Meso-structure and processing of three-dimensional braided material based on space group P 4 symmetry

Abstract

A novel three-dimensional (3D) braided material was obtained using group theory. The 3D braided structural unit was found by means of group elements in the point group 4. Based on the symmetry of space group P4, a novel 3D braided material geometry was deduced by transforming the unit. Analyzing the conceivable motion of the yarns, it was found that the geometric structure corresponding to a novel 3D braided material was feasible. The mathematic model for describing the geometry of this novel material was established. The fiber volume fraction of the material was predicted and its value is similar to the traditional four-step 3D braided material. A first prototype of the fabric is produced using manual production.

Keywords

point group 4 processing structure-properties three-dimensional braided geometry

Three-dimensional (3D) braided composites have been widely used in aerospace applications and other high-temperature resistance structures, extending to the medical and sports and automotive industries, because they have many advanced properties, such as high modulus and damage tolerance and fracture toughness.^1–3 In fact, some new 3D braided material structures are obtained from living things surviving in the natural environment. For example, a trabecula in beetle forewings greatly increases resistance against peeling for laminated composite structures and possesses excellent energy absorption and buffering properties.^4–6 Therefore, the research of 3D braided composites has become a hot research topic.

In 3D braided composites, four-step and two-step 3D braided processes are the most common and useful.⁷ The meso-structure of 3D braided composites has been analyzed by using the point group and the space group.⁸ According to the symmetry theories of traditional crystallography, the space group is an assembly including the geometric symmetry operation that makes a 3D periodic crystal into its own.⁹ For 3D braided materials, the structural unit has been formed by the yarn segment through the geometric symmetry operation. The assembly of the geometric symmetry operation is defined as the space group. A series of new 3D braided geometries was deduced.^10–18 According to the symmetry of space group $P \bar{3}$ , a 3D braided geometry was obtained using translation and symmetry operations of the unit.^11,12 The 3D tubular braided composites based on the space group $P \bar{3}$ symmetry were studied, and the feasibility of the 3D braided process was verified by the literature.^13,14 The plate 3D braided fabric based on the space group $P \bar{3}$ symmetry was discussed, which is verified by the process of industrial production.^15,16 The feasibility of a new type of 3D braided geometry based on the space group R3 symmetry was proposed, and an experiment verified the feasibility of the 3D braided process.^17,18

Traditional 3D braided preforms produced by the four-step 1 × 1 method have been fully discussed.^19–24 Ko¹⁹ first identified a unit cell representing the fiber architecture of traditional 3D braided preforms. Chen et al.²⁰ presented an analytical approach to establish the relationship between the braid structure and the braiding parameters. Fang et al.²¹ employed a representative volume cell to analyze the mechanical properties of traditional 3D braided performs. Li et al.²² proposed three different types of micro-structural unit cell models to analyze the mathematical relationships among the structural parameters. Renkens and Kyosev²³ concentrated on a computation of the geometry of warp knitted structures, which have complex 3D forms or which are draped over similar ones. Xu and Qian²⁴ adopted a multi-unit cell model to analyze the structural geometry parameters of the preforms.

Due to the low production efficiency and high cost, new material with different structures and feasible processes is urgently required. For optimal performance of 3D braided composites, a new braiding process should be developed to obtain the new geometry structures. Most of novel 3D braided geometries deduced from group theory are restricted by the complicated process. Based on the group theory, 3D braided structural units satisfying space point group 4 are deduced. The geometry of 3D braided material is obtained by operations in space group P4 (4 and P4 are the symbolic notation of Hermann-Mauguin in group theory⁶). This novel braided structure belongs to the structure of Cartesian 3D braids, which are novel 3D fully braided fabric structures.^25–27 The processing and properties of the novel material are discussed in this study. The fiber volume fraction of the material is predicted. A first prototype of the fabrics is produced.

