Mechanical Behaviour of Engineering Materials - Metals, Ceramics, Polymers and Composites 2010 Part 9 pot

Typically, the stress-strain curve for a material witha matrix with sufficiently large fracture toughness is similar to that shown infigure 9.3, with the only difference that there is no

xl/2

F dx =

Z l00

Wf = 1

l c/2

Z l c / 2 0

Figure 9.10 schematically shows the stress-strain diagram of a ceramic matrixcomposite First cracks in the matrix occur at a stress of σ0 The load can be

increased beyond that because the bridging fibres can bear larger loads, untilthey finally fracture.

9.3.4 Statistics of composite failure

So far, we considered one fibre of the composite only, assuming it to be resentative of all fibres However, this implies that all fibres have the sameproperties

rep-In reality, fibre properties are statistically distributed This is true for theirgeometry (length and diameter), but also, especially in the case of ceramic fi-bres, for their mechanical properties that are distributed according to Weibullstatistics (see section 7.3) Non-ceramic fibres are also usually not identicalsince they may contain surface defects, for instance Because of this statisticaldistribution of their properties, not all fibres fail simultaneously even in a ho-mogeneously loaded composite Instead, the weakest fibre will fail first Due

to the volume effect (see section 7.3.1), the failure probability of a long fibre

is greater than that of a short one

In the following, we consider the case of long fibres with a length severaltimes larger than the critical length (see equation (9.9)) In this case, the fibre

is loaded in tension over most of its length, for load transfer occurs only nearits end points (see figure 9.6) The fibre will thus fail by fracture

If the load on the composite is increased, the weakest fibre will breakand will thus not transfer any tensile stresses at the position of failure Thisfracture, however, will not unload the whole fibre If it is much longer that thecritical length, the load will be transferred by interfacial shear stresses fromthe matrix to both fibre fragments At some distance from this region, bothfibre fragments bear the same load as before Near to the fracture position, thematerial is weakened and the load is transferred to the surrounding material

If the fracture toughness of the matrix is low, this increase in stress cancause local failure of the matrix, initiating a crack that propagates from thesite of fibre fracture Because fibre properties are statistically distributed, thecrack will usually not cause the next fibre it encounters to fracture and will

be stopped there The increased load is thus distributed to the surroundingfibres

If the load is increased further, the failure behaviour depends mainly onthe fracture toughness of the matrix and the properties of the interface If thematrix is brittle and the fracture toughness of the interface is large, the stressconcentration in front of the crack tip is transferred to the fibre, causing it

to break In this case, the crack propagates on load increase, starting fromthe site of first fibre fracture If, on the other hand, the stress concentration

in front of the crack tip is not sufficient to cause fibre fracture, another weakfibre somewhere else in the material will fail first, at a position that is com-pletely independent of the first failure position Thus, fibres will fracture atarbitrary positions in the material, and the load on the material will increase

homogeneously, with a decrease in stiffness due to the damaged regions Inthis case, the material will fail by a growing number of breaking fibres, even-tually failing completely Typically, the stress-strain curve for a material with

a matrix with sufficiently large fracture toughness is similar to that shown infigure 9.3, with the only difference that there is no distinct kink in the curvebecause the fibres do not fail simultaneously

Because fibre composites frequently fail in this statistical manner by cumulating local damage, the methods of fracture mechanics are often nottoo useful If, on the other hand, a sufficiently long crack in a fibre compositeforms, it may propagate In this case, the fracture toughness KIcof compositeswith ductile matrix is often smaller than in the pure matrix material becausethe fibres cause the stress state to be triaxial (see section 3.5.3) This happens

ac-in some polymer matrix composites, but mostly ac-in metal matrix composites

in which the fracture toughness may be halved compared to the matrix rial [62]

mate-9.3.5 Failure under compressive loads

If a fibre composite is loaded in compression in fibre direction, the deformationmechanism is completely different from the failure behaviour discussed so far

In many fibre composites, the compressive strength is smaller than the tensilestrength, a fact that has to be taken into account when designing with thesematerials Because the fibres are long compared to their diameter, they maybuckle The buckling load of a cylinder with Young’s modulus E loaded incompression is – assuming Euler’s case 2 of buckling [18] – determined by

short-Buckling of the fibres is impeded by the matrix material that has to deformalso when the fibres buckle A single fibre does not form a single large buckle,but buckles in a sine-shaped wave pattern, keeping the deformation of thematrix smaller In a fibre composite, the fibres are usually so close to eachother that neighbouring fibres cannot deform independently There are twodifferent deformations patterns, sketched in figure 9.11: Neighbouring fibresmay deform either in phase or out of phase

¾

¾

¾Fig 9.11 Deformation of a fibre composite under compressive stress The fibrescan bend in an in-phase or out-of-phase pattern

