Key issues in microstructure modeling of 3D braided composites

Numerical analysis based on micromechanics is an effective method to study the mechanical properties of three-dimensional (3D) braided composites, in which the establishment of micromechanics model is the basis of mechanical analysis. The direction cosine of fiber yarns and the characterization of geometric parameters of microstructure are two key issues in the modeling of microstructure. Focusing on these two key issues, the three-cell model of 3D four-directional braided composites are studied to comprehensively analyze various spatial directions of fiber yarns in the 45° division and horizontal division of unit cell, so as to obtain the detailed fiber yarn direction cosine. Aiming at the cross sections of three types of fiber yarns, the complex relationship between the braiding process parameters and the geometric parameters of the unit cell is analyzed and deduced. The unit cell structure is quantitatively characterized by using the braiding process parameters as the initial parameters, and the calculation methods of the volume fraction of fiber yarns and yarn filling coefficient are obtained. Finally, the predicted elastic constants by stiffness-volume averaging method are compared with experimental results, demonstrating the analysis results of fiber yarns directions. This paper can provide a theoretical reference for the microstructure modeling of 3D multi-directional braided composites, and has certain practical value in engineering.


Introduction
3D braided composites are formed by reinforcing long fiber braid into preforms and solidified with the matrix materials, which have high specific strength, high specific modulus, impact resistance, fatigue resistance, optimal structural integrity and strong design ability [1,2]. The fiber yarns of 3D braided materials intertwine with each other with complex spatial topology, and its microstructure has good periodicity. The numerical analysis method based on micromechanics has become the main method to study the mechanical properties of 3D braided composites [3][4][5][6][7][8], that is, to study the interaction of material components from the mesoscopic scale, and to establish the quantitative relationship between effective properties of 3D braided materials, component properties and structure parameters of mesoscopic model. In-depth analysis of yarn interweaving mode and microstructure characteristics, and the establishment of reasonable and effective mesoscopic model are the basis for the analysis of its mechanical properties, which have always been one of the key contents of 3D braided composites research.
Several representative mesoscopic models for the spatial topology of 3D braided composites, such as fiber inclination model [9], 'M' type unit cell model [10], laminate model [11] and three-cell model [12][13][14] have been successively proposed. The three-cell model is composed of interior cells, surface cells and corner cells with periodic characteristics, which can truly reflect the micro-structure and also is widely used [15][16][17][18][19].
There are two key problems in the three-cell model: the direction cosine of fiber yarns and the characterization of the geometric parameters of fiber yarns. 3D braided composites are both a kind of material and a structure. Through designing the material properties of components and the spatial direction of fibers, it can obtain the mechanical properties of nearly isotropic or better mechanical properties in a certain direction.
The transverse isotropy and the braiding direction of fiber yarns are the main factors that make the 3D braided materials anisotropic. For the prediction of the mechanical properties of 3Dcomposites, whether it is the definition of the material properties of the fiber yarn model in the finite element analysis or the coordinate transformation of the stiffness matrix of fiber yarns in the theoretical analysis method, the direction cosine of fiber yarns is involved. Determining the spatial direction of the fiber yarn is the premise for determining the direction cosine. On the other hand, in the case of perform molding or material loading, fiber yarns will be twisted and deformed [20], affecting the macroscopic mechanical properties of the material [8,16,21] and the micro-stress state [15,[22][23][24]. Only properly characterizing this deformation and establishing the quantitative relationship between the geometric parameters of fiber yarns and the braiding process parameters, can the actual mechanical state of the material be more truly reflected. In particular, it is of practical value to establish the model by using the process parameters as the initial parameters.
Taking the three-cell model of 3D four-directional braided composites as the object, the definition of fiber yarn spatial direction and the characterization of geometric parameters of unit cell are studied. The spatial direction of fiber yarns in the three-cell model is analyzed from the 45°scheme [21] and horizontal scheme [7,17,21,22] of periodic division, and the direction cosines of fiber yarns are given. In addition, according to three cross-section shapes of fiber yarns, the complex relationship between the braiding process parameters and the unit cell parameters in the two unit cell division schemes is analyzed and derived. Taking the geometric dimensions of the fiber yarn cross-section and the braiding angle as the characteristic parameters, the parametric modeling of the unit cell is realized.

