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Finite element analysis of 3D braided composites based on three unit-cells models
Composite Structures 98 (2013) 130–142
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Finite element analysis of 3D braided composites based on three unit-cells models Chao Zhang ⇑, Xiwu Xu State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, China
a r t i c l e
i n f o
Article history: Available online 23 November 2012 Keywords: 3D braided composites Three unit-cell Elastic properties Stress distribution Finite element analysis
a b s t r a c t 3D braided composites have a skin-core structure and can be divided into three regions: interior, surface and corner. Each region is built up of identical unit-cells but the yarn configuration in the three regions is different thus must be treated separately. In this paper, three distinct solid structure models of unit-cell located in the interior, surface and corner regions of 3D braided composites are established first. With appropriate boundary conditions, the mechanical responses of 3D braided composites are simulated by using the finite element method. The deformation and stress distribution of the three unit-cell models are presented and the effects of the braiding angle and fiber volume fraction on the elastic constants of 3D braided composites are investigated in detail. Numerical results show that the three unit-cells have distinctive properties and the surface and corner unit cells are stiffer than the interior unit cell. Therefore, the effect of surface and corner regions on the mechanical properties of 3D braided composites must be considered in order to obtain accurate results. The calculated results show good agreement with available experiment data, thus verifies the applicability of the proposed three unit-cells models. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction As a kind of new lightweight and advanced textile material, 3D braided composites have been subjected to special concern due to their excellent performances over the conventional laminated composites, including better structural integrity, more balanced properties, higher damage tolerance and lower production costs. Because of these benefits of mechanical properties, 3D braided composites have tremendous potential applications to primary load–bearing structures in aeronautics and astronautics industries. However, the microstructure of 3D braided composites which determines the macro-mechanical properties of the materials is much more complicated than the conventional laminated composites. Establishing a reasonable microstructure is the premise for analyzing its mechanical properties. In 1980s, some simple geometric models have been proposed to evaluate the mechanical properties of 3D braided composites, such as ‘fabric geometry model’ [1], ‘fiber interlock model’ [2] and ‘fiber inclination model’ [3]. Although these analytical models can calculate the elastic stiffness easily; however, the microstructure of 3D braided composites in these models is over simplified thus less accurate results are predicted. Since 1990s, researchers have devoted to establish more realistic micro-geometric structure models for predicting more accurate mechanical properties of 3D braided composites.
⇑ Corresponding author. Address: Mailbox 261, No. 29, Yudao Street, Nanjing 210016, China. Tel.: +86 25 84891780; fax: +86 25 8489 1422. E-mail address: [email protected] (C. Zhang). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.11.003
Li et al. [4] investigated the internal yarn structure and proposed an interior unit-cell model of 3D braiding composites. In the interior unit-cell model, the braiding yarns were thought to be straight rods with a circular cross-section and there were no yarns interaction and deformation. Wang and Wang [5] analyzed the yarn’s topological structure by tracing the yarn path during the braiding processes. Three different types of unit-cell were identified in the preform interior, surface and corner regions, respectively. Subsequently, a mixed volume averaging method was performed to predict the elastic behavior of 3D braided composites. Chen et al. [6] developed three types of microstructure unitcell models of 3D braided composites by employing an analytical approach combined with experimental observations. The mathematical relationship between braided structure and braiding parameters were also presented. Sun and Sun [7] implemented a digital-element model to simulate the braiding process and investigate the microstructure of 3D braided preform. A comparative analysis was carried out by using digital-element model and topological model on interior unit-cell to calculate the mechanical properties of 3D braided composites. Tang and Postle [8] conducted an analytical and experimental study on the microstructure and elastic behavior of 3D braided composites. In their paper, the normalized pitch length was introduced as a key parameter of 3D braided structures. And the braiding angles and fiber volume fraction can be represented as functions of this key parameter. Chen et al. [9] established a unit-cell model and predicted the effective elastic properties of 3D braid composites using a homogenization method, in which the internal interface condition of heterogeneous material was introduced. Shokrieh and Mazloomi [10]
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proposed multi-unit-cell model, namely interior, interior surface, exterior surface and corner unit-cell models, to calculate the stiffness of 3D four-directional braided composites. The curved yarn paths in the exterior surface were considered as one straight line and corner unit-cell models as two lines for the sake of simplicity. Although theoretical analysis is easy to implement, however, it can only yield the elastic constants and is difficult to exhibit the accurate micro-stress distribution and other mechanical characteristics of the 3D braided composites. On the other hand, analysis by the finite element method can overcome these limitations existing in the theoretical analysis and provide more detailed information needed by investigators. Some early finite element analysis was based on simple structure models of 3D braided composites. Zeng et al. [11] developed a multiphase finite element model to determine the effective properties and the local stress within 3D braided composites under different loading conditions. Sun et al. [12] proposed a new method for calculating mechanical properties of 3D braided composite materials via homogenization theory and incompatible multivariable finite element method. On the basis of three unit-cell models proposed previously, Chen et al. [13] further presented a multiphase finite element method to predict the elastic properties of 3D braided composites. This method uses regular meshes, in which the element comprises more than one material. That is, the element is separated by the interface between yarns and matrix. Thus, the integration points at two sides of the interface have different mechanical properties. Actually, in the application of this method, the location of the interface in the three unit-cells is difficult to determine accurately, especially when the yarn configuration is complex. Recently, with the rapid development of computer simulation technology, more attention has been paid to establish the solid structure model of 3D braided composites and study the mechanical properties by using finite element method. Yu and Cui [14] applied a two-scale finite element analysis to predict the mechanics parameters of 3D four-directional braided composites. Xu and Xu [15] predicted the elastic constants and simulated the stress field of 3D braided composites based on a new finite element model with octagonal yarns cross-section. Fang et al. [16] divided the braiding yarn into seven regions and considered the distortion characteristics by introducing stochastic function theory to elastic properties in each region of yarns, and then investigated the mechanical properties of 3D braided composites with the influences of yarn distortion. Li et al. [17] established a parameterized 3D finite element model to examine the mechanical response of 3D braided composites. This model can precisely simulate the spatial configuration of the braiding yarns and considered the surface contact between the yarns due to mutual squeezing. Previous work has shown that 3D braided composites have a skin-core structure and are composed of three regions: interior, surface and corner. Each region is built up of identical unit-cell with unique yarns configuration and must be treated separately. Up to now, however, the solid finite element modeling mentioned above mostly focused only on the interior unit-cell which constitutes the main part of the whole structure. The accurate microstructure of surface and corner unit-cell needs to be established and their effect on the mechanical properties of 3D braided composites requires to be further studied. The present work is aimed at establishing the three unit-cell solid structure models and analyzing the mechanical properties of 3D braided composites by using finite element method. To establish the three unit-cell structure models, assumptions are made that the cross-section of braiding yarns is elliptical and their respective squeezing condition in different regions is considered. Moreover, the three unit-cell structure models are parameterized with input
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parameters of yarn cross-section dimensions and braiding angles. Based on the volume averaging method, a finite element model is developed for the prediction of elastic constants and the analysis of the mechanical response. The effect of the braiding angle and fiber volume fraction on the mechanical properties of 3D braided composites is investigated. 2. 3D four-step 1 1 braiding process 3D braided composites are a kind of textile composites that are manufactured by braided preforms impregnated and consolidated with resin materials. The four-step 1 1 braiding is the most common manufacturing techniques available to make 3D braiding preforms. Fig. 1 demonstrates the yarn carriers’ pattern on a machine bed (x–y plane) and their movements in one machine cycle. The number of rows and columns of carriers are defined as m and n, respectively. Additional carriers are added to the outside of the arrangement in alternating locations. The size of the braid is denoted by [m n] and the total number of yarns N is given by:
N ¼ mn þ m þ n
ð1Þ
Each machine cycle consists of four movement steps and each carrier moves one position at one step, this is the reason why the process is called four-step 1 1 braiding. At steps 1, the braiding yarn carriers in rows move horizontally one position in an alternating manner. At steps 2, the braiding yarn carriers in columns move one position vertically in an alternating manner. At steps 3 and steps 4, the carrier movements are opposite to those at steps 1 and steps 2. After these four-steps, all the yarn carriers return to their original pattern, completing one braiding cycle [6]. Then a ‘jamming’ action is put on all the braiding yarns, which makes the yarns structure to be compacted. A finite length of preform is realized and called as a braiding pitch, denoted by h. All the subsequent machine movements in producing more preform is just a repeat of the four-step motion and ‘jamming’ action aforementioned. 3. Motion mechanisms of braiding yarns and three unit-cell geometrical models In this section, the geometrical structure of a 3D braided composite is analyzed based on the movements of the carriers on the machine bed and the motion mechanisms of braiding yarns. The three unit-cell geometrical models representing the preforms and the composites are presented, which identify different yarn architectures in the interior, surface, and corners of the preform. 3.1. Planar yarn paths The plane referred to the x–y plane is the machine bed and the movement of yarn carriers takes place on this plane. The traces of three typical yarn carriers in the interior, surface and corners regions are shown in Fig. 2a. For carrier 22 in the interior, it follows a zig–zag way and moves two positions in the x and y direction during a braiding cycle. For carrier 62 on the surface, it moves from the main-part region into the surface and then returns back, holding its position at step J–J0 . For carrier 66 at the corner, it comes into the corner region from the interior, shifts its position according to the braiding process and holds its position at step N–N’ and O–O’, afterward re-enters to the interior. Due to the ‘jamming’ action applied by yarn tension, the yarns will reposition after a braiding cycle, hence, the yarn paths in the x–y plane are different from the traces of carriers. Considering the carrier motion positions as the control points of the yarn path and on the basis of least square principle, the path of braiding yarn in the x–y plane is fitted to be the connecting line of the middle
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Fig. 1. Scheme of 3D four-step braiding process.
Fig. 2. Yarn moment traces in x–y plane: (a) carriers’ movement traces, (b) x–y plane projection of all traces.
points of the carrier positions during the braiding process, as lines A0 E0 , Q–R–S and L–U–V–W in the three regions shown in Fig. 2a. Furthermore, all the yarn paths projected in the x–y plane are
presented in Fig. 2b. The angle between the projection of braiding yarn in x–y plane and y direction of the preform, u, is called as horizontal orientation angle. Ideally, u = ± 45°.