The stress required to form these patterns can be calculated using anenergy balance: The energy to compress the material without buckling is com-pared to that needed for the buckling modes At small stresses, a homogeneouscompression needs less energy, but starting from a certain critical stress value,

it is easier to let the fibres buckle than to homogeneously compress the rial further This critical stress is the compressive stress of the material It isdifferent for the two deformation patterns

mate-In the out-of-phase deformation mode, the matrix is loaded in tension andcompression, in the in-phase mode, it is sheared Because of this, the modesare sometimes called extension mode and shear mode Except at small volumefractions of the fibre, the strength of the composite is smaller in in-phasedeformation which is thus the mode of interest If a purely elastic deformation

of the matrix is assumed, the calculated strength values for the composite arevery large, but the observed values are usually much smaller In metal andpolymer matrix composites, the matrix deforms plastically in the in-phasemode If we make the simplifying assumption that the matrix is perfectlyplastic with a yield strength of σm,F, the compressive strength is [122]

by plastic deformation alone, its Young’s modulus also plays a role Furthereffects that are not considered in the equation and which may reduce thecompressive strength are the fibre orientation, the limited interfacial strengthbetween fibre and matrix, and the possibility that the fibres deform and failnot by buckling, but by kinking The compressive stress calculated with thegiven equation is independent of the fibre diameter and the fibre length Inreality, longer and thicker fibres are advantageous because it is easier to alignthem during processing of the material

9.3.6 Matrix-dominated failure and arbitrary loads

If a composite with unidirectional fibres is loaded in tension or compressionperpendicular to the fibre direction or in axial shear in fibre direction, it canfail without failure of the fibres by fracture, buckling, or kinking These casesare therefore called matrix-dominated failure

In tensile load perpendicular to the fibres, the strengthening effect of thefibres is small If their elastic stiffness is larger than that of the matrix, thefibres constrain the transversal contraction of the matrix and cause a triaxialstress state This may, in a metal matrix composite, for example, shift theyield strength to higher loads If the matrix is brittle, the triaxiality mayfacilitate crack formation If the volume fraction of the fibres is large, thematrix between the fibres has to deform more strongly The exact arrangement

of the fibres plays an important role here, for it determines the geometricallynecessary deformation of the matrix

Under compressive loads perpendicular to the fibre direction, the matrixmay shear on planes parallel to the fibres In this case, the fibres are irrele-vant for the compressive strength Shearing on planes cut by the fibres is notpossible because the fibres impede this If shear occurs in the direction of thefibres, either the matrix itself can shear between the fibres or there may beshearing along the interface The strengthening effect of the fibres is small inthe latter case as well If the interface is weak, the strength of the compositemay even be smaller than that of the pure matrix material [122]

To design components made of fibre composites, for example using thefinite element method [15, 63], it is useful to know yield or failure criteria forthe composite as a whole that can be evaluated for arbitrary stress states.Several such criteria have been suggested, but all of them are of limited appli-cability [29, 72, 122]

9.4 Examples of composites

9.4.1 Polymer matrix composites

Polymers are well-suited as matrix materials due to their low density and theirlow processing temperatures Accordingly, composites with a polymer matrixare of extreme technical importance They are indispensable in aerospace in-dustry and many other areas, for example in sports equipment Polymer ma-trix composites can be used with long and short fibres We will start thissection by discussing long-fibre polymer matrix composites and then studyshort-fibred ones

Long-fibre reinforced polymer matrix composites

Because the strength and elastic stiffness of the fibres used in polymer matrixcomposites is frequently more than a hundred times larger than that of the

Table 9.1 Density and mechanical parameters (Young’s modulus, tensile strength,fracture strain) of some important fibre materials [29, 41, 100, 117, 131, 141]

glass fibre 2.5 2.6 69 85 1 500 4 800 1.8 5.3aramid fibre 1.4 1.5 65 147 2 400 3 600 1.5 4.0polyethylene fibre 0.97 62 175 2 200 3 500 2.7 4.4carbon fibre 1.75 2.2 140 820 1 400 7 000 0.2 2.4silicon carbide fibre 2.4 3.5 180 430 2 000 3 700 1.0 1.5aluminium oxide fibre 3.3 3.95 300 380 1 400 2 000 0.4 1.5

polymer matrix, the mechanical properties of polymer matrix composites aremainly determined by the fibre properties For this reason, the highest possi-ble fibre volume fractions are aimed at, with maximum values in aerospaceindustry of about 60% Nevertheless, the mechanical behaviour of the matrix

is also important because it determines load transfer to the fibres and it mustnot fail if the strength of the fibres is to be exploited fully Accordingly, wewill start this section by discussing the mechanical behaviour of fibres and de-rive the requirements on the matrix material from this Finally, the compositeproperties are discussed

of the composite is the main design variable, but they are less useful forapplications requiring a high stiffness

Carbon fibres are characterised by a high stiffness and strength However,both parameters cannot be maximised simultaneously Figure 9.12 plots thetensile strength and Young’s modulus of several carbon fibres In high-strengthfibres, Young’s modulus does not exceed 400 GPa, in high-stiffness fibres, thetensile strength is reduced

