Modeling of Internal Geometric Variability and Statistical Property Prediction of Braided Composites

Due to the advantages of high specific strength, specific stiffness, and excellent fatigue resistance, carbon fiber reinforced braided composites have been widely applied in engineering. Since the molding process of braided composites is complex and immature, substantial variability of the internal geometry exists in composites, in which the yarn path with uncertainty is a main factor, so it is necessary to establish an uncertainty model to study the influence of randomness of the yarn path on mechanical properties, which is significantly related to the fatigue resistance properties of composite. An uncertain mesoscopic model with uniform distribution of yarn paths is proposed. Assuming the yarn path is spatially varying in interval range, the variability of yarn path is represented geometrically in the unit cell of composite. The three-dimensional coordinates of the yarn trajectory are calculated, the meso-uncertainty models of 2-D and 2.5-D braided composites are established. The equivalent elastic parameters and the thermal expansion coefficients are obtained by applying homogenization method and temperature field boundary conditions to the mesoscopic model. The effect of yarn path uncertainty on the statistical characteristics of elastic and thermal parameters of braided composites was studied by using Monte-Carlo simulation. A simulation method for modeling yarn path uncertainty of braided composites is provided in this paper for predicting the statistical characteristics of the equivalent elastic and thermal parameters.


Introduction
Braided composites have been widely used in aerospace due to their excellent properties, such as high stiffness, high strength to weight ratio, and good fatigue strength [1][2][3]. They can reduce the use of high alloy on the aircraft, save on structural costs, and have a great advantage in weight reduction and efficiency. The research on composite materials is usually based on the elastic stage, and then further study on the mechanical properties of materials in the plastic stage, impact, strength, fatigue, and other load conditions. To make more efficient use of braided composites, we need to further study their mechanical properties.
Obtaining the elastic constants of braided composites is one of the important topics in research. Based on the homogenization theory, 3D composite materials are equivalent to orthotropic materials with 9 independent parameters. Accurate parameters are the basis for studying the mechanical properties of composites [4,5]. However, the yarn path is uncertain due to the complexity of the braided composite forming process. To improve the reliability of numerical analysis of braided composite structures, it is necessary to study the effect of randomness on mechanical properties of composites [6][7][8][9].
In geometric modeling of braided composites, yarn paths are usually represented by specific function curves, which makes modeling difficult [10][11][12]. Tsai K H [13] used one-dimensional spring units to simulate yarn predicts elastic properties. However, the

Methodology
Based on the research literature, a mesoscopic model with uncertain yarn path is proposed. Si/C matrix and Toray T300-1K fiber bundle are used in this model. It is an orthotropic material with nine independent parameters. Assuming the yarn path is spatially varying in an interval range, the variability of yarn path is represented geometrically in the unit cell of composite. First, load boundary conditions are applied to the model to obtain equivalent parameters. Then, the yarn path uncertainty is equivalent and the interval uniform distribution is assumed. Finally, the influence of yarn path change on elastic parameters and thermal expansion coefficient was obtained using finite element analysis software. Conventional finite element modeling software generally has tetrahedron and hexahedron meshes. In addition, Voxel, DST, MFEM, and XFEM meshes can be used to quickly establish finite element models. The Voxel method is used in Texgen to discretely establish the meso model of braided composites. It has two main algorithms to avoid intrusions between yarns. The first method is to determine whether the points on the yarn's surface are contained in any other yarn, but this method does not work if there are not enough points. The second is to judge whether the envelope of each yarn surface intersects.
It is assumed that the macroscopic mechanical properties of braided composites are orthogonal anisotropy and have three orthogonal symmetry planes. The left and right, back and front, and upper and lower planes are labeled as planes A, B, C, D, E and F. The constitutive relation of orthotropic composites can be expressed in Figure 1. parameters and thermal expansion coefficient was obtained using finite element analysis software.
Conventional finite element modeling software generally has tetrahedron and hexa hedron meshes. In addition, Voxel, DST, MFEM, and XFEM meshes can be used to quickly establish finite element models. The Voxel method is used in Texgen to discretely establish the meso model of braided composites. It has two main algorithms to avoid intrusions between yarns. The first method is to determine whether the points on the yarn's surface are contained in any other yarn, but this method does not work if there are not enough points. The second is to judge whether the envelope of each yarn surface intersects.
It is assumed that the macroscopic mechanical properties of braided composites are orthogonal anisotropy and have three orthogonal symmetry planes. The left and right back and front, and upper and lower planes are labeled as planes A, B, C, D, E and F. The constitutive relation of orthotropic composites can be expressed in Figure 1.  Taking the displacement boundary conditions in the x direction as an example, plane A is subjected to A fixed constraint (0, 0, 0), and plane B is subjected to A displacement in the x direction (x1, 0, 0). The freedom of other planes in the x direction is released, and the constraints in the y and z directions are (_, 0, 0). The strain component in the constitutive equation is [ε1 m = 1, 0, 0, 0, 0]. Then the first column elements of the stiffness matrix are obtained by means of the stress average method. The other aspects are shown in Table 1.  The main implementation steps of the method are as follows: TexGen textile material simulation software was used to efficiently study the effec of yarn path uncertainty on the mechanical properties of mesoscopic structures. Due to the need to establish a mesoscopic model of uncertainty [33], according to the help docu Taking the displacement boundary conditions in the x direction as an example, plane A is subjected to A fixed constraint (0, 0, 0), and plane B is subjected to A displacement in the x direction (x 1 , 0, 0). The freedom of other planes in the x direction is released, and the constraints in the y and z directions are (_, 0, 0). The strain component in the constitutive equation is [ε 1 m = 1, 0, 0, 0, 0]. Then the first column elements of the stiffness matrix are obtained by means of the stress average method. The other aspects are shown in Table 1. Table 1. Periodic displacement boundary conditions.

