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COMPOSITE MATERIALS COMPOSITE MATERIALS CHASSIS - carbon FIBER composite materials It is super lightweight. It is super strong. It is super stiff. It it can be easily molded into all kinds of different shapes. What are Composites? What are Composites? The term The term “COMPOSITE COMPOSITEis derived from the latin is derived from the latin “COMPOSITUS COMPOSITUSwhich comes which comes from from COMPONERE COMPONERE COMPONERE COMPONERE is made up of is made up of “COM COMand and PONERE PONEREmeaning meaning “together together” and and to put to put” respectively. respectively. The general definition of an engineering composite is: The general definition of an engineering composite is: THE COMBINATION OF TWO OR MORE DISSIMILAR MATERIALS THAT ARE STRONGER THAN THE INDIVIDUAL MATERIALS” This include both NATURAL and MAN-MADE composites. A more specific definition was given by A more specific definition was given by B. STRONG B. STRONG in his book in his book Fundamentals of Composites Manufacturing Fundamentals of Composites Manufacturing”: The combination of a reinforcement in a matrix material The combination of a reinforcement in a matrix material The term composite also means that the materials ( The term composite also means that the materials (matrix matrix and and reinforcement reinforcement) are identifiable at the macroscopic level. ) are identifiable at the macroscopic level.

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Page 1: CHASSIS - carbon FIBER composite materials It is super ... · 2/9/2011  · from “COMPONERE ” COMPONERE is made up of “COM ” and “PONERE ” meaning “together ” and

COMPOSITE MATERIALSCOMPOSITE MATERIALS

CHASSIS - carbon FIBER composite materials

It is super lightweight.It is super strong.

It is super stiff.It it can be easily molded into all kinds of

different shapes.

What are Composites?What are Composites?

�� The term The term ““COMPOSITECOMPOSITE”” is derived from the latin is derived from the latin ““COMPOSITUSCOMPOSITUS”” which comes which comes

from from ““COMPONERECOMPONERE””

�� COMPONERECOMPONERE is made up of is made up of ““COMCOM”” and and ““PONEREPONERE”” meaning meaning ““togethertogether”” and and

““to putto put”” respectively. respectively.

�� The general definition of an engineering composite is:The general definition of an engineering composite is:

““THE COMBINATION OF TWO OR MORE DISSIMILAR MATERIALS THAT ARE STRONGER THAN THE INDIVIDUAL MATERIALS”

� This include both NATURAL and MAN-MADE composites.

•• A more specific definition was given by A more specific definition was given by B. STRONGB. STRONG in his book in his book

““Fundamentals of Composites ManufacturingFundamentals of Composites Manufacturing””::

““The combination of a reinforcement in a matrix materialThe combination of a reinforcement in a matrix material””

�� The term composite also means that the materials (The term composite also means that the materials (matrixmatrix and and reinforcementreinforcement) are identifiable at the macroscopic level. ) are identifiable at the macroscopic level.

Page 2: CHASSIS - carbon FIBER composite materials It is super ... · 2/9/2011  · from “COMPONERE ” COMPONERE is made up of “COM ” and “PONERE ” meaning “together ” and

�� An engineering composite must meet the following An engineering composite must meet the following criteria:criteria:

1.1. Must contain two or more constituentsMust contain two or more constituents

2.2. Processed in a way that the form, distribution and Processed in a way that the form, distribution and amount of constituents are controlled in a predermined amount of constituents are controlled in a predermined wayway

3.3. Must have unique, useful and superior performance that Must have unique, useful and superior performance that can be predicted from the properties, amounts and can be predicted from the properties, amounts and arrangements of constituents using principles of arrangements of constituents using principles of mechanicsmechanics

What Benefits De We Get From A Composite?What Benefits De We Get From A Composite?