This variation in the mechanical properties is due to the fibre ture There are two different structures (so-called ‘allotropes’) of carbon: Thediamond structure, shown in figure 1.13, only forms at high temperatures andpressures and is in fact metastable at room temperature The stable confor-mation of carbon is graphite In graphite, the carbon atoms are ordered in a

Fig 9.12 Mechanical properties of technically used carbon fibres from differentsuppliers [56, 100, 134, 141] The two types of fibre differ in their manufacturingprocess

(a) Basal planes in graphite (b) Arrangement of the basal planes in

high-strength carbon fibreFig 9.13 The basal planes of graphite are arranged in parallel to the fibre axis incarbon fibres In high-strength fibres, the different regions are connected, renderingslip of the planes past each other more difficult (after [29, 97])

hexagonal lattice The bonds within the hexagonal planes are strong, thosebetween the planes are much weaker (see figure 9.13(a)) The sheets or layerplanes can easily slide apart, explaining why it is possible to draw pictureswith charcoal sticks

The microstructure of the high-stiffness carbon fibres is similar to thatsketched in figure 9.13(a), with the sheets arranged almost perfectly along the

fibre axis Because the covalent C-C bonds within the sheets are extremelystrong, a large Young’s modulus in fibre direction results A strong fibre tex-ture is thus key to the large elastic stiffness.

However, the problem with this microstructure is that the basal planesare only weakly bonded to each other because the bond strength betweenthem is small Accordingly, the stiffness transversally to the fibre direction isvery low (about 6 GPa) Furthermore, this reduces the fibre strength and theinterfacial strength between fibre and matrix To achieve maximal strength,

a microstructure is used where the sheets are interwoven, with cross-linksbetween the sheets hampering shearing (see figure 9.13(b)) Because the sheetsare oriented obliquely to the fibre axis in this configuration, the stiffness isreduced Carbon fibres thus have to be optimised either for strength or forstiffness

These two microstructures are produced in two different processes Oneprocess starts with polymer fibres, usually made of polyacrylonitrile(pan) The other process uses pitch produced during refinement of min-eral oil Accordingly, the fibres are called pan fibres, with high strength,and pitch fibres, with high elastic stiffness (figure 9.12) Although car-bon fibres can be rather cheap at 25AC/kg, high-performance fibres cancost as much as 1000AC/kg due to the involved production process

Because of their high strength, the energy absorption until fracture

of high-strength fibres is rather large For example, a metal with a yieldstrength of 700 MPa has to be plastically deformed by 10% to achievethe same energy absorption as a fibre with Rm = 7000 MPa and afracture strain of 2%.10

The strength of the fibres is also determined by their diameter because athinner fibre contains smaller defects To achieve a strength of 2000 MPa, adiameter of 10µm is required, which has to be reduced to 5 µm for a strength

of Rm= 6000 MPa

Reducing the fibre diameter has some disadvantages as well It eases ling or kinking of the fibres, so that the shear or compressive strength of thecomposite does not increase as much as the tensile strength does or may evendecrease This limits the applicability of thin fibres

buck-A further important point is that the fracture strain of high-strength bon fibres is about 2% although they deform only elastically Consideringthat the strains in the polymer matrix locally exceeds that of the fibre (seefigure 9.4), we see that the fracture strain in the matrix has to be rather large

car-To avoid crack formation in the matrix, its fracture strain should be abouttwice that of the fibre i e., 4% to 5% Currently available duromers do not

10

To arrive at this number, it has to be kept in mind that a perfectly plastic materialcan absorb twice as much energy as a linear-elastic material at identical maximumstress and strain

meet this requirement, reducing the permissible strain Thus, the full strength

of the fibres can often not be exploited (see also exercise 29)

Polymers can also be strengthened using polymer fibres As already plained in section 8.4, high strength polymer fibres can be produced by draw-ing the chain molecules in fibre direction (see figure 8.20 and section 8.5.2).Commonly used fibres are based on aramid or polyethylene As the density ofcarbon bonds can never be as high as that in carbon fibres because of the sidegroups, it is easily understood that the mechanical properties of polyethylenefibres are inferior to that of carbon fibres

ex-Polymer fibres are viscoelastic even at room temperature Strength andstiffness are time- and temperature-dependent, a fact that has to be takeninto account in the design process In glass fibres, this is the case only attemperatures of about 200℃, well beyond the service temperature of polymermatrix composites Carbon fibres are even more stable Time-dependent be-haviour causes a hysteresis between applied load and observed stress that isespecially important under cyclic loading (see section 10.4)