Spatial direction of fiber yarns and their direction cosines
The braiding process determines the fixed braiding direction of fiber yarns of 3D braided materials, and the spatial direction of fiber yarns determines the effective direction of the reinforcement. Since the local coordinate system of fiber yarns does not coincide with the overall coordinate system of the unit cell [10,25], when calculating the micro elastic constants of 3D four-directional braided composites using the stiffness-volume averaging method or homogenization theory, the local stiffness matrix C of fiber yarns needs to be transformed into the overall coordinates. Then the global stiffness matrix ¢ C is obtained, ¢ = s s C T CT .
T s T is the stress space transformation matrix, which is composed of direction cosines of fiber yarns in every direction [5,21]. Determination of the spatial direction and its direction cosine of fiber yarns is the basis of effective mechanical property analysis.
The periodic division of unit cells includes 45°division and horizontal division, as shown in figure 1. In the two division schemes, the directions of fiber yarns in the three unit cells are different, as shown in figure 2 and figure 3. Taking the unit cell as the overall coordinate system, the z-direction is specified as the braiding direction, that is, the direction of cell height; taking the fiber yarn as the local coordinate system, the fiber yarn is a transversely isotropic material [19]. As a result, axis 1 is defined as the fiber yarn axis, that is, the main direction of the material, and axis 2 and axis 3 are transverse and perpendicular to axis 1. The included angle between the fiber yarn axis in the interior cell, surface cell and corner cell, and the z-axis of the global coordinate system is used to characterize the spatial direction of the fiber yarn. The cosine of the included angle is called the direction cosine. The volume of the interior cell and the surface cell usually accounts for more than 90% of the total volume of the three unit cells. This paper mainly focuses on the fiber yarn direction of the interior cell and the surface cell.   Figure 4 is the solid model of interior cell and its fiber yarns. There are 12 straight fiber yarns in the model, which have four directions, as shown in figure 5, and the braiding angle of interior cell g is defined as the angle between the yarn axis and the z-axis. The direction cosines of the four directions in table 1 can be obtained through spatial geometric projection analysis.

Surface cell
The distribution of fiber yarns in the surface cell is shown in figure 2(b). Two fiber yarn axes are helical [7], and the braiding angle q of the surface cell is defined as the angle between the tangent line of the helix and the z-axis. Since the fiber yarn axis is a curve, the elastic constant of fiber yarns under the overall coordinate system is theoretically continuous. This is complex for the definition of the material properties of fiber yarns in coordinate transformation and finite element modeling. To this end, two straight lines are used to fit for each spiral axis, as shown by the red dotted line in figure 2(b). Figure 6 is the solid model of surface cell and its fiber yarns. There are two straight fiber yarns extending from the interior cell in the surface cell, and the fiber yarns in the surface cell have six directions, as shown in figure 7. Directions I and IV are straight fiber yarns extending from the interior cell, and directions II, III, V and VI are four straight fibers used to fit two spiral fibers respectively. The included angle between the projection of fiber yarns on the xoy plane and the positive direction of the x-axis is j = 45°( determined according to the yarn carriers movement).The projection method is used for spatial geometry analysis, and finally the direction cosines of six directions in the surface cell are obtained, as shown in table 2. The cosines of each direction are related to the braiding angles g and q.

Corner cell
For 45°division, there is only one fiber yarn in the corner cell and its axis is helical [7], as shown in figure 2(c). The braiding angle b of the corner cell is defined as the angle between the tangent line of the helix and Z-axis. Three straight lines are used to fit for the spiral axis, as shown the red dotted line in figure 2(c). Accordingly, the fiber yarns in the corner cell have three directions, as shown in figure 8. Through the projection method, the direction cosines of three directions in the corner cell are obtained as shown in table 3, and j is 45°.