StiffnessStiffness

StrengthStrength

ToughnessToughness

Wear ResistanceWear Resistance

Thermal propertiesThermal properties

Low CTE (Coefficient of Thermal Expansion)Low CTE (Coefficient of Thermal Expansion)

Materials ConsiderationsMaterials Considerations

ReinforcementReinforcement

Metal

Ceramic

Polymer

MatrixMatrix

Metal Matrix Composites (MMC)

Ceramic Matrix Composites (CMC)

Polymer Matrix Composites (PMC)

– Continuous (long fibres): e.g; SiC, C,

– Discontinuous� Short fibres: e.g; Al2O3, SiC

� Particulates: e.g; SiC Al2O3

� Whiskers: e.g; SiC

Gained strong market

penetration

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9

• Composites:

-- Multiphase material w/significant

proportions of each phase.

• Dispersed phase:

-- Purpose: enhance matrix properties.MMC: increase σy, TS, creep resist.

CMC: increase KcPMC: increase E, σy, TS, creep resist.

-- Classification: Particle, fiber, structural

• Matrix:-- The continuous phase

-- Purpose is to:- transfer stress to other phases

- protect phases from environment

-- Classification: MMC, CMC, PMC

metal ceramic polymer

Reprinted with permission fromD. Hull and T.W. Clyne, An

Introduction to Composite Materials, 2nd ed., Cambridge University Press,

New York, 1996, Fig. 3.6, p. 47.

woven fibers

cross section view

0.5 mm

0.5 mm

Classification

Particulate WhiskersContinuous Fibre

Reinforcement Properties Dominate

Matrix Properties Dominate

Monofilaments ParticulateWhiskers/Staple Fibres

12

Large-

particle

Dispersion-

strengthened

Particle-reinforced

Continuous

(aligned)

Aligned Randomly

oriented

Discontinuous

(short)

Fiber-reinforced

Laminates Sandwich

panels

Structural

Composites

Adapted from Fig. 16.2, Callister 7e.

Composite Survey

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13

• Examples:Adapted from Fig.

10.19, Callister 7e. (Fig. 10.19 is

copyright United

States Steel Corporation, 1971.)

- Spheroidite steel

matrix: ferrite (α)

(ductile)

particles: cementite(Fe3C)

(brittle)60 µm

Adapted from Fig.

16.4, Callister 7e. (Fig. 16.4 is courtesy Carboloy Systems,

Department, General Electric Company.)

- WC/Co cemented carbide

matrix: cobalt (ductile)

particles: WC (brittle, hard)Vm:

10-15 vol%! 600 µm

Adapted from Fig.

16.5, Callister 7e. (Fig. 16.5 is courtesy Goodyear Tire and

Rubber Company.)

- Automobile tires

matrix: rubber (compliant)

particles: C (stiffer)

0.75 µm

Particle-reinforced Fiber-reinforced Structural

Composite Survey: Particle-I

14

Concrete – gravel + sand + cement- Why sand and gravel? Sand packs into gravel voids

Reinforced concrete - Reinforce with steel rerod or remesh- increases strength - even if cement matrix is cracked

Prestressed concrete - remesh under tension during setting of concrete. Tension release puts concrete under compressive force

- Concrete much stronger under compression. - Applied tension must exceed compressive force

Particle-reinforced Fiber-reinforced Structural

threadedrod

nut

Post tensioning – tighten nuts to put under tension

Composite Survey: Particle-II

15

• Elastic modulus, Ec, of composites:

-- two approaches.

• Application to other properties:-- Electrical conductivity, σe: Replace E in equations with σe.

-- Thermal conductivity, k: Replace E in equations with k.