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3.2. Spatial yarn paths
4. Parameterized structure models of three unit-cells
From the above analysis, it is seen that the carrier’s motion in the interior, surface and corners is different, which leads to unique yarn configuration in each of these regions. However, for every motion step of the carriers, braiding yarns in any regions expand h/4 along z direction simultaneously. The spatial motion trace of carrier 22 experiences four steps from position point A–E, as shown in Fig. 3a. After the jamming action, the yarn axis becomes straight, as depicted by line A0 E0 . The interior yarn paths are all located in two families of orthogonal planes orientated ±45° with respect to the preform surfaces. Moreover, the angle between every interior yarn axis and the z direction is identical, defined as interior braiding angle c. Following the motion trace of carrier 62, it undergoes five steps from point F–K. Among the five steps, carrier stops moving at step J–J0 , which determines the spatial configuration of surface yarn dissimilar to the interior. The spatial motion process is shown in Fig. 3b. After yarn compacting, the actual path of yarn is a spatial curve. To facilitate the modeling, some simplifications on the spatial configuration of surface yarn are made. In this paper, path of surface yarn is modeled as two straight lines QR and RS. The angle of line QR with respect to z direction is identified as surface braiding angle h, as displayed in Fig. 3b. Following the motion trace of carrier 66, it moves eight steps from point L to P, holding its position at step N–N0 and O–O0 . The spatial motion diagram is illustrated in Fig. 3c. Obviously, the spatial configuration of corner yarn is distinct to the spatial configurations of the interior and surface. For the sake of simplicity, the curved path of corner yarn upon jamming is considered as three straight lines: TU, UV and VW, as presented in Fig. 3c. The corner braiding angle is measured from the direction of TU to z axis, denoted by b.
4.1. Basic assumptions
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The following assumptions have been made to establish the solid structure models of three unit-cells of 3D braided composites: 1. The mutual squeezing state of interior and exterior braiding yarns is different. The cross-section of interior braiding yarns is elliptical and the relationship between the major pffiffiffi and minor radii of the ellipse, a and b, is expressed as a ¼ 3b cos c. The cross-sections of surface and corner yarns are also elliptical, but the cross-sectional area reduces the corresponding proportion /f and /c respect to interior. 2. After yarns jamming action, the interior braiding yarn keeps straight and the surface and corner yarns are composed of straight segments in the space. 3. The cross-section shape of braiding yarn changes gradually from the surface and corner regions to the interior. 4. The braiding procedure maintains relatively steady and the braided structure is uniform, at least in a certain length of interest. 4.2. Geometrical models of three unit-cells Fig. 4 shows the topology geometrical models of unit-cells in the three distinct regions of 3D braided composites. Note that these models just illustrate the topology relationships of braiding yarns, without considering their physical cross sectional dimensions. The braiding yarns inside the interior unit-cell which forms the main-part of the composites are inclined in four directions; it is the reason why the composites are called the 3D four-directional braided composites.
3.3. Selecting of three unit-cells
4.3. Structural parameters of the three unit-cells
For studying the mechanical properties of composites with periodic microstructure, a representative volume element is usually picked out and analyzed instead of the whole composites structure. Such a representative volume element, the smallest region that can produce the whole structure by translating its copies, is called a unit-cell. It is observed that 3D braided composite is composed of three different regions with different yarn configurations. Therefore, three unit cells are selected in this paper, as shown in Fig. 2b. These unit cells are oriented in the same reference plane, the cross-section of the preform.
According to the spatial configuration of the braiding yarns in each unit-cell and the basic assumptions stated earlier, the structural parameters of the three unit-cell models can be obtained. They are favorable to calculate the fiber volume fraction in composite structures. Furthermore, they are the basis to establish parameterized solid structure models of three unit-cells. As displayed in Fig. 4, the width and thickness of the three unitcell models are given by:
pffiffiffi pffiffiffi W i ¼ 4 2b T i ¼ 4 2b
Fig. 3. The spatial traces of yarns moment: (a) interior region, (b) surface region, and (c) corner region.
ð2Þ
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Fig. 4. Geometrical models of three different unit-cells, (a) interior unit-cell, (b) surface unit-cell, (c) corner unit-cell.
pffiffiffi pffiffiffi pffiffiffiffiffi W s ¼ 4 2b T s ¼ 2 2b þ /s b
ð3Þ
pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi W c ¼ 2 2b þ /c b T c ¼ 2 2b þ /c b
ð4Þ
V sf ¼
3pabh 2
ei þ/s js 2 cos c
s js þ /cos h
ð13Þ
Us
where sub-index i, s and c refer to the interior, surface and corner unit-cells, respectively. Meanwhile, the three unit-cells have the same height, namely:
where Us, Ys and Vsf are volume of surface unit-cell, volume of braiding yarns in surface unit-cell and fiber volume fraction in the surface region, respectively. And for corner unit-cell, one has:
h ¼ 8b= tan c
pffiffiffiffiffiffiffiffi 3 U c ¼ ð64 þ 8/c þ 32 2/c Þb = tan c
ð5Þ
The braiding angle is an important structural parameter for the 3D braided composites. Generally, the interior braiding angle is difficult to measure since the interior of the preform cannot be observed without cutting the specimen. However, the angle of inclination of the yarns on the surface of composite can be measured readily. The relationship between the interior braiding angle c and the braiding angle a on the surface of composite is expressed as:
pffiffiffi tan a ¼ ð 2=2Þ tan c
ð6Þ
Using trigonometry, the surface and corner braiding angle can be calculated by:
tan h ¼ tan b ¼ b=ðð3=8ÞhÞ ¼ tan c=3
ð7Þ
Eq. (7) shows that the surface braiding angle and corner braiding angle are equal, and smaller than the interior braiding angle. That is to say, the yarns are closer to z direction in the exterior region of the composites, indicating that the mutual squeezing of yarns is more severe. 4.4. Fiber volume fraction As shown in Fig. 4, the volume of interior unit-cell is: 3
U i ¼ 256b = tan c
ð8Þ
The total volume of braiding yarns in this unit-cell model is:
Y i ¼ 4pabh= cos c
ð9Þ
Therefore, the fiber volume fraction in the interior region is written as:
V if ¼ Y i ji =U i ¼
pffiffiffi 3pji =8
Ys ¼
3pabh 1 þ /s / þ s 2 2 cos c cos h
pabh 1 þ /c 3/c þ 2 2 cos c cos b
V cf ¼
ð14Þ
pabh ei þ/c jc 2 2 cos c
c jc þ 3/ cos b
ð15Þ
Uc
ð16Þ
where Uc, Yc and Vcf are volume of corner unit-cell, volume of braiding yarns in corner unit-cell and fiber volume fraction in the corner region, respectively. In Eqs. (10), (13), and (16), ji, js and jc are yarn packing factor of the interior, surface and corner regions, respectively. For the whole structure of 3D braided composites, the total fiber volume fraction, Vf, is obtained by weighted averaging method, namely:
V f ¼ V i V if þ V s V sf þ V c V cf
ð17Þ
where Vi, Vs and Vc are volume proportions of the interior, surface and corner regions to the whole structure of composites and are presented as follows:
2ðm 2Þðn 2Þ pffiffiffi V i ¼ pffiffiffi ð 2m þ 1Þð 2n þ 1Þ
ð18Þ
pffiffiffi ð4 þ 2Þðm þ n 4Þ pffiffiffi V s ¼ pffiffiffi ð 2m þ 1Þð 2n þ 1Þ
ð19Þ
Vc ¼ 1 Vi Vs
ð20Þ
4.5. Solid structural models of three unit-cells
ð10Þ
Similarly, for surface unit-cell, one has:
pffiffiffiffiffiffiffiffi 3 U s ¼ ð128 þ 32 2/s Þb = tan c
Yc ¼
ð11Þ ð12Þ
The solid structural models of these unit-cells are established using TexGen which is open source software developed at the University of Nottingham for geometric modeling of textile structures. TexGen provides a friendly graphical user interface (GUI) through which many functions can be utilized. For more advanced functions, using of scripts written in Python is recommended.
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Fig. 5. Structure models of three different unit-cells, (a) interior unit-cell, (b) surface unit-cell, and (c) corner unit-cell.
Fig. 6. Parameterized model of interior unit-cell: (a) c = 20°, (b) c = 30°, and (c) c = 40°.
The path of braiding yarn is the yarn’s centerline in 3D space and is represented through a series of discreet control points coupling with a suitable interpolation function in TexGen. In this paper, the intersection points of yarns’ straight segments and some other points in yarn’s straight centerline are selected as control points, and the linear interpolation function is adopted. It is known that the squeezing states of yarns in the three regions of the composites vary. Thus, the shape reduction coefficients /s and /c are both set to 0.9. The solid structure models of three unit-cells are shown in Fig. 5. By analyzing the relationship between microstructure parameters of three unit-cells, the input parameters to generate the parameterized unit-cell models are identified as the minor radii of the yarn cross-section, the braiding angle, and the cross-section shape reduction coefficients. Other parameters can be determined by these parameters. Parameterized unit-cell models are beneficial to parameters modification and model regeneration. As an example, Fig. 6 illustrates the solid models of interior unit-cell with the same yarn cross-section size but different braiding angles.
4.6. Visualization of the internal microstructure of three unit-cell models The microstructure of 3D braided composites is complex and cannot be observed from the outside of the specimen. To investigate the internal microstructure, one obvious way is to cut the specimen but is costly and difficult for real materials. With the parameterized unit-cell models established in this paper, cutting them along different cross-sections allows one to observe and visualize the internal microstructure. This helps us to understand the spatial configuration of the unit-cells. Fig. 7c shows the multiple unit-cells model of the interior cutting with ±45° respect to the boundary. In comparisons, the yarn intersection condition in the preform interior concluded by Wang and Wang [5] is depicted in Fig. 7a and b. Obviously, the yarn topology is validly reflected in the unit-cell model. Furthermore, Fig. 8 gives the comparison of 45° cross-section of interior unit-cell and SEM micrograph. It can be seen that the yarn cross-section shape, the arrangement of braiding yarns and space distance
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Fig. 7. Yarn topology in the preform interior region: (a) two families of plates [5], (b) yarn formation and plate intersection [5], and (c) ±45° cross-section of interior unit-cell model.