Horizontal division scheme 2.2.1. Interior cell
The spatial direction of fiber yarns of the interior cell in the horizontal division scheme is shown in figure 3 (a). Figure 9 is the solid model of the interior cell and its fiber yarns. It has 12 straight fiber yarns, with four directions, as shown in figure 10. The direction cosines of fiber yarns in four directions are shown in table 4, where j is 45°.

Surface cell
For horizontal division, the spatial direction of fiber yarns in the surface cell is shown in figure 3(b). Similarly, each spiral is replaced by two straight segments, as shown in figure 3(b), and its solid model and fiber yarn model are shown in figure 11. There are eight directions in the surface cell, as shown in figure 12. Directions II, III, VI and VII are the same as directions II, III, V and VI of surface cell in 45°division respectively. Directions I, IV, V and VIII are straight fibers extending from the interior cell, and their directions are respectively directions IV, II, I and III in figure 10. The same analysis method as that used in the 45°division method is used to obtain the direction cosine values of 8 directions, as shown in table 2 and table 4.

Corner cell
For horizontal division, the spatial direction of fiber yarns in the corner cell is shown in figure 3(c). Similarly, each spiral is replaced by three straight segments, as shown in figure 3 q j cos cos q j cos sin q sin Table 3. Direction cosine of fiber yarn in the corner cell in 45°d ivision mode.
x y z

Comparison of two division methods
Compared with the 45°division method, the volume of interior cell, surface cell and corner cell in horizontal division is small. If the mesh size is the same, the number of meshes in the finite element model is smaller with small calculation. However, in the horizontally divided unit-cells, there are more small volumes in the fibrous bundle solid model, and each small block needs to define its directional cosine separately; the number of fibrous bundle directions in the horizontally divided surface and corner cells is more than the number of 45°divisions, both of which increase the workload of solid modeling. Nevertheless, 45°division will lead to some small areas at the corners of the braided material cannot be covered by unit-cells [7].
Both the 45°division and the horizontal division are periodic, which will not have a substantial impact on the prediction of the macroscopic mechanical properties of materials.

Parametric modeling of unit cells for three fiber yarn cross-sections
The braiding parameters of 3D four-directional braided composites mainly include the diameter d f of fiber monofilament, the fiber monofilament number N f in the fiber yarns, the braiding angle a and the fiber volume fraction V . f For the three-cell model, the fiber volume fraction V f can be expressed as [8]: where V , i V s and V c are respectively the volume ratio of interior cell, surface cell and corner cell accounting for the whole three unit cells model, and can be calculated by the size of woven fabric [20,21]. V , if V sf and V cf are the fiber volume fractions of interior cell, surface cell and corner cell, respectively. It is generally considered that the yarn-packing factor e in three unit cells is the same, then: where V , iy V sy and V cy are the volume fractions of fiber yarns in the interior cell, surface cells and corner cells, representing the total volume of fiber yarns in an unit cell to the volume of its unit cell.
The above process parameters are characterized by the unit cell structure parameters, and the relevant parameters are:   (3) Cross-section geometric dimension of fiber yarns. In the three-dimensional braided structure, the fiber yarns are kinked and squeezed against each other, resulting in non-circular cross-section shapes. The fiber yarn cross-section is mainly assumed to be oval [16,17,20,22,26,27], hexagonal [9,24,28], or octagonal [3,6,8,18,23,25,29]. In recent years, numerical simulations that consider process defects, progressive damage or failure of materials [15,21,22,30] involve local extrusion or torsion of fiber yarns [31] are also based on these three shapes.

Elliptical section
As shown in figure14(a), the geometric parameters describing the elliptical section are the long semi axis a and short semi axis b of the ellipse, and its relationship with the braiding process parameters is: To facilitate the analysis, the fiber yarn is equivalent to a solid cylinder, and its diameter is . W is the actual cross-section area of the fiber yarn.

