Handout Booklet_Composite Materials_ 2013.pdf

8/11/2019 Handout Booklet_Composite Materials_ 2013.pdf

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SHEFFIELD HALLAM UNIVERSITYDepartment of Engineering & MathematicsFaculty of Arts, Computing, Engineering

and Sciences

Composite Materials

Dr Syed T HasanMaterials Engineering

Room 4124 Sheaf Building

[email protected]

Tel: 0114 225 3407 (direct line)
mailto:[email protected]:[email protected]:[email protected]


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POLYMERS:

The word polymer originates from the Greek word polymeros and meansmany-membered. In polymer science it refers to molecules held togetherby covalent bonds and composed of small units which are repeated many

times to form very large molecules.

ALL POLYMERS

THERMOPLASTICS THERMOSETS

CRYSTALLINE AMORPHOUS

MOLECULAR STRUCTURES:

Linear Polymers Branched Polymers Lightly Crosslinked Polymers Network and Heavily Crosslinked Polymers

Molecular weight:

Extremely large molecular weights are to be found in polymers with very longchains. During the polymerisation process in which these largemacromolecules are synthesised from smaller molecules, not all polymerchains will grow to the same length; this results in a distribution of chainlengths or molecules weights.

There are several methods of defining average molecular weight.

i) Number-Average Molecular Weight Mn:

The number average molecular weight Mnis obtained by dividing the chainsinto a series of size ranges and then determining the number fraction ofchains with each size range. This number average molecular weight isexpressed as,

Mn= xiMiwhere Mirepresents the mean molecular weight of size range i, and xiis thefraction of the total number of chains within the corresponding size range.


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ii) Weight-Average Molecular Weight Mw:A weight average molecular weight Mwis based on the weight fraction ofmolecules within the various ranges, expressed as,

Mw= wiMiwhere Miis the mean molecular weight within a size range, where as wi

denotes the weight fraction of molecules within the same size interval.

Degree of Polymerisation, n:

An alternative way of expressing average chain size of a polymer is as thedegree of polymerisation which represents the average number of mer unitsin a chain. Both number-average and weight-average degrees ofpolymerisation can be calculated,

nn= Mn/ m

nw= Mw/ mwhere Mnand Mware the number-average and weight-average molecularweights while m is the mer molecular weight.

Effects of Molar Mass on Properties:

Properties such as, density, transparency and refractive index vary verylittle with molar mass. However, properties such as softening temperature,melting temperature, melt viscosity, tensile strength, elastic modulus and

toughness vary considerably with molar mass. Most of these properties change rapidly with molar mass over the lower

molar mass range but stabilise at higher values.

Polymer Crystallinity:

The crystalline state may exist in polymeric materials. However, it involvesmolecules instead of just atoms or ions, as with metals and ceramics, theatomic arrangements will be more complex for polymers.

Polymer crystallinity can be considered as the packing of molecular chainsso as to produce an ordered atomic array.

The degree of crystallinity may range from completely amorphous toalmost entirely crystalline (95%).

The density of crystalline polymer will be greater than an amorphous oneof the same material and molecular weight.

The degree of crystallinity by weight can be determined from accuratedensity measurements,

%crystallinity c s a

s c a

100


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where sis the density of a specimen for which the percent crystallinity is to

be determined, ais the density of the totally amorphous polymer and cisthe density of the perfectly crystalline polymer.

The degree of crystallinity of a polymer depends on the rate of coolingduring solidification as well as on the chain configuration. The molecularchemistry as well as chain configuration also influence the ability ofpolymer to crystallise.

Polymer Crystals:

A semicrystalline polymer consists of small crystalline regions, having aprecise alignment, which are embedded within the amorphous matrixcomposed of randomly oriented molecules.

These crystals are regularly shaped, thin platelets approximately 10 to 20nm thick and 10 m long.

Most of the polymers when crystallise from melt form spherulites. Thespherulite consists of an aggregate of ribbonlike chain-folded crystallitesthat radiate from the centre to outward.

Mechanical and Thermomechanical Behaviour of Polymers:

The mechanical properties of polymers are specified with many of the

same parameters that are used for metals, modulus of elasticity, tensile,impact and fatigue strengths.

The mechanical characteristics of polymers are highly sensitive to the rateof deformation (strain rate), the temperature and the chemical nature ofthe environment.

Mechanism of Deformation of Semicrystalline Polymers:

Macroscopic Deformation

Crystallisation:

The crystallisation of a molten polymer occurs by nucleation and growthprocesses. For polymers upon cooling through the melting temperaturenuclei form wherein small regions of the tangled and random moleculesbecome ordered and aligned in the manner of chain folded layers.

Melting:

There are several features distinctive to the melting of polymers that arenot normally observed with metal and ceramics.


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Melting of polymers takes place over a range of temperatures. The melting behaviour depends on the history of the specimen, in

particular the temperature at which it crystallised.

The thickness of chain-folded lamellae will depend on crystallisationtemperature, the thicker the lamellea the higher the melting temperature.

Melting behaviour is a function of the rate of heating, increasing the rateresults in an elevation of the melting temperature.

Polymeric materials are also responsive to heat treatments which producestructural and property alterations.

Glass Transition:

The glass transition occurs in amorphous or glassy polymers when uponcooling from liquid crystallisation does not take place, that is polymer

chains are not able to rearrange into a three dimensional long rangeordered structure.

Upon cooling the glass transition corresponds to an increase in viscosityand the gradual transformation from a liquid to a rubbery material andfinally to a rigid solid.

The temperature at which the polymer experiences the transition fromrubbery to rigid states is termed the glass transition temperature.

Thermoplastic and Thermosetting Polymers:

Thermoplasts: Thermoplasts soften when heated and harden

when cooled. These materials are normally fabricated by the simultaneousapplication of heat and pressure.

Thermosetting: Thermosetting polymers become permanently hard whenheat is applied and do not soften upon subsequent heating.

Fracture of Polymers:


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Introduction to Composites:

A composite is considered to be any multiphase material that exhibits a significant

proportion of the properties of both constituents phases.

A composite, in the present context, is a multiphase material that is artificially made,

as opposed to one that occurs or forms naturally. The constituent phases must be

chemically dissimilar and separated by a distinct interface.

Most composites have been created to improve combination of mechanical

characteristics such as stiffness, toughness and ambient and high temperature strength.

The properties of composites are a function of the properties of the constituent phases,

their relative amounts and the geometry of the dispersed phase.

Composite Theory

In its most basic form a composite material is one, which is composed of at least two

elements working together to produce material properties that are different to the

properties of those elements on their own. In practice, most composites consist of a

bulk material (the matrix), and a reinforcement of some kind, added primarily to

increase the strength and stiffness of the matrix. This reinforcement is usually in fibre

form. Today, the most common man-made composites can be divided into three main

groups:

Polymer Matrix Composites (PMCs) These are the most common and willbe discussed here. Also known as FRP - Fibre Reinforced Polymers (or Plastics)

these materials use a polymer-based resin as the matrix, and a variety of fibres such as

glass, carbon and aramid as the reinforcement.

Metal Matrix Composites (MMCs) - Increasingly found in the automotive

industry, these materials use a metal such as aluminium as the matrix, and reinforce it

with fibres such as silicon carbide.

Ceramic Matrix Composites (CMCs) - Used in very high temperature

environments, these materials use a ceramic as the matrix and reinforce it with short

fibres, or whiskers such as those made from silicon carbide and boron nitride.

Polymer Matrix Composites

Polymer matrix composites consists of a polymer resin as the matrix with fibres as the

reinforcement medium. These materials are used in the greatest diversity of composite

applications as well as in the largest quantities in light of their room temperature

properties, ease of fabrication and cost.

Resin systems such as epoxies and polyesters have limited use for the manufacture of

structures on their own, since their mechanical properties are not very high when

compared to, for example, most metals. However, they have desirable properties, most

notably their ability to be easily formed into complex shapes. Materials such as glass,


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aramid and boron have extremely high tensile and compressive strength but in solid

form these properties are not readily apparent. This is due to the fact that when

stressed, random surface flaws will cause each material to crack and fail well below

its theoretical breaking point. To overcome this problem, the material isproduced in

fibre form, so that, although the same number of random flaws will occur, they will be

restricted to a small number of fibres with the remainder exhibiting the materialstheoretical strength. Therefore a bundle of fibres will reflect more accurately the

optimum performance of the material. However, fibres alone can only exhibit tensile

properties along the fibres length, in the same way as fibres in a rope.

