CN102041635B - Three-dimensional woven material based on space group P* - Google Patents

Abstract

The invention relates to a space group based

Symmetrical three-dimensional braided material, the braided geometric structure of the braided material is a continuous yarn structure extended in three-dimensional space, and the yarn segment in the representative volume unit in the braided geometric structure satisfies the spatial point group

Described by the symmetry of the points, the braided structure as a whole presents the representative volume unit with the space group

The described translational symmetry operation performs translation to obtain a three-dimensional woven fabric interwoven with each other in three-dimensional space. The braided material of the present invention satisfies the point group

The representative volume unit of symmetry is derived from the basic structural unit to satisfy the space group Symmetrical new three-dimensional weaving geometric structure, through the study of its technological feasibility and the prediction of the fiber volume percentage of the corresponding three-dimensional weaving, a new three-dimensional weaving material with excellent geometric structure and performance has been obtained. The weaving material array adopts a regular hexagon, which makes the movement track of the yarn carrier easy to realize and easy to organize production.

Description

Based on the symmetric three-dimensional knitting material of space group P*

Technical field

The present invention relates to a kind of three-dimensional knitting material, relate in particular to a kind of based on space group

Symmetric three-dimensional knitting material.

Background technology

D braided composites obtains the extensive concern of the industries such as Aero-Space, national defence and medical treatment with its excellent mechanical property.Because be subject to processing the restriction of the factors such as technique, the kind of D braided composites is very few, working (machining) efficiency is low and process costs is high.Obtain the better D braided composites of combination property, be badly in need of the more 3 D weaving technique of exploitation.At present most researchs concentrate on four step rule and two kinds of braid methods of two step method, and the research of the prediction of relevant three dimensional knitting method also is in the starting stage.

The crystal of different lattice structures shows different performances, adopts the Crystals in Symmetry group crystal geometry can be classified.With reference to the research method of symmetric group, to the in addition inductive research of the cell geometry of braided material, the symmetry operation of describing according to space group and space group pushes away to get a large amount of new three-dimensional yarn crossovers method.Yet technological feasibility and performance study as new three-dimensional knitting material not yet launch.From the meeting spatial point group

(

Be the HM symbolic notation), deriving on the basis of all symmetry operation obtains a kind of unit cell of new yarn crossovers geometry.This unit cell is put into the space lattice that satisfies it can obtain the continuous three-dimensional yarn crossovers geometry of yarn.

Summary of the invention

It is a kind of based on space group that task of the present invention is to provide Symmetric three-dimensional knitting material.

Technical program of the present invention lies in be: based on space group

Symmetric three-dimensional knitting material, the braided geometry of this braided material is for extending the structure that is continuous yarn that is woven into, the yarn line segment meeting spatial point group in the representative volume unit in the braided geometry at three dimensions

The symmetry of the point of describing, braided structure presents on the whole with representative volume unit space group

The symmetrical behaviour of the translation of describing carries out be interweaved together 3 D weaving fabric of three dimensions that translation obtains.

Described have a point group

Symmetric representative volume unit, it is derived based on point group

The symmetry operation described of group element be

(n is 1～6) can be expressed as

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}={\mathrm{<figure-callout\; id="6z"\; label="c"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">c</figure-callout>}}_{6z}{\mathrm{\&sigma;}}_{\mathrm{xy}}=[\frac{x}{2}-\frac{\sqrt{3}}{2}y,\frac{\sqrt{3}}{2}x+\frac{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">y</figure-callout>}}{2},z]$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^2(x,y,z)=[-\frac{x}{2}-\frac{\sqrt{3}}{2}y,\frac{\sqrt{3}}{2}x-\frac{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">y</figure-callout>}}{2},z]$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^3(x,y,z)=(-x,-y,z)$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^4(x,y,z)=[-\frac{x}{2}+\frac{\sqrt{3}}{2}y,-\frac{\sqrt{3}}{2}x-\frac{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">y</figure-callout>}}{2},z]$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^5(x,y,z)=[\frac{x}{2}+\frac{\sqrt{3}}{2}y,-\frac{\sqrt{3}}{2}x+\frac{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">y</figure-callout>}}{2},z]$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^6(x,y,z)=E$

Expression is transformed to the symmetry operation of putting behind the equal sign with the point (x, y, z) on a certain yarn line segment axis.

