1

László M. Vas 1

Polymer Composites I.Fibrous Reinforcements

BMEGEPT9110, 2+0+0 lecture, 3 cr.

Lecturer: Prof. Dr. László M. Vas

Budapest University of Technology and EconomicsDepartment of Polymer Engineering

1. General Properties of Fibrous Structures

2016.10.27.

2016.10.27. 2

Literature

Sources

1. Chou T.-W. and Ko F.K. (edited by): Textile Structural Composites. CompositeMaterials Series 3. Elsevier, New York, 1989.

2. Vas L.M.: Textiltermékek tervezése. Szerkezeti és makrotulajdonságok. BME PT Tanszék, Bp. 2000.

3. Stoyan D. und Mecke J. Stochastische Geometrie – eine Einführung. Akademie-Verlag, Berlin, 1983.

4. Zurek W.: The Structure of Yarn. Warsaw (Poland), Springfield (USA), 1975.

5. Hearle J.W.S, Thwaites J.J., and Amirbayat J. (editors): Mechanics of Flexible Fiber Assemblies. Sijthoff&Noordhoff, (NATO ASI Series) Alphen a.d. Rijn (Ned.), Germantown (USA), 1980.

Literature recommended

6. Vas L.M.: Idealized statistical fiber bundle cells and their application to modeling fibrous structures andcomposites (in Hungarian), (DSc), Budapest 2007. (http://pt.bme.hu/~vas/HAS_DSc_Thesis/)

7. Bolotin V.V.: Statisztikai módszerek a szerkezetek mechanikájában. Műszaki Könyvkiadó Bp. 1970.

8. Álló G., Főglein J., Hegedűs Gy.Cs., Szabó J.: Bevezetés a számítógépes képfeldolgozásba. Kézirat. BME MTKI. Bp. 1993.

9. Neckar B. and Ibrahim S.: Structural Theory of Fibrous Assemblies and Yarns. TU of Liberec, 2003.

10. Vetier A.: Szemléletes mérték- és valószínűségelmélet. Tankönyvkiadó Bp. 1991.

11. Gibson R.F.: Principles of Composite Material Mechanics. McGraw-Hill, New York, 1994.

12. Wulfhorst B.: Textile Fertigungsverfahren . Eine Einführung. Carl Hanser Verlag, München, 1998.

22016.10.27. László M. Vas

2

3László M. Vas

Multiphase structures

Phase morphology of multiphase structures – in the case of

two components

Multiphase composed material structures:

• Polymer blends, alloys

• Filled polymers

• Composites: reinforced, fiber reinforced structures

2016.10.27.

4László M. Vas

Reinforced composite

Composites*:

Multiphase (its parts are separated by phase boundaries), composed (is built up of

several types of material) structural material, the components of which are:

- reinforcement (fibrous structure of usually high strength and modulus ) and

- matrix (embedding material of smaller strength but high toughness ),

and it is characterized by the excellent connection (adhesion, high interfacial

strength) between fiber and that is maintained at high levels of deformation

and load.

*Czvikovszky T., Nagy P., Gaál J.: A polimertechnika alapjai. Műegyetemi Kiadó, Budapest, 2000. 368. old.

2016.10.27.

3

2016.10.27. 5

Composites

Creating composite materials

• Metals (M)• Ceramics (C)• Polymers (organic) (P)• Composites created

M→M: steel fiber →Al (MMC – Al-foam composite);

C →C: glass fiber →cement (CMC – glass-concrete);

P →P: PET-fiber →PVC (PMC – roof-membrane)

M →C: steel →concrete (steel-concrete); P →C: cellulose fiber →clay (adobe)

C →M: ceramic →Al (ceramic foam comp.); M →P: steel →rubber (steel-radial tyre)

C →P: glass fiber →UP (UP resin comp.); P →M: ??? (carbon/PBO fiber+metallic

foam???)