A novel three-dimensional braided geometry for space group P4 symmetry

The 3D braided geometry unit satisfied with point group 4 symmetry

In the Cartesian coordinate system xyz, the z-axis is a rotation axis. The group elements of point group 4 are ${1, 4, 4^{2} = 2, 4^{3}}$ (or ${E, c_{4}, c_{4}^{2} = c_{2}, c_{4}^{3}}$ ). The operation of point group 4 is the rotation operation, which is the motion around the z-axis with $π / 2$ rotation angle, and $c_{4}$ represents their generator. The matrices corresponding to the elements of point group 4 can be expressed as

W_{c_{4}} = (0 - 1 0100001), W_{c_{4}^{2}} = (- 1 000 - 1 0001), W_{c_{4}^{3}} = (010 - 1 00001), W_{E} = (100010001)

The rotation operation corresponding to the elements of point group 4 can be expressed as

c_{4} (x, y, z) = (- y, x, z), c_{4}^{2} (x, y, z) = (- x, - y, z),

c_{4}^{3} (x, y, z) = (y, - x, z), c_{4}^{4} (x, y, z) = E

The rotation operations that transform a point $(x, y, z)$ on the yarn segment into the point after the equal are denoted by $c_{4}^{n}$ .

The combination of the yarn segment (Figure 1) can be deduced by the symmetry operation of yarn segment 1, which can be used as a representative volume unit to derive a novel 3D braided geometry that is satisfied with space group P4 symmetry. The unit (Figure 1) from the middle part of the preform can represent the preforms, which is straight, as shown in Figure 1. In the Cartesian coordinate system, the translational symmetry operations can be expressed as

T_{i} = x_{i} u + y_{i} v + z_{i} w

(1)

where

x_{i}

y_{i}

and

z_{i}

are integers, u , v and w are base vectors and

T_{i}

is the translation vector. Putting the unit into a simple rectangular lattice B-oP, the 3D braided inner geometry contented with the braided space group P4 can be obtained (Figure 2). In Figure 2, the symbol h stands for the thickness of the combination of the single layer yarn segment in the z direction. In the xoy coordinate plane and the plane clusters parallel to it, the translation distance of the unit is

z_{i}

times of h along the z direction.

Figure 1.

Combination to meet point group 4 of the yarns.

Figure 2.

Rectangular lattice B-oP and unit combination of the yarn.

The novel 3D braided geometry corresponding to space group symmetry

The unit of 3D braided geometry

The simple rectangular lattice B-oP¹⁵ (as shown in Figure 3) is described by crystal symmetric group P4, which is harmonized with space point group 4. The unit for space group 4 can be described by a lattice. Putting the unit (Figure 1) into the rectangular lattice and verifying the continuity of the yarns, the usual unit can be obtained (as shown in Figure 2).

Figure 3.

A novel three-dimensional braided geometry.

A novel 3D braided geometry for space group P4 symmetry

The novel 3D braided inner geometry can be obtained by transforming the unit, which is deduced from the symmetry operation of space point group 4. During the process, taking into account the continuity of the yarns and studying its regularity, a novel 3D braided geometry will be obtained (as shown in Figure 3).

Research on the process of the novel three-dimensional braided material geometry

The novel 3D braided material geometry corresponds to a novel 3D braided method. The cross-movement rules of the yarn to meet the fabric geometry have to be discussed.

The movement rules of the carrier carrying the yarn are illustrate in Figure 4. Each symbol represents a carrier, and the arrow points denote the direction of movement. The different arrows are used to classify the carrier of the different moving directions. In the yarn array, carriers with the same trajectory are classified as one group, which is divided into two groups (as shown in Figures 4(a) and (b)). The movement trajectory of the yarn in the same group does not change during the process, and all the carriers alternately move once along the direction of the arrow to complete a braided cycle.

Figure 4.

The carrier array of the novel braided process (a) carriers rotatingcounterclockwise in horizontal direction (b) carriers rotating counterclockwise in vertical direction.

Each group yarn of the novel 3D braided material is approximately straight in the fabric, only bending at the edge of the boundary (as shown in Figure 5). The novel process is as follows.

Step 1: The yarns of group (a) circle counterclockwise. The carrier with the yarns moves to the next station along the counterclockwise direction.