Fig. 8. Comparison of cross-section of interior model and SEM micrograph: (a) SEM micrograph [6], (b) cross-section of Interior model.
between the yarns are almost identical with the experimental observations. This indicates that the established interior unit-cell model is correct. Due to the absence of experimental results, the internal microstructures of surface and corner unit-cells are not given. 5. Finite element models 3D braided composites are composed of braiding yarns and the resin matrix pockets. Consequently, the mechanical behavior of the braided composites is determined by the elastic constitutive relationships of the constituents. In this work, the resin matrix is assumed to be isotropic; the braiding yarns containing thousands of fibers and matrix are modeled as transversely isotropic and unidirectional composites in local coordinate system. The stiffness properties of the braiding yarns are obtained using the micromechanics formulae given by Chamis [18]. For the local coordinate definition of yarn in a specify orientation, local 1-axis follows the yarn path line and local 3-axis is in the upright plane perpendicular to the x–y plane of the global coordinate system. 5.1. Boundary conditions If the composite materials are regarded as perfect periodic structures consisting of periodic array of unit-cells and the analysis is based on the unit-cell modeling, the periodic boundary conditions should be applied to replicate its repeating nature. Furthermore, displacement continuity condition and traction continuity condition must be satisfied at the opposite boundaries of the neighboring unit-cells. As proposed by Suquet [19], the displacement field for the periodic structure is:
ui ¼ eik xk þ ui
ð21Þ
In the above, eik are the average strains of the unit-cell, xk is the Cartesian coordinate of a unit-cell point and ui is the periodic part of the displacement components on the boundary surfaces. Eq. (21) is difficult to be imposed to the unit-cell boundaries because of the unknown periodic part. Fortunately, for most unitcells, the boundary surfaces always appear in parallel pairs. The displacements on a pair of parallel opposite boundary surfaces (with their normal along the Xj axis) can be given as [20]:
uijþ ¼ eik xkjþ þ ui
ð22Þ
uij ¼ eik xkj þ ui
ð23Þ
where the index ‘‘j+’’ means along the positive Xj direction and ‘‘j’’ means along the negative Xj direction. Since ui is the same at the two parallel boundaries, the difference between the above two equations is:
uijþ uij ¼ eik ðxkjþ xkj Þ ¼ eik Dxkj
ð24Þ
The right side of the above Equation becomes constant once eik is specified, because Dxjk are constants for each pair of boundary surfaces. Thus Eq. (24) can be rewritten as [20]:
uijþ ðx; y; zÞ uij ðx; y; zÞ ¼ cij
ði; j ¼ 1; 2; 3Þ
ð25Þ
The above equations do not contain the periodic part of the displacement components and can be carried out easily in the finite element analysis by setting the nodal displacement linear constraint equations. It was proved by Xia et al. [20] that the application of this boundary condition ensures not only the displacement continuity conditions, but also the traction continuity conditions.
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For the finite element modeling of interior unit-cell, it has periodicity in all the three directions in space, therefore, the deformation of its opposite paired faces is completely governed by Eq. (25). It is, however, that the surface unit-cell has periodicity only in two directions (x and z) and the corner unit-cell is periodical just in direction z. This is because that the outer boundaries of both surface and corner unit-cells are free. To take account into the aperiodicity of surface and corner unit-cells in the finite element simulation, the internal connected face is completely fixed in its normal direction, and the outer free face is forced to deform as a plane parallel to the connected face. Certainly, the periodical boundary condition should be applied to the pair of faces with periodicity. 5.2. Mesh discretization Due to the complicated material arrangements of 3D braided composites, mesh generation may be a large obstacle for predicting mechanical properties by finite element method. In this study, the 3D voxel-based meshing technique available in TexGen is adopted for the discretization of yarns and matrix. This technique allows the mesh generation automated but the element faces can only match the geometry boundaries of the constituents in the unit-cell approximately. However, the voxel meshing technique is sufficient for stiffness prediction of textile composites and even suitable for damage simulations with the usage of adaptive mesh refinement proposed by Kim and Swan [21]. In this paper, the 8-nodes reduced integration element (C3D8R) available in ABAQUS is used for mesh discretization. In order to guarantee the convergence of the solution, the analysis of mesh sensitivity is conducted. Smaller meshing size can provide more accurate stress distributions, but the computer memory increase sharply. If only the global stiffness is concerned, relative coarse mesh size can still give satisfactory results. According to a compromise consideration, the mesh size of the models in this study is fine enough to provide the stress state and predict stiffness properties.
In order to calculate the elastic constants of the three unit-cells, the homogenization method is employed in this study. The heterogeneous composites in the micro-scale are considered as a homogeneous material in the macro-scale. For each unit-cell, the constitutive relationship is:
ð26Þ
i and ej are the global where Cij is the effective stiffness matrix. r average stress and global average strain defined by:
r ij ¼ eij ¼
1 V 1 V
Pn e
Z
e k¼1 V k
V
Z V
ðreij V ek Þ
k¼1 rij dV ¼ P ne
Pn e
ðeeij V ek Þ
k¼1 eij dV ¼ P ne
e k¼1 V k
ðno summation over eÞ
ðno summation over eÞ
ð27Þ
ð28Þ
where reij and eeij are the stress and strain of element. V ek , ne and V are element volume, total number of element and volume of the unitcell, respectively. ¼ ½r 11 ; r 22 ; r 33 ; r 23 ; r 13 ; r 12 T and ½e ¼ ½e11 ; e22 ; e33 ; e23 ; Let ½r e13 ; e12 T . In the 3D case, give six independent sets of global strain vector in the finite element analysis and obtain six sets of global stress vector. Then, the effective compliance matrix, Sij, the inverse of effective stiffness matrix, can be worked out by:
1 Sij ¼ ei r j
¼ ½C
c X V n ½C ij n
ðn ¼ i; s; cÞ
ð30Þ
n¼i
Similarly, sub-index i, s and c refer to the interior, surface and corner unit-cells; Vn are the volume of the three unit-cells respectively and have been given in Eqs. (18)–(20). Correspondingly, the compliance matrix of the whole composite 1 ; and then the effective structure can be calculated by ½ S ¼ ½C elastic constants of 3D braided composites can be obtained. 6. Results and discussions 6.1. Comparison with experimental data In order to verify the effectiveness of the proposed three unitcells model, predicated results are compared to the experimental data cited from Ref. [23]. The elastic properties of the component materials, including AS4 carbon fiber and epoxy resin, are listed in Table 1. The geometric parameters of the specimens and the structure parameters of three unit-cell models used in the calculation are summarized in Table 2. Tables 3 and 4 present the measured moduli in longitudinal and transverse directions against the predicted results from Weighted Average Model (WAM) of Kalindidi and Abusafieh [23], a theoretical model based on a single idealized unit-cell, and from the proposed finite element model of three unit-cells. It can be found that the predicted results by the two methods agree well with experimental data. Furthermore, the proposed method can give accurate micro-stress distribution and deformation of the three unit-cells. 6.2. The effect on elastic properties of surface and corner unit-cells
5.3. Calculation of effective elastic constants
r i ¼ C ij ej
The effective elastic constants of each unit-cell can be obtained from the calculated compliance matrix. Details on this procedure can be found in Ref. [22]. Next, stiffness matrix from each unit-cell is summed up to calculate the stiffness matrix of the whole structure of braided composites by volume averaging method, namely:
ð29Þ
Most often, only the interior unit-cell is concerned for the prediction of stiffness by numerical method previously. However, the real 3D braided composite usually has three different unit cells, namely, the interior, surface and corner unit cells. Table 5 displays the predicted elastic constants for each unit-cell and the overall composite structure (specimen number 4). For specimen 4, there are 117,936 nodes and 110,250 elements in the interior unit-cell, 68,796 nodes and 63,000 elements in the surface unit-cell, and 116,281 nodes and 108,000 elements in the corner unit-cell, respectively. Surely, the meshing of corner unit-cell is refined here. From Table 5, it is seen that the interior unit-cell is perfect transversely isotropic, but the surface and corner unit-cells are only approximate transversely isotropic. In the calculation of stiffness properties of the whole composite, both the surface and corner unit-cells are also regarded as transversely isotropic by averaging operations. That is, the 3D braided composites are considered to be transversely isotropic macroscopically. It is also seen that the surface and corner regions are much stiffer than the interior region. Since the braiding pattern of the specimens in the analysis is
Table 1 Elastic properties of AS4 carbon fiber and epoxy resin. Material
E11 (GPa)
E22 (GPa)
G12 (GPa)
G23 (GPa)
l12
AS4 Epoxy
234.6 2.94
13.8 2.94
13.8 1.7
5.5 1.7
0.2 0.35
138
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Table 2 The geometric parameters of the specimens and the structure parameters of three unit-cell models. Specimen number
Geometric parameters of specimens
Structural parameters of three unit-cell models
a
Vf
mn
c (°)
h (mm)
ji
Wi (mm)
Tf = Wc (mm)
0.38 0.40 0.46 0.44 0.29
12 12 12 12 12 12 12 12 12 12
23.38 23.38 27.24 29.74 33.40
5.63 5.46 4.35 4.03 4.37
0.54 0.57 0.64 0.63 0.43
1.72 1.67 1.58 1.63 2.04
1.14 1.11 1.06 1.09 1.36
(°) 1 2 3 4 5
17 17 20 22 25
Table 3 Comparison between experiment data and predicted results of WAM [23] and present finite element model in longitudinal direction. Sample number
Experimental data (GPa)
WAM (GPa)
FEM (GPa)
1 2 3 4 5
43.6 ± 1.9 45.9 ± 1.2 48.0 ± 3.0 38.5 ± 2.4 21.2 ± 1.8
48.3 50.7 47.0 38.6 21.6
48.47 50.94 46.34 40.19 22.54
Table 4 Comparison between experiment data and predicted results of WAM [23] and present finite element model in transverse direction. Sample number
Experimental data (GPa)
WAM (GPa)
FEM (GPa)
1 4
6.21 ± 0.41 6.02 ± 0.30
5.74 6.21
5.56 6.03
Table 5 Predicted values of elastic constants for each unit-cell and the overall composite structure.