It is when the resin systems are combined with reinforcing fibres such as glass, carbon

and aramid that exceptional properties can be obtained. The resin matrix spreads the

load applied to the composite between each of the individual fibres and also protects

the fibres from damage caused by abrasion and impact. High strengths and stiffnesses,

ease of moulding complex shapes, high environmental resistance all coupled with low

densities, make the resultant composite superior to metals for many applications.

Since PMCs combine a resin system and reinforcing fibres, the properties of the

resulting composite material will combine something of the properties of the resin on

its own with that of the fibres on their own, as surmised in Figure 1.

Figure: 1Illustrating the combined effect on Modulus of the addition of fibres to a

resin matrix.

Overall, the properties of the composite are determined by:

The properties of the fibre

The properties of the resin

The ratio of fibre to resin in the composite (Fibre Volume Fraction (FVF))

The geometry and orientation of the fibres in the composite

The ratio of the fibre to resin derives largely from the manufacturing process used to

combine resin with fibre. However, it is also influenced by the type of resin system

used, and the form in which the fibres are incorporated. In general, since the

mechanical properties of fibres are much higher than those of resins, the higher the

fibre volume fraction the higher will be the mechanical properties of the resultant

composite. In practice there are limits to this, since the fibres need to be fully coatedin resin to be effective, and there will be an optimum packing of the generally circular


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cross-section fibres. In addition, the manufacturing process used to combine fibre with

resin leads to varying amounts of imperfections and air inclusions.

Typically, with a common hand lay-up process as widely used in the boat-building

industry, a limit for FVF is approximately 30-40%. With the higher quality, more

sophisticated and precise processes used in the aerospace industry, FVFs approaching70% can be successfully obtained.

The geometry of the fibres in a composite is also important since fibres have their

highest mechanical properties along their lengths, rather than across their widths. This

leads to the highly anisotropic properties of composites, where, unlike metals, the

mechanical properties of the composite are likely to be very different when tested in

different directions. This means that it is very important when considering the use of

composites to understand at the design stage, both the magnitude and the direction of

the applied loads. When correctly accounted for, these anisotropic properties can be

very advantageous since it is only necessary to put material where loads will be

applied, and thus redundant material is avoided.

It is also important to note that with metals the material supplier largely determines

the properties of the materials, and the person who fabricates the materials into a

finished structure can do almost nothing to change those in-built properties.

However, a composite material is formed at the same time, as the structure is itself

being fabricated. This means that the person who is making the structure is creating

the properties of the resultant composite material, and so the manufacturing processes

they use have an unusually critical part to play in determining the performance of the

resultant structure.

Loading

There are four main direct loads that any material in a structure has to withstand:

tension, compression, shear and flexure.

Tension

Figure 2 shows a tensile load applied to a composite. The response of a composite to

tensile loads is very dependent on the tensile stiffness and strength properties of the

reinforcement fibres, since these are far higher than the resin system on its own.

Figure 2Illustrates the tensile load applied to a composite body.

Compression

Figure 3 shows a composite under a compressive load. Here, the adhesive and

stiffness properties of the resin system are crucial, as it is the role of the resin to

maintain the fibres as straight columns and to prevent them from buckling.


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Figure 3- Illustrates the compression load applied to a composite body.

Shear

Figure 4 shows a composite experiencing a shear load. This load is trying to slide

adjacent layers of fibres over each other. Under shear loads the resin plays the major

role, transferring the stresses across the composite. For the composite to perform well

under shear loads the resin element must not only exhibit good mechanical properties

but must also have high adhesion to the reinforcement fibre. The interlaminar shear

strength (ILSS) of a composite is often used to indicate this property in a multiplayer

composite (laminate).

Figure 4- Illustrates the shear load applied to a composite body.

Flexure

Flexural loads are really a combination of tensile, compression and shear loads. Whenloaded as shown (Figure 5), the upper face is put into compression, the lower face into

tension and the central portion of the laminate experiences shear.

Figure 5- Illustrates the loading due to flexure on a composite body.

Comparison with Other Structural Materials

Due to the factors described above, there is a very large range of mechanical

properties that can be achieved with composite materials. Even when considering one

fibre type on its own, the composite properties can vary by a factor of 10 with the

range of fibre contents and orientations that are commonly achieved. The comparisons

that follow therefore show a range of mechanical properties for the composite

materials. The lowest properties for each material are associated with simplemanufacturing processes and material forms (e.g. spray lay-up glass fibre), and the


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higher properties are associated with higher technology manufacture (e.g. autoclave

moulding of unidirectional glass fibre prepreg), such as would be found in the

aerospace industry.

For the other materials shown, a range of strength and stiffness (modulus) figures are

also given to indicate the spread of properties associated with different alloys, forexample.

Figure 6Tensile Strength of Common Structural Materials


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Figure 7Tensile Modulus of Common Structural Materials

The above Figures (6 and 7) clearly show the range of properties that different

composite materials can display. These properties can best be summed up as high

strengths and stiffnesses combined with low densities. It is these properties that giverise to the characteristic high strength and stiffness to weight ratios that make

composite structures ideal for so many applications. This is particularly true of

applications, which involve movement, such as cars, trains and aircraft, since lighter

structures in such applications play a significant part in making these applications

more efficient. The strength and stiffness to weight ratio of composite materials can

best be illustrated by the following graphs that plot specific properties. These are

simply the result of dividing the mechanical properties of a material by its density.

Generally, the properties at the higher end of the ranges illustrated in the previous

graphs (Figures 6 and 7) are produced from the highest density variant of the material.

The spread of specific properties shown in the following graphs (Figures 8 and 9)takes this into account.


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Figure 8Specific Tensile Strength of Common Structural Materials

Figure 9- Specific Tensile Modulus of Common Structural Materials

Continuous Fibre-Reinforced Composites:


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The most important composites are those in which the dispersed phase is in the form

of a fibre. Design goals of fibre reinforced composites often include high strength and

stiffness on a weight basis and these characteristics are expressed in terms of specific

strength and specific modules parameters.

Fibre reinforced composites are sub-classified by fibre length.

Influence of Fibre Length:

The mechanical characteristics of a fibre reinforced composite depend not only on the

properties of the fibre but also on the degree to which an applied load is transmitted to

the fibres by the matrix phase. The extent of the load transmittance is the magnitude

of the interfacial bond between the fibre and matrix phases.

A critical fibre length is necessary for effective strengthening and stiffening of thecomposite material.The critical fibre length lcis dependent on the fibre diameter dand

its ultimate tensile strength fand on the fibre-matrix bond strength cas,

lcfd

c

2

For a number of glass and carbon fibre-matrix combinations, the critical length is the

order of 1 mm, which is 20 - 150 times the fibre diameter.If the fibre length is (l > 15

lc), it is termed as continuous fibre reinforcement.

For discontinuous fibres of lengths significantly less than lcthe matrix deforms around

the fibre such that there is virtually no stress transference and little reinforcement bythe fibre.

Influence of Fibre Orientation and Concentration:

Longitudinal Loading:

The properties of a composite having its fibres aligned are highly anisotropic.

Let us consider the deformation of a composite in which a stress is applied along the

direction of alignment. The total load sustained by the composite Fcis equal to the

loads carried by the matrix phase Fmand the fibre phase Ff,

Fc = Fm+ Ff

or

cAc = mAm+ fAfc = m(Am/Ac) + f(Af/Ac)

where Am/Ac and Af/Acare the area fractions of the matrix and fibre phases

respectively.