Adapted from Fig. 16.3, Callister 7e. (Fig. 16.3 is

from R.H. Krock, ASTM Proc, Vol. 63, 1963.)

lower limit:

1

Ec

=Vm

Em

+Vp

Ep

c m m

upper limit:

E = V E + VpEp

“rule of mixtures”

Particle-reinforced Fiber-reinforced Structural

Data:

Cu matrix

w/tungsten particles

0 20 40 60 80 100

150

200

250

300

350

vol% tungsten

E(GPa)

(Cu) (W)

Composite Survey: Particle-III

16

• Fibers very strong

– Provide significant strength improvement to material

– Ex: fiber-glass

• Continuous glass filaments in a polymer matrix

• Strength due to fibers

• Polymer simply holds them in place

Particle-reinforced Fiber-reinforced Structural

Composite Survey: Fiber-I

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17

• Fiber Materials

– Whiskers - Thin single crystals - large length to diameter ratio

• graphite, SiN, SiC

• high crystal perfection – extremely strong, strongest known

• very expensive

Particle-reinforced Fiber-reinforced Structural

– Fibers

• polycrystalline or amorphous

• generally polymers or ceramics

• Ex: Al2O3 , Aramid, E-glass, Boron, UHMWPE

– Wires

• Metal – steel, Mo, W

Composite Survey: Fiber-II

18

aligned

continuous

aligned random

discontinuous

Adapted from Fig.

16.8, Callister 7e.

Fiber Alignment

19

• Aligned Continuous fibers• Examples:

From W. Funk and E. Blank, “Creep deformation of Ni3Al-Mo in-situ

composites", Metall. Trans. A Vol. 19(4), pp. 987-998, 1988. Used with permission.

-- Metal: γ'(Ni3Al)-α(Mo)by eutectic solidification.

Particle-reinforced Fiber-reinforced Structural

matrix: α (Mo) (ductile)

fibers: γ’ (Ni3Al) (brittle)

2 µm

-- Ceramic: Glass w/SiC fibersformed by glass slurry

Eglass = 76 GPa; ESiC = 400 GPa.

(a)

(b)

fracture surface

From F.L. Matthews and R.L. Rawlings, Composite Materials;

Engineering and Science, Reprint ed., CRC Press, Boca Raton, FL,

2000. (a) Fig. 4.22, p. 145 (photo by

J. Davies); (b) Fig. 11.20, p. 349 (micrograph by H.S. Kim, P.S.

Rodgers, and R.D. Rawlings). Used with permission of CRC

Press, Boca Raton, FL.

Composite Survey: Fiber-III

20

• Discontinuous, random 2D fibers• Example: Carbon-Carbon

-- process: fiber/pitch, thenburn out at up to 2500ºC.

-- uses: disk brakes, gas turbine exhaust flaps, nose

cones.

• Other variations:-- Discontinuous, random 3D

-- Discontinuous, 1DAdapted from F.L. Matthews and R.L. Rawlings,

Composite Materials; Engineering and Science, Reprint ed., CRC Press, Boca Raton, FL, 2000. (a) Fig. 4.24(a), p. 151; (b) Fig. 4.24(b) p. 151.

(Courtesy I.J. Davies) Reproduced with permission of CRC Press, Boca Raton, FL.

Particle-reinforced Fiber-reinforced Structural

(b)

fibers lie in plane

view onto plane

C fibers: very stiff very strong

C matrix: less stiff less strong

(a)

Composite Survey: Fiber-IV

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21

• Critical fiber length for effective stiffening & strengthening:

• Ex: For fiberglass, fiber length > 15 mm needed

Particle-reinforced Fiber-reinforced Structural

c

f d

τ

σ> 15length fiber

fiber diameter

shear strength of

fiber-matrix interface

fiber strength in tension

• Why? Longer fibers carry stress more efficiently!