László M. Vas

Composite:

X(fiber)→Y(matrix)

6László M. Vas

Composite structures

Fiber Matrix

Glass fiber

Carbon fiber

Aramide (Kevlar™), PBO (ZylonTM) fiber

Boron fiber

Ceramic fiber

Natural fiber

Thermoplastics

Duromer

Elastomer

CeramicsMetals

*Czigány T.: Polimer kompozitok. Előadások. BME Polimertechnika Tanszék, Budapest, 2009.

Short fiber Filament fiber

ordered

disordered

ordered

disordered

Material combinations:

Fiber-orientation combination:

2016.10.27.

4

7László M. Vas

Producing reinforcing structures,

reinforcements

Indirect reinforcing (regular or irregular textile

reinforcements):

Direct reinforcing (irregular reinforcing structure):

Composite

production

(Mixing)

Composite

production

(Embedding)

Textile

production

Composite

CompositeReinforcing

textiles

Fibers

Fibers

Matrix

Matrix

2016.10.27.

8László M. Vas

Textile production and textiles

Textiles:

The primary output products of the textile industry, which are fibrous

materials made of fibers by textile technological operations such as

•spinning /forming of yarns/threads/rovings (opening /carding (to

flocks/fibers), ordering, uniting, drawing/drafting, twisting), and

•producing of fabrics/textile sheets (web-forming, weaving, knitting,

braiding).

2016.10.27.

5

9László M. Vas

Human- and technical textiles

Human textiles• Clothing/garment textiles (work-, free-time-, and fashion-textiles);

• Home/Domestic-textiles (carpet, curtain, table-cloth, coverlet, bedcloths, etc.);

Technical textiles• Composite reinforcements;

• Traffic vehicles (covering, carpet), transportation;

• Indutech (industrial textiles: filter fabrics, lifting textiles, conveyor belts);

• Buildtech (construction textiles);

• Geotech (geotextiles);

• Agrotech (agro-textiles);

• Oeko/Ekotech (environmentally friendly textiles)

• Space research (spacesuits);

• Army/Military (camouflage, bulletproof vests)

• Protech (protective textile – personal/object e.g. against fire);

• Sporttech (sports textiles);

• Packtech (packaging textiles);

• Medtech (medical textiles)

2016.10.27.

10László M. Vas

Structural

graph of

textiles

FiberYarnSheet

2016.10.27.

6

11László M. Vas

Chapters for Fibrous Reinforcements

General properties of fibrous structures; classification, structure; properties of fibers; dimension, skeleton-space, density- and porosity-properties.

Fiber flows and fiber bundles, fiber flow types; fiber-diagrams, staple (hauteur) and beard (barbe) diagrams. SSTM fiber flow and the Martindale inequality.

Textile sheets of irregular structure; Poisson fiber mat model, linear vicinity, blindespot and pore size, properties of fibers intersecting a convex sample, area density; mechanical properties, deformation of fibers, energy based equations.

Bundle structure of textile samples, effect of gauge length, idealized statistical fiber bundles and expected tensile force processes. Estimation of tensile strength by Peirce.

Twisted structures, twist count, helix model, effect of twisting; Spun and filament yarns, thread, ropes; Probability of breakage, selecting yarn more suitable to given load.

Textile sheets of regular structure Binding cell, binding elements. Woven, knitted and braided structures. Description of regularity by plain patterns.

Structure and geometry of woven fabrics, basic weaves and their relation. Derived weaves and special technical fabrics.

Structure and geometry of knitted fabrics, special binding elements, knitted sheets of weft- and warp systems, reinforced structures. Properties of relaxed fabrics.

Strength properies of textile sheets and membranes. Evaluation of tensile test results. Linear orthotropic, monotropic and isotropic sheet models. Basics of designing technical textiles, layered- and cell-models. Special mechanical tests.

Nonlinear models of sheets. Kawabata’s models for woven and knitted fabrics. Bundle model of woven fabric sample. 3D reinforcing structures and applications.