Step 2: The yarns of group (b) circle counterclockwise for a step.

Figure 5.

The representative yarn of different groups in the three-dimensional braided material.

Repeat Steps 1 and 2. Two sets of different yarns cross in the space step by steps to form a novel 3D braided geometry (shown in Figure 5).

The geometrical model of the unit in three-dimensional braided material

The novel material can be used to produce the reinforced phase of 3D braided composite material, using the analysis model of the unit geometry to predict its performance. The novel material geometrical model is divided into units with the full yarn, and the adjacent units are beset by the full yarn. The full symmetry of units is preserved, and its mechanical properties are discussed using group theory.

Basic assumptions

The section of the yarn is assumed to be a rhombus (Figure 6) and flexible enough, producing geometrical distortion under random loading.

The braided process is sufficiently stable, and the braided geometry is consistent within some range of the material.

The influence of the surface structure can be ignored with the increase of the cross-sectional dimensions.

In the inner units, each yarn is subjected to the load from different directions, and the final extruded yarn is represented in Figure 7.

Figure 6.

Diagram of internal yarn suffering lateral extrusion.

Figure 7.

The approximate geometry of internal yarn.

The units of the novel 3D braided material

The novel 3D braided material is divided into the interior area, surface area and corner area. Respectively, the three kinds of units are the interior unit, face unit and corner unit, as shown in Figure 8.

Figure 8.

The division of braided material.

Geometric parameters for describing the novel 3D braided material

The number of braided material units in a single layer

Set n represents the numbers of one side units in a single layer. The total number of single layer units N is

N = n^{2}

(2)

The numbers of corner unit $N_{c}$ and face unit $N_{f}$ are

N_{c} = 4, N_{f} = 4 (n - 2)

(3)

The number of interior unit $N_{i}$ is

N_{i} = N - N_{c} - N_{f}

(4)

The geometrical model and fiber volume fraction of the novel 3D braided material

Unit volume $U_{r}$

In the Cartesian coordinate system, the simplified structure of the unit is as shown in Figure 9. Set a stands for the unit side length, the pitch length is h and the braid angle is α (i.e. the angle between the yarn axis and the z-axis). The unit volume is

U_{r} = a^{2} h

(5)

Figure 9.

The relationship of simplified yarn geometrical parameters.

The relationship of geometrical parameters $a, h, α$ is

\cos α = \frac{h}{\sqrt{a^{2} + h^{2}}}

(6)

The cross-sectional area $A_{m}$ of single yarn on the unit surface

From Figure 10, the cross-sectional area $A_{m}$ can be deduced

A_{m} = \frac{1}{8} a^{2}

(7)

Figure 10.

The unit geometrical model of simplified yarn.

The reduction coefficient λ

The equivalent cross-section area of the yarn can be described by the front $A_{0}$ . According to the geometrical relationship of Figures 9 and 10, the equivalent cross-section area of the yarn is

A_{0} = \frac{1}{8} a^{2} \cos α

(8)

Due to the influence of yarn deformation on the geometry of the unit, a reduction coefficient λ of the yarn section is introduced. The equivalent area after deformation is A₀

λ = \frac{A_{0}}{π R^{2}}

(9)

where

R = \sqrt{\frac{Γ}{π ρ}}

(10)

R represents the equivalent radius of the yarn. It is determined by the yarn linear density Γ (

Texy / 1000

) and the density of fiber material ρ (

g / c m^{3}

).¹⁴

The extent of λ is

1 > λ > \frac{a^{2} \cos α}{8 π R^{2}}

(11)

The yarn volume and fiber volume of the braided material

The total volume of simplified yarn shown in Figure 10 is

U_{s} = 4 h A_{m} = \frac{1}{2} a^{2} h

(12)

The total volume of the original yarn is

U_{y} = \frac{U_{s}}{λ}

(13)

The yarn fiber volume fraction is the assemble factor ɛ. The maximum value is¹⁴

ɛ_{m} = \sqrt{3} π / 6 \approx 0.9064

(14)