The Boundary of a Hexahedral Cell
The main implementation steps of the method are as follows: TexGen textile material simulation software was used to efficiently study the effect of yarn path uncertainty on the mechanical properties of mesoscopic structures. Due to the need to establish a mesoscopic model of uncertainty [33], according to the help document, Python script language was used to establish 2D and 2.5D mesoscopic uncertainty models of braided composite materials, and equivalent elastic parameters could be obtained based on the homogenization theory. (1) The three-dimensional coordinates of the yarn paths are numerically calculated, and the coordinate values are obtained by cubic B-spline curve interpolation to obtain smooth curve coordinates.
(2) The equivalent stiffness is obtained according to the homogenization theory, and the uncertainty model is established without considering the special probability distribution of the characteristic quantity.
(3) The equivalent elastic parameters and thermal expansion coefficient are obtained by applying the boundary conditions of concentrated force and temperature field load to the model. Monte Carlo equivalent method was used to obtain the effect of uncertainty on the thermoelastic parameters of braided composites.

Obtaining Equivalent Parameters by Loading Boundary Conditions
Composite material is composed of multiple components, and the non-uniform continuum is represented by M, as shown in Figure 2. According to the knowledge of solid mechanics, this heterogeneous continuum satisfies the equilibrium equation and the motion equation in the R domain: σ ij and ε kl are stress and strain tensors, f is the strength received in the R domain, ρ is the volume density, and C ijkl is the elastic tensor of the constitutive equation. On the boundary S σ and S u , the continuum satisfies the following equation under the basic and natural boundary conditions: In Figure 3, n j is the unit vector perpendicular to the S σ boundary, t i is the traction. The key to realize the homogenization is to obtain the equivalent stiffness of the composite. For the periodic continuum structure, the uniform displacement boundary conditions can ensure that the displacement of the whole structure is continuous, and the distribution of traction force at the relative parallel boundary is the same. Therefore, combined with the above formula, the problem of obtaining macroscopic global elastic parameters of reactive periodic structures from the unit cell model can be solved by applying specific displacement boundary conditions and concentrating forces.
Hill [19] pointed out from the strain energy perspective that when given the boundary condition u, σ ij,j = 0 in the volume of a unit cell. The average strain energy in the region can be obtained from the average stress and strain. Similarly, it can be proved that the average strain energy can be obtained when the boundary is subjected to a traction which produces non-uniform internal stress in the material.
Formula (3) is Hill energy averaging theorem, which proves that the strain energy of homogenized macro continuum is equal to that of microscale unit cell. σ m and ε m are macroscopic stress and strain tensors. σ and ε are the stress and strain tensors of unit cell, respectively.
When a force is applied in each direction of a unit cell, the following relation can be obtained: V is the volume of a unit cell. The macroscopic stress can be expressed by the force applied to the unit cell model and the equivalent material parameters can easily be expressed by the unit cell strain degree of freedom ε 1 m , ε 2 m , ε 3 m , γ m 23 , γ m 31 , γ m 12 , and the loads are

