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

9.3 Plasticity and fracture of comp os ites 311 F x l/2 d ¿ i ¿ i Fig. 9.9. Pull-Out of a fibre from the ma- trix fracture of matrix fracture of fibre fibre pull-out ¾ ¾ 0 " Fig. 9.10. Schematic stress-strain diagram of a fibre-reinforced ceramic (after [29]) W f,l  = Z l  0 F dx = Z l  0 τ i xπddx = 1 2 πdτ i l 2 . In a simple approximation, we can assume that the pull-out length varies between 0 and half of the critical length l c . The mean energy dissipation per fibre is thus W f = 1 l c /2 Z l c /2 0 1 2 πdτ i l 2 dl  = 1 24 πdτ i l 2 c . The fracture toughness of a fibre composite is not determined by the dissipation of one single fibre, but by the total dissipation. To estimate this, we have to take into account how many fibres bridge the crack and can dissipate energy by pull-out. Their number is, if the volume fraction is constant, i nversely proportional to the square of the fibre diameter. Using this, the total energy dissipation in the composite is prop ortional to τ i l 2 c /d. Figure 9.10 schematically shows the stress-strain diagram of a ceramic matrix composite. First cracks in the matrix occur at a stress of σ 0 . The load can be 312 9 Mechanical behaviour of fibre reinforced composites increased beyond that because the bridging fibres can bear larger loads, until they finally fracture. 9.3.4 Statistics of composite failure So far, we considered one fibre of the composite only, assuming it to be rep- resentative of all fibres. However, this implies that all fibres have the same properties. In reality, fibre properties are statistically distributed. This is true for their geometry (length and diameter), but also, especially in the case of ceramic fi- bres, for their mechanical properties that are distributed according to Weibull statistics (see section 7.3). Non-ceramic fibres are also usually not identical since they may contain surface defects, for instance. Because of this statistical distribution 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 several times 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 near its end points (se e figure 9.6). The fibre will thus fail by fracture. If the load on the composite is increased, the weakest fibre will break and will thus not transfer any tensile stresses at the position of failure. This fracture, however, will not unload the whole fibre. If it is much longer that the critical length, the load will be transferred by interfacial shear stresses from the matrix to both fibre fragments. At some distance from this region, both fibre fragments bear the same load as before. Near to the fracture position, the material is weakened and the load is transferred to the surrounding material. If the fracture toughness of the matrix is low, this increase in stress can cause local failure of the matrix, initiating a crack that propagates from the site of fibre fracture. Because fibre properties are statistically distributed, the crack will usually not cause the next fibre it encounters to fracture and will be stopped there. The increased load is thus distributed to the surrounding fibres. If the load is increased further, the failure behaviour depends mainly on the fracture toughness of the matrix and the properties of the interface. If the matrix is brittle and the fracture toughness of the interface is large, the stress concentration in f ront of the crack tip is transferred to the fibre, causing it to break. In this case, the crack propagates on load increase, starting from the 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 weak fibre somewhere else in the material will fail first, at a position that is com- pletely inde pendent of the first f ailure position. Thus, fibres will fracture at arbitrary positions in the material, and the load on the material will increase 9.3 Plasticity and fracture of comp os ites 313 homogeneously, with a decrease in stiffness due to the damaged regions. In this 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 in figure 9.3, with the only difference that there is no distinct kink in the curve because the fibres do not fail simultaneously. Because fibre composites frequently fail in this statistical manner by ac- cumulating local damage, the methods of fracture mechanics are often not too useful. If, on the other hand, a sufficiently long crack in a fibre composite forms, it may propagate. In this case, the fracture toughness K Ic of composites with ductile matrix is often smaller than in the pure matrix material because the fibres cause the stress state to be triaxial (see section 3.5.3). This happens in some polymer matrix composites, but mostly in metal matrix composites in which the fracture toughness may be halved compared to the matrix mate- rial [62]. 9.3.5 Failure under compressive loads If a fibre composite is loaded in compression in fibre direction, the deformation mechanism is completely different from the failure behaviour discussed so far. In many fibre composites, the compressive strength is smaller than the tensile strength, a fact that has to be taken into account when designing with these materials. Because the fibres are long compared to their diameter, they may buckle. The buckling load of a cylinder with Young’s modulus E loaded in compression is – assuming Euler’s case 2 of buckling [18] – determined by σ b = π 2 E 16  d l  2 , (9.10) with d and l denoting diameter and length of the fibre [29]. Even in short- fibre reinforced composites with typical fibre lengths of a few millimetres, we usually find l /d > 100. If we consider the e xample of a glass fibre with Young’s modulus of 80 GPa and l /d = 100, we find in the ideal case of a perfectly straight fibre a buckling strength of only 5 MPa. Without the presence of the matrix, the compressive strength of the material would thus be vanishingly small. Buckling of the fibres is impeded by the matrix material that has to deform also when the fibres bu ckle. A single fibre does not form a single large buckle, but buckles in a sine-shaped wave pattern, keeping the deformation of the matrix smaller. In a fibre composite, the fibres are usually so close to each other that neighbouring fibres cannot deform independently. There are two different deformations patterns, sketched in figure 9.11: Neighbouring fibres may deform either in phase or out of phase. 