Described space group The symmetrical behaviour of the translation of describing is in the xyz coordinate system, with point group

Described yarn line segment is combined in and makes following mode in the three dimensions

T _i=ux _i+ vy _i+ wz _i(u, v, w are base vector)

The translation symmetry operation.

The yarn line segment combination of described representative volume unit has z to thickness t, reaches in the parallel with it bundle of planes space group of representative volume unit at the xoy coordinate surface

The symmetrical behaviour of the translation of describing space group

The symmetrical behaviour of the translation of describing space group

The symmetrical behaviour of the translation of describing space group

The symmetrical behaviour of the translation of describing cross section is hexagonal section, the translation of representative volume unit is with the cornerwise integral multiple translation of the hexagonal section of positive six prisms, along z to the integral multiple translation with 2t, the corresponding dot matrix in representative volume unit is simple hexagonal lattice in the three-dimensional knitting material, and final formation has space group

Symmetric a kind of new space continuous yarn intersection geometry.

Described each yarn line segment combination unit cell has surperficial representative volume unit, representative volume unit, bight and inner representative volume unit, individual layer knitted body representative volume unit sum

N＝＝3n ²-3n+1 (1)

Representative volume unit, bight N _cSpace group

The symmetrical behaviour of the translation of describing space group

The symmetrical behaviour of the translation of describing space group

The symmetrical behaviour of the translation of describing space group

The symmetrical behaviour of the translation of describing space group

The symmetrical behaviour of the translation of describing and facial representative volume unit number N _fBe respectively

N _c＝6，N _f＝6(n-2) (2)

Inner representative volume unit number N _iFor

N _i＝N-N _f-N _c (3)

N is regular hexagon limit, braided fabric cross section representing gonosome product unit number.

Described each establishment yarn line segment combination center is provided with axial yarn.

Braided material of the present invention is to satisfy point group

Symmetric representative volume unit is the meeting spatial group that basic structural unit is derived

Symmetric new 3 D weaving geometry by to the research of its technological feasibility with to the prediction of the fiber volume percentage composition of corresponding three-dimensional braid, has obtained the new three-dimensional knitting material kind of geometry and function admirable.This braided material array has adopted regular hexagon, makes the movement locus of taking the yarn device be convenient to realize, easily tissue is produced.

Description of drawings

Fig. 1 is the front view of braided material section of the present invention;

Fig. 2 is the right view of Fig. 1;

Fig. 3 is the top view of Fig. 1;

Fig. 4 is the stereogram of Fig. 1;

Fig. 5 is the three-dimensional cutaway view of Fig. 1;

Fig. 6 is the structural representation of minimum yarn line segment among Fig. 1;

Fig. 7 is the yarn line segment combination single cell structure schematic diagram by yarn line segment translation design shown in Figure 6;

Fig. 8 is the residing hexagonal-lattice schematic diagram of each unit cell in the braided material;

Fig. 9 is each unit cell regional structure schematic diagram of living in the braided material;

Figure 10 is surperficial single cell structure schematic diagram;

Figure 11 is bight single cell structure schematic diagram;

Figure 12 is inner single cell structure schematic diagram;

Figure 13 is the view that interweaves of the part braiding yarn that surrounds of three adjacent yarn;

Figure 14 is that the yarn braid angle of braided geometry is the structural representation of β;

Figure 15 is the variation diagram of the deformation in the braid angle of the fiber percentage composition of braided fabric and weaving textile yarn and cross section.

The specific embodiment

Concrete structure of the present invention is described below:

1, yarn line segment combination single cell structure is described

Space group

Group element be

(n is 1～6), group's generator is s ₆Corresponding symmetry operation can be expressed as

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}={\mathrm{<figure-callout\; id="6z"\; label="c"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">c</figure-callout>}}_{6z}{\mathrm{\&sigma;}}_{\mathrm{xy}}=[\frac{x}{2}-\frac{\sqrt{3}}{2}y,\frac{\sqrt{3}}{2}x+\frac{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">y</figure-callout>}}{2},z]$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^2(x,y,z)=[-\frac{x}{2}-\frac{\sqrt{3}}{2}y,\frac{\sqrt{3}}{2}x-\frac{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">y</figure-callout>}}{2},z]$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^3(x,y,z)=(-x,-y,z)$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^4(x,y,z)=[-\frac{x}{2}+\frac{\sqrt{3}}{2}y,-\frac{\sqrt{3}}{2}x-\frac{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">y</figure-callout>}}{2},z]$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^5(x,y,z)=[\frac{x}{2}+\frac{\sqrt{3}}{2}y,-\frac{\sqrt{3}}{2}x+\frac{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">y</figure-callout>}}{2},z]$

${\mathrm{<figure-callout\; id="6z"\; label="s"\; filenames="HSA00000397628600023.png"\; state="\{\{state\}\}">s</figure-callout>}}_{6z}^6(x,y,z)=E$

Expression is transformed to the symmetry operation of putting behind the equal sign with the point (x, y, z) on a certain yarn line segment axis.