2016.10.27.

2016.10.27. 12László M. Vas

Fiber types applied to reinforcements

NATURAL FIBER:

Plant origin: (bast-, seed-, leaf-, fruit-, stalk-fibers)

Bast fibers: flax, hemp, jute

Animal origin: (wool or hair, worm/insect-silk, sea/shell-silk)

Gland secretions: natural silk (silkworm) (chord); <spider-silk>

Mineral origin:

Asbestos!!!

MAN-MADE FIBERS:

Natural based:

Plant origin: viscose (cord); <chitin, protein>

Mineral origin: basalt

Artificial based (synthetic):

Organic polymer: HPPE, polyester, polyamide, aramid (Kevlar),

PBO (Zylon)

Inorganic polymer: glass fiber, carbon fiber, ceramic fiber

7

13László M. Vas

Basic properties and types fibers

Fibrous structures: 1D, 2D, 3D

Linear density: q=m(l)/l, 1 tex=1 g/km =1 mg/m

Slenderness ratio: λ=l/d Definition of textile fiber: 1D, λ=1000…5000…,

and they can be manufactured by textile technologies

Strength properties of fibers: specific strength [N/tex], breaking length [km]

Types of textile fibers:

• Filament – mono-

and multifilament

(length undefined)

• Staple/short-fiber

(length well defined)

2016.10.27.

14László M. Vas

Geometrical properties of fibers 1.

Cross section of fibers• Convex shaped

• Concave shaped

• Hollow

Density properties of fibers• Volume and linear density

Properties of fiber length• Arc-length, chord-length, extent (projected length)

• Statistical properties (mean, standard deviation, beard-diagram, beard-length, short- and long fiber content,…)

Types of fiber shape• Straight, wavy, crimped, curly/kinky, hooked/looped/coiled – wavy shaped fibers, waviness/crimp

Properties of fiber surface• Even, uneven/rugged, grooved, slotted, pitted/broken, segmented

2016.10.27.

8

15László M. Vas

Geometrical properties of fibers 2.

Cross section of fibers

Homogeneous fibers:

• Convex, concave, hollow

Inhomogeneous fibers:

• Bilateral (a), core/sheath (b),

fiber/matrix (c)

Natural fibers

Artificial/Man made fibers

Recently: application of

hollow glass fibers to self-

curing composites

2016.10.27.

16László M. Vas

Geometrical properties of fibers 3.

Density properties of fibers:

Ultra-rough: > 10 dtex > 100 µm

Rough: 5…10 dtex 22…100 µm

Normal, medium 2…5 dtex 15…22 µm

Fine: 1…2 dtex 10…15 µm

Microfibers: 0,1…1 dtex 3…10 µm

Ultrafine: < 0,1 dtex 0,5…3 µm

Nanofibers < 0,01 dtex < 500 nm

Volume density, porosity

Linear density

Fiber Relativepore

volume%

Meanporesize

[nm]

CottonRamiWoolNatural silkViscoseAcetatePolyamide

2

7562

8…13,56556

Fiber fineness Lin. dens. Diameter

2016.10.27.

9

17László M. Vas

Mechanical properties 1.Calculated strength properties of textile fibers

Specific tensile force (Q)

Specific breaking force (Qs)

Tensile stress (σσσσ)

Tensile- (σσσσB) and breaking strength (σσσσS)

Initial tensile stiffness (K)

Specific initial tensile stiffness (κκκκ)

K=AE[N]

Initial tensile modulus (E)

Formulae of Hooke’ law (tensile load andsmall deformation)

F =KεQ = κεσ = Eε

Breaking length (R)

2016.10.27.

18László M. Vas

Breaking length of technical fibers

Weak LDPE foil Super-strong (high

performance)

HPPE: 400 km

Aramid (Kevlar):

235 km

Zylon (PBO): 450 km(E=270 GPa; σ

B=5,8 GPa

Tb=650oC; LOI=68)

Steel: 25-35 km(E=210 GPa,

σB=1,9 GPa; T

o=1425oC)

2016.10.27. www.dsm.com

Mechanical properties 2.