Shorter, thicker fiber:

c

f d

τ

σ< 15length fiber

Longer, thinner fiber:

Poorer fiber efficiency

Adapted from Fig. 16.7, Callister 7e.

c

f d

τ

σ> 15length fiber

Better fiber efficiency

σ(x) σ(x)

Composite Survey: Fiber-V

22

Continuous fibers - Estimate fiber-reinforced composite

strength for long continuous fibers in a matrix

• Longitudinal deformation

σc = σmVm + σfVf but εc = εm = εf

volume fraction isostrain∴ Ece = Em Vm + EfVf longitudinal (extensional)

modulus

mm

ff

m

f

VE

VE

F

F= f = fiber

m = matrix

Composite Strength:

Longitudinal Loading

23

• In transverse loading the fibers carry less of the load - isostress

σc = σm = σf = σ εc= εmVm + εfVf

f

f

m

m

ct E

V

E

V

E+=

1transverse modulus

Transverse Loading

24

• Estimate of Ec and TS for discontinuous fibers

-valid when-- Elastic modulus in fiber direction:

-- TS in fiber direction:

efficiency factor:-- aligned 1D: K = 1 (aligned )

-- aligned 1D: K = 0 (aligned )

-- random 2D: K = 3/8 (2D isotropy)

-- random 3D: K = 1/5 (3D isotropy)

(aligned 1D)

c

f d

τ

σ> 15length fiber

Particle-reinforced Fiber-reinforced Structural

(TS)c = (TS)mVm + (TS)fVf

Ec = EmVm + KEfVf

Composite Strength

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• Large particle

- Particle-matrix interactions cannot be treated on the atomic or

molecular level.

- Most of the composite, particulate is harder and stiffer than matrix

- Theses particles tend to restrain movement of the matrix

- The matrix transfers some of the applied stress to the particles

- The degree of reinforcement depends on strong bonding at the

matrix-particle interface.

• Dispersion-strengthened composites

- Particles are normally much smaller (diameter: 10-100 nm)

- Particle-matrix interactions that lead to strengthening occur on

the atomic or molecular level.

- The mechanism of strengthening is similar to precipitation

hardening

- The small dispersed particles hinder or impede the dislocation

motions.

Particle-Reinforced Composite

• The major difference in strengthening mechanism between large-particle and dispersion-strengthened particle-reinforced composites is that for large-particle the particle-matrix interactions are not treated on the molecular level, whereas, for dispersion-strengthening these interactions are treated on the molecular level.

Question: Cite the general difference in strengthening mechanism between large-particle

and dispersion-strengthened particle-reinforced composites.

• The design goals : high strength and /or stiffness over weight basis.

• Fiber length:

- Longer fibers carry stress more efficiently (above critical length)

• Fiber orientation:

- The properties of a composite having its fibers aligned are highly anisotropy, means dependent on the direction in which they are measured, max strength and reinforcement along the alignment (longitudinal) direction.

In transverse, fiber reinforcement is virtually nonexistent.

- For applications of involving multidirectional usually use discontinuous fibers which are randomly oriented in the matrix material.

Fiber-Reinforced Composites

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Question: Cite one desirable characteristic and one

less desirable characteristic foreach of (1) discontinuous-oriented (aligned), and (2)

discontinuous-random fiber-reinforced composites.

Answer: For discontinuous-oriented fiber-reinforced composites one desirable characteristic is that the composite is relativelystrong and stiff in one direction; a less desirable characteristic is that the mechanical properties are anisotropic.

For discontinuous and random fiber-reinforced, one desirable characteristic is that the properties are isotropic; a less desirable characteristic is there is no single high-strength direction.

Example

A metal matrix composite is made from a boron (B)

reinforced aluminum alloy (figure). To form the boron fiber

a tungsten wire (W) (r = 10 µm) is coated with boron,

giving a final radius of 75 µm. The aluminum alloy is then

bonded around the boron fibers, giving a volume fraction of

0.65 for the aluminum alloy. Assuming that the rule of

binary mixtures applies also to ternary mixtures, calculate

the effective tensile elastic modulus of the composite

material under isostrain conditions.

