¼ k=q

ð2Þ

where k (g/m) is the yarn linear density, and q (g/cm3) is the fiber density. Defining the yarn packing factor e as the fiber volume fraction in a fiber bundle, thus the cross-sectional area of a fiber bundle is:

Ab ¼ Af =e

ð3Þ

It is known from the assumptions that the cross-section of a fiber bundle is a regular octagon before extrusion. The octagon contains an inscribed circle with radius b, Assuming the area of the regular octagon is A1, the relationship between A1 and Ab is:

a Ab ¼ b A1

ð4Þ

The area of a regular octagon can be simply calculated by:

A1 ¼ 8b tan 22:5 2

ð5Þ

Combining Eqs. (1), (4) and (5), yields:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A pffiffiffi pffiffiffi b b¼ 8 3ð 2  1Þ cos c

ð6Þ

Fig. 2 shows the cross-section of a unit cell, the dimension of the unit cell in the braiding direction is called the braiding pitch, denoted by h. The width Wi, the thickness Ti and the pitch length of the unit cell are, respectively

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(a) fiber tow

(b) hexagonal distribution

(c) RVE model

Fig. 1. RVE of the fiber tows.

Fig. 2. Distribution of unit cells on the cross-section of braided preforms.

b a

Fig. 3. section of braiding yarn.

pffiffiffi W i ¼ T i ¼ 4 2b h¼

8b tan c

ð7Þ ð8Þ

By analyzing the track movements of the braiding yarn carriers during a machine cycle according to reference [6], the spatial locations of the yarn axis in a unit cell can be obtained. So the topological relationship of the main yarns in a unit cell is shown in Fig. 4 (a). Assuming the cross-section of the yarn is octagonal and calculating the parameter a and b according to Eqs. (1) and (6), the geometric model of unit cell is established using the software ANSYS. Fig. 4(b), (c) and (d) show the geometric model of fiber tows, matrix and whole structure, respectively. 2.3. Finite element model of RVE with random pore defects During the molding process of 3D braided composites, due to insufficient resin infiltration, pore defects are inevitably occurred

after curing. Most experimental investigations indicated that voids and dry patches are the most common types of defects [26] existing in the 3D braided structures. It is well known that flow will choose the easier direction with higher permeability. Therefore, if the permeability of flow along the fiber tows is higher than that across the fiber tows, voids defects will be formed in the matrix. Otherwise, the pore defects along the fiber tows which are called dry patches will be formed. Fig. 5 shows the two types of defects. Considering geometrical regularity of the RVE of the fiber tow, the hexahedral element SOLID186 in ANSYS 12.0 was selected for meshing the model (shown in Fig. 6). However, the geometry of the RVE of 3D braided structures is relatively irregular, so the tetrahedral element SOLID187 was chosen to mesh this model. Meanwhile, for meshing the RVE of the braided structures, local coordinate systems were defined to represent four directions of fiber tows. Fig. 7 shows the finite element model of this model. It is noted both these FE models require that the nodes of the opposite surfaces of the unit cell were identical because periodical boundary conditions would be applied later. Voids and dry patches are two kinds of defects which are randomly distributed in the 3D braided composites. The void elements are regarded to have no stiffness. However, to guarantee the convergence of the numerical calculation, the stiffness of each void element can’t decrease to zero, but to a very small positive value. As dry patches exist in fiber tows, the RVE model of the fiber tows was used to simulate dry patches. In this model, defining Pfv is the void volume fraction of the tows, and the method of defining dry patches is as follows: Firstly, a random element is selected from matrix by using the rand function in ANSYS. Next, the elastic modulus Em and Poisson’s ratio mm of the element is set to a very small value (here Em = 1 Pa and mm = 106). In this way, continue to select

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(a) topological relationship

(b) fiber tows

(c) matrix

(d) whole structure

Fig. 4. RVE of the 3D braided structures.

(a) voids in matrix

(b) dry patches in fiber tows

Fig. 5. two types of defects.

(a) Fiber tows

(b) Matrix

Fig. 7. Finite element model of RVE of 3D braided structures.

3. Two-scale FE analysis on stiffness 3.1. Periodical boundary conditions

Fig. 6. Finite element model of RVE of the fiber tows.

other matrix elements randomly and reduce their stiffness, until the volume fraction of all selected elements is greater than Pfv. Similarly, defining Pmv as the void fraction of the resin matrix pocket, and the RVE model of 3D braided structures with voids can also be established by the above method. Fig. 8 shows RVE models with these two kinds of defects (here Pfv = Pmv = 0.02).