10

19László M. Vas

Mechanical properties 3.

2016.10.27.

Fiber paradoxes

Material Tensile strength, σσσσB[MPa]

Bulk form Fiber form Theoretical max.

Aluminum (Al)Iron, steel (Fe)

6001400

8004100

380011200

Polyethylene (HDPE)Polyethylene (HPPE)Polyamide (PA)Aramid (Kevlar)

303080-

10002000-3500

8503000

25000250002500025000

CarbonGraphite

(100)(100)

300020000

3500035000

GlassCeramics (Al2O3)

(100)200

40001600

1100026000

(1) Solid paradox: The tensile strength (σ

B) of fiber shaped materials is greater than that in bulk form, but less

than the theoretical one :

20László M. Vas

Mechanical properties 4.

2016.10.27.

Fiber paradoxes

(2) Fiber shape paradox:

While increasing the fiber diameter (d) the tensile

breaking force (FB) increases, but the tensile

strength (σB) decreases, that is if d1<d2 :

(3) Fiber length paradox:

While increasing the fiber gauge length (lo) the mean

tensile breaking force decreases, that is if lo1<lo2 :

11

21László M. Vas

Mechanical properties 5.

2016.10.27.

Fiber paradoxes

(4) Paradox of fiber blends :

Certain strength properties (X=S) of fiber

blends or hybrid fiber reinforced composites

may be better than those of the components.

Hence, if Si is the ith (i=1;2) component, S(α) is

the considered strength property of blend,

where α1=α and α2=1-α are the volume or

mass fraction of components, then for certain

blend ratios (0<α<1) it can stand that:

(Synergetic or hybrid effect)

22László M. Vas

Mechanical properties 6.

2016.10.27.

Fiber paradoxes

(5) Fiber bundle-paradox:

Because of the consecutiveness of the breakages taking place in a fiber bundle of n

fibers subjected to tensile load the measured bundle breaking force represented by the

maximum tensile force (Fn,max) normalized by the number of fibers (n) is smaller than

the mean tensile force of single fibers (FS). Hence the utilization factor of fiber

strength in the fiber bundle (ηn) can be defined :

12

23László M. Vas

Fibers, fiber models 1.

Fiber shape properties

Straight (a), wavy (b), hooked (c) and

coiled (d) fiber shapes Arc length (lo) and chord length (l), chord

center (C) and extent (l1) of fibers of

different shapes

l1=l l1>l

Waviness/crimp #1:

Waviness/crimp #2:

2016.10.27.

24László M. Vas

Fibers, fiber models 2.

Description of fiber curve/path, fiber surface

Center line of fiber and its

accompanying vector triple

Fiber as point set:

S = P(x,y,z)∈R3: r(s)=(x(s),y(s),z(s))∈Ck, s∈[so,so+lo]

Centerline vector-function:

r(s) = r(s;ω), ω∈Ω, so=so(ω), lo= lo(ω)

Vector of fiber surface point:

r(s,ϕ) = ro(s) + R(s,ϕ)[no(s)cosϕ + bo(s)sinϕ]

Chord length of fiber:

Fiber mass (q – linear density)

Possible stochastic variables (ω):

• starting point of fiber (so)

• fiber length (lo)

• fiber shape (r)

(tangent, normal, binormal)

2016.10.27.

13

25László M. Vas

Fibers, fiber models 3.

Fiber orientation

Vector polygon approximating the

fiber curve

Orientation I: Defined by unit chord vectors of chain elements

Isotropic Uniaxial Biaxial

(planar)

Endpoints of fiber segment unit vectors

(ei) on the unit sphere:

Unit vector of ai is ei

2016.10.27.