Obtaining Equivalent Parameters by Loading Boundary Conditions
Composite material is composed of multiple components, and the non-uniform continuum is represented by M, as shown in Figure 2. According to the knowledge of solid mechanics, this heterogeneous continuum satisfies the equilibrium equation and the motion equation in the R domain: In Figure 3, nj is the unit vector perpendicular to the Sσ boundary, t ̅ i is the traction.
The key to realize the homogenization is to obtain the equivalent stiffness of the composite. For the periodic continuum structure, the uniform displacement boundary can ensure that the displacement of the whole structure is continuous, and the distr of traction force at the relative parallel boundary is the same. Therefore, combin the above formula, the problem of obtaining macroscopic global elastic paramete active periodic structures from the unit cell model can be solved by applying spec placement boundary conditions and concentrating forces. Hill [19] pointed out from the strain energy perspective that when given the ary condition u, σij,j = 0 in the volume of a unit cell. The average strain energy in th can be obtained from the average stress and strain. Similarly, it can be proved average strain energy can be obtained when the boundary is subjected to a tractio produces non-uniform internal stress in the material. (3) is Hill energy averaging theorem, which proves that the strain en homogenized macro continuum is equal to that of microscale unit cell. σ m and ε m a roscopic stress and strain tensors. σ and ε are the stress and strain tensors of u respectively.
When a force is applied in each direction of a unit cell, the following relation obtained:   23 , v 31 meets the following conditions: α 1 , α 2 , α 3 meets the following conditions: When obtaining each of these attributes, the corresponding load conditions must be satisfied. It can be observed that if the composites are orthotropic, the above properties are sufficient to describe them in most cases. However, the orthogonal anisotropy equivalent parameters mentioned above will not be enough if irregular geometry or local microcracks are involved in the composite cells, i.e., the uncoupled parts of the cell produce shear stress and strain components, resulting in the general anisotropy of the material. For such materials, more material properties are needed to describe their mechanical properties.

Meso-Geometric Model of Plain Weave Composites
To further characterize the uncertainty of plain weave composite yarn amplitude, the number of warp and weft yarns is also taken as parameterized input, and the modeling idea is as follows: First, several weft yarns, which are uncertain in the z direction, are generated according to the number of warp yarns and weft yarns. Since warp and weft yarns fit each other, it is easier to obtain the corresponding coordinates of warp yarns and generate uncertainty modeling of plain weave composite materials. Figure 4 shows the established parameterized mesoscopic uncertainty parameterized model of plain weave composite material. Yellow is weft yarn, red is warp yarn, and each yarn fluctuates within a certain range.  There was serious matrix intrusion between yarns during modeling. Therefore, in this example, the yarn path follows uniform distribution changes in a certain range without considering the special probability distribution in the geometric model. In fact, there is a very small amount of matrix filling between weft and warp. Ideally, the unit cell model of plain weave composite is assumed to be the trig function of the weft path, and the warp cross section fits the contour of the trig function, but this problem is relatively complicated. In this section, the analysis model is simplified as shown in Figure 5, and the mesouncertain geometric model of plain braided composite material is further established. To simplify the problem, the established model makes the following assumptions: (ⅰ) The yarn section is simplified to a rectangle; (ⅱ) The yarn path is assumed to be a broken line; (ⅲ) There is no matrix filling between yarns; (ⅳ) It is only considered that the yarn amplitude in the plane follows uniform distribution, yarn torsion, deformation in other directions and defects are not considered. According to Ref [34], the geometric parameter design description of the uncertainty model of plain braided composite material and the values of the model in this paper are shown in Table 2. The geometric characteristics of the yarn are parameterized as input. Where JL, Jh, WL and Wh are the length and width of the rectangular warp and weft sections respectively. GL is the distance between two adjacent yarns. The height of yarn is uniformly distributed within the interval [0,0.028]. The distance between yarn and outer matrix is constant value 0.04. Outer matrix height 0.04 There was serious matrix intrusion between yarns during modeling. Therefore, in this example, the yarn path follows uniform distribution changes in a certain range without considering the special probability distribution in the geometric model. In fact, there is a very small amount of matrix filling between weft and warp. Ideally, the unit cell model of plain weave composite is assumed to be the trig function of the weft path, and the warp cross section fits the contour of the trig function, but this problem is relatively complicated. In this section, the analysis model is simplified as shown in Figure 5, and the meso-uncertain geometric model of plain braided composite material is further established. To simplify the problem, the established model makes the following assumptions: (i) The yarn section is simplified to a rectangle; (ii) The yarn path is assumed to be a broken line; (iii) There is no matrix filling between yarns; (iv) It is only considered that the yarn amplitude in the plane follows uniform distribution, yarn torsion, deformation in other directions and defects are not considered.  There was serious matrix intrusion between yarns during modeling. Therefore, in this example, the yarn path follows uniform distribution changes in a certain range without considering the special probability distribution in the geometric model. In fact, there is a very small amount of matrix filling between weft and warp. Ideally, the unit cell model of plain weave composite is assumed to be the trig function of the weft path, and the warp cross section fits the contour of the trig function, but this problem is relatively complicated. In this section, the analysis model is simplified as shown in Figure 5, and the mesouncertain geometric model of plain braided composite material is further established. To simplify the problem, the established model makes the following assumptions: (ⅰ) The yarn section is simplified to a rectangle; (ⅱ) The yarn path is assumed to be a broken line; (ⅲ) There is no matrix filling between yarns; (ⅳ) It is only considered that the yarn amplitude in the plane follows uniform distribution, yarn torsion, deformation in other directions and defects are not considered. According to Ref [34], the geometric parameter design description of the uncertainty model of plain braided composite material and the values of the model in this paper are shown in Table 2. The geometric characteristics of the yarn are parameterized as input. Where JL, Jh, WL and Wh are the length and width of the rectangular warp and weft sections respectively. GL is the distance between two adjacent yarns. The height of yarn is uniformly distributed within the interval [0,0.028]. The distance between yarn and outer matrix is constant value 0.04. Outer matrix height 0.04 According to Ref. [34], the geometric parameter design description of the uncertainty model of plain braided composite material and the values of the model in this paper are shown in Table 2. The geometric characteristics of the yarn are parameterized as input. Where J L , J h , W L and W h are the length and width of the rectangular warp and weft sections respectively. G L is the distance between two adjacent yarns. The height of yarn is uniformly distributed within the interval [0,0.028]. The distance between yarn and outer matrix is constant value 0.04.