If the composite, matrix and fibre lengths are all equal, Am/Acis equivalent to the

volume fraction of the matrix Vmand likewise for the fibres Vf.c = m Vm + fVf


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Now if we assume an isostrain state,

c = m = fwhen each term in equation -- is divided by its respective strain and if composite,

matrix and fibre deformations are all elastic then,

Ecl = EmVm+ Ef Vf

where, Ec= c/c etc. and since composite consists of only matrix and fibre phases,that is

Vm+ Vf= 1 then

Ecl = Em(1+ Vf)+ Ef Vf

The ratio of the load carried by the fibres to that carried by the matrix is,

(Ff/ Fm) = (EfVf) / (EmVm)

Transverse Loading:

A continuous and oriented fibre composite may be loaded in the transverse direction,

the load is applied at a 90oangle to the direction of fibre alignment. The stress to

which the composite as well as both phases are exposed is the same,c = m = f =

This is termed as isostress state. The strain of the entire composite cis,c = mVm+ fVf

but, since = /E,(/Ect) = (/Em)Vm+ (/Ef)Vf

where Ectis the modules of elasticity in the transverse direction and dividing by yield,

Ect

EmEf

V

f

E

f

V

f

E

m

1

The Fibre Phase:

On the basis of diameter and character fibres are grouped into three different

classifications: whiskers, fibres, and wires.

Whiskers are very thin single crystals that have extremely large length to diameter

ratios.

Materials that are classified as fibres are either polycrystalline or amorphous and have

small diameters.

Fine wires have relatively large diameters.

The Matrix Phase:

The matrix phase of fibrous composites may be a metal, polymer or ceramic. In

general metals and polymers are used as matrix materials because some ductility is

desirable. For fibre reinforced composites the matrix phase serves several functions.

First it binds the fibres together and act as the medium by which an externally applied

stress is transmitted and distributed to the fibres. The second function of the matrix isto protect the individual fibres from surface damage as a result of mechanical abrasion


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or chemical reactions with the environment.Finally the matrix separates the fibres and

by virtue of its relative softness and plasticity prevents the propagation of brittle

cracks from fibre to fibre, which could result in catastrophic failure.It is essential that

adhesive bonding forces between fibre and matrix be high to minimise fibre pull out.

The most widely utilised and least expensive polymer resins are the polyesters andvinyl esters, these matrix materials are used primarily for glass fibre reinforced

composites. The epoxies are more expensive are utilised extensively in aerospace

applications. For high temperature applications polyimide resins are used and their

upper temperature limit is approximately 230oC. High temperature thermoplastic

resins offer the potential to be used in future aerospace applications.

Glass fibre reinforced polymer composites:

Fibreglass is simply a composite consisting of glass fibres, either continuous or

discontinuous, contained within a polymer matrix.Fibre diameter normally rangebetween 3 and 20 m.

Glass is popular as a fibre reinforcement material for several reasons:

1. it is easily drawn into high-strength fibres from the molten state.

2. it is readily available and may be fabricated into a glass reinforced plastic

economically using a wide variety of composite manufacturing techniques.

3. as a fibre, it is relatively strong and when embedded in a plastic matrix it

produces a composite having a very high specific strength.

4. when coupled with the various plastics it possesses a chemical inertness that

renders the composite useful in a variety of corrosive environments.

Carbon fibre reinforced polymer composite:

Carbon is a high performance fibre material that is the most commonly used

reinforcement in advanced polymer matrix composites. The reasons for this are as

follows: Carbon fibres have the highest specific modules and specific strength of all

reinforcing fibre materials. They retain their high tensile modules and high strength at

elevated temperatures, high temperature oxidation may be a problem. At room

temperature carbon fibres are not affected by moisture nor a wide variety of solvents,

acids and bases. These fibres exhibit a diversity of physical and mechanicalcharacteristics allowing composites incorporating these fibres to have specific

engineered properties. Fibre and composite manufacturing processes have been

developed that are relatively inexpensive and cost effective.

Production and Properties of Commonly Used Fibres:

(i) Graphite/Carbon.Polymer fibres e.g. PAN, stretched, heated to about 220C to

cross-link and stabilise. Further heated to 900C to carbonise and then to 1300C+ to

produce correct graphite structure. High modulus or high strength fibres can be

produced.


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(ii) Glass. Homogeneous melt of high purity. Spun from a high temperature, rapidly

chilled and immediately given a protective coating.

(iii) Kevlar.Concentrated solution of an aramid polymer in concentrated sulphuric

acid. Spun into a neutralising bath. Jet shape and degree of stretch are critical

parameters to achieve required properties. Fibre washed dried and heated to 550C in anitrogen atmosphere.

(iv)Boron.Core of tungsten or graphite is heated to about 1000-1300C in atmosphere

containing volatile compound of boron e.g. boron chloride. CVP to produce pure

boron with few flaws. Careful cleaning of base fibre in hydrogen is carried out. This

prevents excessive crystal growth which would reduce strength.

PROPERTIES

(i) Very high specific strengths. Strain to failure dependent on flaw population. 2%strain to failure is possible. Tensile modulus in excess of 250GPa depending on

production route. Stiffness of Strength can be maximised but not at same time. This

situation is improving. Pitch fibres give high Modulus but lower strength. Vice versa

for PAN.

(ii) Glass fibres have significantly lower modulus e.g. 40-120GPa. Good strength of 3-

4.5GPa. Higher density giving considerably lower specific strengths.

(iii)Kevlar fibres have modulus values between 50-130GPa with strengths up to 3.1

GPa High specific properties. High strain to fracture and hence good damage

tolerance.

(iv) Boron fibres. Low density metal, which is weak in bulk. High modulus fibre,

370-410GPa. Good in compression. Strength up to 5GPa. Good specific properties.

Applications in terms of requirements e.g. cost versus weight. High specific properties

e.g. E/in aerospace. Glass for lower cost land based applications. Kevlar where highdamage tolerance required. Hybrids e.g. Glass/carbon, carbon/Kevlar are common to

produce required property combinations.


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Mechanics of Composite Materials

Version 2.1

Bill Clyne,University of Cambridge

Boban Tanovic,MATTER Project

Assumed Pre-knowledge

It is assumed that the student is familiar with simple concepts of mechanical behaviour, such as the broad meanings of

stress and strain. It would be an advantage for the student to understand that these are really tensor quantities,

although this is by no means essential. All of the terms associated with the assumed pre-knowledge are defined in the

glossary, which can be consulted by the student at any time.

Most of the material in this package is based on a recently published book. This is:

"An Introduction to Composite Materials", D.Hull and T.W.Clyne, Cambridge University Press (1996) Order!

This source should be consulted for background to the treatments in this module, particularly mathematical details.

What is a Composite Material?

Most composites have strong, stiff fibresin a matrixwhich is weaker and less stiff. The objective is usually to make

a component which is strong and stiff, often with a low density. Commercial material commonly has glassor carbon

fibres in matrices based on thermosetting polymers, such as epoxyor polyesterresins. Sometimes, thermoplastic

polymersmay be preferred, since they are mouldable after initial production. There are further classes of composite

in which the matrix is a metalor a ceramic. For the most part, these are still in a developmental stage, with problems

of high manufacturing costs yet to be overcome. Furthermore, in these composites the reasons for adding the fibres

(or, in some cases, particles) are often rather complex; for example, improvements may be sought in creep, wear,

fracture toughness, thermal stability, etc. This software package covers simple mechanics concepts of stiffness and

strength, which, while applicable to all composites, are often more relevant to fibre-reinforced polymers.

Module Structure

The module comprises three sections:

Load Transfer

Composite Laminates

Fracture Behaviour

Brief descriptions are given below of the contents of these sections, covering both the main concepts involved and thestructure of the software.
mailto:[email protected]:[email protected]://www.amazon.co.uk/exec/obidos/ASIN/0521388554/matterhttp://www.amazon.co.uk/exec/obidos/ASIN/0521388554/matterhttp://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435309#_Toc360435309http://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435309#_Toc360435309http://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435310#_Toc360435310http://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435310#_Toc360435310http://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435311#_Toc360435311http://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435311#_Toc360435311http://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435311#_Toc360435311http://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435310#_Toc360435310http://www.matter.org.uk/matscicdrom/manual/co.html#_Toc360435309#_Toc360435309http://www.amazon.co.uk/exec/obidos/ASIN/0521388554/mattermailto:[email protected]


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Load Transfer

Summary

This section covers basic ideas concerning the manner in an applied mechanical load is shared between the matrix

and the fibres. The treatment starts with the simple case of a composite containing aligned, continuous fibres. This

can be represented by the slab model. For loading parallel to the fibre axis, the equal strain condition is imposed,

leading to the Rule of Mixtures expression for the Young's modulus. This is followed b y the cases of transverse

loading of a continuous fibre composite and axial loading with discontinuous fibres.