The matrix

Although most of the mechanical load is borne by the fibres, there are stillseveral requirements for the mechanical properties of the matrix Its fracturestrain should be sufficiently large to avoid premature damage of the compos-ite by crack formation in the matrix Its elastic stiffness should be as large

as possible to achieve a sufficient support of the fibres under compressiveloads and to avoid buckling or kinking of the fibres Finally, its mechanicalbehaviour should remain unchanged under different environmental conditions(humidity, temperature, irradiation) Unfortunately, these requirements arepartially contradictory The fracture strain of a duromer matrix, for exam-ple, can be increased by decreasing the cross-linking density This, however,reduces the elastic stiffness Large fracture strains can also be achieved byusing thermoplastic matrices which are considered for aerospace applicationsfor this reason However, they are less temperature-resistant than duromersand are more difficult to manufacture because they cannot be produced bycuring a resin and thus have to be processed at higher temperatures

Depending on the application, different matrix materials are used Amongthe duromers, most common are polyester and epoxy resins Thermoplasticmatrix materials are polyethylene (pe) and polypropylene (pp), but the use

of thermoplastics with aromatic rings on the chain and thus with increasedtemperature stability also grows One example is polyetheretherketone (peek),characterised by high toughness and a glass temperature of about 150℃.Composite properties

It was already stressed that the properties of fibre and matrix have to becarefully adjusted to obtain optimal properties of the component under me-chanical loads Under tensile loads, the fracture strain of the matrix has to be

Table 9.2 Increase of Young’s modulus and tensile strength of a duromer matrix(polyester resin) by addition of glass fibres with a volume fraction of 65% to 70% [77]

short fibres, irregular 20 190

short fibre, oriented at ±7◦ 35 520

continuous fibres, uniaxial 38 1 300

sufficient for the chosen fibre material Although cracks in the matrix do notreduce the strength of the component significantly, they can cause consequen-tial damage by penetration of water or other media In applications with highsafety requirements, for example in aerospace industry, the permitted totalstrain of the composite is limited to a value well below the fracture strain ofthe fibres for this reason Because duromer matrix composites are viscoelasticand have no plastic regime, this reduces the permitted stress accordingly If,for example, the permitted strain is limited to half of the fracture strain, only50% of the fracture strength can be exploited This limitation is a crucialreason for the high interest in matrix materials with large fracture strain andtemperature stability

Humidity also has a strong influence on the composite’s mechanical haviour because it changes the properties of the matrix as already discussed

be-in section 8.8 The strength of the matrix decreases whereas its failure strabe-inincreases with increasing water content Some residual humidity can therefore

be advantageous in composites with a duromer matrix Glass or carbon fibres

do not absorb any water If the polymer matrix swells, large residual stressescan be generated This can also happen in polymer fibres Aramid fibres, forexample, do absorb water, but due to their anisotropic microstructure, theyswell mainly in radial direction, also causing large residual stresses

Short-fibre reinforced polymer matrix composites

The strength and stiffness that can be obtained in short-fibre reinforced mer matrix composites are well below that of long-fibre reinforced materials.Depending on the chosen processing route, the fibres can be oriented in loadingdirection or irregularly (see section 9.1.1)

poly-The influence of the fibre direction on the mechanical properties can beseen from table 9.2 for the example of a glass-fibre reinforced duromer matrix.Young’s modulus is strongly increased even when irregularly oriented fibresare added Directing the fibres further increases the stiffness Using continuousinstead of directed short fibres has no significant effect

Relations are different concerning the tensile strength: Although larly oriented short fibres significantly increase the tensile strength, their effect

irregu-is much smaller than that of directed fibres The strength further increases

by more than a factor of two when continuous fibres are used because thelength of the short fibres is below the critical length.11 Even if the fibres arelarger than the critical length, it is experimentally observed that a furtherincrease in fibre length increases the tensile strength [122] because local weakpoints, caused by irregularities in the fibre distribution, determine the tensilestrength.

Mechanically, it is thus best to use fibres that are as long as possible This,however, is limited by processing technology For example, long fibres maybreak or clog the nozzles in injection moulding Processing technology alsolimits the volume fraction of short fibres, usually to values that are smallerthan in long-fibre reinforced composites

The same materials can be used as in long-fibre reinforced polymer trix composites Short-fibre reinforced polymers are useful in many applica-tions where unreinforced polymers are not sufficient The design of injectionmoulded components made of short-fibre reinforced polymers is complicated

ma-by the fact that the orientation of the fibre is determined ma-by the fluid flow(see section 9.1.1) and can be irregular within the material

9.4.2 Metal matrix composites

Metals are especially attractive as matrix material in a composite As the ture strain of the matrix is larger than that of common fibre materials, thefibre strength can be fully exploited, and the local strain concentration nearthe interface (see section 9.3.2) is irrelevant for the composite strength Sincethe adhesion between fibre and matrix is frequently strong in metal matrixcomposites, the maximum interfacial shear stress is usually limited by themetal’s yield strength and is correspondingly large The critical fibre length

frac-is thus small and even short fibres result in a high strengthening effect Thelarge Young’s modulus and yield strength of the matrix also lead to a highcompressive strength because bending or kinking of the strengthening fibres

is avoided (see section 9.3.5) Metal matrix composites can be used at highertemperatures than polymer matrix composites because the temperature sta-bility of the matrix is larger