Hexagonal section
As shown in figure14 (b), the geometric parameters describing the hexagon section are the hexagon top angle d, hexagon bevel c, hexagon vertical side d, and the hexagon inscribed ellipse short half axis b as auxiliary parameters. The relationship between the above geometrical parameters and braiding process parameters is as follows:

Octagonal section
As shown in figure14(c), the geometric parameters describing the octagon section are: the long semi axis a and the short semi axis b of the inscribed ellipse, the vertical edge L , a and the bottom edge L b of the octagon. The relationship between the above geometrical parameters and braiding process parameters is as follows: Substituting equations (14)-(16) into equation (17), the expression of the short semi axis b of the inscribed ellipse of the octagon section is obtained as: 1 cos 18 The above is the relationship between the geometric characteristic parameters of the three sections and the process parameters. It is found that the determination of the geometric parameters of the section requires obtaining the yarn packing factor e. According to equations (1)-(2), e can be obtained by calculating the volume fractions of fiber yarns of the three unit cells. Therefore, the volume fractions of fiber yarns for the three unit cells under two division modes are derived, respectively, and the results are listed in table 5 and table 6. It shows that the volume fraction of fiber yarns of the interior cell has nothing to do with the division method, only depending on the cross-section shape, with fixed values. Whereas, the volume fraction of fiber yarns of the surface cell is both related to the division method and cross-section shape. The volume fraction V sy of fiber yarns of the surface cell is a function of g and q, and that V cy of the corner cell is a function of g and b. g, q and b can be calculated from the known braiding angle a according to equation (6). Through table 5 or table 6, V , iy V sy and V cy can be solved, which is substituted into equations (1) and (2) to calculate the yarn filling coefficient e. Then the geometric parameters of various sections can be determined according to the above formulas.
It should be noted that the elastic constants of fiber yarns should be determined when using the stiffnessvolume averaging method, the finite element method or the homogenization theory to predict the mechanical properties of 3D braided materials. Generally, the prediction method of elastic constants of unidirectional composite materials [8] is adopted, which requires the determination of the yarn-packing factor e. The above analysis process also indicates the determination method of e.

Results and discussion
To verify the correctness of the directional cosine derived in this paper, the stiffness volume averaging method [5] is used to predict the elastic constants of three-dimensional braided materials. Taking the 3D fourdirectional composites as the test specimen, the material properties of the substrate and fiber are shown in table 7, and the weaving process parameters are shown in table 8.
Case 1 in table 9 shows the calculation results of each helix in the surface and corner cells using 2 and 3 straight-line segments, respectively, and Case 2 is the calculation results of fitting 4 and 8 straight-line segments, respectively. Table 9 shows that the predicted results are in good agreement with the experimental data. For the same number of fitting segments, the elastic constants obtained by horizontal division and 45°division are the same. This proves that the two divisions are equivalent for the analysis of mechanical properties of materials. The experimental results show that fitting helical fiber bundles with few polylines will lead to large elastic constant errors. The greater the number of polylines for fitting the helical fiber bundle, the more realistic the fiber bundle direction can be simulated, and the more accurate the calculation results.

Conclusion
In this paper, the three-cell model of 3D four-directional braided composites is studied. Focusing on the spatial direction of fiber yarns and the parametric modeling of unit cells, the following conclusions are drawn: (1) The direction cosines of fiber yarns in the interior cell and surface cell are obtained under two division modes.
(2) The quantitative relationship between braiding process parameters and geometric parameters of unit cell model is established.
(3) The calculation formulas of volume fraction of fiber yarns for three cross-section shapes are obtained. The volume fraction of fiber yarns of the interior cell has nothing to do with the division mode, but has something to do with the cross-section shape. Whereas, the volume fraction of fiber yarns of the surface cell or the corner cell is both related to the division mode and the section shape, and is a function of the braiding angle.
(4)The method of calculating yarn-packing factor by using braiding process parameters is obtained.  Table 8. Braiding parameters of 3D braided composites [16].