What is meant by Load Transfer?

The concept of load sharing between the matrix and the reinforcing constituent (fibre) is central to an understanding

of the mechanical behaviour of a composite. An external load (force) applied to a composite is partly borne by the

matrix and partly by the reinforcement. The load carried by the matrix across a section of the composite is given by

the product of the average stress in the matrix and its sectional area. The load carried by the reinforcement is

determined similarly. Equating the externally imposed load to the sum of these two contributions, and dividing

through by the total sectional area, gives a basic and important equation of composite theory, sometimes termed the

"Rule of Averages".

(1)

which relates the volume-averaged matrix and fibre stresses ( ), in a composite containing a volume (or

sectional area) fractionfof reinforcement,to theapplied stress sA. Thus, a certain proportion of an imposed load will

be carried by the fibre and the remainder by the matrix. Provided the response of the composite remains elastic, this

proportion will be independent of the applied load and it represents an important characteristic of the material. It

depends on the volume fraction, shape and orientation of the reinforcement and on the elastic properties of both

constituents. The reinforcement may be regarded as acting efficiently if it carries a relatively high proportion of the

externally applied load. This can result in higher strength, as well as greater stiffness, because the reinforcement is

usually stronger, as well as stiffer, than the matrix.

What happens when a Composite is Stressed?

Figure 1

Consider loading a composite parallel to the fibres. Since they are bonded together, both fibre and matrix will stretch

by the same amount in this direction, i.e. they will have equal strains, e(Fig. 1). This means that, since the fibres are

stiffer (have a higher Young modulus, E), they will be carrying a larger stress. This illustrates the concept of load

transfer, orload partitioningbetween matrix and fibre, which is desirable since the fibres are better suited to bear

high stresses. By putting the sum of the contributions from each phase equal to the overall load, the Young modulus

of the composite is found (diagram). It can be seen that a "Rule of Mixtures"applies. This is sometimes termed the

"equal strain"or "Voigt"case. Page 2 in the section covers derivation of the equation for the axial stiffness of a

composite and page 3 allows the effects on composite stiffness of the fibre/matrix stiffness ratio and the fibre volume

fraction to be explored by inputting selected values.

What about the Transverse Stiffness?

Also of importance is the response of the composite to a load applied transverse to the fibre direction. The stiffness


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and strength of the composite are expected to be much lower in this case, since the (weak) matrix is not shielded from

carrying stress to the same degree as for axial loading. Prediction of the transverse stiffness of a composite from the

elastic properties of the constituents is far more difficult than the axial value. The conventional approach is to assume

that the system can again be represented by the "slab model". A lower bound on the stiffness is obtained from the

"equal stress"(or "Reuss") assumption shown in Fig. 2. The value is an underestimate, since in practice there are

parts of the matrix effectively "in parallel" with the fibres (as in the equal strain model), rather than "in series" as is

assumed. Empirical expressions are available which give much better approximations, such as that of Halpin-Tsai.There are again two pages in the section covering this topic, the first (page 4) outlining derivation of the equal stress

equation for stiffness and the second (page 5) allowing this to be evaluated for different cases. For purposes of

comparison, a graph is plotted of equal strain, equal stress and Halpin-Tsai predictions. The Halpin-Tsai expression

for transverse stiffness (which is not given in the module, although it is available in the glossary) is:

(2)

in which

Figure 2

The value of x may be taken as an adjustable parameter, but its magnitude is generally of the order of unity. The

expression gives the correct values in the limits off=0 andf=1 and in general gives good agreement with experiment

over the complete range of fibre content. A general conclusion is that the transverse stiffness (and strength) of an

aligned composite are poor; this problem is usually countered by making a laminate(see section on "composite

laminates").

How is Strength Determined?

There are several possible approaches to prediction of the strength of a composite. If the stresses in the two

constituents are known, as for the long fibre case under axial loading, then these values can be compared with the

corresponding strengths to determine whether either will fail. Page 6 in the section briefly covers this concept. (More

details about strength are given in the section on "Fracture Behaviour".) The treatment is a logical development from

the analysis of axial stiffness, with the additional input variable of the ratio between the strengths of fibre and matrix.

Such predictions are in practice complicated by uncertainties about in situ strengths, interfacial properties,

residual stressesetc. Instead of relying on predictions such as those outlined above, it is often necessary to measure

the strength of the composite, usually by loading parallel, transverse and in shear with respect to the fibres. This

provides a basis for prediction of whether a component will fail when a given set of stresses is generated (see section

on "Fracture Behaviour"), although in reality other factors such as environmental degradationor the effect of

failure mode on toughness, may require attention.

What happens with Short Fibres?

Short fibres can offer advantages of economy and ease of processing. When the fibres are not long, the equal strain


20/75

condition no longer holds under axial loading, since the stress in the fibres tends to fall off towards their ends (see

Fig. 3). This means that the average stress in the matrix must be higher than for the long fibre case. The effect is

illustrated pictorially in pages 7 and 8 of the section.

Figure 3

This lower stress in the fibre, and correspondingly higher average stress in the matrix (compared with the long fibrecase) will depress both the stiffness and strength of the composite, since the matrix is both weaker and less stiff than

the fibres. There is therefore interest in quantifying the change in stress distribution as the fibres are shortened.

Several models are in common use, ranging from fairly simple analytical methods to complex numerical packages.

The simplest is the so-called "shear lag" model. This is based on the assumption that all of the load transfer from

matrix to fibre occurs via shear stresses acting on the cylindrical interface between the two constituents. The build-up

of tensile stress in the fibre is related to these shear stresses by applying a force balance to an incremental section of

the fibre. This is depicted in page 9 of the section. It leads to an expression relating the rate of change of the stress in

the fibre to the interfacial shear stress at that point and the fibre radius, r.

(3)

which may be regarded as the basic shear lag relationship. The stress distribution in the fibre is determined by

relating shear strains in the matrix around the fibre to the macroscopic strain of the composite. Some mathematical

manipulation leads to a solution for the distribution of stress at a distancexfrom the mid-point of the fibre which

involves hyperbolic trig functions:

(4)

where e1is the composite strain,sis the fibre aspect ratio (length/diameter) and nis a dimensionless constant given

by:

(5)

in which nmis the Poisson ratio of the matrix. The variation of interfacial shear stress along the fibre length is

derived, according to Eq.(3), by differentiating this equation, to give:

(6)

The equation for the stress in the fibre, together with the assumption of a average tensile strain in the matrix equal to

that imposed on the composite, can be used to evaluate the composite stiffness. This leads to:


21/75

(7)

The expression in square brackets is the composite stiffness. In page 10 of the section, there is an opportunity to

examine the predicted stiffness as a function of fibre aspect ratio, fibre/matrix stiffness ratio and fibre volumefraction. The other point to note about the shear lag model is that it can be used to examine inelastic behaviour. For

example, interfacial sliding (when the interfacial shear stress reaches a critical value) or fibre fracture (when the

tensile stress in the fibre becomes high enough) can be predicted. As the strain imposed on the composite is

increased, sliding spreads along the length of the fibre, with the interfacial shear stress unable to rise above some

critical value, ti*. If the interfacial shear stress becomes uniform at ti*along the length of the fibre, then a critical

aspect ratio,s*, can be identified, below which the fibre cannot undergo fracture. This corresponds to the peak

(central) fibre stress just attaining its ultimate strengthsf*, so that, by integrating Eq.(3) along the fibre half-length:

(8)

It follows from this that a distribution of aspect ratios betweens*ands*/2 is expected, if the composite is subjected toa large strain. The value ofs*ranges from over 100, for a polymer composite with poor interfacial bonding, to about

2-3 for a strong metallic matrix. In page 10, the effects of changing various parameters on the distributions of

interfacial shear stress and fibre tensile stress can be explored and predictions made about whether fibres of the

specified aspect ratio can be loaded up enough to cause them to fracture.

Conclusion

After completing this section, the student should:

Appreciate that the key issue, controlling both stiffness and strength, is the way in which an applied load isshared between fibres and matrix.

Understand how the slab model is used to obtain axial and transverse stiffnesses for long fibre composites.