Interior Surface Corner Overall
Ex (GPa)
Ey (GPa)
Ez (GPa)
Gxz (GPa)
Gyz (GPa)
Gxy (GPa)
lxz
lyz
lxy
5.77 6.30 6.51 6.03
5.77 6.42 6.41 6.03
36.95 42.45 59.49 40.19
9.62 6.21 3.65 8.22
9.62 6.34 3.53 8.22
3.35 2.37 2.18 2.97
0.81 0.56 0.42 0.72
0.81 0.51 0.44 0.72
0.33 0.35 0.35 0.34
guarantee the displacement continuity between the neighboring unit-cells and provide a reasonable stress distribution. Note that the warping extents of the three sets of opposite boundary faces are various under all the loading cases. These are attributed to that the unit-cell model of 3D braided composites does not have the symmetries of geometrical structure and physical properties. In Fig. 9b, the surface unit-cell does not have periodicity in the y direction. The opposite boundary face in y direction is confined to deform in a plane. In Fig. 9c, the corner unit-cell only has periodicity in z direction. Therefore, the opposite boundary faces in the x and y direction are confined to deform in a plane under z tension. However, for xy shear or xz shear, the plane deformation constraint in x direction is removed in order for the implementation of shear deformation. Fig. 10 illustrates the stress nephogram of the three unit-cells on un-deformed shape. In Fig. 10a, the traction continuity at the opposite boundary faces of the interior unit-cell has been guaranteed and satisfied the periodic boundary condition. Under z tension, the stresses in the braiding yarns are almost 15 times than that in the matrix pocket region on the average. While under xy shear and xz shear, the stresses in the yarns become 7 and 9 times approximatively than that in the matrix. Obviously, the mechanical response characteristic of the model subjected to different loadings varies, since the load carried by yarns is different. In Fig. 10b and c (the matrix pockets have been removed for clarity), the stresses in the yarns of surface and corner unit-cells are higher than that of interior unit-cell. This is because that the braiding yarns are closer to z direction in the exterior regions and the yarns will bear more loads. Moreover, in these regions, the squeezing state of yarns is more severe and the fiber volume faction is larger than it in interior region. The stress concentration is a concerned issue for composite designers. As shown in Fig. 10, for interior unit-cell, stress concentration occurs in the contact area between yarns and matrix under tension load, while the stress concentration is more obvious and arises in the interlaced region of braiding yarns under shear load. For surface and corner unit-cell, the braiding yarns are curved in these regions. They are inclined to be straight under tension and shear loads, and therefore the stress peaks occur in the squeezing positions. Obviously, the contact zones between yarns are the significant parts in the strength analysis problems since micro-cracks and damage always initially generated in these regions. 6.4. Parametric study on the effective elastic properties
[12 12], thus the interior, surface and corner unit-cells account for 62.36%, 33.22% and 4.42% of the whole structure, respectively. Conclusion may be drawn that even though the interior unit-cell comprises the main-part of the composite structure, consideration of the effect of surface and corner unit-cells is also important, especially when the number of rows and columns of carriers are relatively small. 6.3. Deformation and stress distributions of unit cells The advantage of the proposed modeling strategy is that the detailed information on the deformation and stress distribution in the 3D braided composites under different loadings can be obtained. Three typical loading cases, namely, z tension (e33 ¼ 0:002), xy shear (e12 ¼ 0:002) and xz shear (e13 ¼ 0:002), are selected to demonstrate the mechanical behavior of 3D braided composites (specimen number 4). Fig. 9 depicts the deformation of the three unit-cells when the deformation scale factor is 300. From Fig. 9a, it can be observed that the boundary faces are no longer plane and warped. The two opposite boundary faces have the same deformation which can
Braiding angle and fiber volume fraction are two important structural parameters which directly control and influence the overall mechanical properties of the 3D braided composites. In this section, the effects of the two parameters on the effective elastic properties are studied by using the proposed modeling strategy. Note that the change of fiber volume fraction of the models with a certain braiding angle is realized by setting different yarn packing factors within a reasonable range. Fig. 11 shows the variation of the predicted elastic constants with the braiding angle for three different fiber volume fractions. It is clear seen from Fig. 11a that the elastic modulus Ez decreases sharply as the braiding angle a increases. This is because that with the increasing of a, the braiding yarn is inclined to deviate from the z axis thus causes the decreasing of rigidity in the z direction. Meanwhile, with the increase of the fiber volume fraction Vf, the elastic modulus Ez increases and the increment is relatively significant when a is small. Fig. 11b shows that the elastic moduli, Ex and Ey (Ex = Ey), increase gradually as the braiding angle increases. When a is greater than 30°, the increment becomes obvious. With the increase of the fiber volume fraction, the elastic modulus Ex and Ey increase in a similar tendency. From Fig. 11c, it can be found
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Fig. 9. Deformation of the finite element models: (a) deformation of the interior unit-cell under z tension, xy shear and xz shear load, (b) deformation of the surface unit-cell under z tension, xy shear and xz shear load, and (c) deformation of the corner unit-cell under z tension, xy shear and xz shear load.
that the transverse shear modulus Gxy increases monotonically with the increasing of a. When a is less than 25°, the increase is not obvious, but greater than that, Gxy increases much rapidly. This
is mainly due to the simultaneous increasing reinforcement of the braiding yarns in the x and y direction with the increase of a. Similarly, with the increase of Vf, Gxy also increases in a similar
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Fig. 10. Stress nephogram of the finite element models, (a) stress nephogram of the interior unit-cell under z tension, xy shear and xz shear load on un-deformed shape, (b) stress nephogram of the surface unit-cell under z tension, xy shear and xz shear load on un-deformed shape, and (c) stress nephogram of the corner unit-cell under z tension, xy shear and xz shear load on un-deformed shape.
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Fig. 11. Variation of effective elastic constants with braiding angle and fiber volume faction.
tendency. Fig. 11d describes that the longitudinal shear moduli, Gxz and Gyz (Gxz = Gyz), increase steadily as a and Vf increase. When a is small, the increase is significant. When a is greater than 25°, the rate of increment slows down, but the change of Gxz and Gyz caused by the increment of Vf becomes comparatively larger. Fig. 11e and f show that a and Vf also affect the Poisson’s ratios, mxy and mxz (mxz = myz). It can be found that a is the main factor that affects Poisson’s ratios. When a increase, mxy firstly decreases and then increases, it reaches the minimum value at a = 30°; mxz and myz firstly increase and then decrease, they reach the maximum value at a = 25°. As Vf increases, the Poisson’s ratio mxy decreases for a less than 25°and then increases for a greater than 25°, however, the Poisson’s ratios mxz and myz, always increase. Generally, the effect of the fiber volume fraction on the Poisson’s ratios is relatively weak. 7. Conclusion The mechanical behavior of the 3D four-step braided composite is investigated. The topology geometrical models in the interior,
surface and corner regions of the preform are established. And then, the spatial configuration of the braiding yarns as well as the squeezing conditions in different regions is considered to establish the parameterized three unit-cell structure models. Based on these structure models in conjunction with finite element method, the mechanical properties of 3D braided composites are predicted. It is found that the three regions have distinctive properties and the surface and corner unit-cells are stiffer than the interior region. Hence, the surface and corner regions should be considered in order to predict the elastic properties more precisely. Good agreement is obtained between the calculated results and experimental data. Thus the proposed simulation method is validated. Since the deformation and stress distribution in the unit cells can be obtained, the proposed method is useful for a better understanding of mechanical response of constituent materials under various loading cases. Furthermore, the effects of braiding angle and fiber volume fraction on the effective elastic constants of 3D braided composites are discussed in detail. The results may be helpful for designers to optimize the braided composite structures.
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References [1] Ko FK, Pastore CM. Structure and properties of integrated 3-D fabric for structural composites. In: Vinson JR, Taya M, editors. Recent advances in composites in the United States and Japan. Philadelphia: American Society for Testing and Materials; 1985. p. 428–39. [2] Ma CL, Yang JM, Chou TW. Elastic stiffness of three-dimensional braided textile structure composites. In: Composite Materials: Testing and Design. ASTM 893; 1986, p. 404–21. [3] Yang JM, Ma CL, Chou TW. Fiber inclination model of three dimensional textile structural composites. J Compos Mater 1986;20(5):472–84. [4] Li W, Hammad M, El-Shiekh A. Structure analysis of 3-D braided preforms for composites. J Text Inst 1990;81(4):491–514. [5] Wang YQ, Wang A. Microstructure/property relationships in threedimensionally braided fiber composites. Compos Sci Technol 1995;53(2): 213–32. [6] Chen L, Tao XM, Choy CL. On the microstructure of three-dimensional braided preforms. Compos Sci Technol 1999;59(3):391–404. [7] Sun XK, Sun CJ. Mechanical properties of three-dimensional braided composites. Compos Struct 2004;65(3–4):485–92. [8] Tang ZX, Postle R. Mechanics of three-dimensional braided structures for composite materials part I: fabric structure and fiber volume fraction. Compos Struct 2000;49(4):451–9. [9] Chen ZR, Zhu D, Lu M, Lin Y. A homogenization scheme and its application to evaluation of elastic properties of three-dimensional braided composites. Composites: Part B 2001;32(1):67–86. [10] Shokrieh M, Mazloomi M. A new analytical model for calculation of stiffness of three-dimensional four directional braided composites. Compos Struct 2012;94(3):1005–15. [11] Zeng T, Wu LZ, Guo LC. Mechanical analysis of 3D braided composites: a finite element model. Compos Struct 2004;64(3–4):399–404.