Equivalent Results
The fiber bundle used in the calculation example is Toray T300-1K with a density of 1.76 g/cm 3 . The matrix of composite material is Si/C material, and the elastic parameters of each component of braided composite material are shown in Tables 3 and 4:  Table 4. The thermal expansion coefficients of components.

Thermal Expansion
The finite element model of braided composite material established by Voxel technology is shown in Figure 6. The element divided by this method is C3R8, which is a regular grid of regular hexahedron element, and appropriate local mechanical properties and material direction are defined at the center of the element. The yarn simulated by this element is serrated, so the mesh density needs to be increased to describe the yarn path more accurately and reduce the influence of mesh shape on the equivalent result. The number of yarns was reduced to reduce the influence of independent variables on elastic parameter response, the numerical was 2 × 2. Although the parametric geometric model is established, the influence of the number of derived meshes on the computational efficiency still needs to be considered. The number of grids established after the above consideration is 104,400, and the number of nodes is 117,660. The volume fraction of carbon fiber in warp and weft yarn was 0.82%, the yarn accounted for 64.46% of the total volume, and the carbon fiber content in the model was about 50%.

Equivalent Results
The fiber bundle used in the calculation example is Toray T300-1K with a density of 1.76 g/cm 3 . The matrix of composite material is Si/C material, and the elastic parameters of each component of braided composite material are shown in Tables 3 and 4: The finite element model of braided composite material established by Voxel technology is shown in Figure 6. The element divided by this method is C3R8, which is a regular grid of regular hexahedron element, and appropriate local mechanical properties and material direction are defined at the center of the element. The yarn simulated by this element is serrated, so the mesh density needs to be increased to describe the yarn path more accurately and reduce the influence of mesh shape on the equivalent result. The number of yarns was reduced to reduce the influence of independent variables on elastic parameter response, the numerical was 2 × 2. Although the parametric geometric model is established, the influence of the number of derived meshes on the computational efficiency still needs to be considered. The number of grids established after the above consideration is 104,400, and the number of nodes is 117,660. The volume fraction of carbon fiber in warp and weft yarn was 0.82%, the yarn accounted for 64.46% of the total volume, and the carbon fiber content in the model was about 50%.   x direction y direction z direction According to the constitutive equation of stress-strain relationship, the equivalent stiffness matrix C can be obtained as follows: The flexibility matrix can be obtained by inverting the equivalent elastic matrix. According to the relationship between flexibility matrix and elastic parameters, the equivalent elastic parameters can be obtained. It can be seen from Table 5 that the arrangement of in-plane yarns in two directions is similar due to the structure of orthogonal plain weave composite material. Therefore, the equivalent elastic parameter E11 is close to E22.