Both the fiber tows and the braided structures are regarded as periodical structures consisting of a periodical array of their RVEs. It is assumed that each RVE in the composites has the same deformation mode and there is no separation or overlap between neighboring RVEs. Therefore, the periodical boundary conditions must be applied to the RVE models. Fig. 9 shows an anisotropic cubic RVE model of a material. In order to calculate all the effective modulus of the model, four kinds of periodical boundary conditions were applied as Table 1(here assuming z direction is longitudinal). These periodical boundary conditions described above can be applied using nodes coupling (CP) or constraint equations (CE) defining in ANSYS. If a set of nodes have some identical degrees of freedom, the coupling command should be used to constraint these nodes. On the other hand, if there is a certain relationship between the degrees of freedom of some nodes, a constraint equa-

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(a) dry patches in fiber tows

(b) voids in matrix

Fig. 8. RVE models with two kinds of defects.

G

Table 2 Elastic properties of carbon fiber and resin matrix.

C

H

D

Carbon fiber Epoxy

y

E1/GPa

E2/GPa

G12/GPa

G23/GPa

m12

230 3.5

40

24

14.3

0.25

mm 0.35

x B

F E

z

A Fig. 9. A cubic RVE model.

Table 1 The periodical boundary conditions. Loading case

Surfaces normal to x

Surfaces normal to y

Surfaces normal to z

Longitudinal tension

vx+  vx- = 0

uy+  uy- = 0 wy+  wy- = 0

uz+  uz- = 0 vz+  vz- = 0 wz+  wz- = e L

Transverse tension

ux+  ux- = e W vx+  vx- = 0 wx+  wx- = 0

uy+  uy- = 0 wy+  wy- = 0

uz+  uz- = 0 vz+  vz- = 0

uy+  uy- = 0

uz+  uz- = 0 vz+  vz- = 0.5e L wz+  wz- = 0

Longitudinal shear Transverse shear

wx+  wx- = 0

ux+  ux- = 0

vx+  vx- = 0

vy+  vy- = 0

wx+  wx- = 0

wy+  wy- = 0.5e T

ux+  ux- = 0 vx+  vx- = 0 wx+  wx- = 0.5e W

vy+  vy- = 0

uy+  uy- = 0.5e T

wy+  wy- = 0

uz+  uz- = 0 vz+  vz- = 0 wz+  wz- = 0

tion can be defined. Correspondingly, if the right of an equation is equal to zero, the nodes coupling will be used, otherwise, the constraint equation defining will be used. Although there are lots of nodes on three pairs of opposite surfaces, APDL language in ANSYS can be written to define these boundary conditions easily. 3.2. Effective elastic properties prediction by two-scale analyses By using the four kinds of periodical boundary conditions shown in Table 1, the effective elastic properties can be predicted by a homogenization approach. This approach is employed by con-

Table 3 Braiding parameters and structural parameters of RVE models. Number

Dimensions

c /(°)

Vf /%

e/%

h /mm

Number 1 Number 2

20.6  6.32  250 20.6  8.58  250

26.2 46.4

60.69 52.04

75.6 69.5

5.02 2.02

sidering the heterogeneous composites in the mesoscale to be a homogeneous material in the macro-scale. Given the periodical cubic RVE, the global stain-stress relation under small deformation assumption can be written as

eij ¼ S ijkl rkl

ð9Þ

where Sijkl is the flexibility matrix, eij is the volume averaging strain, and rkl is the volume averaging stress. The effective elastic constants can be calculated from the flexibility matrix. In order to predict the effective elastic properties of 3D braided composites, both the FE models on the micro scale and the meso scale model in the last section were used. The process of simulation includes the following four steps: Step 1. Create the microscopic FE model of the RVE of the fiber tows with random dry patches as shown in Fig. 8(a). Step 2. Apply the periodical boundary conditions shown in Table 1 on the microscopic FE model and calculate the effective elastic constants of the fiber tows by the homogenization approach via Eq. (9). Step 3. Regard the fiber tows in 3D braided composites as homogeneous materials, and define their material properties by using the elastic constants in step 2, then create the mesoscopic FE model of the RVE of 3D braided composites with random voids as shown in Fig. 8(b). Step 4. Similar to step 2, calculate the effective elastic constants of the braided composites based on the periodical boundary conditions.

