Meso-structure and modeling of a new three-dimensional braided tubular material

Abstract

A new three-dimensional braided tubular preform was introduced in this study. The new preform structure can be derived from the representative volume unit which was deduced by the symmetry operations of space group P4. The braiding process of the tubular preform has been discussed. A mathematical model was established to analyze the structural properties of the three-dimensional braided tubular preform. The interrelation of geometrical parameters is analyzed. The fiber volume fraction of the preform was predicted. The new tubular preform was obtained in laboratory to verify the feasibility of the braiding process.

Keywords

Braiding process modeling of tubular preform geometric structure structural properties

Introduction

Three-dimensional (3D) braided composites as a sort of high-performance material have an expansive application, which has high specific strength and stiffness, high damage tolerance and excellent design ability of material properties.¹ Three-dimensional tubular braided reinforcements which are lightweight and advanced textile material are fabricated by a modified flat weaving loom or in some cases by specially designed automated looms and manufactured to a near-net shape to reduce scrap.² The products of 3D tubular braided composites may be involved in the architectural skeleton, transmission shaft, missile’s fuel duct and injection pipe. They are mainly applied in the astronautics, space, marine and medical fields.

The mechanical properties of braided composites largely depend on the meso-structure of reinforcement. It is critical to study the effect of meso-structure on the macroscopic characteristics of braided materials with the mechanical method.³ There are large numbers of geometrical models and mechanical analysis methods for 3D braided composites. The geometrical models and methods can be divided into two categories. One category is established to study the physical properties of different meso-structures. Fabric geometry model (FGM) by Ko and Pastore,⁴ fiber inclination model by Yang et al.⁵ and Wu’s three cell model⁶ belong to this category for theoretical analysis. Wang and Wang presented an analysis method of the topological structure.⁷ Byun and Chou presented the geometric model of unit cells and identified several types of unit cells of two-step and four-step braided composites.⁸ Chen et al. reported the interior cell comprising 12 straight yarns and oriented 45° in the preform cross section.⁹ The other category is mainly to predict the mechanical properties of 3D braided composites. Yang et al. proposed the mechanical model based on the fiber-inclined laminate.⁵ Ma et al.^10,11 presented the elastic strain energy method to facilitate the analysis of elastic properties for 3D braided composites. Wu improved the three-cell model to analyze the mechanical performance,⁶ including bi-modular and elastic-plastic behavior. Liang and Du¹² proposed the equivalent inclusion method to investigate the elastic-plastic problem of composite materials. Finite element analysis is proposed to study the progressive damage and failure analysis of 3D braided composites, in which the elaborated braided structures in meso-scale can be considered.^13–15 All the methods mainly focus on the rectangular braided composites.

Wang and Wang¹⁶ presented an analysis method and a suitable micro-mechanics model to study the spatial yarn distribution in the composite and properties of the components. Ma and Feng³ provided a sort of hexahedral micro-structure unit-cell geometric model to analysis the gradient characteristics in the tubular composites components. Calme et al.¹⁷ implemented an analytical elastic stress modeling of composite cylinders. Xiao et al.¹⁸ developed an analog model to describe the unloading path for the compressively damaged composites. Cagri and Jason¹⁹ presented the curved unit-cell geometry to predict the longitudinal elastic modulus of two-dimensional braided tubular composites. Hiroyuki et al.²⁰ proposed a step-response model to predict the temporary change in the braiding angle on a cylindrical braided preform. Ma et al.²¹ established the unit geometric models corresponding to different zones to describe the characteristics of braided fabric. Hwan et al.²² conducted a spatial spring model to investigate the effective elastic moduli of four-step 3D tubular braided composites. Tuba²³ established a simple versatile 3D model to simulate three-dimensional yarn paths of the braided structures. Wang et al.²⁴ presented a modeling approach based on Free Form Deformation theory focusing on the micro-structure of 3D four-directional braided tubular composites.

In summary, processing methods of 3D braided composite materials are mainly concentrated in four-step method, two-step method and multidimensional braided are based on the four-step method. For optimal performance of 3D braided composites, new braiding process should be developed to obtain the new geometric structures.

To meet the demand for tubular braided composite materials, a new meso-structure of 3D braided tubular material for space group P4 symmetry is elaborated in this study. The main purpose of this work is to find this new material available and valuable. The geometrical feature of this material is analyzed by using the translational symmetry operations of unit for point group ‘4’.²⁵ The new braiding process including the movement rules of orbits and carriers is illustrated in detail. A geometric model for this new tubular preform is established, and the fiber volume fraction is analyzed. To verify the feasibility of the braiding process, the new tubular preform is obtained by experiment.