(Given : EW = 410 GPa, EB = 379 GPa, and EAl = 68.9 GPa)

Solution:

EC = fWEW + fBEB + fAlEAl fW+B = 0.35

fW / fW+B = (area of W wire) / (area of B fiber(W+B))

;1022.6)35.0()75(

)10( 3

2

2−== xx

m

mfW

µπ

µπ

65.0=Alf

344.0)35.0()75(

)10()75(

)(

2

22

=−

=

−= +

−−

xm

mm

fxarea

areaareaf BW

fiberBoron

wireWfiberBoron

B

µπ

µπµπ

EC = fWEW + fBEB + fAlEAl

EC = (6.22x10-3)(410 GPa) + (0.344)(379 GPa) + (0.65)(68.9 GPa)=178 GPa

Note: the tensile modulus (stiffness) of the composite is about 2.5 times that of the unreinforced aluminum alloy

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Matrix Materials for Engineering CompositesMatrix Materials for Engineering Composites Metal Matrix CompositesMetal Matrix Composites

•• MMCMMC’’s consist of a low density metal reinforced with a ceramic s consist of a low density metal reinforced with a ceramic

materialmaterial

•• The most common metal alloys used as matrix are:The most common metal alloys used as matrix are:

•• Aluminium (Al), Magnesium (Mg), and Titanium (Ti)Aluminium (Al), Magnesium (Mg), and Titanium (Ti)

•• The ceramic reinforcement can be The ceramic reinforcement can be

•• Continuous: long fibresContinuous: long fibres

•• Discontinuous: particulates or whiskersDiscontinuous: particulates or whiskers

�� E.g. Continuous B fiber E.g. Continuous B fiber –– aluminum alloy matrix compositealuminum alloy matrix composite

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Metal Matrix CompositesMetal Matrix Composites

1.1. Continuous Fibres:Continuous Fibres:

�� Good propertiesGood properties

�� ExpensiveExpensive

�� Difficult to processDifficult to process

�� No secondary processingNo secondary processing

�� Have larger defense Have larger defense

demand, but little demand, but little

commercial demandcommercial demand

2.2. Discontinuous Fibres:Discontinuous Fibres:

�� Poor propertiesPoor properties

�� Lower costLower cost

�� Easier to processEasier to process

�� Secondary processing possibleSecondary processing possible

�� Have considerable defense and Have considerable defense and

commercial demandcommercial demand

•• The reinforcement type determines the mechanical properties, cosThe reinforcement type determines the mechanical properties, cost t and performance of the composite material producedand performance of the composite material produced

MMC offer:MMC offer:

MMC disadvantages include:MMC disadvantages include:

•• Higher specific stiffnessHigher specific stiffness

•• Higher operating temperaturesHigher operating temperatures

•• Greater wear resistanceGreater wear resistance

•• Possibility to tailor the properties for a Possibility to tailor the properties for a specific application.specific application.

•• Higher cost of materials and processingHigher cost of materials and processing

•• Lower ductility and toughness Lower ductility and toughness

•• Currently the focus is on two main types of MMCCurrently the focus is on two main types of MMC’’s:s:

1.1. High performance compositesHigh performance composites reinforced with expensive fibres which reinforced with expensive fibres which require the use of expensive processing techniques: (military anrequire the use of expensive processing techniques: (military and space d space applications)applications)

2.2. Low cost and low performanceLow cost and low performance composites reinforced with relatively composites reinforced with relatively cheaper particulates (commercial applications)cheaper particulates (commercial applications)

Applications of MMCApplications of MMC’’ss

1. Transport: Automotive, Aerospace (Al matrix-B fiber, TiAl matrix-SiC fiber), and Marine

2. Sport

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•• MMCMMC’’s can be manufactured by several production processes which s can be manufactured by several production processes which

can be divided into:can be divided into:

1.1. Primary processing methodsPrimary processing methods

•• Liquid State ProcessesLiquid State Processes

•• Squeeze castingSqueeze casting Squeeze casting: Molten metal is injected into a form with

fibers preplaced inside it.,,

•• Stir CastingStir Casting Discontinuous reinforcement is stirred into molten metal, which is

allowed to solidify.