[12] Sun HY, Di SL, Zhang N, Pan N, Wu CC. Micromechanics of braided composites via multivariable FEM. Compos Struct 2003;81(20):2021–7. [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(16): 2383–91. [14] Yu XG, Cui JZ. The prediction on mechanical properties of 4-step braided composites via two-scale method. Compos Sci Technol 2007;67(3–4):471–80. [15] Xu K, Xu X. On the microstructure model of four-step 3D rectangular braided composites. Acta Mater Compos Sin 2006;23(5):154–60. [16] Fang GD, Liang J, Wang Y, Wang BL. The effect of yarn distortion on the mechanical properties of 3D four-directional braided composites. Composites: Part A 2009;40(4):343–50. [17] Li D S, Fang DN, Lu ZX, Yang ZY, Jiang N. Finite element analysis of mechanical properties of 3D four directional rectangular braided composites part 2: validation of 3D finite element model. Appl Compos Mater 2010;17(4): 389–404. [18] Chamis CC. Mechanics of composites materials: past, present and future. J Compos Technol Res 1989;11(1):3–14. [19] Suquet P. Elements of homogenization theory for inelastic solid mechanics. In: Sanchez-Palencia E, Zaoui A, editors. Homogenization techniques for composite media. Berlin: Springer-Verlag; 1987. p. 194–275. [20] Xia ZH, Zhou CW, Yong QL. On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites. Int J Solids Struct 2006;43(2):266–78. [21] Kim HJ, Swan CC. Voxel-based meshing and unit-cell analysis of textile composites. Int J Numer Meth Eng 2003;56(7):977–1006. [22] Xu K, Xu XW. Finite element analysis of mechanical properties of 3D fivedirectional braided composites. Mater Sci Eng A 2008;487(1–2):499–509. [23] Kalidindi SR, Abusafieh A. Longitudinal and transverse moduli and strengths of low angle 3-D braided composites. J Compos Mater 1996;30(8):885–905.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Finite element analysis of 3D braided composites based on three unit-cells models Chao Zhang ⇑, Xiwu Xu State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, China
a r t i c l e
i n f o
Article history: Available online 23 November 2012 Keywords: 3D braided composites Three unit-cell Elastic properties Stress distribution Finite element analysis
a b s t r a c t 3D braided composites have a skin-core structure and can be divided into three regions: interior, surface and corner. Each region is built up of identical unit-cells but the yarn configuration in the three regions is different thus must be treated separately. In this paper, three distinct solid structure models of unit-cell located in the interior, surface and corner regions of 3D braided composites are established first. With appropriate boundary conditions, the mechanical responses of 3D braided composites are simulated by using the finite element method. The deformation and stress distribution of the three unit-cell models are presented and the effects of the braiding angle and fiber volume fraction on the elastic constants of 3D braided composites are investigated in detail. Numerical results show that the three unit-cells have distinctive properties and the surface and corner unit cells are stiffer than the interior unit cell. Therefore, the effect of surface and corner regions on the mechanical properties of 3D braided composites must be considered in order to obtain accurate results. The calculated results show good agreement with available experiment data, thus verifies the applicability of the proposed three unit-cells models. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction As a kind of new lightweight and advanced textile material, 3D braided composites have been subjected to special concern due to their excellent performances over the conventional laminated composites, including better structural integrity, more balanced properties, higher damage tolerance and lower production costs. Because of these benefits of mechanical properties, 3D braided composites have tremendous potential applications to primary load–bearing structures in aeronautics and astronautics industries. However, the microstructure of 3D braided composites which determines the macro-mechanical properties of the materials is much more complicated than the conventional laminated composites. Establishing a reasonable microstructure is the premise for analyzing its mechanical properties. In 1980s, some simple geometric models have been proposed to evaluate the mechanical properties of 3D braided composites, such as ‘fabric geometry model’ [1], ‘fiber interlock model’ [2] and ‘fiber inclination model’ [3]. Although these analytical models can calculate the elastic stiffness easily; however, the microstructure of 3D braided composites in these models is over simplified thus less accurate results are predicted. Since 1990s, researchers have devoted to establish more realistic micro-geometric structure models for predicting more accurate mechanical properties of 3D braided composites.
⇑ Corresponding author. Address: Mailbox 261, No. 29, Yudao Street, Nanjing 210016, China. Tel.: +86 25 84891780; fax: +86 25 8489 1422. E-mail address: [email protected] (C. Zhang). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.11.003
Li et al. [4] investigated the internal yarn structure and proposed an interior unit-cell model of 3D braiding composites. In the interior unit-cell model, the braiding yarns were thought to be straight rods with a circular cross-section and there were no yarns interaction and deformation. Wang and Wang [5] analyzed the yarn’s topological structure by tracing the yarn path during the braiding processes. Three different types of unit-cell were identified in the preform interior, surface and corner regions, respectively. Subsequently, a mixed volume averaging method was performed to predict the elastic behavior of 3D braided composites. Chen et al. [6] developed three types of microstructure unitcell models of 3D braided composites by employing an analytical approach combined with experimental observations. The mathematical relationship between braided structure and braiding parameters were also presented. Sun and Sun [7] implemented a digital-element model to simulate the braiding process and investigate the microstructure of 3D braided preform. A comparative analysis was carried out by using digital-element model and topological model on interior unit-cell to calculate the mechanical properties of 3D braided composites. Tang and Postle [8] conducted an analytical and experimental study on the microstructure and elastic behavior of 3D braided composites. In their paper, the normalized pitch length was introduced as a key parameter of 3D braided structures. And the braiding angles and fiber volume fraction can be represented as functions of this key parameter. Chen et al. [9] established a unit-cell model and predicted the effective elastic properties of 3D braid composites using a homogenization method, in which the internal interface condition of heterogeneous material was introduced. Shokrieh and Mazloomi [10]
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proposed multi-unit-cell model, namely interior, interior surface, exterior surface and corner unit-cell models, to calculate the stiffness of 3D four-directional braided composites. The curved yarn paths in the exterior surface were considered as one straight line and corner unit-cell models as two lines for the sake of simplicity. Although theoretical analysis is easy to implement, however, it can only yield the elastic constants and is difficult to exhibit the accurate micro-stress distribution and other mechanical characteristics of the 3D braided composites. On the other hand, analysis by the finite element method can overcome these limitations existing in the theoretical analysis and provide more detailed information needed by investigators. Some early finite element analysis was based on simple structure models of 3D braided composites. Zeng et al. [11] developed a multiphase finite element model to determine the effective properties and the local stress within 3D braided composites under different loading conditions. Sun et al. [12] proposed a new method for calculating mechanical properties of 3D braided composite materials via homogenization theory and incompatible multivariable finite element method. On the basis of three unit-cell models proposed previously, Chen et al. [13] further presented a multiphase finite element method to predict the elastic properties of 3D braided composites. This method uses regular meshes, in which the element comprises more than one material. That is, the element is separated by the interface between yarns and matrix. Thus, the integration points at two sides of the interface have different mechanical properties. Actually, in the application of this method, the location of the interface in the three unit-cells is difficult to determine accurately, especially when the yarn configuration is complex. Recently, with the rapid development of computer simulation technology, more attention has been paid to establish the solid structure model of 3D braided composites and study the mechanical properties by using finite element method. Yu and Cui [14] applied a two-scale finite element analysis to predict the mechanics parameters of 3D four-directional braided composites. Xu and Xu [15] predicted the elastic constants and simulated the stress field of 3D braided composites based on a new finite element model with octagonal yarns cross-section. Fang et al. [16] divided the braiding yarn into seven regions and considered the distortion characteristics by introducing stochastic function theory to elastic properties in each region of yarns, and then investigated the mechanical properties of 3D braided composites with the influences of yarn distortion. Li et al. [17] established a parameterized 3D finite element model to examine the mechanical response of 3D braided composites. This model can precisely simulate the spatial configuration of the braiding yarns and considered the surface contact between the yarns due to mutual squeezing. Previous work has shown that 3D braided composites have a skin-core structure and are composed of three regions: interior, surface and corner. Each region is built up of identical unit-cell with unique yarns configuration and must be treated separately. Up to now, however, the solid finite element modeling mentioned above mostly focused only on the interior unit-cell which constitutes the main part of the whole structure. The accurate microstructure of surface and corner unit-cell needs to be established and their effect on the mechanical properties of 3D braided composites requires to be further studied. The present work is aimed at establishing the three unit-cell solid structure models and analyzing the mechanical properties of 3D braided composites by using finite element method. To establish the three unit-cell structure models, assumptions are made that the cross-section of braiding yarns is elliptical and their respective squeezing condition in different regions is considered. Moreover, the three unit-cell structure models are parameterized with input
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parameters of yarn cross-section dimensions and braiding angles. Based on the volume averaging method, a finite element model is developed for the prediction of elastic constants and the analysis of the mechanical response. The effect of the braiding angle and fiber volume fraction on the mechanical properties of 3D braided composites is investigated. 2. 3D four-step 1 1 braiding process 3D braided composites are a kind of textile composites that are manufactured by braided preforms impregnated and consolidated with resin materials. The four-step 1 1 braiding is the most common manufacturing techniques available to make 3D braiding preforms. Fig. 1 demonstrates the yarn carriers’ pattern on a machine bed (x–y plane) and their movements in one machine cycle. The number of rows and columns of carriers are defined as m and n, respectively. Additional carriers are added to the outside of the arrangement in alternating locations. The size of the braid is denoted by [m n] and the total number of yarns N is given by:
N ¼ mn þ m þ n
ð1Þ
Each machine cycle consists of four movement steps and each carrier moves one position at one step, this is the reason why the process is called four-step 1 1 braiding. At steps 1, the braiding yarn carriers in rows move horizontally one position in an alternating manner. At steps 2, the braiding yarn carriers in columns move one position vertically in an alternating manner. At steps 3 and steps 4, the carrier movements are opposite to those at steps 1 and steps 2. After these four-steps, all the yarn carriers return to their original pattern, completing one braiding cycle [6]. Then a ‘jamming’ action is put on all the braiding yarns, which makes the yarns structure to be compacted. A finite length of preform is realized and called as a braiding pitch, denoted by h. All the subsequent machine movements in producing more preform is just a repeat of the four-step motion and ‘jamming’ action aforementioned. 3. Motion mechanisms of braiding yarns and three unit-cell geometrical models In this section, the geometrical structure of a 3D braided composite is analyzed based on the movements of the carriers on the machine bed and the motion mechanisms of braiding yarns. The three unit-cell geometrical models representing the preforms and the composites are presented, which identify different yarn architectures in the interior, surface, and corners of the preform. 3.1. Planar yarn paths The plane referred to the x–y plane is the machine bed and the movement of yarn carriers takes place on this plane. The traces of three typical yarn carriers in the interior, surface and corners regions are shown in Fig. 2a. For carrier 22 in the interior, it follows a zig–zag way and moves two positions in the x and y direction during a braiding cycle. For carrier 62 on the surface, it moves from the main-part region into the surface and then returns back, holding its position at step J–J0 . For carrier 66 at the corner, it comes into the corner region from the interior, shifts its position according to the braiding process and holds its position at step N–N’ and O–O’, afterward re-enters to the interior. Due to the ‘jamming’ action applied by yarn tension, the yarns will reposition after a braiding cycle, hence, the yarn paths in the x–y plane are different from the traces of carriers. Considering the carrier motion positions as the control points of the yarn path and on the basis of least square principle, the path of braiding yarn in the x–y plane is fitted to be the connecting line of the middle
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Fig. 1. Scheme of 3D four-step braiding process.