Geometric structure of the new material

The rectangular braided material for space group P4 symmetry was obtained by transforming the representative volume unit (RVU) as shown in Figure 1. The geometry of this plate material has been analyzed in Ma et al.²⁶

Considering the deformation of the RVU, the units in the tubular model can be divided into three types. It contains the units in the inner surface, outer surface and interior. Figure 2 shows the three kinds of units in the model. The units in the inner surface, outer surface and interior are analyzed. The geometric structure of this material is satisfied with space group P4 symmetry. With the increase of the diametric size, the RVU in the next circular rows can be deduced by the same operation. With the increase of the size, the interior units in different rows are different. In Figure 3, the architecture is obtained by the polar array operation of the units.

Braiding process

The braiding process and distribution of carriers

The new braided structure closely relates to its braiding process. Figure 4 shows a schematic illustration of the circular orbits. These circular orbits are numbered using the word ‘g₁-g₂’. The size of the braided machinery is characterized as follows: let A be the number of rows in circle and B be the number of yarn carriers in each row. The braided array of this machine is expressed as A × B. According to the size of the braided machine, the numbers of rows can be increased to a certain extend. Set R_i stand the mandrel of radius. The parameters A, B and R_i will define the size of the braided machine. The total number N_i of carriers in A x B machine is

N_{i} = A B

(1)

The new tubular preform being braided is hung above the machine bed, on which yarn carriers are arranged in a pattern consisting of circumferential rows ‘I’ and radial columns ‘II’ (as shown in Figure 4). Circles that can appear filled or empty use as different group carriers on a feature comparison chart. The filled circle represents the carriers moving along the circumferential direction and the empty circle represents the carriers of radial direction motion. The original arrangement of these carriers is illustrated in Figure 4.

Braiding steps

The new tubular preform is braided through movements of the yarn carriers along the row and column tracks. The circular orbits of ‘ $g_{3}$ ’, ‘ $g_{5}$ ’ and ‘ $g_{7}$ ’ are keeping stable during the processing. Each machine cycle consists of three movement steps:

At step 1, the braided yarn carriers in ‘ $g_{2}$ ’ and ‘ $g_{6}$ ’ rows move one position along the clockwise paths. On the contrary, the carriers in ‘ $g_{4}$ ’ and ‘ $g_{8}$ ’ rows move in counterclockwise ones.

At step 2, the braided yarn carriers in radial columns ‘II’ move one position vertically in an alternating manner.

At step 3, the braided yarn carriers in ‘ $g_{1}$ ’ and ‘ $g_{9}$ ’ rows move one position along the circle path in an alternating manner.

The carriers return to their original pattern (Figure 4) after the three movement steps. Then, the jamming action is run to tighten the braided structure. For now, most of the jamming actions are completed manually. Figure 5 shows the three-step process for the new tubular material. All the subsequent machine movements are just a repeat of the three-step motion and action aforementioned.

Space structure of yarns

In order to validate the feasibility of the braiding process, the space structure of the tubular preform needs to be studied in this section. As shown in Figure 6, the yarn path in the new tubular preform is illustrated. The topological structure of the ‘II’ yarn is represented by number ‘1–4’. The topological structures of braided yarns in outer surface and inner surface are expressed via 1 and 2 separately. Numbers 3 and 4 show the ‘II’ yarn in the preform. The topological structure of the ‘I’ yarn is represented by numbers ‘5–6’. Repeating the three-step and packing, the new tubular preform can be finally obtained.

Considering the continuity of the braided yarns, the yarn path of the new tubular preform is analyzed. As shown in Figure 7, the braided yarns of group ‘I’ and ‘II’ are strongly emphasized by thick line. Figure 7(a) shows the axial observation of the preform. The amplifier section can illustrate the space structure correctly. Figure 7(b) shows the 3D view of the preform.

The geometric model

The yarn structure of the new tubular preform have been fully described. A mathematical model is introduced to predict its characteristic. Some basic assumptions are proposed in this section. A sort of hexahedral unit geometric model is used; whose performances are variable.

Basic assumptions

In order to simplify the calculation, some basic assumptions are proposed:

In the inner units, each yarn is subjected to the loads from different directions, and the cross section of the yarns can be assumed to be a rhombus.

If the preform is braided with large value of B, the surface layers will be thin compared to the inner; then the influence of the surface layers could be disregarded.