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Table 4 Comparison of the elastic constants between predicted and experimental data. Number

Model

Number 1

Fiber tow 3D4DB

Number 2

Fiber tow 3D4DB

Ex/GPa

Ez/GPa

Predicted

Predicted

13.609 9.345

174.299 67.761

10.831 8.579

160.520 18.111

Gxz/GPa

mxy

Experiment

Predicted

Predicted

Predicted

Experiment

69.33

6.393 13.184

0.3657 0.3718

0.2145 0.6784

0.69

18.30

5.217 16.675

0.3732 0.3184

0.2429 0.5717

0.67

4. Results and discussion 4.1. Comparison of effective elastic constants with experimental results In order to verify the validity the FEM based on the software ANSYS, two tensile test examples with typical braiding angles are selected from the available experiments studied by Xiu [27]. All the analyses herein were done for the 3D 4-directional braided (3D4DB) composites by the 4-step 1  1 rectangular braided procedures. The elastic properties of the component materials, including T300 carbon fiber and TDE-85 epoxy resin, are listed in Table 2. According to the braiding parameters of two specimens by reference [27], the main microstructure parameters of RVE models used in the calculation are shown in Table 3. The FE software ANSYS 12.0 was adopted to predict the effective elastic constants. The method and the process of simulation

(a) Fiber tows

were introduced in the third section. Periodical boundary conditions were applied on the microscopic and mesoscopic models to predict the effective constants of fiber tows and 3D braided composites, respectively. According to the definition of engineering elastic constants, the FEM based on the RVE models under uniaxial tensile loading along x-, y-, and z- axes was established to obtain elastic modulus and Poisson’s ratio. Meanwhile, the pure shear loading was applied to the corresponding models to calculate the shear modulus. Therefore, all the effective elastic constants can be predicted by FEM with periodical boundary conditions listed in Table 1. Table 4 gives the predicted elastic constants of fiber tows and 3D braided composites with internal defects (supposing Pfv = Pmv = 0.5%). As both the fiber tows and the braided composites are transversely isotropic in macroscale, five independent constants are listed in the table. It can be seen there is a good agree-

(b) Matrix

(c) Unit cell

Fig. 10. Stress nephogram of RVE under z tension load.

(a) Fiber tows

mzx

(b) Matrix

(c) Unit cell

Fig. 11. Stress nephogram of RVE under yz shear load.

J. Dong, N. Huo / Composite Structures 152 (2016) 1–10

Fig. 12. Influences of braiding angle and fiber volume fraction on the elastic constants.

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Table 5 Elastic constants of the fiber tows with different void fraction.

4.4. Discussion on pore defects

Pfv/(%)

Ex/GPa

Ez/GPa

Gxz/GPa

mxy

mzx

0 1 2 3 4

14.608 12.289 9.726 6.897 3.220

174.517 174.206 174.011 173.817 173.687

6.949 5.899 4.301 2.940 1.666

0.3723 0.3564 0.3458 0.2929 0.2489

0.2211 0.2133 0.1936 0.1747 0.1382

ment of the longitudinal elastic module Ez and the longitudinal Poisson’s ratio mxz between the predicted and experimental constants for the two samples studied. 4.2. Micro stress and deformation simulation Based on the periodical boundary conditions shown in Table 1, the deformation and the micro stress distribution in the RVE of 3D braided composites can be simulated, which reflects the heterogeneity of the materials. The effects of the voids and the dry patches on the micro stress were discussed herein. Fig. 10 and 11 shows the Von Mises stress nephogram of the whole FEM subjected to z tension and yz shear loads, respectively. Obviously, micro stress in fiber tows is much higher than that in matrix region, and the interfaces between fiber tows and matrix have high stress gradients. Although the cross sections are warping and no longer retain planes, the opposite surfaces of the RVE model have the same deformations, which verify the periodical boundary conditions. 4.3. Discussion on braided parameters The influences of braiding angle and fiber volume fraction on engineering elastic constants of 3D braided composites are shown in Fig. 12. Fig. 12(a) depicts that with the increase of the braiding angle, the elastic modulus Ez decreases and the change is gradually slowing down. Meanwhile, Ez increases as the fiber volume fraction increases and the change is comparatively significant when the braiding angle is small. Fig. 12(b) presents that the elastic moduli, Ex and Ey (Ex = Ey), vary with the braided parameters. The elastic modulus Ex increases with the increase of the braiding angle and almost retain a constant when the braiding angle is less than 30°. As the fiber volume fraction increases, Ex also increases. Fig. 12(c) describes that the variation of the transverse Poisson’s ratio, mxy, with the braided parameters. It can be seen that the braiding angle is the main factor affecting the transverse Poisson’s ratio. The transverse Poisson’s ratio mxy decreases with the increase of the braiding angle. It also decreases as the fiber volume fraction increases, but the change is not obvious. Fig. 12(d) shows the variation of the longitudinal Poisson’s ratios, mxz and myz (mxz = myz), with the braiding angle and the fiber volume fraction. It is observed that the longitudinal Poisson’s ratio mxz increases first but then decreases with the increase of the braiding angle. mxz increases with the increase of the fiber volume fraction. Fig. 12(e) depicts that the transverse shear modulus Gxy increases monotonically as the braiding angle increases and the increase amplitude is also increasing. At the same time, as the fiber volume fraction increases, the transverse shear modulus also increases. Fig. 12(f) gives that the longitudinal shear moduli, Gxz and Gyz (Gxz = Gyz), increases with the crease of the braiding angle. But the increase becomes weak gradually as the braiding angle increases. And Gzx or Gzy also increases with the increase of the fiber volume fraction.