•• Solid State ProcessesSolid State Processes

•• Powder Metallurgy, Diffusion BondingPowder Metallurgy, Diffusion Bonding

2.2. Secondary processing methodsSecondary processing methods

•• ExtrusionExtrusion

Monofilaments

Continuous Fibre

Whiskers

Particulates Liquid Metal

Low

High

$$$$

Powder Metallurgy

Spray Deposition

Diffusion Bonding

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Furnace

Crucible

Reinforcement

Particles

Matrix

Metal

Stirrer

Argon Gas Inlet

Stir Casting of Particulate Reinforced MMCStir Casting of Particulate Reinforced MMC’’ss

– This method is currently the most common commercial technique ofThis method is currently the most common commercial technique of

producing Alproducing Al--based MMC reinforced with ceramic based MMC reinforced with ceramic particulatesparticulates

–– It involves mixing of particulates into a liquid or semiIt involves mixing of particulates into a liquid or semi--solid metal matrixsolid metal matrix

–– When the matrix is in the semiWhen the matrix is in the semi--solid condition the method is generally known solid condition the method is generally known

as as ““compocastingcompocasting””

–– Issues in Liquid State Processing of MMCIssues in Liquid State Processing of MMC’’ss

–– Wetting between liquid matrix (liquid) and reinforcement (soliWetting between liquid matrix (liquid) and reinforcement (solid)d)

–– Interfacial reaction between matrix and reinforcementInterfacial reaction between matrix and reinforcement

Stir Casting of Particulate Reinforced MMCStir Casting of Particulate Reinforced MMC’’ss

Cast MMC’s have several characteristics:

– Stir casting is simple and cheap

– Wetting between matrix and reinforcement is ensured by pretreatment of the particulates (heat treatment or coating)

– High volume production possible

– Chemical reaction at the matrix / reinforcement interface is a problem.

Stir Casting of Particulate Reinforced MMCStir Casting of Particulate Reinforced MMC’’ss

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Uniform distribution of SiC particles Uniform distribution of SiC particles

Cast Particulate AlCast Particulate Al--SiC MMCSiC MMC

Preform

Liquid Metal

Pressure

Squeeze Casting of MMCSqueeze Casting of MMC’’ss

MMC

MMC

Billet

Mix Raw MaterialsCold Compaction + Sintering

Secondary Process

(Extrusion)

SiCp

Al-Powder

+

Powder Metallurgy of MMCPowder Metallurgy of MMC’’ss • MMC produced by PM is characterised by:

1. Unique MMC material chemistry, with very fine microstructure and uniform distribution of reinforcement

2. 10 – 40 % of reinforcement is possible

3. High properties: Strength, Ductility, Toughness, Fatigue resistance

4. Little chemical reaction at the matrix / reinforcement interface.

5. High quality products (aerospace applications)

6. Higher cost

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Diffusion Bonding of MMCDiffusion Bonding of MMC’’ss

MMC’s produced by

Diffusion Bonding SiC fibre with vapour

deposited layer of Ti-5Al-

5V alloy

Ti-5Al-5V alloy/ 80vol% SiC

fibres consolidated by hot

pressing or HIP

Vapour Deposition of MMCVapour Deposition of MMC’’ss

CoCo--Spray of MMCSpray of MMC’’ss Applications of MMCApplications of MMCApplications of MMCApplications of MMC’’’’ssss

Reinforcement Aerospace Automotive Advantage

Continuous

Fibres

Fins, compressor

blades, aircraft structure, engine

components

High thrust to

weight ratio, high stiffness, low

density, controlled

CTE

Discontinuous Reinforcement

Particulate and

whiskers

Wing panels,

precision components, engine

components

Piston, connecting

rods, bearings, cylinder liners,

brake parts, drive shafts,

Wear resistance,

low cost, elevated temperature

properties, fatigue strength

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Mechanical Behaviour of CompositesMechanical Behaviour of Composites