Realise why the slab model (equal stress) expression for transverse stiffness is an underestimate and be ableto obtain a more accurate estimate by using the Halpin-Tsai equation.

Understand broadly why the axial stiffness is lower when the fibres are discontinuous and appreciate thegeneral nature of the stress field under load in this case.

Be able to use the shear lag model to predict axial stiffness and to establish whether fibres of a given aspectratio can be fractured by an applied load.

Note that the treatments employed neglect thermal residual stresses, which can in practice be significant insome cases.


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Composite Laminates

Summary

This section covers the advantages of lamination, the factors affecting choice of laminate structure and the approach

to prediction of laminate properties. It is first confirmed that, while unidirectional plies can have high axial stiffness

and strength, these properties are markedly anisotropic. With a laminate, there is scope for tailoring the properties in

different directions within a plane to the requirements of the component. Both elastic and strength properties can be

predicted once the stresses on the individual plies have been established. This is done by first studying how the

stiffness of a ply depends on the angle between the loading direction and then imposing the condition that all the

individual plies in a laminate must exhibit the same strain. The methodology for prediction of the properties of any

laminate is thus outlined, although most of the mathematical details are kept in the background.

What is a Laminate?

High stiffness and strength usually require a high proportion of fibres in the composite. This is achieved by aligning a

set of long fibres in a thin sheet (a laminaor ply). However, such material is highly anisotropic, generally being

weak and compliant (having a low stiffness) in the transverse direction. Commonly, high strength and stiffness are

required in various directions within a plane. The solution is to stack and weld together a number of sheets, each

having the fibres oriented in different directions. Such a stack is termed a laminate. An example is shown in the

diagram. The concept of a laminate, and a pictorial illustration of the way that the stiffness becomes more isotropic

as a single ply is made into a cross-ply laminate, are presented in page 1 of this section.

What are the Stresses within a Crossply Laminate?

The stiffness of a single ply, in either axial or transverse directions, can easily be calculated. (See the section on Load

Transfer). From these values, the stresses in a crossply laminate, when loaded parallel to the fibre direction in one of

the plies, can readily be calculated. For example, the slab model can be applied to the two plies in exactly the same

way as it was applied in the last section to fibres and matrix. This allows the stiffness of the laminate to be calculated.

This gives the strain (experienced by both plies) in the loading direction, and hence the average stress in each ply, for

a given applied stress. The stresses in fibre and matrix within each ply can also be found from these average stresses

and a knowledge of how the load is shared. In page 2 of this section, by inputting values for the fibre/matrix stiffness

ratio and fibre content, the stresses in both plies, and in their constituents, can be found. Note that, particularly with

high stiffness ratios, most of the applied load is borne by the fibres in the "parallel" ply (the one with the fibre axis

parallel to the loading axis).

What is the Off-Axis Stiffness of a Ply?

For a general laminate, however, or a crossply loaded in some arbitrary direction, a more systematic approach is

needed in order to predict the stiffness and the stress distribution. Firstly, it is necessary to establish the stiffness of a

ply oriented so the fibres lie at some arbitrary angle to the stress axis. Secondly, further calculation is needed to find

the stiffness of a given stack. Consider first a single ply. The stiffness for any loading angle is evaluated as follows,

considering only stresses in the plane of the ply The applied stress is first transformed to give the components parallel

and perpendicular to the fibres. The strains generated in these directions can be calculated from the (known) stiffnessof the ply when referred to these axes. Finally, these strains are transformed to values relative to the loading direction,

giving the stiffness.


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Figure 4

These three operations can be expressed mathematically in tensorequations. Since we are only concerned with

stresses and strains within the plane of the ply, only 3 of each (two normal and one shear) are involved. The first step

of resolving the applied stresses, sx, syand txy, into components parallel and normal to the fibre axis, s1, s2and t12(seeFig. 4), depends on the angle, f between the loading direction (x) and the fibre axis (1)

(9)

where the transformation matrixis given by:

(10)

in which c= cosf and s = sinf. For example, the value of s1would be obtained from:

(11)

Now, the elastic response of the ply to stresses parallel and normal to the fibre axis is easy to analyse. For example,

the axial and transverse Youngs moduli (E1andE2) could be obtained using the slab model or Halpin-Tsai

expressions (see Load Transfer section). Other elastic constants, such as the shear modulus (G12) and Poissons ratios,

are readily calculated in a similar way. The relationship between stresses and resultant strains dictated by these elastic

constants is neatly expressed by an equation involving the compliance tensor, S, which for our composite ply, has

the form:

(12)

in which, by inspection of the individual equations, it can be seen that


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Application of Eq.(12), using the stresses established from Eq.(9), now allows the strains to be established, relative to

the 1 and 2 directions. There is a minor complication in applying the final stage of converting these strains so that

they refer to the direction of loading (xandyaxes). Because engineering and tensorial shear strains are not quite the

same, a slightly different transformation matrix is applicable from that used for stresses

(13)

in which,

and the inverse of this matrix is used for conversion in the reverse direction,

(14)

in which,

The final expression relating applied stresses and resultant strains can therefore be written,

(15)

The elements of | |, the transformed compliance tensor, are obtained by concatenation(the equivalent of

multiplication) of the matrices | T '|-1

, | S | and | T |. The following expressions are obtained

(16)


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Figure 5

The final result of this rather tedious derivation is therefore quite straightforward. Eq.(16), together with the elasticconstants of the composite when loaded parallel and normal to the fibre axis, allows the elastic deformation of the ply

to be predicted for loading at any angle to the fibre axis. This is conveniently done using a simple computer program.

The results of such calculations can be explored using pages 4 and 5 in this section. As an example, Fig. 5 shows the

Young's modulus for the an polyester-50% glass fibre ply as the angle, f between fibre axis and loading direction

rises from 0 to 90. A sharp fall is seen as f exceeds about 5-10.

How is the Stiffness of a Laminate obtained?

Once the elastic response of a single ply loaded at an arbitrary angle has been established, that of a stack bonded

together (i.e. a laminate) is quite easy to predict. For example, the Young's modulus in the loading direction is given

by an applied normal stress over the resultant normal strain in that direction. This same strain will be experienced by

all of the component plies of the laminate. Since every ply now has a known Young's modulus in the loading

direction (dependent on its fibre direction), the stress in each one can be expressed in terms of this universal strain.Furthermore, the force (stress times sectional area) represented by the applied stress can also be expressed as the sum

of the forces being carried by each ply. This allows the overall Young's modulus of the laminate to be calculated. The

results of such calculations, for any selected stacking sequence, can be explored using pages 4 and 5.

Are Other Elastic Constants Important?

There are several points of interest about how a ply changes shape in response to an applied load. For example, the

lateral contraction (Poisson ratio, n) behaviour may be important, since in a laminate such contraction may be

resisted by other plies, setting up stresses transverse to the applied load. Another point with fibre composites under

off-axis loading is that shear strains can arise from tensile stresses (and vice versa). This corresponds to the elements

of Swhich are zero in Eq.(12) becoming non-zero for an arbitrary loading angle (Eq.(16)). These so-called "tensile-

shear interactions" can be troublesome, since they can set up stresses between individual plies and can cause the

laminate to become distorted. The value of , for example, represents the ratio between g12and s1. Its value can be

obtained for any specified laminate by using page 6 of this section. It will be seen that, depending on the stacking

sequence, relatively high distortions of this type can arise. On the other hand, a stacking sequence with a high degree

of rotational symmetry can show no tensile-shear interactions. When the tensile-shear interaction terms contributed

by the individual laminae all cancel each other out in this way, the laminate is said to be "balanced". Simple crossply

and angle-ply laminates are notbalanced for a general loading angle, although both will be balanced when loaded at

f=0 (i.e. parallel to one of the plies for a cross-ply or equally inclined to the +q and -q plies for the angle-ply case). If

the plies vary in thickness, or in the volume fractions or type of fibres they contain, then even a laminate in which the

stacking sequence does exhibit the necessary rotational symmetry is prone to tensile-shear distortions and

computation is necessary to determine the lay-up sequence required to construct a balanced laminate. The stacking

order in which the plies are assembled does not enter into these calculations.