There is point group in the yarn line segment that is in position shown in Figure 6 in yarn line segment combination shown in Figure 7 and the coordinate system

Symmetry operation relation.This combination pattern is as the representative volume unit (abbreviation unit cell) of new 3 D weaving geometry.

2 meeting spatial groups

Symmetric 3 D weaving geometry

2.1 corresponding space group

The simple hexagonal lattice in space

The corresponding simple hexagonal lattice (as shown in Figure 8) of the space lattice that the Crystals in Symmetry group describes, space group

Coordinate with it.To be expressed with a lattice point by the representative volume unit (as shown in Figure 7) that this point group is derived, put into hexagonal lattice.Consider the continuity of yarn, namely obtain a kind of meeting spatial group

Symmetric new space continuous yarn intersection geometry (as shown in Figure 4).

Consider the feasibility that technique realizes, the z in above-mentioned figure is to having added axial yarn.

2.2 space group

Corresponding structure (obtaining by translation representative volume unit)

Set up coordinate system xyz, in three-dimensional translation symmetry operation be

T _i=ux _i+ vy _i+ wz _i(u, v, w are base vector)

The yarn line segment combination shown in Figure 7 through translation can obtain Fig. 1 to three-dimensional yarn crossovers geometry shown in Figure 5.

If the combination z of individual layer yarn line segment shown in Figure 7 is t to thickness.At the xoy coordinate surface and with it in the parallel bundle of planes, the translation of unit cell is the cornerwise integral multiple translation of hexagonal section of positive six prisms that relied on point group yarn changing section, along z to the integral multiple translation with 2t.

2.3 space group

Corresponding new 3 D weaving geometry

Use space group

Symmetry operation push away to get the unit cell of this new 3 D weaving geometry, with the translation symmetry operation this unit cell is carried out translation, can obtain possible 3 D weaving interior geometry.Consider the continuity requirement of border yarn in the actual braiding process, and its rule is studied, and then obtain a kind of brand-new 3 D weaving geometry (as shown in Figure 4).

3, the geometrical analysis model of new three-dimensional knitting material

New 3 D weaving geometry is expected to be used to the prefabricated component of making new D braided composites material.As a kind of new three-dimensional knitting material, its performance prediction is the important content of basic research.

3.1 basic assumption

(1) cross section of fabric braiding yarn is assumed to be ellipse, and two-semiaxle is respectively a/2, b/2;

(2) z is regular hexagon to yarn cross section, can be covered with the hexagon space with the extruding of braiding yarn;

(3) all braiding yarns have identical performance, do not consider to weave the factor such as yarn physical damnification and the difference of the performance that causes;

(4) geometry with stable and consistent of inner, the surface of braided structure and folding corner region.

3.2 the zone of 3 D weaving fabric is divided and unit cell

Braided fabric is divided into interior zone, surf zone and folding corner region.Corresponding unit cell is called inner unit cell (Interior Unit), surperficial unit cell (Face Unit) and bight unit cell (Corner Unit) (as shown in Figure 9).

Inner unit cell is cut apart by shown in Figure 12; Surface unit cell and bight unit cell are cut apart shown in Figure 10,11.