Equivalent Results of Yarn Path Uncertainty
It can be seen from Table 2 that the height difference between warp and weft yarns is between [0,0.028], and the distance between adjacent yarns is between [0,0.02]. The process of obtaining equivalent elastic parameters based on TexGen software and the equivalent results of elastic parameters E11, E22, E33, G12, ν12 and thermal expansion coefficient α11 were calculated by Abaqus software for 1000 s, as shown in Figure 8. As can be seen from the distribution of scattered points, the equivalent parameters will be affected to varying degrees as the yarn path changes within the interval. The total distance R of the elastic moduli of E11, E22 and E33 were 49.65 Gpa, 46.95 Gpa and 27.42 Gpa, respectively. The polar distance of Poisson's ratio ν12 is 0.0368. The shear modulus G12 mainly varies in the range of 20.597 Gpa. R is the difference value between the maximum and minimum of the equivalent parameter. According to the constitutive equation of stress-strain relationship, the equivalent stiffness matrix C can be obtained as follows: The flexibility matrix can be obtained by inverting the equivalent elastic matrix. According to the relationship between flexibility matrix and elastic parameters, the equivalent elastic parameters can be obtained. It can be seen from Table 5 that the arrangement of in-plane yarns in two directions is similar due to the structure of orthogonal plain weave composite material. Therefore, the equivalent elastic parameter E 11 is close to E 22 .