Fig. 2. Yarn moment traces in x–y plane: (a) carriers’ movement traces, (b) x–y plane projection of all traces.
points of the carrier positions during the braiding process, as lines A0 E0 , Q–R–S and L–U–V–W in the three regions shown in Fig. 2a. Furthermore, all the yarn paths projected in the x–y plane are
presented in Fig. 2b. The angle between the projection of braiding yarn in x–y plane and y direction of the preform, u, is called as horizontal orientation angle. Ideally, u = ± 45°.
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3.2. Spatial yarn paths
4. Parameterized structure models of three unit-cells
From the above analysis, it is seen that the carrier’s motion in the interior, surface and corners is different, which leads to unique yarn configuration in each of these regions. However, for every motion step of the carriers, braiding yarns in any regions expand h/4 along z direction simultaneously. The spatial motion trace of carrier 22 experiences four steps from position point A–E, as shown in Fig. 3a. After the jamming action, the yarn axis becomes straight, as depicted by line A0 E0 . The interior yarn paths are all located in two families of orthogonal planes orientated ±45° with respect to the preform surfaces. Moreover, the angle between every interior yarn axis and the z direction is identical, defined as interior braiding angle c. Following the motion trace of carrier 62, it undergoes five steps from point F–K. Among the five steps, carrier stops moving at step J–J0 , which determines the spatial configuration of surface yarn dissimilar to the interior. The spatial motion process is shown in Fig. 3b. After yarn compacting, the actual path of yarn is a spatial curve. To facilitate the modeling, some simplifications on the spatial configuration of surface yarn are made. In this paper, path of surface yarn is modeled as two straight lines QR and RS. The angle of line QR with respect to z direction is identified as surface braiding angle h, as displayed in Fig. 3b. Following the motion trace of carrier 66, it moves eight steps from point L to P, holding its position at step N–N0 and O–O0 . The spatial motion diagram is illustrated in Fig. 3c. Obviously, the spatial configuration of corner yarn is distinct to the spatial configurations of the interior and surface. For the sake of simplicity, the curved path of corner yarn upon jamming is considered as three straight lines: TU, UV and VW, as presented in Fig. 3c. The corner braiding angle is measured from the direction of TU to z axis, denoted by b.
4.1. Basic assumptions
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The following assumptions have been made to establish the solid structure models of three unit-cells of 3D braided composites: 1. The mutual squeezing state of interior and exterior braiding yarns is different. The cross-section of interior braiding yarns is elliptical and the relationship between the major pffiffiffi and minor radii of the ellipse, a and b, is expressed as a ¼ 3b cos c. The cross-sections of surface and corner yarns are also elliptical, but the cross-sectional area reduces the corresponding proportion /f and /c respect to interior. 2. After yarns jamming action, the interior braiding yarn keeps straight and the surface and corner yarns are composed of straight segments in the space. 3. The cross-section shape of braiding yarn changes gradually from the surface and corner regions to the interior. 4. The braiding procedure maintains relatively steady and the braided structure is uniform, at least in a certain length of interest. 4.2. Geometrical models of three unit-cells Fig. 4 shows the topology geometrical models of unit-cells in the three distinct regions of 3D braided composites. Note that these models just illustrate the topology relationships of braiding yarns, without considering their physical cross sectional dimensions. The braiding yarns inside the interior unit-cell which forms the main-part of the composites are inclined in four directions; it is the reason why the composites are called the 3D four-directional braided composites.
3.3. Selecting of three unit-cells
4.3. Structural parameters of the three unit-cells
For studying the mechanical properties of composites with periodic microstructure, a representative volume element is usually picked out and analyzed instead of the whole composites structure. Such a representative volume element, the smallest region that can produce the whole structure by translating its copies, is called a unit-cell. It is observed that 3D braided composite is composed of three different regions with different yarn configurations. Therefore, three unit cells are selected in this paper, as shown in Fig. 2b. These unit cells are oriented in the same reference plane, the cross-section of the preform.
According to the spatial configuration of the braiding yarns in each unit-cell and the basic assumptions stated earlier, the structural parameters of the three unit-cell models can be obtained. They are favorable to calculate the fiber volume fraction in composite structures. Furthermore, they are the basis to establish parameterized solid structure models of three unit-cells. As displayed in Fig. 4, the width and thickness of the three unitcell models are given by:
pffiffiffi pffiffiffi W i ¼ 4 2b T i ¼ 4 2b
Fig. 3. The spatial traces of yarns moment: (a) interior region, (b) surface region, and (c) corner region.
ð2Þ
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Fig. 4. Geometrical models of three different unit-cells, (a) interior unit-cell, (b) surface unit-cell, (c) corner unit-cell.
pffiffiffi pffiffiffi pffiffiffiffiffi W s ¼ 4 2b T s ¼ 2 2b þ /s b
ð3Þ
pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi W c ¼ 2 2b þ /c b T c ¼ 2 2b þ /c b
ð4Þ
V sf ¼
3pabh 2
ei þ/s js 2 cos c
s js þ /cos h
ð13Þ
Us
where sub-index i, s and c refer to the interior, surface and corner unit-cells, respectively. Meanwhile, the three unit-cells have the same height, namely:
where Us, Ys and Vsf are volume of surface unit-cell, volume of braiding yarns in surface unit-cell and fiber volume fraction in the surface region, respectively. And for corner unit-cell, one has:
h ¼ 8b= tan c
pffiffiffiffiffiffiffiffi 3 U c ¼ ð64 þ 8/c þ 32 2/c Þb = tan c
ð5Þ
The braiding angle is an important structural parameter for the 3D braided composites. Generally, the interior braiding angle is difficult to measure since the interior of the preform cannot be observed without cutting the specimen. However, the angle of inclination of the yarns on the surface of composite can be measured readily. The relationship between the interior braiding angle c and the braiding angle a on the surface of composite is expressed as:
pffiffiffi tan a ¼ ð 2=2Þ tan c
ð6Þ
Using trigonometry, the surface and corner braiding angle can be calculated by:
tan h ¼ tan b ¼ b=ðð3=8ÞhÞ ¼ tan c=3
ð7Þ
Eq. (7) shows that the surface braiding angle and corner braiding angle are equal, and smaller than the interior braiding angle. That is to say, the yarns are closer to z direction in the exterior region of the composites, indicating that the mutual squeezing of yarns is more severe. 4.4. Fiber volume fraction As shown in Fig. 4, the volume of interior unit-cell is: 3
U i ¼ 256b = tan c
ð8Þ
The total volume of braiding yarns in this unit-cell model is:
Y i ¼ 4pabh= cos c
ð9Þ
Therefore, the fiber volume fraction in the interior region is written as:
V if ¼ Y i ji =U i ¼
pffiffiffi 3pji =8
Ys ¼
3pabh 1 þ /s / þ s 2 2 cos c cos h
pabh 1 þ /c 3/c þ 2 2 cos c cos b
V cf ¼
ð14Þ
pabh ei þ/c jc 2 2 cos c
c jc þ 3/ cos b
ð15Þ
Uc
ð16Þ
where Uc, Yc and Vcf are volume of corner unit-cell, volume of braiding yarns in corner unit-cell and fiber volume fraction in the corner region, respectively. In Eqs. (10), (13), and (16), ji, js and jc are yarn packing factor of the interior, surface and corner regions, respectively. For the whole structure of 3D braided composites, the total fiber volume fraction, Vf, is obtained by weighted averaging method, namely:
V f ¼ V i V if þ V s V sf þ V c V cf
ð17Þ
where Vi, Vs and Vc are volume proportions of the interior, surface and corner regions to the whole structure of composites and are presented as follows:
2ðm 2Þðn 2Þ pffiffiffi V i ¼ pffiffiffi ð 2m þ 1Þð 2n þ 1Þ
ð18Þ
pffiffiffi ð4 þ 2Þðm þ n 4Þ pffiffiffi V s ¼ pffiffiffi ð 2m þ 1Þð 2n þ 1Þ
ð19Þ
Vc ¼ 1 Vi Vs
ð20Þ
4.5. Solid structural models of three unit-cells
ð10Þ
Similarly, for surface unit-cell, one has:
pffiffiffiffiffiffiffiffi 3 U s ¼ ð128 þ 32 2/s Þb = tan c
Yc ¼
ð11Þ ð12Þ
The solid structural models of these unit-cells are established using TexGen which is open source software developed at the University of Nottingham for geometric modeling of textile structures. TexGen provides a friendly graphical user interface (GUI) through which many functions can be utilized. For more advanced functions, using of scripts written in Python is recommended.