The rectangular unit discussed above can reflect the topological features of 3D new tubular braiding structure.

The topological structure of the yarns in both the rectangular and the tubular units are the same.

The volume of the RVU in rectangular and tubular preform are the same, the braiding pitch h and the reduction coefficient $λ$ remain unchanged.

The cross section of yarns

Due to the loading from four-direction, the cross-sectional shape of internal yarns in the new tubular preform is assumed to be a rhombus (Figure 8). As shown in Figure 8, the length of diagonal in the rhombus is represented by front ‘l’ and ‘m’. The area of the equivalent cross section is $A_{f}'$ ; the area of the yarns in original form is $A_{f}$

{A^{'}}_{f} = \frac{1}{2} l m A_{f} = π r^{2}

(2)

where

r = \sqrt{\frac{Γ}{π ρ}}

(3)

The font ‘r’ represents the diameter of the original yarns. It is determined by the yarn linear density $Γ$ ( $T e x y / 1000$ ) and the density of fiber material $ρ$ ( $g / {cm}^{3}$ ).²⁷

The reduction coefficient $λ$ is introduced to simplify the calculation.

{A^{'}}_{f} = λ A_{f}

(4)

Rectangular unit and tubular unit

Rectangular unit

The RVU of rectangular preform is established before analyzing the characteristics of the tubular preform. As shown in Figure 9(a), the interior unit of the rectangular preform is illustrated.

α

denotes the braiding angle along the circumference,

β

denotes the braiding angle along the radial direction, h denotes the pitch length, L and W denote the length and width in the bottom,

γ

is the interlaced angle which equals to 90°.

2. Tubular unit

There are four parameters to describe the meso-structure of the new 3D braided tubular material: angle $α'$ , $β'$ , $γ'$ and radial interval distance $Δ R$ . In the braiding process, the position of the braiding yarns in the space is stable. So the pitch length h is unchanged from inside out. In Figure 9(b), the symbols $α'$ , $β'$ , $h'$ and $γ'$ represent the parameters of the RVU in tubular material. $Δ R$ denotes the increment in radius, $Δ θ$ denotes the central angle of each unit.

According to the basic assumption, the fiber volume fraction of the tubular unit is equal to the rectangular unit ones.²⁸ Only the orientation angle and the increments in radius are changed in the process.

Geometric model of the tubular composites

According to the basic assumptions, the braided yarns in the new tubular material are simplified. The properties of the tubular unit can be represented by the rectangular unit. In each rectangular unit, the crossection of yarns are represented by quadrangular. Each unit is composed of braided yarns and matrix. In the Cartesian coordinate system, the simplified structure of the rectangular unit is showed in Figure 10(a). The simplified section of the new preform is expressed in Figure 10(b). In order to reflect the actual tubular unit, the simplified section of the rectangular unit should be changed to the sector as shown in Figure 10(c). In order to investigate the properties of the tubular preform, the variable microstructure unit (VMU) approach is adopted.

Unit volume

Ignoring the difference between the surface units and the interior units of the new tubular material, the characterization of the interior units alone would be sufficient. The unit volume of an RVU contains the volume of fiber and matrix. It can be obtained by using zonal divided method

U_{r} = π h (2 R_{i} Δ R + Δ R^{2}) / A, 1 \leq i \leq B

(5)

2. The relationship of geometric parameters

The relationship between the parameters are shown by the following equations

{\begin{matrix} W_{i} = \frac{2 π R_{i}}{A}, l \times m = 2 {A^{'}}_{f}, \tan α = \frac{W_{i}}{h} \\ \tan β = \frac{L^{'}}{h}, W_{i} = 2 l, L^{'} = 2 m = Δ R \end{matrix}

(6)

Equation (7) can be obtained

{\begin{matrix} l = \frac{W_{i}}{2} \\ m = \frac{4 {A^{'}}_{f}}{W_{i}} \\ α = \arctan \frac{W_{i}}{h} \\ β = \arctan \frac{8 {A^{'}}_{f}}{h W_{i}} \end{matrix}

(7)

Put $W_{i} = 2 π R_{i} / A$ into the equation (7). The functions, which describe the yarn cross-section parameters and the variation of the yarn path, can be obtained

l = ξ (R_{i}), m = ζ (R_{i}), α = η (R_{i}), β = μ (R_{i})

(8)

Variable unit thickness in radius is

Δ R = \frac{8 {A^{'}}_{f}}{W_{i}}

(9)