Equivalent Results of Yarn Path Uncertainty
It can be seen from Table 2 that the height difference between warp and weft yarns is between [0,0.028], and the distance between adjacent yarns is between [0,0.02]. The process of obtaining equivalent elastic parameters based on TexGen software and the equivalent results of elastic parameters E 11 , E 22 , E 33 , G 12 , ν 12 and thermal expansion coefficient α 11 were calculated by Abaqus software for 1000 s, as shown in Figure 8. As can be seen from the distribution of scattered points, the equivalent parameters will be affected to varying degrees as the yarn path changes within the interval. The total distance R of the elastic moduli of E 11 , E 22  The mean value only reflects the midpoint value of a set of data, while the mean square error can reflect the dispersion degree of a set of data. In this paper, the influence degree of yarn path on elastic parameters can be reflected side by side. According to the data in Table 6 and Table 7, it can be concluded that the influences on the elastic moduli E11 and E22 are relatively large. The standard deviation of E33 and G12 was 5.440 and 3.704, respectively. For the elastic parameters ν12, ν23, ν31, G23, G31 have relatively little effect. The pole distances of shear moduli G23 and G31 are relatively small around 10 3 Mpa. In addition, the expansion coefficient of carbon fiber itself is small, so it has little influence on the thermal expansion coefficient. It can be concluded that the elastic parameters E11, E22, E33, and G12 are greatly affected when the yarn path of plain weave composite is changed. The mean value only reflects the midpoint value of a set of data, while the mean square error can reflect the dispersion degree of a set of data. In this paper, the influence degree of yarn path on elastic parameters can be reflected side by side. According to the data in Tables 6 and 7, it can be concluded that the influences on the elastic moduli E 11 and E 22 are relatively large. The standard deviation of E 33 and G 12 was 5.440 and 3.704, respectively. For the elastic parameters ν 12 , ν 23 , ν 31 , G 23 , G 31 have relatively little effect. The pole distances of shear moduli G 23 and G 31 are relatively small around 10 3 Mpa. In addition, the expansion coefficient of carbon fiber itself is small, so it has little influence on the thermal expansion coefficient. It can be concluded that the elastic parameters E 11 , E 22 , E 33 , and G 12 are greatly affected when the yarn path of plain weave composite is changed. For further analysis, the influence of yarn trend amplitude on elastic parameters. In this paper, the correlation coefficient between elastic parameters and fiber amplitude is solved, and the correlation coefficient between elastic parameters is given to study the quantitative variation relationship between elastic parameters. According Table 8, the correlation coefficients of elastic modulus, shear modulus and Poisson's ratio of ν 12 are all positive. The correlation coefficients of E 33 , ν 12 and ν 23 are all greater than 0.652, which indicates that these parameters are highly correlated. It shows that the above equivalent parameters increase or decrease together with the change in yarn path. Only Poisson's ratio ν 23 , ν 31 and other parameters appear negative correlation, and the correlation appears below 0.5. Compared with plain weave, 2.5D braided composite has a more complex structure and mechanical mechanism, so further research is needed. To facilitate calculation, the uniform distribution of the weft in the z direction. When the warp yarn changes in the direction of height, the cross section also changes constantly. To ensure the fit of warp yarn, we determine the contour of the weft yarn according to the changing height of warp yarn.
In this paper, according to the number of weft yarns in x and z directions and warp yarns in y direction, a relatively simple mesoscopic parametric model of 2.5D braided composites was established to study the effects of yarn height fluctuation in z direction and transverse yarn distance on elastic parameters of 2.5D braided composites. Figure 9 shows the 2.5D weaving meso model, in which yellow is warp yarn and red is weft yarn. This part is more difficult to achieve, is the use of MATLAB numerical calculation to ensure. In addition, the height of two adjacent warp yarns in the y direction is different, so the cross section of the weft yarns is certain. Parametric modeling is carried out in the y direction according to the number of warp yarns. After the above two steps of modeling, according to the number of z direction weft yarn, the lateral fluctuation of weft yarn is not considered. The warp yarn and weft yarn contact each other, layer upon layer of each other. To sum up, when the yarn position in x direction is determined, parametric modeling in the other two directions is equivalent to translating the yarn in each direction by a certain distance.
For the 2.5D braided composite with layer and shallow layer crossbending connection, the model is established as shown in Figure 10. The path is assumed to be a broken line to avoid the difficulty of modeling yarn as a curve. To simplify the modeling problem, the following assumptions are made: (ⅰ) the warp section is simplified as rectangle and the weft section as flat hexagon. (ⅱ) the warp path is a broken line, the weft path is a straight line. (ⅲ) Fit between yarns without matrix filling. (ⅳ) Consider only the uniform distribution of the warps in the topper vertical direction, excluding the transversal fluctuations of the weft, deformations in other directions, and substrate defects. The yarns fit together without matrix filling which means that when the warp changes in height, the height of the filling changes with it. In addition, when the warp height of rectangular section changes, the inclination angle of the four edges of the filling hexagon should also change. Since the warp is a broken line, the slope can be found from the distance between its height and the warp. In addition, a weft is attached to two spatially intersecting warp yarns. P1-P6 are the intersections of the warps. Therefore, the dip Angle of the side edge of the weft section needs to consider the effect of the uncertainty of the two weft paths. When the transverse distance of the weft is constant, the slope of the four edges can be obtained according to the height of two warp yarns crossed in space. After a series of numerical operations to obtain the weft section of the profile and assigned to the yarn, this model can avoid overlapping area between warp and weft yarns and the uncontrollable shape of weft section. Finally, the uncertainty modeling of path height and spacing of 2.5D braided composite yarn was realized.  