Bodor G.-Vas L.M. Polimer anyagszerkezettan

Műegyetemi K. Bp. 2000.

26László M. Vas

Fibers, fiber models 3a.

Fiber orientation

Measurement and evaluation using gradient method

E.g. fibers of a glass fiber mat (machine direction: 90o)

Orientation I: Orientation angle distribution

of fiber-element-vectors detected in an

digital image of the fibrous structure

2016.10.27.

Vas, L.M., Balogh, K.: Investigating Damage Processes of Glass Fiber

Reinforced Composites Using Image Processing, Journal of Macromolecular

Sciences Part B – Physics, Vol. B 41(4-6), 977-989 (2002)

(Laplace-Gauss)

14

27László M. Vas

Fibers, fiber models 4.

Fiber orientation

Orientation II: Defined by the chord vector of the fiber

Joint probability density function of the

independent angle co-ordinates of the fiber chord

vector:

The small surface element, dA=sinv dudv, on the

surface (∂Go) of unit sphere Go=G(0,1) defines a

space-angle and the probability that a unit chord

vector falls in this infinitesimally small space-angle

is proportional to q(φ,θ)(u,v)dA. Hence the

distribution function of the fiber orientation is as

follows:

2016.10.27.

28László M. Vas

Fibers, fiber models 5.

Fiber orientation – orientation tensor

Orientation II: Defined by fiber chord vector

Orientation vector of a fiber: p=(pi) unit vector:

A simpler parametric characterization of the fiber orientation is the expected value of

the orientation tensor (see: covariance matrix) than the joint distribution of the

orientation angles

• Orientation tensor, P, is the tensorial

or diadic self-product of orientation

vector, p.

• The expected value of P tensor

is given by the expected value of the

tensor-elements:

2016.10.27.

(E(p)=0T

on the whole sphere)

[E(P)=D2(p) és E(pipj)=cov(pi,pj) on the whole sphere]

15

29László M. Vas

Fibers, fiber models 6.

Fiber orientation – orientation tensor

Measuring orientation: on polished cross sections of an injection molded plate

Measuring by image processing software: • Sizes of the small (d=2b) and large (2a≥d ) axes

of intersection ellipse of fibers and the inclination

angle (α) of the large axis related to axis y. Calculated from these:

• Inclination angle (β) of fibers related to axis z :

•Supposing that a fiber takes up the orientation angles

β or -β with the same probability hence the measured

β∈ [0, π/2] angles can be extended by mirroring over

the domains [0, π] or [-π/2, π/2].

• The measured α angles falling into interval [0, π], can

be extended by translation over [0, 2π] corresponding

to π -period. α↔φ, β↔θ

2016.10.27.(x↔1, y↔2, z↔3)

Sample: 80x80x2 mm

Polished: 3 at edge, 3 in the middle

Image: 10 image/polished section

Polished-sections

30László M. Vas

Fibers, fiber models 7. Fiber orientation – orientation tensor

Measuring orientation: by image analysis of polished cross sections (PCS)

Joint angle distribution (injection molded composite):

α↔φ, β↔θ

2016.10.27.

PCS: 3rd position (in the middle (z); at the edge (x))

Frequency

θ [degree]ϕ [degree]

Length

16

31László M. Vas

Fibers, fiber models 8. Fiber orientation – orientation tensor

Measuring orientation: by image analysis of polished cross sections (PCS)

Orientation tensor (x↔1, y↔2, z↔3):

Main diagonal elements along the thickness (y)

Edge-distributions:

2016.10.27.

In the sheath: p33 is large, p11 is small, p22i is very small

In the core: p33 is small, p11 is large, p22 is very small

Position 3: in the middle, at the edge

Position 3: in the middle, at the edge

Place of measurement, y [µm]

ϕ [degree]

32László M. Vas

Fiber sets, fiber flows 1.