Conclusion


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Appreciate that, while individual plies are highly anisotropic, they can be assembled into laminates having aselected set of in-plane properties.

Understand broadly how the elastic properties of a laminate, and the partitioning of an applied load betweenthe constituent plies, can be predicted.

Be able to use the software package to predict the characteristics of specified laminate structures.

Understand the meaning of a "balanced" laminate.


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Fracture Behaviour

Summary

This section covers simple approaches to prediction of the failure of composites from properties of matrix and fibre

and from interfacial characteristics. The axial strength of a continuous fibre composite can be predicted from

properties of fibre and matrix when tested in isolation. Failures when loaded transversely or in shear relative to the

fibre direction, on the other hand, tends to be sensitive to the interfacial strength and must therefore be measured

experimentally. An outline is given of how these measured strengths can be used to predict failure of various laminate

structures made from the composite concerned. Finally, a brief description is given of what is meant by the toughness

(fracture energy) of a material. In composites the most significant contribution to the fracture energy usually comes

from fibre pullout. A simple model is presented for prediction of the fracture energy from fibre pullout, depending on

fibre aspect ratio, fibre radius and interfacial shear strength.

How do Composites Fracture?

Figure 6

Fracture of long fibre composites tends to occur either normal or parallel to the fibre axis. This is illustrated on page

1 of this section - see Fig. 6. Large tensile stresses parallel to the fibres, s1, lead to fibre and matrix fracture, with thefracture path normal to the fibre direction. The strength is much lower in the transverse tension and shear modes and

the composite fractures on surfaces parallel to the fibre direction when appropriate s2or t12stresses are applied. In

these cases, fracture may occur entirely within the matrix, at the fibre/matrix interface or primarily within the fibre.

To predict the strength of a lamina or laminate, values of the failure stresses s1*, s2*and t12*have to be determined.

Can the Axial Strength be Predicted?

Understanding of failure under an applied tensile stress parallel to the fibres is relatively simple, provided that both

constituents behave elastically and fail in a brittle manner. They then experience the same axial strain and hence

sustain stresses in the same ratio as their Young's moduli. Two cases can be identified, depending on whether matrix

or fibre has the lower strain to failure. These cases are treated in pages 2 and 3 respectively.


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Figure 7

Consider first the situation when the matrix fails first (em*


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Figure 8

Alternatively, if the fibres break before matrix cracking has become sufficiently extensive to transfer all the load to

them, then the strength of the composite is given by:

(18)

where sfm*is the fibre stress at the onset of matrix cracking (e1=em*). The composite failure stress depends therefore

on the fibre volume fraction in the manner shown in Fig. 8. The fibre volume fraction above which the fibres can

sustain a fully transferred load is obtained by setting the expression in Eq.(18) equal tof sf*, leading to:

(19)

If the fibres have the smaller failure strain (page 3), continued straining causes thefibresto break up into

progressively shorter lengths and the load to be transferred to the matrix. This continues until all the fibres have

aspect ratios below the critical value (see Eq.(8)). It is often assumed in simple treatments that only the matrix is

bearing any load by the time that break-up of fibres is complete. Subsequent failure then occurs at an applied stress of

(1-f) sm*. If matrix fracture takes place while the fibres are still bearing some load, then the composite failure stress is:

(20)

where smfis the matrix stress at the onset of fibre cracking. In principle, this implies that the presence of a small

volume fraction of fibres reduces the composite failure stress below that of the unreinforced matrix. This occurs up to

a limiting valuef' given by setting the right hand side of Eq.(20) equal to (1-f) sm*.

(21)

The values of these parameters can be explored for various systems using pages 2 and 3. Prediction of the values of

s2*and t12*from properties of the fibre and matrix is virtually impossible, since they are so sensitive to the nature of

the fibre-matrix interface. In practice, these strengths have to be measured directly on the composite material

concerned.

How do Plies Fail under Off-axis Loads?


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Failure of plies subjected to arbitrary (in-plane) stress states can be understood in terms of the three failure

mechanisms (with defined values of s1*, s2*and t12*) which were depicted on page 1. A number of failure criteria

have been proposed. The main issue is whether or not the critical stress to trigger one mechanism is affected by the

stresses tending to cause the others - i.e. whether there is any interactionbetween the modes of failure. In the simple

maximum stress criterion, it is assumed that failure occurs when a stress parallel or normal to the fibre axis reaches

the appropriate critical value, that is when one of the following is satisfied:

(22)

For any stress system (sx, syand txy) applied to the ply, evaluation of these stresses can be carried out as described in

the section on Composite Laminates (Eqs.(9) and (10)).


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Figure 9

Monitoring of s1, s2and t12as the applied stress is increased allows the onset of failure to be identified as the point

when one of the inequalities in Eq.(22) is satisfied. Noting the form of | T | (Eq.(10)), and considering applied

uniaxial tension, the magnitude of sxnecessary to cause failure can be plotted as a function of angle f between stressaxis and fibre axis, for each of the three failure modes.

(23)

(24)

(25)

The applied stress levels at which these conditions become satisfied can be explored using page 5. As an example, the

three curves corresponding to Eqs.(23)-(25) are plotted in Fig. 9, using typical values of s1*, s2*and t12*. Typically,

axial failure is expected only for very small loading angles, but the predicted transition from shear to transverse

failure may occur anywhere between 20 and 50, depending on the exact values of t12*and s2*.

In practice, there is likely to be some interaction between the failure modes. For example, shear failure is expected to

occur more easily if, in addition to the shear stress, there is also a normal tensile stress acting on the shear plane. The

most commonly used model taking account of this effect is the Tsai-Hill criterion. This can be expressed

mathematically as


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(26)

This defines an envelope in stress space: if the stress state (s1, s2and t12) lies outside of this envelope, i.e. if the sum

of the terms on the left hand side is equal to or greater than unity, then failure is predicted. The failure mechanism isnot specifically identified, although inspection of the relative magnitudes of the terms in Eq.(26) gives an indication

of the likely contribution of the three modes. Under uniaxial loading, the Tsai-Hill criterion tends to give rather

similar predictions to the Maximum Stress criterion for the strength as a function of loading angle. The predicted

values tend to be somewhat lower with the Tsai-Hill criterion, particularly in the mixed mode regimes where both

normal and shear stresses are significant. This can be explored on page 6.

What is the Failure Strength of a Laminate?

The strength of laminates can be predicted by an extension of the above treatment, taking account of the stress

distributions in laminates, which were covered in the preceding section. Once these stresses are known (in terms of

the applied load), an appropriate failure criterion can be applied and the onset and nature of the failure predicted.

Figure 10

However, failure of an individual ply within a laminate does not necessarily mean that the component is no longer

usable, as other plies may be capable of withstanding considerably greater loads without catastrophic failure.

Analysis of the behaviour beyond the initial, fully elastic stage is complicated by uncertainties as to the degree to

which the damaged plies continue to bear some load. Nevertheless, useful calculations can be made in this regime

(although the major interest may be in the avoidance of anydamage to the component).In page 7, a crossply (0/90)

laminate is loaded in tension along one of the fibre directions. The stresses acting in each ply, relative to the fibre

directions, are monitored as the applied stress is increased. Only transverse or axial tensile failure is possible in either

ply, since no shear stresses act on the planes parallel to the fibre directions. The software allows the onset of failure to

be predicted for any given composite with specified strength values. Although the parallel ply takes most of the load,

it is commonly the transverse ply which fails first, since its strength is usually very low.


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In page 8, any specified laminate can be subjected to an imposed stress state and the onset of failure predicted. An

example of such a calculation is shown in Fig. 10.

What is the Toughness (Fracture Energy) of a Composite?

The fracture energy, Gc, of a material is the energy absorbed within it when a crack advances through the section of a

specimen by unit area. Potentially the most significant source of fracture work for most fibre composites is interfacial

frictional sliding. Depending on the interfacial roughness, contact pressure and sliding distance, this process can

absorb large quantities of energy. The case of most interest is pull-out of fibres from their sockets in the matrix. Thisprocess is illustrated schematically in page 9.

The work done as a crack opens up and fibres are pulled out of their sockets can be calculated in the following way.