This part is more difficult to achieve, is the use of MATLAB numerical calculation to ensure. In addition, the height of two adjacent warp yarns in the y direction is different, so the cross section of the weft yarns is certain. Parametric modeling is carried out in the y direction according to the number of warp yarns. After the above two steps of modeling, according to the number of z direction weft yarn, the lateral fluctuation of weft yarn is not considered. The warp yarn and weft yarn contact each other, layer upon layer of each other. To sum up, when the yarn position in x direction is determined, parametric modeling in the other two directions is equivalent to translating the yarn in each direction by a certain distance.
For the 2.5D braided composite with layer and shallow layer crossbending connection, the model is established as shown in Figure 10. The path is assumed to be a broken line to avoid the difficulty of modeling yarn as a curve. To simplify the modeling problem, the following assumptions are made: (i) the warp section is simplified as rectangle and the weft section as flat hexagon. (ii) the warp path is a broken line, the weft path is a straight line. (iii) Fit between yarns without matrix filling. (iv) Consider only the uniform distribution of the warps in the topper vertical direction, excluding the transversal fluctuations of the weft, deformations in other directions, and substrate defects. The yarns fit together without matrix filling which means that when the warp changes in height, the height of the filling changes with it. In addition, when the warp height of rectangular section changes, the inclination angle of the four edges of the filling hexagon should also change. Since the warp is a broken line, the slope can be found from the distance between its height and the warp. In addition, a weft is attached to two spatially intersecting warp yarns. P 1 -P 6 are the intersections of the warps. Therefore, the dip Angle of the side edge of the weft section needs to consider the effect of the uncertainty of the two weft paths. When the transverse distance of the weft is constant, the slope of the four edges can be obtained according to the height of two warp yarns crossed in space. After a series of numerical operations to obtain the weft section of the profile and assigned to the yarn, this model can avoid overlapping area between warp and weft yarns and the uncontrollable shape of weft section. Finally, the uncertainty modeling of path height and spacing of 2.5D braided composite yarn was realized. This part is more difficult to achieve, is the use of MATLAB numerical calculation to ensure. In addition, the height of two adjacent warp yarns in the y direction is different, so the cross section of the weft yarns is certain. Parametric modeling is carried out in the y direction according to the number of warp yarns. After the above two steps of modeling, according to the number of z direction weft yarn, the lateral fluctuation of weft yarn is not considered. The warp yarn and weft yarn contact each other, layer upon layer of each other. To sum up, when the yarn position in x direction is determined, parametric modeling in the other two directions is equivalent to translating the yarn in each direction by a certain distance.
For the 2.5D braided composite with layer and shallow layer crossbending connection, the model is established as shown in Figure 10. The path is assumed to be a broken line to avoid the difficulty of modeling yarn as a curve. To simplify the modeling problem, the following assumptions are made: (ⅰ) the warp section is simplified as rectangle and the weft section as flat hexagon. (ⅱ) the warp path is a broken line, the weft path is a straight line. (ⅲ) Fit between yarns without matrix filling. (ⅳ) Consider only the uniform distribution of the warps in the topper vertical direction, excluding the transversal fluctuations of the weft, deformations in other directions, and substrate defects. The yarns fit together without matrix filling which means that when the warp changes in height, the height of the filling changes with it. In addition, when the warp height of rectangular section changes, the inclination angle of the four edges of the filling hexagon should also change. Since the warp is a broken line, the slope can be found from the distance between its height and the warp. In addition, a weft is attached to two spatially intersecting warp yarns. P1-P6 are the intersections of the warps. Therefore, the dip Angle of the side edge of the weft section needs to consider the effect of the uncertainty of the two weft paths. When the transverse distance of the weft is constant, the slope of the four edges can be obtained according to the height of two warp yarns crossed in space. After a series of numerical operations to obtain the weft section of the profile and assigned to the yarn, this model can avoid overlapping area between warp and weft yarns and the uncontrollable shape of weft section. Finally, the uncertainty modeling of path height and spacing of 2.5D braided composite yarn was realized.  Figure 10. The uncertain geometry of 2.5D braided composite. Figure 10. The uncertain geometry of 2.5D braided composite.
Description of geometric parameter design of 2.5D braided uncertainty model. Parameter values are shown in Table 9. The geometric features in the model are also taken as parameterized inputs to achieve uncertainty modeling. Where J L and J h are the length and width of the rectangular warp and the filling section, W L and W h are the length and height of the short side of the flat hexagon, respectively, and G L is the distance between the filling axes. The same height changes in uniform distribution between the interval [0,0.028]. The distance between the highest and lowest position of yarn and the outer matrix is constant value 0.04.