The fibres determine the mechanical properties of the composite not only

by load transfer, but also by additional effects: Strengthening particles orfibres can pin grain boundaries during processing of the material and thusreduce grain size This increases the strength by grain boundary strengthening(see section 6.4.2) at low temperatures The dislocation density can also beincreased by adding fibres: If the composite is cooled from the required highprocessing temperatures, differences in the coefficient of thermal expansioncan cause plastic deformation in the vicinity of the fibre This increases thestrength, but also causes residual stresses which may reduce the strength

11

The maximum interfacial shear stress in polymer matrix composites is determined

by the adhesion between fibre and matrix, not by the yield strength of the matrix

A further increase in dislocation density occurs during plastic deformationbecause plastic deformation is usually limited to the matrix, leading to aformation of dislocation loops around the fibres (see also section 6.4.4) TheOrowan mechanism (see section 6.3.1 and figure 6.45), which would impededislocation movement, is not relevant, though, because the fibre diameter anddistance are too large.

Fibre materials in metal matrix composites are limited to those with asufficiently high melting temperature because they have to withstand highprocessing temperatures Possible materials are carbon, ceramics (for examplealuminium oxide or silicon carbide), and high-melting point metals like boron

or tungsten Suitable matrix materials are mainly the light metals aluminium,titanium, and magnesium

Aluminium is the most frequently used matrix material due to its ratherlow melting point (depending on the alloy, about 600℃ to 660℃) whicheases the processing, but also because of its high ductility In applications,

it is not only the strengthening, but also the increase in stiffness that is tractive since Young’s modulus of aluminium is rather low (approximately

at-70 GPa) Adding Al2O3 long fibres with a volume fraction of 50% increasesits value to 200 GPa [121]; by using carbon fibres, a stiffness of 400 GPa can

425℃ decreases only to 1050 MPa [49] If titanium is used as matrix materialinstead, the strength at room temperature does not increase much because it

is determined by the fibre material However, due to the high melting point

of titanium, the material can be used at higher temperatures and the tensilestrength at 600℃ is still about 1000 MPa [49]

Due to their high specific strength and stiffness, long-fibre reinforced minium matrix composites are attractive in aerospace applications The high-gain antenna boom of the Hubble Space Telescope, for example, is made from

alu-a calu-arbon-fibre reinforced alu-aluminium malu-atrix composite [114] Aluminium ide reinforced aluminium matrix composites are also suitable for push rods

ox-in motorcycle engox-ines and for electrically conductive and mechanically loadedconnectors on power poles [1]

Short-fibre reinforced metal matrix composites are significantly less pensive than long-fibre reinforced materials and can thus be used in automo-tive engineering or in sports equipment For example, short-fibre reinforcedaluminium-silicon carbide composites can be used as pistons in diesel engines

ex-at elevex-ated temperex-atures [49] Golf clubs and bicycle components can also

be manufactured from aluminium matrix composites Frequently, whiskers(see section 6.2.8) are used as short fibres because of their high strength andfavourable aspect ratio

The stiffness and strength of metals can be increased not only by addingfibres, but also using particles In contrast to fibres, load is transferred also

at the front and back end of the particle, not only by shear stresses In analuminium-silicon carbide composite, for example, the tensile strength can be

as high as 700 MPa

Metal matrix composites can be interesting due to other properties as well:The coefficient of thermal expansion of a metal can be strongly reduced byadding carbon fibres and may even become negative.12This is important if thecomponent may not distort on thermal loading or when the material has to bejoined to a ceramic because the coefficient of thermal expansion of ceramics

is usually much smaller than that of metals (see section 2.6) The thermalproperties are also of interest in copper-carbon composites because copperhas a large thermal conductivity, but is mechanically rather weak Carbon isespecially suited as fibre material not only due to its stiffness and strength,but also because of its high thermal conductivity that may even exceed that

of copper.13

9.4.3 Ceramic matrix composites

As we saw in chapter 7, ceramics have the attractive properties of high ature resistance, high strength and stiffness, low density, and high resistanceagainst many aggressive media Their main disadvantage is their low fracturetoughness and the resulting sensitivity to small defects The main objective

temper-in strengthentemper-ing ceramics with fibres is thus to temper-increase the fracture ness It can take values of up to 30 MPa√

tough-m [25,149], approxitough-mately ten titough-meslarger than in most unreinforced ceramics Furthermore, using a fibre compos-ite can also increase the Weibull modulus to about 30, reducing the scatter ofstrength and thus easing component design

Suitable fibre materials in ceramic matrix composites are ceramics (forexample aluminium oxide or silicon carbide), carbon, and high-melting pointmetals like boron or tungsten (see table 9.1) In short-fibre reinforced ceramics,whiskers are commonly used because longer irregular short fibres may decreasethe tensile strength, though they increase the fracture toughness [25] Themost frequently used matrix materials are aluminium oxide, silicon carbide,

ensure a sufficient energy dissipation during pull-out of the fibres Chemicalbonding between fibre and matrix is therefore usually not desired because itwould produce a high-strength interface Fibre and matrix material thus have

to be adjusted to ensure that no chemical reactions occur even at the ratherhigh processing temperatures required