A simple shear lag approach is used. Provided the fibre aspect ratio,s(=L/r), is less than the critical value,s*

(=sf*/2ti*), see page 10 of the Load Transfer section, all of the fibres intersected by the crack debond and are

subsequently pulled out of their sockets in the matrix (rather than fracturing). Consider a fibre with a remaining

embedded length ofxbeing pulled out an increment of distance dx. The associated work is given by the product of

the force acting on the fibre and the distance it moves

dU= (2prxti*) dx (27)

where ti*is the interfacial shear stress, taken here as constant along the length of the fibre. The work done in pulling

this fibre out completely is therefore given by

(28)

wherex0is the embedded length of the fibre concerned on the side of the crack where debonding occurs (x0=L). The

next step is an integration over all of the fibres. If there areNfibres per m2, then there will be (Ndx0/L) per m

2with

an embedded length betweenx0and (x0+ dx0). This allows an expression to be derived for the pull-out work of

fracture, Gc

(29)

The value ofNis related to the fibre volume fraction,f, and the fibre radius, r

N=(30)

Eq.(29) therefore simplifies to


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(31)

This contribution to the overall fracture energy can be large. For example, takingf=0.5,s=50, r=10 m and

ti*=20 MPa gives a value of about 80 kJ m

-2

. This is greater than the fracture energy of many metals. Since sf*wouldtypically be about 3 GPa, the critical aspect ratio,s* (=sf*/2ti*), for this value of ti*, would be about 75. Since this is

greater than the actual aspect ratio, pull-out is expected to occur (rather than fibre fracture), so the calculation should

be valid. The pull-out energy is greater when the fibres have a larger diameter, assuming that the fibre aspect ratio is

the same. In page 10, the cumulative fracture energy is plotted as the crack opens up and fibres are pulled out of their

sockets. The end result for a particular case is shown in Fig. 11.

Figure 11

Conclusion


Appreciate that a unidirectional composite tends to fracture axially, transversely or in shear relative to thefibre direction.

Be able to use simple expressions for axial composite strength, based on fibre and matrix fracturing similarlyin the composite and in isolation.

Understand what is meant by "mixed mode" failure and be able to use Maximum Stress or Tsai-Hill criteriato predict how a unidirectional composite will fail under multi-axial loading.

Be able to use measured strength values for a unidirectional composite to predict how ply damage willdevelop in a laminate.

Understand the concept of the fracture energy of a composite and be able to use the software package topredict the contribution to this from fibre pull-out.

Bibliography


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36/75

Content:

Introduction to Metal Matrix Composites (MMC)

Theoretical Analysis of MMCs

Production of Reinforcements

Production and Applications of MMCs


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Metal Matrix Composite:

Metal matrix composites materials have been active subjects of scientificinvestigation and applied research for more than three decades.

A significant volume fraction of a stiff non-metallic phase in a ductile metal matrixresults in phenomena that are specific to reinforced metals.

In the past few years, we have seen significant advances in our understanding ofthese materials and of phenomena specific to their fabrication and behaviour.

Scientific investigation have addressed the governing principles of theirprocessing and general laws have been identified for the influence exerted by thereinforcement on the microstructural evolution of the matrix.

Advances in computational mechanics have brought to light practically importantmicromechanical phenomena that were often ignored in analytic treatments.

A composite material can be described as a mixture of component materialsdesigned to meet a specific engineering role by exploiting the desirable properties ofthe components, whilst minimising the harmful effects of their less desirableproperties.

It must be man made

It must be a combination of at least two chemically distinct materials

It should be created to obtain the properties which were not otherwise be

achieved by any of the individual constituents

Metal matrix composites consist of at least two components, one is the matrix(usually a metal or an alloy) and the second a reinforcement

The distinction of metal matrix composites from other two or more phasealloys comes from the processing of the composite

Types of Metal Matrix Composites:

Dispersion Strengthened: Microstructure consisting of an elemental matrixwithin which fine particles are uniformly dispersed. The particle diameter ranges

from 0.01 to 0.1 m and the volume fraction 1 to 15%.

Reinforcing particles are much smaller having diameters between 0.01 and 0.1

m (10 and 100 nm). The mechanism of strengthening is similar to that forprecipitation hardening, the matrix bears the major portion of applied load and thesmall dispersed particles hinder or impede the motion of dislocations. Whichresults in restricted plastic deformation and leads to improved yield, tensilestrength and as well as hardness of the composite.

Particle Reinforced: Dispersed particles of greater than 1.0 m diameter

with a vol. frac. of 5 to 40%. There are further two subdivisions of particlereinforced composites as large particle and dispersion strengthen composites.


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The distinction between these is based upon reinforcement or strengtheningmechanism. The hard and stiffer particles tend to restrain movement of the matrixphase in the vicinity of each particle. The matrix (softer phase) transfers some ofthe applied stress to the particles which bear a fraction of the load.The degree ofreinforcement or improvement of mechanical behaviour depends on strongbonding at the matrix-particle interface.

Fibre Reinforced: The reinforcing phase in fibre composite materials span theentire size range and spans the entire range of volume concentrations.

The distinguishing micro-structural feature of fibre-reinforced materials is that thereinforcing fibre has one long dimension, whereas the reinforcing particles of theother two types do not.

Basic Mechanical Behaviour:

A stress-strain curve of a metal matrix composite can conveniently represent itsbasic mechanical behaviour: stiffness, yield flow and fracture stresses.

The stress-strain curve of a typical continuous fibre metal matrix composite consistsat most of three stages.

First stage both matrix and fibre deform elastically,

Second stage the matrix deform plastically while the fibre remains elastic,

and the third stage both matrix and fibre deform plastically.

The existence of the second and third stages of the stress-strain curve of a metalmatrix composite depends on the types of the matrix metal and fibre.

Short Fibre Metal Matrix Composites:

The stress-strain curve of these compositions can be divided into two classes.

First there is a distinct linear region in the initial portion of the stress-strain curvecorresponding to stage I in the continuous fibre composites followed by a parabolicstress-strain curve corresponding to stage III in the continuous fibre composites.

Second the stress-strain curve does not have the linear region, the stress-straincurve are parabolic in shape over the entire range.


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Deformation Characteristics of MMC Containing Second Phase Particles:

The strengthening of a pure metal is carried out by alloying and supersaturating toa extent that on suitable heat treatment the excess alloying addition precipitateout (generally known as ageing process).

To study the deformation behaviour of precipitate hardened alloy or particulatereinforced metal matrix composites,

the interaction of dislocation with reinforcing particles is much moredependent on particle size, spacing and density than on the composition.

When a particle is introduced in a matrix, an additional barrier to the movement ofdislocation is created and the dislocation must behave one of two ways,

cut through the particles or

take a path around the obstacles

Mott and Nabarro Theory:

Mott and Nabarro considered an alloy with spherical solute atoms or groups of soluteatoms, which by virtue of their different atomic size from that of the solvent atoms,caused internal stresses in the matrix.

If the atomic radius of the solvent atom is Rs

then the atomic radius of the solute atom is Rs(1+) where is the misfit parameterdefined as

1a

dadc

. (1)

abeing the lattice parameter and cthe atomic concentration of solute. The sameconcept can be applied to groups of solute atoms.

The internal stress fields caused by the elastic strain between the group of particles

and the matrix are an average distance apart which is the wavelength of theinternal stress field.

From elasticity theory, the shear strain in the internal stress field at distance l fromthe centre of a spherical particle of radius rowhen l>>rois given as:

r

l

o3

3 (2)


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They then calculated a mean shear strain Mfrom equation 2 by assuming that lcould be defined as half the distance between particles:

l N 12

13 (3)

where N is the number of particles per unit volume, so

M o or N r N ( )( )3

13 3 32 8 (4)

The concentration of solute C r No o43

3 . The critical shear stress of the

system is defined in terms of the mean elastic strain as,

o M o oG G r N G C 8 23

(5)

This expression is independent of the particle spacing, the yield stress dependingonly on the mismatch function and the solute concentration.

The experimental results shows that for incoherent particles the yield stress isrelated inversely to the particle spacing.

In the above analysis a rigid dislocation line was assumed but later modification tothe theory introduced the idea of flexible dislocation movement.

The movement would then be dependent on the distance between internal stresscentres.

A dislocation line always tends to reduce its energy by shortening, i-e it tries tostraighten itself. So we can introduce the concept of tension T along the line.