The geometrical shape of the unit is supposed to be a hexahedron. The length of the unit is the arc length and the width is $L'$ . In order to coincide with the actual tubular unit, the rectangular unit should be changed into the actual situation. The geometry of the unit is updated according to the actual situation. According to the basic assumptions, three parameters are updated.³

The updated increment of radius is

Δ R^{'} = \sqrt{R_{i}^{2} + 2 R_{i} Δ R} - R_{i}

(10)

The updated braiding angle $α'$ is

\tan α^{'} = [2 R_{i} \tan (Δ θ / 2) + 1] / h

(11)

The updated braiding angle $β'$ is

\tan β^{'} = (\sqrt{R_{i}^{2} + 2 R_{i} Δ R} - R_{i}) / h

(12)

3. The yarn length $L_{t}$ , the volume of yarns $U_{y}$ and the volume of fiber $U_{f}$

The length of yarns in RVU can be expressed as

L_{t} = 2 \sqrt{W_{i}^{2} + h^{2}} + 2 \sqrt{Δ R^{2} + h^{2}}

(13)

According to the Figure 9, two kinds of yarns are equal to each other in the RVU. The total volume of yarns is

U_{y} = A_{f} L_{t} = {A^{'}}_{f} L_{t} / λ

(14)

The yarn fiber volume fraction is the assemble factor $ε$ . The maximum value is²⁵

ε_{m} = \sqrt{3} π / 6 \approx 0.9064

(15)

The total volume of fiber in the unit is

U_{f} = ε U_{y}

(16)

4. The fiber volume fraction and the matrix fraction

In the braided tubular material, the reinforce phase is composed of the braided yarns. Each yarn is compound of a certain amount of fibers. The fiber volume fraction $V_{f}$ in the representative unit can be deduced as

V_{f} = f (R_{i}) = \frac{U_{f}}{U_{r}} \times 100 %

(17)

Theoretical results

The radius increment $Δ R$

In order to observe the changes in detail, multiple data sets of

A_{f}

are adopted. As shown in Figure 11,

Δ R

decreases with the increase of the radius

R

and tends to be stable.

2. Braiding angle $α$ , $β$

The braiding angle $α$ and $β$ are variable during the process. In Figure 12, the tendency of braiding angles $α$ and $β$ are expressed with the length of radius. Different values of pith length h are introduced to make accurate comparisons. As shown in Figure 12(a), the braiding angle $α$ increases with the radius increasing and tends to be a stable value when the radius is large enough. In Figure 12(b), the braiding angle along the radial direction $β$ decreases with increasing radius and tends to be stable.

3. Fiber volume fraction

As shown in Figure 13, the value of fiber volume fraction is reduced with the radius increasing. Results correspond with the local production practice. The fiber volume fraction has a continuous gradient change in the case of the variable radius. The rationalization of the VMU approach is to be proved.

Experiment of the process feasibility

To validate the feasibility of the new braiding process, the experiment of 3D braided tubular preform is introduced. The preform is braided by 16 × 4 arrays. The number of rows in circle is 16 and the number of yarn carriers in each row is not truly identical.

Flexible material is used to realize the braiding process. To distinguish the space structure of these two group yarns, different color of cable is adopted. The cross-section area of ‘I’ and ‘II’ yarns is 1.5 mm².

The two sets of carrier movement rules have been illustrated in Figure 5. Repeating three-step for several times, a new tubular preform which satisfies the space group P4 symmetry can be obtained (Figure 14). Figure 14 shows the tubular preform in the lab. The yellow yarns represent the yarns in circumferential rows, and the red yarns illustrate the yarns in radial columns. The geometry of the tubular preform is similar to the simulation model, and the yarn path is identical.

Conclusion

This paper aims at introducing the 3D braided tubular preform based on space group P4 symmetry. Analyzing the unit of the rectangular preform, the geometrical structure of the tubular preform was obtained. The rational braiding process is illustrated in detail. The space structure of the new tubular material has been studied to validate the feasibility of the braiding process. RVU, which is closely related to the actual braiding structure, was introduced to establish the geometric model. To simplify the calculation, some basic assumptions were proposed. VMU approach was introduced to investigate the meso-structure properties of the tubular preform. The interaction of all parameters was analyzed. It is found that the fiber volume fraction of this new tubular preform is similar to 3D four-direction materials in theoretical analysis. The braiding process has been proved to be feasible using flexible material to realize. The new braided material needs to be further studied regarding the mechanical properties.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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