To design the interfacial properties, the fibres can be coated before thecomposite is produced by applying thin coatings with a thickness between0.1µm and 1 µm A graphite layer of 1 µm thickness on a fibre based on siliconcarbide (called Nicalon), for example, can reduce the interfacial shear strengthfrom 400 MPa to 100 MPa [28]

Furthermore, care has to be taken to ensure a smooth surface of the fibre.Even without chemical bonding between fibre and matrix, a rough surface mayimpede the pull-out of the fibre by mechanical clamping in the matrix [28].The coefficient of thermal expansion of fibre and matrix should also not betoo different to avoid large thermal stresses during cooling from the processingtemperature Especially problematic is the case of the coefficient of thermalexpansion of the matrix being larger than that of the fibre, for the matrix willthen shrink onto the fibre and mechanically clamp it, making pull-out difficult

If, on the other hand, the coefficient of thermal expansion of the fibre is larger,the matrix will be under compressive stress in axial fibre direction This can

be advantageous because it impedes the propagation of cracks, as long as thestresses in the fibre do not become too large To avoid local thermal stresses,

a coating interlayer between fibre and matrix may be helpful

In ceramic matrix composites, the fracture strain of the matrix is usuallysmaller than that of the fibre, resulting in the matrix to fail first The stress-strain diagram (figure 9.10) is more similar to that of a material with anapparent yield point (figure 3.5(b)) than to that of a standard ceramic Todesign with the composite, the fracture strength of the matrix can thereforesafely be used to determine the maximum permissible stress in the compo-nent because no catastrophic failure will ensue if the load is exceeded Thecomposite thus has a higher failure tolerance

Due to the excellent high-temperature properties of ceramics, ceramic trix composites are mainly used in aerospace industry and in power engineer-ing For example, components for gas turbines, rocket engines, or heat shields(e g., in the Space Shuttle) can be made of ceramic matrix composites Theymay also be used in brake discs in aeroplanes or in upmarket cars One exam-ple are the brake discs of the Boeing 767, manufactured from a carbon-carboncomposite Compared to a conventional brake disc, the mass could be reduced

ma-by almost 40% [28]

If market volume is taken as a measure, the most important application ofceramic matrix composites are cutting tools made of SiC-whisker reinforcedaluminium oxide for cutting of hard-to-machine materials, especially nickel-base superalloys and hardened steels [25] Compared to tungsten carbide re-inforced hard metals, their wear and temperature resistance is larger In ma-

chining steels, one problem is that carbon may diffuse from the silicon carbideinto the steel, causing eventual failure of the tool.

∗ 9.4.4 Biological composites

Composites are frequently used by organisms in nature to meet the ments of the environment In this section, we will discuss three naturallyoccurring composites

require-Different from most man-made materials, biological materials are oftencharacterised by their water content The mechanical properties of wood orbone in the natural i e., humid, state are vastly different from that of the driedmaterials This requires some effort in testing biological materials because it isdifficult to control the water content in the laboratory with sufficient precision

∗ Wood

Wood is made of plant cells elongated in the axial direction of the tree orbranch The mechanical properties of wood are determined by the cell wallswhich are a composite of a natural polymer matrix with cellulose fibres [9,144] Cellulose is a polysaccharide, a chain molecule with sugar molecules asmonomers.14The cellulose molecules have a degree of polymerisation of about

104 and are arranged in microfibrils with a diameter of 10 nm to 20 nm, with

a high crystallinity The bonds between the cellulose molecules are hydrogenbonds and are very strong due to the ordered structure in the crystallineregions Up to now, Young’s modulus of the crystalline regions can only beestimated theoretically, taking a value of about 250 GPa, whereas the modulus

of the amorphous regions is about 50 GPa The surrounding matrix comprises

an amorphous phenylpropanol duromer, called lignin, hemicellulose, a chained cellulose variant, water, oils, and salts The volume fraction of cellulose

short-in wood is about 45%, the lignshort-in and hemicellulose content is about 20% each.The cellulose fibres are situated in the cell walls of long, tube-shaped cells,directed in the axial direction of the tree Within the cell walls, they arearranged in different layers (see figure 9.14) The outer, primary cell wall,contains irregularly arranged fibres The next layer, the secondary cell wall,consists of three layers The cellulose fibres in the outer and inner layer areoriented transversally to the cell direction (and the main loading direction),

in the medial layer of the secondary cell wall, they are arranged helically,slightly inclined to the longitudinal direction This helical arrangement of thefibres in the medial cell wall increases the strength because, under tensileloads, the fibres are straightened and have to slide against each other This issimilar to the carbon fibres discussed above, where the non-perfect alignment

in loading direction also serves to increase the strength (see section 9.4.1, page