Equivalent Results
The time factor and the calculation efficiency of the finite element model still need to be considered. Figure 11 shows the finite element model of 2.5D braided composite. The composite material is the toray T300-3K fiber bundle. Its linear density is 198 g/km, and its density is 1.76 g/cm 3 . The material parameters of each component are the same as those in Table 3. The number of voxel grids in x, y and z directions is [150,75,75], the number of grids is 843,750, and the number of nodes is 872,176. The volume fraction of warp fiber is 69%, weft fiber is 82%, and the volume fraction of yarn of the whole model is 43.18%. Description of geometric parameter design of 2.5D braided uncertainty model. Parameter values are shown in Table 9. The geometric features in the model are also taken as parameterized inputs to achieve uncertainty modeling. Where JL and Jh are the length and width of the rectangular warp and the filling section, WL and Wh are the length and height of the short side of the flat hexagon, respectively, and GL is the distance between the filling axes. The same height changes in uniform distribution between the interval [0,0.028]. The distance between the highest and lowest position of yarn and the outer matrix is constant value 0.04. Outer matrix height 0.04

Equivalent Results
The time factor and the calculation efficiency of the finite element model still need to be considered. Figure 11 shows the finite element model of 2.5D braided composite. The composite material is the toray T300-3K fiber bundle. Its linear density is 198 g/km, and its density is 1.76 g/cm 3 . The material parameters of each component are the same as those in Table 3. The number of voxel grids in x, y and z directions is [150, 75, 75], the number of grids is 843,750, and the number of nodes is 872,176. The volume fraction of warp fiber is 69%, weft fiber is 82%, and the volume fraction of yarn of the whole model is 43.18%. The specific x, y and z axial tensile and xy, yz, zx axial shear loads were applied to the 2.5D braided composite meso model, and the statics analysis was carried out. Figure 12 shows the stress results under the boundary conditions of six loads. The specific x, y and z axial tensile and xy, yz, zx axial shear loads were applied to the 2.5D braided composite meso model, and the statics analysis was carried out. Figure 12 shows the stress results under the boundary conditions of six loads.
The stress vectors in the constitutive equation are 1, respectively, by applying specific loading boundary conditions, i.e., the equation F i /V = 1. The equivalent stiffness matrix C in the constitutive equation is easily represented by elements in the strain component.

Shear load condition in yz direction
Shear load condition in zx direction Shear load condition in xy direction When specific loading boundary conditions are applied to the mesoscopic model in finite element, the strain component can be expressed by the displacement calculated statically. According to Formulas (5)- (8), when all the elements in the stress component are 1, the equivalent elastic parameter is equal to 1/εi, where ε = [ε1 m , ε2 m , ε3 m , γ m 23, γ m 31, γ m 12]. Table 10 shows the values of equivalent thermoelastic parameters.   Table 11, it can be concluded that the yarn path uncertainty of 2.5D braided composite mainly has a relatively large impact on the elastic parameters E11, E22, E33, G23 and G12. Scatter diagrams of elastic parameters E11, E22, E33, G12, G23 and G31 with large standard deviations in Table 11 are shown in Figure 13.  Table 10 shows the values of equivalent thermoelastic parameters.

Equivalent Results of Yarn Path Uncertainty
The center lines of two adjacent weft yarns of the 2.5D braided composite mesoscopic model change at [0,0.16], and the height difference between the weft yarns changes between [0,0.3]. Combined with the extreme values of the sample data and the mid-scale measurement in Table 11, it can be concluded that the yarn path uncertainty of 2.5D braided composite mainly has a relatively large impact on the elastic parameters E 11 , E 22 , E 33 , G 23 and G 12 . Scatter diagrams of elastic parameters E 11 , E 22 , E 33 , G 12 , G 23 and G 31 with large standard deviations in Table 11 are shown in Figure 13.   According to the distribution of scattered points, the total distance R of the sample points of the above elastic parameters are 25.01 Gpa, 15.45 Gpa, 11.51 Gpa, 9.736 Gpa, 12.023 Gpa, and 6.695 Gpa, respectively.
Combined with the extreme values of sample data and the middle measurement in Table 11 and Table 12, it can be concluded that the yarn path uncertainty of 2.5D braided composite mainly has a relatively large impact on the elastic parameters E11, E22, E33, G23 and G12. According to the distribution of scattered points, the total distance R of the sample points of the above elastic parameters are 25.01 Gpa, 15.45 Gpa, 11.51 Gpa, 9.736 Gpa, 12.023 Gpa, and 6.695 Gpa, respectively.
Combined with the extreme values of sample data and the middle measurement in Tables 11 and 12, it can be concluded that the yarn path uncertainty of 2.5D braided composite mainly has a relatively large impact on the elastic parameters E 11 , E 22 , E 33 , G 23 and G 12 . The correlation between elastic constant samples can be obtained from the data in Table 13. By analyzing the data in the table, it can be concluded that the main direction elastic modulus E 11 is only relatively correlated with the parameters ν 12 and G 12 samples. At the same time, the correlation between E 22, E 33, G 12, G 23 and G 31 and the sample of parameter ν 12 is small. The negative correlation between ν 23 , ν 31 and other equivalent parameter sample data are consistent with the plain weave simulation example. The correlation between special ν 23 and other equivalent parameter samples is negative, and the correlation between E 11 , ν 12 parameter samples is less than 0.018.