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Fig. 5. Structure models of three different unit-cells, (a) interior unit-cell, (b) surface unit-cell, and (c) corner unit-cell.
Fig. 6. Parameterized model of interior unit-cell: (a) c = 20°, (b) c = 30°, and (c) c = 40°.
The path of braiding yarn is the yarn’s centerline in 3D space and is represented through a series of discreet control points coupling with a suitable interpolation function in TexGen. In this paper, the intersection points of yarns’ straight segments and some other points in yarn’s straight centerline are selected as control points, and the linear interpolation function is adopted. It is known that the squeezing states of yarns in the three regions of the composites vary. Thus, the shape reduction coefficients /s and /c are both set to 0.9. The solid structure models of three unit-cells are shown in Fig. 5. By analyzing the relationship between microstructure parameters of three unit-cells, the input parameters to generate the parameterized unit-cell models are identified as the minor radii of the yarn cross-section, the braiding angle, and the cross-section shape reduction coefficients. Other parameters can be determined by these parameters. Parameterized unit-cell models are beneficial to parameters modification and model regeneration. As an example, Fig. 6 illustrates the solid models of interior unit-cell with the same yarn cross-section size but different braiding angles.
4.6. Visualization of the internal microstructure of three unit-cell models The microstructure of 3D braided composites is complex and cannot be observed from the outside of the specimen. To investigate the internal microstructure, one obvious way is to cut the specimen but is costly and difficult for real materials. With the parameterized unit-cell models established in this paper, cutting them along different cross-sections allows one to observe and visualize the internal microstructure. This helps us to understand the spatial configuration of the unit-cells. Fig. 7c shows the multiple unit-cells model of the interior cutting with ±45° respect to the boundary. In comparisons, the yarn intersection condition in the preform interior concluded by Wang and Wang [5] is depicted in Fig. 7a and b. Obviously, the yarn topology is validly reflected in the unit-cell model. Furthermore, Fig. 8 gives the comparison of 45° cross-section of interior unit-cell and SEM micrograph. It can be seen that the yarn cross-section shape, the arrangement of braiding yarns and space distance
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Fig. 7. Yarn topology in the preform interior region: (a) two families of plates [5], (b) yarn formation and plate intersection [5], and (c) ±45° cross-section of interior unit-cell model.
Fig. 8. Comparison of cross-section of interior model and SEM micrograph: (a) SEM micrograph [6], (b) cross-section of Interior model.
between the yarns are almost identical with the experimental observations. This indicates that the established interior unit-cell model is correct. Due to the absence of experimental results, the internal microstructures of surface and corner unit-cells are not given. 5. Finite element models 3D braided composites are composed of braiding yarns and the resin matrix pockets. Consequently, the mechanical behavior of the braided composites is determined by the elastic constitutive relationships of the constituents. In this work, the resin matrix is assumed to be isotropic; the braiding yarns containing thousands of fibers and matrix are modeled as transversely isotropic and unidirectional composites in local coordinate system. The stiffness properties of the braiding yarns are obtained using the micromechanics formulae given by Chamis [18]. For the local coordinate definition of yarn in a specify orientation, local 1-axis follows the yarn path line and local 3-axis is in the upright plane perpendicular to the x–y plane of the global coordinate system. 5.1. Boundary conditions If the composite materials are regarded as perfect periodic structures consisting of periodic array of unit-cells and the analysis is based on the unit-cell modeling, the periodic boundary conditions should be applied to replicate its repeating nature. Furthermore, displacement continuity condition and traction continuity condition must be satisfied at the opposite boundaries of the neighboring unit-cells. As proposed by Suquet [19], the displacement field for the periodic structure is:
ui ¼ eik xk þ ui
ð21Þ
In the above, eik are the average strains of the unit-cell, xk is the Cartesian coordinate of a unit-cell point and ui is the periodic part of the displacement components on the boundary surfaces. Eq. (21) is difficult to be imposed to the unit-cell boundaries because of the unknown periodic part. Fortunately, for most unitcells, the boundary surfaces always appear in parallel pairs. The displacements on a pair of parallel opposite boundary surfaces (with their normal along the Xj axis) can be given as [20]:
uijþ ¼ eik xkjþ þ ui
ð22Þ
uij ¼ eik xkj þ ui
ð23Þ
where the index ‘‘j+’’ means along the positive Xj direction and ‘‘j’’ means along the negative Xj direction. Since ui is the same at the two parallel boundaries, the difference between the above two equations is:
uijþ uij ¼ eik ðxkjþ xkj Þ ¼ eik Dxkj
ð24Þ
The right side of the above Equation becomes constant once eik is specified, because Dxjk are constants for each pair of boundary surfaces. Thus Eq. (24) can be rewritten as [20]:
uijþ ðx; y; zÞ uij ðx; y; zÞ ¼ cij
ði; j ¼ 1; 2; 3Þ
ð25Þ
The above equations do not contain the periodic part of the displacement components and can be carried out easily in the finite element analysis by setting the nodal displacement linear constraint equations. It was proved by Xia et al. [20] that the application of this boundary condition ensures not only the displacement continuity conditions, but also the traction continuity conditions.
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For the finite element modeling of interior unit-cell, it has periodicity in all the three directions in space, therefore, the deformation of its opposite paired faces is completely governed by Eq. (25). It is, however, that the surface unit-cell has periodicity only in two directions (x and z) and the corner unit-cell is periodical just in direction z. This is because that the outer boundaries of both surface and corner unit-cells are free. To take account into the aperiodicity of surface and corner unit-cells in the finite element simulation, the internal connected face is completely fixed in its normal direction, and the outer free face is forced to deform as a plane parallel to the connected face. Certainly, the periodical boundary condition should be applied to the pair of faces with periodicity. 5.2. Mesh discretization Due to the complicated material arrangements of 3D braided composites, mesh generation may be a large obstacle for predicting mechanical properties by finite element method. In this study, the 3D voxel-based meshing technique available in TexGen is adopted for the discretization of yarns and matrix. This technique allows the mesh generation automated but the element faces can only match the geometry boundaries of the constituents in the unit-cell approximately. However, the voxel meshing technique is sufficient for stiffness prediction of textile composites and even suitable for damage simulations with the usage of adaptive mesh refinement proposed by Kim and Swan [21]. In this paper, the 8-nodes reduced integration element (C3D8R) available in ABAQUS is used for mesh discretization. In order to guarantee the convergence of the solution, the analysis of mesh sensitivity is conducted. Smaller meshing size can provide more accurate stress distributions, but the computer memory increase sharply. If only the global stiffness is concerned, relative coarse mesh size can still give satisfactory results. According to a compromise consideration, the mesh size of the models in this study is fine enough to provide the stress state and predict stiffness properties.
In order to calculate the elastic constants of the three unit-cells, the homogenization method is employed in this study. The heterogeneous composites in the micro-scale are considered as a homogeneous material in the macro-scale. For each unit-cell, the constitutive relationship is:
ð26Þ
i and ej are the global where Cij is the effective stiffness matrix. r average stress and global average strain defined by:
r ij ¼ eij ¼
1 V 1 V
Pn e
Z
e k¼1 V k
V
Z V
ðreij V ek Þ
k¼1 rij dV ¼ P ne
Pn e
ðeeij V ek Þ
k¼1 eij dV ¼ P ne
e k¼1 V k
ðno summation over eÞ
ðno summation over eÞ
ð27Þ
ð28Þ
where reij and eeij are the stress and strain of element. V ek , ne and V are element volume, total number of element and volume of the unitcell, respectively. ¼ ½r 11 ; r 22 ; r 33 ; r 23 ; r 13 ; r 12 T and ½e ¼ ½e11 ; e22 ; e33 ; e23 ; Let ½r e13 ; e12 T . In the 3D case, give six independent sets of global strain vector in the finite element analysis and obtain six sets of global stress vector. Then, the effective compliance matrix, Sij, the inverse of effective stiffness matrix, can be worked out by:
1 Sij ¼ ei r j
¼ ½C
c X V n ½C ij n
ðn ¼ i; s; cÞ
ð30Þ
n¼i
Similarly, sub-index i, s and c refer to the interior, surface and corner unit-cells; Vn are the volume of the three unit-cells respectively and have been given in Eqs. (18)–(20). Correspondingly, the compliance matrix of the whole composite 1 ; and then the effective structure can be calculated by ½ S ¼ ½C elastic constants of 3D braided composites can be obtained. 6. Results and discussions 6.1. Comparison with experimental data In order to verify the effectiveness of the proposed three unitcells model, predicated results are compared to the experimental data cited from Ref. [23]. The elastic properties of the component materials, including AS4 carbon fiber and epoxy resin, are listed in Table 1. The geometric parameters of the specimens and the structure parameters of three unit-cell models used in the calculation are summarized in Table 2. Tables 3 and 4 present the measured moduli in longitudinal and transverse directions against the predicted results from Weighted Average Model (WAM) of Kalindidi and Abusafieh [23], a theoretical model based on a single idealized unit-cell, and from the proposed finite element model of three unit-cells. It can be found that the predicted results by the two methods agree well with experimental data. Furthermore, the proposed method can give accurate micro-stress distribution and deformation of the three unit-cells. 6.2. The effect on elastic properties of surface and corner unit-cells
5.3. Calculation of effective elastic constants
r i ¼ C ij ej
The effective elastic constants of each unit-cell can be obtained from the calculated compliance matrix. Details on this procedure can be found in Ref. [22]. Next, stiffness matrix from each unit-cell is summed up to calculate the stiffness matrix of the whole structure of braided composites by volume averaging method, namely:
ð29Þ
Most often, only the interior unit-cell is concerned for the prediction of stiffness by numerical method previously. However, the real 3D braided composite usually has three different unit cells, namely, the interior, surface and corner unit cells. Table 5 displays the predicted elastic constants for each unit-cell and the overall composite structure (specimen number 4). For specimen 4, there are 117,936 nodes and 110,250 elements in the interior unit-cell, 68,796 nodes and 63,000 elements in the surface unit-cell, and 116,281 nodes and 108,000 elements in the corner unit-cell, respectively. Surely, the meshing of corner unit-cell is refined here. From Table 5, it is seen that the interior unit-cell is perfect transversely isotropic, but the surface and corner unit-cells are only approximate transversely isotropic. In the calculation of stiffness properties of the whole composite, both the surface and corner unit-cells are also regarded as transversely isotropic by averaging operations. That is, the 3D braided composites are considered to be transversely isotropic macroscopically. It is also seen that the surface and corner regions are much stiffer than the interior region. Since the braiding pattern of the specimens in the analysis is
Table 1 Elastic properties of AS4 carbon fiber and epoxy resin. Material
E11 (GPa)
E22 (GPa)
G12 (GPa)
G23 (GPa)
l12
AS4 Epoxy
234.6 2.94
13.8 2.94
13.8 1.7
5.5 1.7
0.2 0.35
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Table 2 The geometric parameters of the specimens and the structure parameters of three unit-cell models. Specimen number
Geometric parameters of specimens
Structural parameters of three unit-cell models
a
Vf
mn
c (°)
h (mm)
ji
Wi (mm)
Tf = Wc (mm)
0.38 0.40 0.46 0.44 0.29
12 12 12 12 12 12 12 12 12 12
23.38 23.38 27.24 29.74 33.40
5.63 5.46 4.35 4.03 4.37
0.54 0.57 0.64 0.63 0.43
1.72 1.67 1.58 1.63 2.04
1.14 1.11 1.06 1.09 1.36
(°) 1 2 3 4 5
17 17 20 22 25
Table 3 Comparison between experiment data and predicted results of WAM [23] and present finite element model in longitudinal direction. Sample number
Experimental data (GPa)
WAM (GPa)
FEM (GPa)
1 2 3 4 5
43.6 ± 1.9 45.9 ± 1.2 48.0 ± 3.0 38.5 ± 2.4 21.2 ± 1.8
48.3 50.7 47.0 38.6 21.6
48.47 50.94 46.34 40.19 22.54
Table 4 Comparison between experiment data and predicted results of WAM [23] and present finite element model in transverse direction. Sample number
Experimental data (GPa)
WAM (GPa)
FEM (GPa)
1 4
6.21 ± 0.41 6.02 ± 0.30
5.74 6.21
5.56 6.03
Table 5 Predicted values of elastic constants for each unit-cell and the overall composite structure.