The total volume of the fiber in the unit is

U_{f} = ɛ U_{y} = \frac{ɛ a^{2} h}{2 λ}

(15)

The fiber volume fraction of the novel braided material

The fiber volume fraction V_f can be deduced by equations (5), (14) and (15)

V_{f} = \frac{U_{f}}{U_{r}} \times 100 % = \frac{ɛ a^{2} h}{2 λ a^{2} h} = \frac{ɛ}{2 λ} \times 100 % < ɛ_{m} = \frac{\sqrt{3} π}{6} = V_{f max}

(16)

When the cross-section of the braided yarn is circular (under ideal conditions), the fiber volume percent of the braided material is the minimum

V_{f min} = \frac{U_{s}}{U_{r}} \times 100 % = \frac{ɛ a^{2} h / 2}{a^{2} h} = \frac{ɛ}{2}

(17)

The fiber volume fraction $V_{f}$

The reduction coefficient λ affects the fiber volume fraction. Under ultimate limit states, the fiber volume fraction $V_{f}$ cannot achieve the maximum $ɛ_{m}$ . The braided angle influences the reduction coefficient λ and the fiber volume fraction. The larger the braided angle, the smaller the value of λ, and vice versa. According to equation (11), the fiber volume fraction is the minimum because of its loose structure when λ has become maximum. On the contrary, the percentage reaches a maximum. The novel 3D braided material is still in theoretical research, which can only be predicted and analyzed qualitatively.

Experimental verification of the braided process based on space group P4 symmetry

To verify the feasibility of the novel braided process, the following experiment was done on the basis of theoretical research. The sample is braided by 4 × 4 arrays. The distribution of the carriers is illustrated in Figure 4. As show in Figure 5, the two sets of yarns intercross in the space and form a novel braided material.

A novel 3D braided geometry, which satisfies the symmetry of space group P4, is derived by using the symmetry operation of point group 4. The novel 3D braided preforms can be obtained in the laboratory based on the definitive array of the yarn and motion method (shown in Figure 11). It is clear that the geometry of the preforms is similar to the simulation model, and the yarn path is identical.

Figure 11.

The preform of novel three-dimensional braided material: (a) experimental results; (b) simulation model.

Conclusion

A novel 3D braided geometry, which satisfied the space group P4 symmetry, was derived by using the symmetry operation of point group 4. The movement rule of yarns was analyzed, and the process was proved to be feasible. Instead of the traditional four-step braided material, the novel process and the structure will be widely used when the technology is mature. The fiber volume fraction of the novel material was predicted. Its fraction corresponds to the reduction coefficient of the cross-section and the assemble factor. The novel 3D braided material, for which fiber the volume fraction is similar to the traditional ones, could be obtained by optimization design of the geometrical parameters. Using manual production of some prototype machine, a first prototype of the fabrics is produced. The mechanical properties and the equipment of the novel material need to be further studied in the future.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

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Meso-structure and processing of three-dimensional braided material based on space group P 4 symmetry

Abstract

Keywords

A novel three-dimensional braided geometry for space group P4 symmetry

The 3D braided geometry unit satisfied with point group 4 symmetry

The novel 3D braided geometry corresponding to space group symmetry

The unit of 3D braided geometry

A novel 3D braided geometry for space group P4 symmetry

Research on the process of the novel three-dimensional braided material geometry

The geometrical model of the unit in three-dimensional braided material

Basic assumptions

The units of the novel 3D braided material

Geometric parameters for describing the novel 3D braided material

The number of braided material units in a single layer

The geometrical model and fiber volume fraction of the novel 3D braided material

Unit volume U r

The cross-sectional area A m of single yarn on the unit surface

The reduction coefficient λ

The yarn volume and fiber volume of the braided material

The fiber volume fraction of the novel braided material

The fiber volume fraction V f

Experimental verification of the braided process based on space group P4 symmetry

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Unit volume $U_{r}$

The cross-sectional area $A_{m}$ of single yarn on the unit surface

The fiber volume fraction $V_{f}$