14

There is another polysaccharide, chitin, that is used as ‘engineering material’ innature Most biological polymers, however, are proteins

primary cell wallsecondary cell wall

}

Fig 9.14 Arrangement of cellulose fibres in the cell wall of wood cells The diameter

of the cells lies between 20µm and 40 µm, their length between 2 mm and 4 mm.Simplified illustration after [9, 144]

316) Nevertheless, it is much easier to split wood parallel to the fibre thantransversally because the crack can run between the cells

As all fibres, cellulose fibres can bear higher loads in tension than in pression because the fibres can buckle or kink under compressive loads Thearrangement of the cellulose fibres in the outer and inner layer of the sec-ondary cell wall ensures that they are loaded in tension if the wood as awhole is loaded in compression and thus increase the compressive strength.The compressive strength of wood, however, is about 30 MPa, approximatelyone third of its tensile strength

com-The elastic stiffness of wood is much smaller than the theoretical stiffness

of single cellulose fibres This is due to the orientation of the fibres whichdiffers from the loading direction as explained above, but also to the volumefraction of the cell walls which comprise only about 25% of the total volume.Young’s modulus of wood is thus only about 10 GPa in longitudinal direction.Although this is a rather low value, wood is an attractive material for light-weight applications because its density is rather small (with values between0.2 g/cm3 and 1.4 g/cm3) The anisotropy of wood can be avoided by usingplywood or flake boards

Trees can react to external stresses by adapting their growth If a tree

is loaded asymmetrically (in bending), for example by wind loads ordue to growth on inclined ground, it will form so-called reaction wood

In softwoods (as found in conifers), the reaction wood forms on theside that is under compressive loads, in hardwoods on the tensile side.This reaction wood creates residual (compressive or tensile) stressesthat tend to straighten the tree [96]

Even in a straightly grown tree, the wood is pre-stressed: In thecentre of the tree, stresses are compressive, near the bark, they are

Fig 9.15 Structure of mother-of-pearl (nacre) Flat aragonite platelets are stacked

in a staggered way The organic matrix lies between the platelets (after [145])

tensile This has the advantage that these stresses are superimposed toexternal stresses under bending (for example due to wind loads) Theresidual stresses thus increase the tensile and decrease the compressivestresses Because the compressive strength of wood is smaller than thetensile strength, this results in a higher load capacity of the tree

∗ Nacre

Bivalves, snails, and cephalopods, biologically united as molluscs, often tect themselves with hard shells These shells are a composite, comprising anorganic matrix with included ceramic particles, with a particle volume fraction

pro-of 95% or even more [144]

Because of the high ceramic volume fraction, the mechanical properties

of the shells are mainly determined by those of the ceramic The ceramiccomponent is aragonite, a rhombic crystal modification of calcium carbonateCaCO3, forming prismatic crystals Young’s modulus of aragonite is approx-imately 100 GPa, its fracture toughness is rather low, with a value of about0.5 MPa√

m

There are different shell microstructures in different species In this section,

we only discuss the so-called nacre or mother-of-pearl structure, found, forexample, in the pearl oyster In nacre, the ceramic takes the shape of polygonalaragonite platelets with a diameter of approximately 5µm (see figure 9.15).The thickness of the aragonite platelets is only 400 nm The matrix in betweenthe platelets is organic and is very thin, with a typical thickness of only 20 nm.The mechanical properties of nacre are highly anisotropic due to the lay-ered structure If Young’s modulus of a shell is measured in the plane of theplatelets using a three-point bending test, the result is about 50 GPa Of muchhigher interest is the fracture toughness, for it can be as high as 10 MPa√

m,twenty times larger than that of the ceramic component, if the direction of

crack propagation is perpendicular to the platelets (vertical direction in thefigure).

This high fracture toughness is caused by several mechanisms: Single gonite platelets are thinner than the critical crack length of aragonite At

ara-a stress of ara-about 150 MPara-a, the tensile strength, the criticara-al crara-ack length isabout 3.5µm, according to equation (5.2) Therefore, they cannot containcritical cracks The low fracture toughness of the organic matrix causes acrack to be deflected on reaching a platelet and to propagate around them.15

Additionally, there may be pull-out of the platelets Nano-asperities on theplatelets cause additional dissipation during sliding of the platelets In total,the work needed to create fresh surface in nacre is about 1600 J/m2 if thecrack propagates perpendicularly to the platelets; if it propagates in parallel

to the platelets, it is only 100 J/m2, but still larger than in pure aragonite,where the value is about 2 J/m2, according to equation (5.17)

If we compare the increase in fracture toughness that has been achieved innacre to those obtained in technical ceramics (see table 7.4), it is rather obvi-ous that it would be highly desirable to technically exploit the same strength-ening mechanisms This is one reason for the strong scientific interest in nacre.The main aim of these studies is to create artificial materials with similar prop-erties Such materials, which mimic the properties of biological materials, arecalled biomimetic materials