Interior Surface Corner Overall
Ex (GPa)
Ey (GPa)
Ez (GPa)
Gxz (GPa)
Gyz (GPa)
Gxy (GPa)
lxz
lyz
lxy
5.77 6.30 6.51 6.03
5.77 6.42 6.41 6.03
36.95 42.45 59.49 40.19
9.62 6.21 3.65 8.22
9.62 6.34 3.53 8.22
3.35 2.37 2.18 2.97
0.81 0.56 0.42 0.72
0.81 0.51 0.44 0.72
0.33 0.35 0.35 0.34
guarantee the displacement continuity between the neighboring unit-cells and provide a reasonable stress distribution. Note that the warping extents of the three sets of opposite boundary faces are various under all the loading cases. These are attributed to that the unit-cell model of 3D braided composites does not have the symmetries of geometrical structure and physical properties. In Fig. 9b, the surface unit-cell does not have periodicity in the y direction. The opposite boundary face in y direction is confined to deform in a plane. In Fig. 9c, the corner unit-cell only has periodicity in z direction. Therefore, the opposite boundary faces in the x and y direction are confined to deform in a plane under z tension. However, for xy shear or xz shear, the plane deformation constraint in x direction is removed in order for the implementation of shear deformation. Fig. 10 illustrates the stress nephogram of the three unit-cells on un-deformed shape. In Fig. 10a, the traction continuity at the opposite boundary faces of the interior unit-cell has been guaranteed and satisfied the periodic boundary condition. Under z tension, the stresses in the braiding yarns are almost 15 times than that in the matrix pocket region on the average. While under xy shear and xz shear, the stresses in the yarns become 7 and 9 times approximatively than that in the matrix. Obviously, the mechanical response characteristic of the model subjected to different loadings varies, since the load carried by yarns is different. In Fig. 10b and c (the matrix pockets have been removed for clarity), the stresses in the yarns of surface and corner unit-cells are higher than that of interior unit-cell. This is because that the braiding yarns are closer to z direction in the exterior regions and the yarns will bear more loads. Moreover, in these regions, the squeezing state of yarns is more severe and the fiber volume faction is larger than it in interior region. The stress concentration is a concerned issue for composite designers. As shown in Fig. 10, for interior unit-cell, stress concentration occurs in the contact area between yarns and matrix under tension load, while the stress concentration is more obvious and arises in the interlaced region of braiding yarns under shear load. For surface and corner unit-cell, the braiding yarns are curved in these regions. They are inclined to be straight under tension and shear loads, and therefore the stress peaks occur in the squeezing positions. Obviously, the contact zones between yarns are the significant parts in the strength analysis problems since micro-cracks and damage always initially generated in these regions. 6.4. Parametric study on the effective elastic properties
[12 12], thus the interior, surface and corner unit-cells account for 62.36%, 33.22% and 4.42% of the whole structure, respectively. Conclusion may be drawn that even though the interior unit-cell comprises the main-part of the composite structure, consideration of the effect of surface and corner unit-cells is also important, especially when the number of rows and columns of carriers are relatively small. 6.3. Deformation and stress distributions of unit cells The advantage of the proposed modeling strategy is that the detailed information on the deformation and stress distribution in the 3D braided composites under different loadings can be obtained. Three typical loading cases, namely, z tension (e33 ¼ 0:002), xy shear (e12 ¼ 0:002) and xz shear (e13 ¼ 0:002), are selected to demonstrate the mechanical behavior of 3D braided composites (specimen number 4). Fig. 9 depicts the deformation of the three unit-cells when the deformation scale factor is 300. From Fig. 9a, it can be observed that the boundary faces are no longer plane and warped. The two opposite boundary faces have the same deformation which can
Braiding angle and fiber volume fraction are two important structural parameters which directly control and influence the overall mechanical properties of the 3D braided composites. In this section, the effects of the two parameters on the effective elastic properties are studied by using the proposed modeling strategy. Note that the change of fiber volume fraction of the models with a certain braiding angle is realized by setting different yarn packing factors within a reasonable range. Fig. 11 shows the variation of the predicted elastic constants with the braiding angle for three different fiber volume fractions. It is clear seen from Fig. 11a that the elastic modulus Ez decreases sharply as the braiding angle a increases. This is because that with the increasing of a, the braiding yarn is inclined to deviate from the z axis thus causes the decreasing of rigidity in the z direction. Meanwhile, with the increase of the fiber volume fraction Vf, the elastic modulus Ez increases and the increment is relatively significant when a is small. Fig. 11b shows that the elastic moduli, Ex and Ey (Ex = Ey), increase gradually as the braiding angle increases. When a is greater than 30°, the increment becomes obvious. With the increase of the fiber volume fraction, the elastic modulus Ex and Ey increase in a similar tendency. From Fig. 11c, it can be found
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Fig. 9. Deformation of the finite element models: (a) deformation of the interior unit-cell under z tension, xy shear and xz shear load, (b) deformation of the surface unit-cell under z tension, xy shear and xz shear load, and (c) deformation of the corner unit-cell under z tension, xy shear and xz shear load.
that the transverse shear modulus Gxy increases monotonically with the increasing of a. When a is less than 25°, the increase is not obvious, but greater than that, Gxy increases much rapidly. This
is mainly due to the simultaneous increasing reinforcement of the braiding yarns in the x and y direction with the increase of a. Similarly, with the increase of Vf, Gxy also increases in a similar
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Fig. 10. Stress nephogram of the finite element models, (a) stress nephogram of the interior unit-cell under z tension, xy shear and xz shear load on un-deformed shape, (b) stress nephogram of the surface unit-cell under z tension, xy shear and xz shear load on un-deformed shape, and (c) stress nephogram of the corner unit-cell under z tension, xy shear and xz shear load on un-deformed shape.
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Fig. 11. Variation of effective elastic constants with braiding angle and fiber volume faction.
tendency. Fig. 11d describes that the longitudinal shear moduli, Gxz and Gyz (Gxz = Gyz), increase steadily as a and Vf increase. When a is small, the increase is significant. When a is greater than 25°, the rate of increment slows down, but the change of Gxz and Gyz caused by the increment of Vf becomes comparatively larger. Fig. 11e and f show that a and Vf also affect the Poisson’s ratios, mxy and mxz (mxz = myz). It can be found that a is the main factor that affects Poisson’s ratios. When a increase, mxy firstly decreases and then increases, it reaches the minimum value at a = 30°; mxz and myz firstly increase and then decrease, they reach the maximum value at a = 25°. As Vf increases, the Poisson’s ratio mxy decreases for a less than 25°and then increases for a greater than 25°, however, the Poisson’s ratios mxz and myz, always increase. Generally, the effect of the fiber volume fraction on the Poisson’s ratios is relatively weak. 7. Conclusion The mechanical behavior of the 3D four-step braided composite is investigated. The topology geometrical models in the interior,
surface and corner regions of the preform are established. And then, the spatial configuration of the braiding yarns as well as the squeezing conditions in different regions is considered to establish the parameterized three unit-cell structure models. Based on these structure models in conjunction with finite element method, the mechanical properties of 3D braided composites are predicted. It is found that the three regions have distinctive properties and the surface and corner unit-cells are stiffer than the interior region. Hence, the surface and corner regions should be considered in order to predict the elastic properties more precisely. Good agreement is obtained between the calculated results and experimental data. Thus the proposed simulation method is validated. Since the deformation and stress distribution in the unit cells can be obtained, the proposed method is useful for a better understanding of mechanical response of constituent materials under various loading cases. Furthermore, the effects of braiding angle and fiber volume fraction on the effective elastic constants of 3D braided composites are discussed in detail. The results may be helpful for designers to optimize the braided composite structures.
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