3.3 describe the geometric parameter of 3 D weaving fabric

(1) the unit cell number of individual layer braided fabric

If n is unit cell number on the regular hexagon limit, braided fabric cross section.Individual layer knitted body unit cell sum

N＝＝3n ²-3n+1 (1)

Bight unit cell N _cCount N with facial unit cell _fBe respectively

N _c＝6，N _f＝6(n-2) (2)

Inner unit cell is counted N _IFor

N _i＝N-N _f-N _c (3)

(2) braiding yarn and the axially geometrical relationship of yarn

By pass of trying to achieve between axial yarn and the knitting yarn geometric parameter shown in Figure 13 be

$R=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">b</figure-callout>}}^2+3a^2}-\frac{\sqrt{3}}{3}a---\left(4\right)$

(3) braid angle β

The yarn braid angle of describing braided geometry is β, can be got by geometrical relationship shown in Figure 14

$\mathrm{tg\&beta;}=t/(R+\frac{a}{\sqrt{3}})---\left(5\right)$

Pushed away by formula (4) and (5)

$t=\mathrm{tg\&beta;}\sqrt{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">b</figure-callout>}}^2+3a^2}---\left(6\right)$

(4) cross-sectional area A of 3 D weaving fabric and one deck unit-cell volume U

If the length of side of the regular hexagonal section of axial yarn is R; The cornerwise length in the cross section of three-dimensional knitting material is D, and the thickness of individual layer unit cell is 2t.Can get

$D=2D_{d}R+(N_d+3)a/\cos \frac{\mathrm{\&pi;}}{6}2(N_{d}R+\frac{N_d+3}{\sqrt{3}}a)---\left(7\right)$

$A=\frac{\sqrt{3}}{2}{(5\sqrt{3}R+8a)}^2---\left(8\right)$

The cumulative volume of 3 D weaving fabric one deck unit cell is

$U=2\mathrm{At}=\sqrt{3}{(5\sqrt{3}R+8a)}^{2}t---\left(9\right)$

(5) volume of yarn calculates in the 3 D weaving fabric

Being located at thickness is in the 2t braided fabric, and axially the volume of yarn is U _CyThe volume of inner single unit cell braiding yarn is

The volume of facial single unit cell braiding yarn is

The volume of the single unit cell braiding in bight yarn is , can push away

$U_{\mathrm{cy}}=N\&times;6\&times;\frac{1}{2}\&times;\frac{\sqrt{3}}{2}{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">R</figure-callout>}}^2\&times;2t=57\sqrt{3}{R}^{2}t---\left(10\right)$

Can get simultaneously

$U_{\mathrm{by}}^i=6\mathrm{\&pi;<figure-callout\; id="2"\; label="ab\; t"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">ab</figure-callout>}\sqrt{t^{}}\sqrt{{}^2+\frac{4}{3}{a}^2},$ $U_{\mathrm{by}}^f=7\mathrm{\&pi;<figure-callout\; id="2"\; label="ab\; t"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">ab</figure-callout>}\sqrt{t^{}}\sqrt{{}^2+\frac{4}{3}{a}^2},$

$U_{\mathrm{by}}^c=8\mathrm{\&pi;<figure-callout\; id="2"\; label="ab\; t"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">ab</figure-callout>}\sqrt{t^{}}\sqrt{{}^2+\frac{4}{3}{a}^2}---\left(11\right)$

The cumulative volume U of knitting yarn in the individual layer unit cell braided fabric _By

$U_{\mathrm{by}}=N_I{U}_{\mathrm{by}}^l+N_F{U}_{\mathrm{by}}^F+N_C{U}_{\mathrm{by}}^C=132\mathrm{\&pi;<figure-callout\; id="2"\; label="ab\; t"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">ab</figure-callout>}\sqrt{t^{}}\sqrt{{}^2+\frac{4}{3}{a}^2}---\left(12\right)$

The cumulative volume U of yarn in the individual layer unit cell braided fabric _y

$U_y=U_{\mathrm{by}}+U_{\mathrm{cy}}=132\mathrm{\&pi;<figure-callout\; id="2"\; label="ab\; t"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">ab</figure-callout>}\sqrt{t^{}}\sqrt{{}^2+\frac{4}{3}{a}^2}+57\sqrt{3}{R}^{2}t---\left(13\right)$

(6) the fiber volume percentage composition of 3 D weaving fabric:

$V=\frac{U_y}{U}\&times;100\%---\left(14\right)$

Formula (9) and (13) substitution following formula are got

$V=\frac{\mathrm{\&pi;}(44\mathrm{\&pi;<figure-callout\; id="2"\; label="ab\; t"\; filenames="HSA00000397628600023.png,HSA00000397628600042.png"\; state="\{\{state\}\}">ab</figure-callout>}\sqrt{t^{}}\sqrt{{}^2+4a^2/3}+19\sqrt{3}{R}^{2}t)}{2{(5\sqrt{3}R+8a)}^{2}t}---\left(15\right)$

By formula (5), (6) and (15) can push away

V＝f(a，b，β) (16)

Make λ=a/b, can get V=f (λ, β).