Since the most important impact factor is the void volume fraction of 3D braided composites, the effects of the void volume fraction of the tows Pfv and the void volume fraction of the matrix Pmv on the elastic properties distribution are discussed by analyzing the two scale FEMs. As dry patches exist in fiber tows, the comparison on the elastic constants of the fiber tows of specimen number 1 with different void fraction of the tows Pfv are listed in Table 5. From the table, it can be seen that all the elastic constants decrease as the dry patches increase. Besides the longitudinal elastic modulus Ez, other elastic moduli decrease more and more sharply with the increase of the void volume fraction of the tows, which indicates the existence of dry patches has great effect on the deterioration of the fiber tows’ properties. Variations of the calculated elastic properties of 3D braided specimen number 1 with the two parameters, Pfv and Pmv, are shown in Fig. 13. From these figures, it is observed that the effects of the dry patches in fiber tows are more significant than the effects of voids in matrix on the elastic properties of 3D braided composites. Besides the longitudinal Poisson’s ratios, mxz and myz (mxz = myz), other elastic constants decrease with the increases of Pfv or Pmv. For the elastic moduli and the shear moduli, their values descend more sharply with the ascent of the void fraction of the tows, Pfv. But the decreases of these constants are slight and almost even as the void fraction of matrix, Pmv, increase. The transverse Poisson’s ratio, mxy, decreases evenly with the increase of Pfv and drops slightly as Pmv increase. However, the longitudinal Poisson’s ratios, mxz and myz (mxz = myz), increase with the increase of Pfv or Pmv, which indicates that the existence of pore defects promote the transverse deformation of 3D braided composites under longitudinal load. From Table 5 and Fig. 13, it can be concluded the stiffness degradation of 3D braided composites is mainly due to the rapid reduction of the elastic constants of fiber tows caused by the dry patches. 5. Conclusions Based on the periodical boundary conditions theory, the RVE models of the fiber tows and 3D braided composites on two scales are proposed to predict the effective elastic properties and the micro stress of 3D braided composites. The two-scale FE models based on the two geometric models take into account the voids in the resin matrix and the dry patches in the fiber tows by defining very small stiffness elements. The predicted effective elastic constants agree well with the experimental data, demonstrating the applicability of the two-scale FEMs. The effects of braiding angle and fiber volume fraction on engineering elastic constants of 3D braided composites are discussed and some useful conclusions are drawn. At the same time, the effects of the two types of defects, dry patches in fiber tows and voids in matrix, on the effective elastic properties of 3D braided composites are also discussed in detail. The calculated results show that the elastic constants besides the longitudinal Poisson’s ratios, mxz and myz, decrease with the increase of the void volume fractions, Pfv and Pmv. And the void volume fraction of the fiber tows has more significant effect on the elastic constants than that of the matrix. The longitudinal Poisson’s ratios increase with the increase of the void volume fractions, which proves that the existence of pore defects promote the transverse deformation of 3D braided composites under longitudinal load. Future work will focus on the effect of internal defects on the strength and failure mechanism under tension, shear, bending and other loading conditions of the composites in the subsequent research.

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Fig. 13. Influences of the void volume fraction Pfv and Pmv on the elastic constants.

Acknowledgment The authors would like to acknowledge the support provided by the Fundamental Research Funds for the Central Universities (Grant no. 2012QNA54).

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