•• Mechanical behaviour of Mechanical behaviour of composites depend on several composites depend on several factors:factors:

�� Stress Stress –– Strain behaviors of matrix Strain behaviors of matrix and reinforcementand reinforcement

�� Phase volume fractionsPhase volume fractions

�� Direction in which the stress is Direction in which the stress is appliedapplied

1.1. Stress strain behaviour of Stress strain behaviour of compositecomposite

•• Properties of a composite Properties of a composite represent an average of the represent an average of the properties of the matrix and properties of the matrix and reinforcementreinforcement

•• However, this also depends on the However, this also depends on the geometrygeometry

2.2. Direction of Applied Stress (Continuous Fibres):Direction of Applied Stress (Continuous Fibres):

i) Loading Parallel to Reinforcing Fibres i) Loading Parallel to Reinforcing Fibres -- IsostrainIsostrain

�� If the matrix is intimately bonded to the fibres, the strain of If the matrix is intimately bonded to the fibres, the strain of both both

matrix and fibres is the same, but the load carried by the compomatrix and fibres is the same, but the load carried by the composite is site is

equal to the loads carried by the matrix and fibers (reinforcemeequal to the loads carried by the matrix and fibers (reinforcement)nt)

�� Since: Since: σσ = F/A,= F/A,

rmc FFF +=

rrmmcc AAA σσσ +=

rrmmc VV σσσ +=

rmc εεε ==

c

r

r

c

m

mcA

A

A

Aσσσ +=

Area fraction Area fraction

of matrix of matrix Area fraction Area fraction

of fibreof fibre

r

r

r

m

m

m

c

c VVε

σ

ε

σ

ε

σ+=Dividing by Dividing by εε, ,

•• Dividing by ADividing by Acc (area of composite)(area of composite)

•• If the composite, matrix and If the composite, matrix and

reinforcement have the same length, then reinforcement have the same length, then

the area fractions are equivalent to the the area fractions are equivalent to the

volume: Avolume: Amm/A/Acc= V= Vmm and Aand Arr/A/Acc= V= Vrr , we get: , we get:

•• Under Under isostrainisostrain conditions:conditions:

•• If the composite, matrix and reinforcement are elastic E = If the composite, matrix and reinforcement are elastic E = σσ//εε

•• Another important parameter which is significant to the contribuAnother important parameter which is significant to the contribution of the tion of the

reinforcement to the composite modulus is the fraction of the toreinforcement to the composite modulus is the fraction of the total composite tal composite

load, Pload, Pcc, ,

•• The ratio of the load carried by the fibers to that carried by tThe ratio of the load carried by the fibers to that carried by the matrix is:he matrix is:

ccc

rrr

cc

rr

c

r

AE

AE

A

A

P

P

ε

ε

σ

σ==

ororrrmmc VEVEE += ( ) rrrmc VEVEE +−= 1

mm

rr

m

r

VE

VE

P

P=

Under Under

isostrainisostrain

conditionsconditions

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2.2. Direction of Applied Stress (Continuous Fibres):Direction of Applied Stress (Continuous Fibres):

i) Loading Normal to Reinforcing Fibres i) Loading Normal to Reinforcing Fibres -- IsostressIsostress

�� Under these conditions:Under these conditions:

�� The strain of the composite is:The strain of the composite is:

�� But since But since εε ==σσ/E, /E,

rmc σσσ ==

rrmmc VV εεε +=

r

r

m

mC

VE

VEE

σσσ+=

r

r

m

m

C E

V

E

V

E+=

1

( ) mrrr

rm

mrfm

rm

cEVEV

EE

EVEV

EEE

+−=

+=

1

•• Dividing by Dividing by σσ, ,

•• This also gives:This also gives:

IsostressIsostress