∗ Bone

Bone is a biological material of special importance On the one hand, bonesare the characteristic trait of vertebrates which almost exclusively occupy allecological niches for large animals Thus, it is of biological interest to under-stand why having bones is evolutionary advantageous Even more important

is that understanding bone structure enables us to treat or heal bone illnesses

or injuries For these reasons, the structure and mechanical behaviour of boneshave been intensely studied [36]

Bone has a complex hierarchical structure on several different length scales.The main components of bone are a ceramic, (modified) hydroxyapatite, and

a polymer, the protein collagen Furthermore, bone contains other proteins,protein-sugar compounds, and, as all biological materials do, water

Collagen is a protein containing about 1100 amino acids in an exactlydefined sequence This sequence is determined by the genetic code within thedna If we consider that there are 20 different amino acids used in commonproteins, we see that the number of possible proteins is huge and that anexact control of the amino acid sequence is extremely important to ensurethe correct spatial structure of the macromolecule Collagen molecules form

a helical structure, a long, almost straight helix Three of these helices are

15

This is similar to the crack propagation mechanism in sintered silicon nitride, seepage 249

bloodvessel

intertwined to form a larger component, the tropocollagen molecule, with alength of 296 nm

The tropocollagen molecules themselves are aligned in parallel in bonesand tendons, being shifted by 67 nm in adjacent layers Within each layer,there are gaps between the molecules that serve as nucleation sites for thecrystallisation of the ceramic hydroxyapatite crystals (see figure 9.16).Hydroxyapatite has the chemical composition Ca10(PO4)6(OH)2 In thebody, its composition slightly differs from this formula (with the resultingmaterial frequently called being dahllite), for some calcium ions are replaced

by other ions, and fluorine ions replace some of the (OH)− ions.16 Because

of their small size, it is rather difficult to determine the exact shape of thehydroxyapatite crystals in bone, which may also be different in different bones.Typically, they are platelet-shaped, with a thickness of only 5 nm and an edgelength between 20 nm and 100 nm These platelets are situated between thetropocollagen molecules (see figure 9.16)

16

These fluorine ions reduce the solubility of hydroxyapatite in acidic media Toprotect our teeth, which have a microstructure very similar to that of bone, toothpaste contains fluorides that improve the acid resistance of the tooth enamel

This composite of tropocollagen and hydroxapatite forms fibres that unite

to form fibre bundles The fibre bundles are the building blocks of the nexthierarchical layer Depending on the bone type, the fibre bundles may bearranged irregularly, uniaxially, or in a lamellar structure, the latter structurebeing the one most common in adult humans Within the lamellar bone, thefibres are arranged in parallel in layers; the fibre bundles in adjacent layersare rotated relative to each other, similar to the fibre layers in a laminate (seesection 9.1.1)

In adult humans, these lamellae form tube-shaped structures, called teons or Haversian systems A single osteon has a diameter of about 200µmand a length of a few millimetres or centimetres In long bones, like limb bones,they are parallel to the bone axis In the centre of each osteon, there is a bloodvessel that supplies the cells within the bone with nutrients

os-How the fibres in the lamellae of an osteon are arranged depends on themechanical load applied to the bones Long bones are mainly loaded in bend-ing.17 On the tensile side of the bone, the fibres are oriented in longitudinaldirection, on the compressive side, they are arranged either in circumferentialdirection or alternating between longitudinal and circumferential direction.The arrangement of the lamellae, like that of the osteons, is thus optimised

to the external load

Young’s modulus of bone depends on the volume fraction of the apatite and on the osteon structure In the stiffest direction, it is between

hydroxy-12 GPa and 25 GPa If the strains exceed values of about 0.5%, bone starts todeform by microcracking The fracture strain is usually 2%, but in some spe-cialised bones that are loaded in impact (for example, in the antlers of deer),

it may be as high as 10% The strength of normal bone is approximately

150 MPa in tension and 250 MPa in compression The peak loads under mal loading (walking, running, climbing a staircase) are approximately two tofour times smaller than this value

nor-To ensure a favourable orientation of the fibres, bone is permanently built and adapted to the actual loads Specialised cells within the bone, theosteocytes, measure the loads and initiate the rebuilding The old bone isremoved by acid-excreting cells (osteoclasts) and is then rebuild by othercells (osteoblasts), forming new osteons The rebuilding of the bone not onlyensures its adaptation to changing load patterns, but it also serves to healmicrocracks that may have been formed during excess loading As long asliving bone is not overloaded, it is therefore completely fatigue resistant

re-If the load on a bone is changed compared to the load patterns previouslyencountered, bone material is added or removed New bone is formed, forexample, when a new sports training is begun; bone is removed if it is notloaded anymore, for instance due to long-time illness or to the insertion ofimplants Because Young’s modulus of bone is markedly smaller than that

17

This is the reason why long bones are hollow and are filled with bone marrow– in mammals – or with air sacs – in birds –, for weight can be saved this way