Fiber set: fiber assembly of same properties

Fiber space: spatial structure created by fibers

Unoriented and oriented fiber sets or spaces

Isotropic (a) and uniaxially (b), as well as biaxially (c)

oriented anisotropic fiber sets or spaces

2016.10.27.

17

33László M. Vas

Fiber sets, fiber flows 2.

Fiber flow - types

Types of uniaxial fiber flows (a): scheme of linear (b) and elementary

linear (c), simple linear (d), evenly continuous linear (e), elementary

contonuous linear (f), regular (g) and Zotyikov-type (h) fiber flows

Fiber flow: It is an oriented fiber space where the chord vectors of fibers create a

statistical stream-space that is they follow certain spatial stream-lines tangentially or

regarding their position and orientation they scatter about those trend-like lines.

Fiber: In general

it is modeled by

its chord vector.

2016.10.27.

34László M. Vas

Fiber sets, fiber flows 3.

Fiber bundle: a set of fibers related to one another in some way

Definition of fiber bundle types:

Set of contacting fibers (a, b) and set of

fibers intersecting a given cross section (c, d)

2016.10.27.

18

35László M. Vas

Fiber sets, fiber flows 4.

Twist-oriented fiber flows

Yarn (a) and solenoid (b)

as twist-oriented (circular) fiber space

2016.10.27.

36László M. Vas

Properties of convex ranges 1.

Mean diameter

Diameter of convex range A

measured in direction of vector a

in the plane (k=2)

The mean diameter of convex range A∈ℜk is the average of diameters

mesured in every possible direction (like by caliper rule) (k=1,2,3) :

•paA = Projection of range A to

straight line of orientation vector a

• Go = unit sphere

• ∂Go = surface of unit sphere

• λk=k-dimensional volume (k=1,2,3)

2016.10.27.

19

37László M. Vas

Properties of convex ranges 2.

Mean diameter

Application of LUCIA image analysator

software to marked fiber cross sections

PETP fiber cross section shapes

2016.10.27.

Concave range mean diameter?Convexizing e.g.: by envelope ellipse, area-

equivalent circle/ellipse

38László M. Vas

Properties of convex ranges 3.

Spherical vicinity (G)

Spherical vicinity of point (a) and convex range (b)

in one- and two-dimensional spaces

2016.10.27.

ρ(P,Q)= distance of points P and Q (here: euklidean distance)

20

39László M. Vas

Properties of convex ranges 4.

Linear/Fiber vicinity of point P (a)

and range A (b) in plane

Oriented, linear, or fiber-vicinity

Components of linear vicinity (B) of a 2D

range (A):

• Ao=fiber core (often empty)

• B\intAo=H(r,α,β,∂A)=edge-vicinity

(∂A is the edge of A)

Linear vicinity H(r,α,β,P)⊂Rk of orientation eo(α,β) and radius r for point P :

For a range A:

2016.10.27.

40László M. Vas

Dimension of textiles 1.

Convex hull/envilope: the tightest convex set containing textile Γ

Convex hull (a), ε-body (b) of textiles and

representation in case of yarns (c)

Construction of εεεε-body: from convex hull, or given

formation 2016.10.27.

[0.01 mm]

Gray level density histogram

21

41László M. Vas

Dimension of textiles 2.

Convex hull – measurement of ε-body

Diameter of

non-transparent

(e.g. carbon)

fiber or yarn

2016.10.27.

εεεε-body :

coaxial cylinder

Diameter of

transparent

(e.g. glass)

fiber or yarn

42László M. Vas

Dimension of textiles 3.