314 9 Mechanical behaviour of fibre reinforced composites ¾ ¾ ¾ ¾ Fig. 9.11. Deformation of a fibre composite under compressive stress. The fibres can b end in an in-phase or out-of-phase pattern The stress required to form these patterns can be calculated using an energy balance: The energy to compress the material without buckling is com- pared to that needed for the bu ckling modes. At small stresses, a homogeneous compression needs less energy, but starting from a certain critical stress value, it is easier to let the fibres buckle than to homogeneously compress the mate- rial fu rther. This critical stress is the compressive stress of the material. It is different for the two deformation patterns. In the out-of-phase deformation mode, the matrix is loaded in ten sion and compression, in the in-phase mode, it is sheared. Because of this, the modes are sometimes called extension mode and shear mode. Except at small volume fractions of the fibre, the strength of the composite is smaller in in-phase deformation which is thus the mode of interest. If a purely elastic deformation of the matrix is assumed, the calculated strength values for the composite are very large, but the observed values are usually much smaller. In metal and polymer matrix composites, the matrix deforms plastically in the in-p hase mode. If we make the simplifying assumption that the matrix is perfectly plastic with a yield strength of σ m,F , the compressive strength is [122] R c,in phase =  f f σ m,F E f 3(1 −f f ) . (9.11) This equation is valid only within certain limits. If the volume fraction of the fibres approaches one, the calculated strength becomes infinite, which is obviously not realistic. If the deformation of the matrix is not determined by plastic deformation alone, its Young’s modulus also plays a role. Further effects that are not considered in the equation and which may reduce the compressive strength are the fibre orientation, the limited interfacial strength between fibre and matrix, and the possibility that the fibres deform and fail not by buckling, but by kinking. The compressive stress calculated with the given equation is independent of the fibre diameter and the fibre length. In reality, longer and thicker fibres are advantageous because it is easier to align them during processing of the material. 9.4 Examples of composites 315 9.3.6 Matrix-dominated failure and arbitrary loads If a composite with unidirectional fibres is loaded in tension or compression perpendicular to the fibre direction or in axial shear in fibre direction, it can fail without failure of the fibres by fracture, buckling, or kinking. These cases are therefore called matrix-dominated failure. In tensile load perpendicular to the fibres, the strengthening effect of the fibres is small. If their elastic stiffness is larger than that of the matrix, the fibres constrain the transversal contraction of the matrix and cause a triaxial stress state. This may, in a metal matrix composite, for example, shift the yield strength to higher loads. If the matrix is brittle, the triaxiality may facilitate crack formation. If the volume fraction of the fibres is large, the matrix between the fibres has to deform more strongly. The exact arrangement of the fibres plays an important role here, for it determines the geometrically necessary deformation of the matrix. Under compressive loads perpendicular to the fibre direction, the matrix may 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 not possible because the fibres impede this. If shear occurs in the direction of the fibres, either the matrix itself can shear between the fibres or there may be shearing along the interface. The strengthening effect of the fibres is small in the latter case as well. If the interface is weak, the strength of the composite may even be smaller than that of the pure matrix material [122]. To design components made of fibre composites, for example using the finite element method [15, 63], it is useful to know yield or failure criteria for the 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 their low processing temperatures. Accordingly, composites with a polymer matrix are 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 this section by discussing long-fibre polymer matrix composites and then study short-fibred ones. Long-fibre reinforced polymer matrix composites Because the strength and elastic stiffness of the fibres used in polymer matrix composites is frequently more than a hundred times larger than that of the 316 9 Mechanical behaviour of fibre reinforced composites 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] material /(g/cm 3 ) E/GPa R m /MPa ε B /− glass fibre 2.5 . . . 2.6 69 . . . 85 1 500 . . . 4 800 1.8 . . . 5.3 aramid fibre 1.4 . . . 1.5 65 . . . 147 2 400 . . . 3 600 1.5 . . . 4.0 p ol yethylene fibre 0.97 62 . . . 175 2 200 . . . 3 500 2.7 . . . 4.4 carb on fibre 1.75 . . . 2.2 140 . . . 820 1 400 . . . 7 000 0.2 . . . 2.4 silicon carbide fibre 2.4 . . . 3.5 180 . . . 430 2 000 . . . 3 700 1.0 . . . 