(7) relation of braided geometry and fiber volume percentage composition

The oval cross section that the braiding yarn of same cross section is corresponding different with the difference of urgent degree, the fiber percentage composition of braided fabric is relevant with the deformation in the braid angle of weaving textile yarn and cross section.

Get ab=1, a/b ∈ [1/4,4] can get the fiber volume percentage composition V of 3 D weaving fabric shown in Figure 15 with the Changing Pattern of λ and β.

Upper figure can find out, the excursion of fabric fibre percentage composition is 34～68%.When λ value was constant, the fabric fibre percentage composition increased with the increase of angle of weave β, and amplitude of variation is not obvious.And the variation that shrivels coefficient lambda=a/b of yarn is very large on its percentage composition impact; As seen, the knitting yarn in the braiding process z to press more " flat " then more might form fine and close braided fabric.

Claims (6)

1. Based on space group Symmetrical three-dimensional braided material, characterized in that the braided geometric structure of the braided material is a continuous yarn structure extended and woven in three-dimensional space, and the yarn segments in the representative volume unit in the braided geometric structure satisfy the spatial point group

Described by the symmetry of the points, the braided structure as a whole presents the representative volume unit with the space group

The described translational symmetry operation performs translation to obtain a three-dimensional woven fabric intertwined with each other in three-dimensional space, and the fiber percentage of the three-dimensional woven fabric varies from 34% to 68%. 2. The three-dimensional braided material according to claim 1, wherein said point group

Symmetric representative volume elements whose derivation is based on point groups

The symmetry operation described by the group elements of is

(n is 1~6), it can be expressed as ${\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; z</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; z</span>}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; xy</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>\frac{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; the\; y</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; ]</span>$ ${\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; z</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; the\; y</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; ]</span>$ ${\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; z</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span>$ ${\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; z</span>}}^{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; the\; y</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; ]</span>$ ${\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; z</span>}}^{\mathrm{<span\; class="notranslate">\; 5</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>\frac{\mathrm{<span\; class="notranslate">\; x</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; the\; y</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; ]</span>$ ${\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; z</span>}}^{\mathrm{<span\; class="notranslate">\; 6</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; E.</span>}$

Indicates the symmetric operation of transforming the point (x, y, z) on the axis of a certain yarn segment into a point after the equal sign. 3. The three-dimensional braided material according to claim 2, wherein the space group

The translational symmetry operation described is in the xyz coordinate system, the point group

The yarn segments are combined in three-dimensional space in the following manner T _i =ux _i +vy _i +wz _i (u, v, w are base vectors) translational symmetry operation. 4. The three-dimensional braided material according to claim 3, wherein the yarn segment combination of the representative volume unit has a z-direction thickness t, and in the xoy coordinate plane and the plane clusters parallel thereto, the representative volume unit space group

The translational symmetry operation space group described The translational symmetry operation space group described

The translational symmetry operation space group described The described translational symmetry operation section is a hexagonal section, and the translation of the representative volume unit is an integral multiple of the diagonal line of the hexagonal section of a regular hexagonal prism, and an integral multiple of 2t along the z direction. The representative volume of the three-dimensional braided material The lattice corresponding to the unit is a simple hexagonal lattice, and finally forms a space group

A new space-continuous yarn crossing geometry for symmetry. 5. The three-dimensional braided material according to claim 4, characterized in that, each of the combined unit cells of the yarn segments has a surface representative volume unit, a corner representative volume unit and an internal representative volume unit, and the single-layer braided body Total number of representative volume elements N==3n ² -3n+1 (1) Corner representative volume unit N _c space group

The translational symmetry operation space group described

The translational symmetry operation space group described

The translational symmetry operation space group described The translational symmetry operation space group described

The described translational symmetry operation and the number of facial representative volume units N _f are respectively N _c =6, N _f =6(n-2) (2) The number of internal representative volume units N _i is N _i =NN _f -N _c (3) n is the number of representative volume units on a side of a regular hexagon in the cross-section of the woven fabric. 6. The three-dimensional braided material according to claim 5, characterized in that, an axial yarn is arranged at the center of each braided yarn segment combination.

Images