Mott and Nabarro have shown that T~ Gb2.

So if we have a curved dislocation, it can only be in equilibrium if acted on by a

stress. Let us assume that ois the stress needed to maintain the dislocation in acurvature of radius r.

If we consider a small arc of s of a dislocation of strength b. The angle subtendedby the arc at the centre of curvature O is = s/r.

There is an outward force along OA due to the applied stress equal to obs, and anopposing inward force due to the line tension of


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22

T Tsin (6)

In equilibrium

T b so . (7)

so,

0 Tb s

Tb r

Gbr.

(8)

Thus the radius of curvature of a dislocation is inversely proportional to the appliedstress.

Production of Metal Matrix Composites:

Matrices and Reinforcements for MMCs.

Matrices:

Aluminium alloys

Copper alloys

Iron and steel alloys

Magnesium alloys

Nickel based alloys

Titanium alloys

Zinc alloys

Reinforcements:

Non-metallic

Alumina BoronBoron carbide Boron nitrideGraphite Niobium carbideSilica Silicon nitrideTantalum carbide Titanium borideTitanium carbide Titanium aluminideTungsten carbide Vanadium carbide


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Zircon ZirconiaZirconium

Metal Wires

MolybdenumStainless steelTitaniumTungsten

Production of a component from MMCs involves:

bulk production of the composite from its component materials

secondary working of the composite material into some form

joining of the composite materials, leading to final fabrication of the engineered

component

Essential requirements for any production route:

the reinforcement must be distributed in a controlled manner in the metal matrix, i-e eitheruniformly distributed throughout or placed in designated locations of the component

minimal porosity and full density should result in the final component

typically, volume fractions of 0.1 - 0.6 of reinforcement need to be incorporated in the

matrix

reactions at the reinforcement/matrix interface should be controlled to promote optimumbond strength and avoid reinforcement degradation

the reinforcement should be incorporated into the matrix without breakage. This is aparticularly important factor when processing continuous fibre and whisker reinforced

MMCs

during composite joining and forming, minimal reinforcement degradation of eitherchemical or physical means should result. Reinforcement alignment and distribution

should be maintained

the route should be as flexible as possible in terms of matrices and reinforcements towhich it can be applied

subsequent post-fabrication heat treatments should be allowed for

the route should be capable of producing components with a high degree ofreproducibility at minimum product variability at minimum cost and maximum

productivity

flexibility in the range of shapes capable of being produced is highly desirable

any proposed process route should be amenable to scale-up.


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Processing of metal matrix composites can be broadly divided into two categories of

fabrication techniques:

Solid state including powder metallurgy and diffusion bonding and

Liquid state A majority of the commercially viable applications are nowproduced by liquid state processing technique over solid-state techniques.

Liquid-State Processing:

Liquid-state processing technologies can be divided into four major categories:

Infiltration

Dispersion

Spraying and

In-situ fabrication


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Infiltration Processes:

Infiltration processes involve holding a porous body of reinforcing phase within a mould and

infiltrating it with molten metal that flows through interstices to fill the pores and produce a

composite.

The mechanical properties of any given reinforcement-matrix system are highly dependent on

the matrix microstructure. The rules developed for microstructural control in the

solidification of unreinforced metal are generally not directly applicable to metal matrix

composites because the reinforcing phase influences matrix solidification.

The reinforcement act as a barrier to mass transfer during solidification. This causes an

alloyed matrix to grow in avoidance of the fibres because the fibres present a barrier to solute

diffusion. Thus in most cases the last phase to solidify is found in the vicinity of the

reinforcement surface.

If the interfibre spacing is smaller or comparable to the size of the solidification structure

found in the unreinforced metal, the transitions from plane front to cellular growth and fromcellular to dendritic growth are shifted.

When solidification is dendritic in the unreinforced alloy, ripening of dendrite arms ceases in

the composite when the arm spacing reaches the fibre spacing, further coarsening of the

structure occurs by coalescence.

Matrix microsegregation is also strongly affected by the reinforcement, which places an

upper limit on diffusion distance in the solid phase during solidification.

Thus microsegregation can be reduced by solid-state diffusion during solidification to a much

greater extent than that which occurs in unreinforced metals.

If high pressure is applied during solidification, the matrix exhibits a finer grain than would

be obtained at atmospheric pressure.

Dispersion Processes:

In dispersion processes the reinforcement is incorporated in loose form into the metal matrix.

Due to poor wetting characteristic of metal -reinforcement systems, mechanical force is

required to combine these phases.

The simplest dispersion process is the Vortex method, which consists of vigorous stirring of

the liquid metal and the addition of particles in the vortex.

Mixing of particles and metal can also be achieved while the alloyed metal is kept between

solidus and liquidus temperature. This process is known as Compocasting.

The advantage of using semi-solid metal is the increase in the apparent viscosity of the slurry.

Another method to mix semi-solid metal and particulates is the Thi xomoulding process,

whereby metal pallets and particles are extruded through an injection moulding apparatus.

Critical to the success of dispersion processes is the control over generally undesirable

features such as porosity resulting from gas entrapment during mixing, oxide inclusions,

reaction between reinforcement and metal matrix, particle migration and clustering duringand after mixing.


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When solidification takes place, particle migration caused by solidification effects competes

with migration caused by gravity.

Spray Processes:

In these processes droplets of molten metal are sprayed together with the reinforcing phase

and collected on a substrate where metal solidification is completed.

Alternatively the reinforcement may be placed on the substrate and molten metal may be

sprayed onto it.

The critical parameters in spray processing are the initial temperature, size distribution and

velocity of the metal drops, the velocity, temperature and feeding rate of the reinforcement.

Most spray deposition processes use gases to atomise the molten metal into fine droplets.

The particles can be injected within the droplet stream or between the liquid stream and theatomising gas.

The advantage of spray-deposition techniques resides in the resulting matrix microstructure

that features fine grain size and low segregation.

One of the drawback of the process is the amount of residual porosity and the resulting need

to further process the materials.

Osprey Process:

The osprey process is a rapid solidification spray process developed by Osprey Metals for theproduction of both unreinforced and particulate reinforced.

The process involved;

the production of controlled stream of molten metal

the conversion of the stream to spray of molten droplets by means of inert gas atomisation.

impacting of droplets on to a collecting surface and re-coalescence.

injection of reinforcement particles directly into the atomised spray permits MMC

production.

In-Situ Processes:

In-situ processes are the ideal techniques to produce MMC directly in a single step without

the need for intermediate formation of the reinforcement and also a capability to produce near

net shape components.

The reinforcements are formed in-situ in the metal matrix and the surfaces of reinforcements

generated tend to remain clean i-e free from gas absorption oxidation or other detrimental

surface reactions and the matrix-reinforcement interface bond tends to be stronger.


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Optimisation of Liquid-Metal Processing:

Reinforcement:

A number of problems associated with liquid-state processes arise from the reinforcement.

In infiltration processes a preform of the reinforcing phase is often prepared prior to

infiltration, by dispersing the reinforcement in a solution containing an inorganic binder,

pressing to the desired volume fraction and drying the cake composed of fibres and residual

binder material.

When short fibres are used, fibre alignment is difficult to achieve, unless the composite is

extruded, which generally damages the fibres.


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Solid-State Processing:

Solid-state processes are generally used to obtain the highest mechanical properties inMMCs, particularly in discontinuous MMCs.

Because segregation effects and brittle reaction product formation are at a minimum forthese processes, especially when compared with liquid-state processes.

Powder Consolidation:

Powder metallurgy is the most common method for fabricating metal-ceramic and metal-metal composites.

With the advent of rapid solidification technology, the matrix alloy is produced in aprealloyed powder form rather than starting from elemental blends.

After blending the powder with ceramic reinforcement particulates, cold isostatic pressingis utilised to obtain a green compact that is then thoroughly degassed and forged or

extruded.

In some cases hot isostatic pressing of the powder blend is required, prior to whichcomplete degassing is essential.

Consolidation of matrix powder with ceramic fibres has also been achieved but thedifficulties encountered when attempting to maintain uniform fibre spacing.

Powder based routes for MMC production tend to be more expensive than liqu

Handout Booklet_Composite Materials_ 2013.pdf

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