DimensionThe dimension of textile ΓΓΓΓ is 1≤k≤2 if the tightest real subspace W⊂R3, that is the

skeleton space of the textile is of k-dimensions, and for the convex hull of ΓW ,

denoted by WΓ, where ΓW is the projection of Γ into W (ΓW⊂WΓ⊂W), the following

conditions stand:

(1) Textile Γ can be laid on the skeleton space W without cutting up. This means that

textile Γ can be moved in a position without cutting up that W forms a kind of middle

surface (skeleton space) of Γ and there can be found such a (minimum) real number,

δ>0, for which the δ/2 radius spherical vicinity (G) of WΓ covers textile Γ:

Γ⊂G(δ/2, WΓ)

(2) Textile Γ is concentrated about skeleton space W meaning that δ is negligibly small to

the sizes of textile Γ in W, di(Γ) (i=1,...,k), (δ is less by at least 1-3 magnitudes) that is

for textile Γ of finite sizes it stands:

min d1(Γ), ... ,dk(Γ) >> δ

Textile Γ is of 3 dimensions, that is k=3, if there cannot be found such true W subspace

in R3. In this case W=R3.

2016.10.27.

22

43László M. Vas

Dimension of textiles 4.

Density properties of textile structures

• Density of textiles (apparent density)

• Characteristic density

• ε-body density

λk = k-dimensional volume (k=1, 2, 3)

2016.10.27.

KΓ = convex hull of Γ

WΓ = convex hull of ΓW

KΓε = ε-body of Γ

44László M. Vas

Dimension of textiles 5.

Porosity properties of a textile structure

• Porosity of textiles (apparent porosity)

• ε-body-porosity

2016.10.27.

ρo = volume density of the fiber material

23

45László M. Vas

Dimension of textiles 6.

Definition of a sample

Convex sample cut out of a textile structure (real or model)

2016.10.27.

Real sample:

Model sample (projection):

46László M. Vas

Dimension of textiles 7.

1D Textiles

2016.10.27.

Fiber

Sliver

Yarn

Thread

String

Rope

Strap, belt

24

47László M. Vas

Dimension of textiles 8. 1D textiles

3.8-micron diameter carbon nanotube yarn that

functions as a torsional muscle when filled with an

ionically conducting liquid and electrochemically

charged

2016.10.27.

48László M. Vas

Linear density spectrum of fibers and linear textiles

2016.10.27.

FIBERS YARNS STRINGS/ROPES

Linear density

Dimension of textiles 9.

25

49László M. Vas

Dimension of textiles 10.

2D textiles

2016.10.27.

Web, fleece Woven fabric Braided sheet

Fiber mat Knitted sheet Knotted net

50László M. Vas

Dimension of textiles 11.

2D textiles

2016.10.27.

26

51László M. Vas

Dimension of textiles 12.

Area density ranges of 2D textiles

2016.10.27.

Area density

52László M. Vas

Dimension of textiles 13.

3D textile products

3D textile products made by tailoring (confectioned) (a)

and without tailoring (b)

2016.10.27.

27

53László M. Vas

Dimension of textiles 14.

3D textiles and composite products

2016.10.27.

54László M. Vas

Dimension of textiles 15.

Volume density ranges of 1-3D textiles

2016.10.27.

Volume density

28

55László M. Vas

Dimension

of textiles

16.

Geometrical

and mechanical

quantities in

case of

1D, 2D, and

3D structures

(Practical switch-number of

density and breaking

length: 103)

Properties Dimension of textiles

k = 1 k = 2 k = 3

Geometrical properties

Sizes of brick-shaped sample (lx)

l=length l=lengthb=width

l=lengthb=width

h=thickness

Cross section area (Ak) in skeleton space (SS) ⊥⊥⊥⊥to x

A1=0 A2=b A3=A=bh

Volume in SS (Vk) V1=l V2=bl V3=V=bhl

Characteristic density (ρρρρk) and measure unit

Strength properties

Characteristic specific force in direction x

Fx=F [N]

Specific force related density (Qx)

Hooke’s law for tensile load in x-direction Fx=K1εx fx=K2εx σx=K3εx

Characterizing tensile-stiffness (Kk)

K1= bhE =AE [N] K2=hE [N/m] K3=E [N/m2]

Specific tensile stiffness related to density (κκκκk)

Breaking length inx-direction (Rx)

2016.10.27.