1.5 aluminium 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 are mainly determined by the fibre properties. For this reason, the highest possi- ble fibre volume fractions are aimed at, with maximum values in aerospace industry of about 60%. Nevertheless, the mechanical behaviour of the matrix is also important because it determines load transfer to the fibres and it must not fail if the strength of the fibres is to be exploited fully. Accordingly, we will start this section by discussing the mechanical behaviour of fibres and de- rive the requirements on the matrix material from this. Finally, the comp osite properties are discussed. The fibres Table 9.1 contains a survey of some mechanical parameters of c ommonly used fibre materials. Because glass fibres can have a very high strength of up to 4800 MPa and can also be manufactured inexpensively, it is e asy to understand why they are widespread. Their Young’s modulus is rather low, with values comparable to that of aluminium. It can be increased somewhat by changing the composition of the glass. However, the Si-O bond is less strong than a C-C bond, and the density of bonds in an amorphous material is always smaller than in a crystalline one. This explains why glass fibres cannot be as stiff as carbon fibres. Accordingly, glass fibres are a reasonable choice if the strength of the composite is the main design variable, but they are less useful for applications 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 the tensile strength and Young’s modulus of several carbon fibres. In high-strength fibres, Young’s mo d ulus does not exceed 400 GPa, in high-stiffness fibres, the tensile strength is reduced. This variation in the mechanical properties is due to the fibre microstruc- ture. There are two different structures (so-called ‘allotropes’) of carbon: The diamond structure, shown in figure 1.13, only forms at high temperatures and pressures 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 9.4 Examples of composites 317 3 000 3 500 4 000 4 500 5 000 5 500 6 000 6 500 200 400 600 800 E/GPa R m MPa pan Pitch Fig. 9.12. Mechanical properties of technically used carbon fibres from different suppliers [56, 100, 134, 141]. The two types of fibre differ in their manufacturing pro cess (a) Basal planes in graphite (b) Arrangement of the basal planes in high- strength carbon fibre Fig. 9.13. The basal planes of graphite are arranged in parallel to the fibre axis in carb on fibres. In high-strength fibres, the different regions are connected, rendering slip of the planes past each other more difficult (after [29,97]) hexagonal lattice. The bonds within the hexagonal planes are strong, those between the planes are much weaker (see figure 9.13(a)). Th e sheets or layer planes can easily slide apart, explaining why it is possible to draw pictures with charcoal sticks. The microstructure of the h igh-stiffne ss carbon fibres is similar to that sketched in figure 9.13(a), with the sheets arranged almost perfectly along the 318 9 Mechanical behaviour of fibre reinforced composites fibre axis. Because the covalent C-C bonds within the sheets are extremely strong, 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 planes are only weakly bonded to each other because the bond strength between them is small. Accordingly, the stiffness transversally to the fibre direction is very low (about 6 GPa). Furthermore, this reduces the fibre strength and the interfacial strength between fibre and matrix. To achieve maximal strength, a microstructure is used where the sheets are interwoven, with cross-links between the sheets hampering shearing (see figure 9.13(b)). Because the sheets are oriented obliquely to the fibre axis in this configuration, the stiffness is reduced. Carbon fibres thus have to be optimised either for strength or for stiffness. These two microstructures are produced in two different processes. One pro cess 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- b on fibres can be rather cheap at 25 AC/kg, high-performance fibres can cost as much as 1000 AC/kg due to the involved production pro cess . Because of their high strength, the energy absorption until fracture of high-strength fibres is rather large. For example, a metal with a yield strength of 700 MPa has to be plastically deformed by 10% to achieve the same energy absorption as a fibre with R m = 7000 MPa and a fracture strain of 2%. 10 The strength of the fibres is also determined by their diameter because a thinner fibre contains smaller defects. To achieve a strength of 2000 MPa, a diameter of 10 µm is required, which has to be reduced to 5 µm for a strength of R m = 6000 MPa. Reducing the fibre diameter has some disadvantages as well. It eases buck- ling or kinking of the fibres, so that the shear or compressive strength of the composite does not increase as much as the tensile strength does or may even decrease. This limits the applicability of thin fibres. A further important point is that the fracture strain of high-strength car- bon fibres is about 2% although they deform only elastically. Considering that the strains in the polymer matrix locally exceeds that of the fibre (see figure 9.4), we see that the fracture strain in the matrix has to be rather large. To avoid crack formation in the matrix, its fracture strain should be about twice 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 material can absorb twice as much energy as a linear-elastic material at identical maximum stress and strain. 9.4 Examples of composites 319 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 ex- 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 of carbon bonds can never be as high as that in carbon fibres because of the side groups, it is easily understood that the mechanical properties of polyethylene fibres are inferior to that of carbon fibres. Polymer fibres are viscoelastic even at room temperature. Strength and stiffness are time- and temperature-dependent, a fact that has to be taken into account in the design process. In glass fibres, this is the case only at temperatures of about 200℃, well beyond the service temperature of polymer matrix composites. Carbon fibres are even more stable. Time-dependent be- haviour causes a hysteresis between applied load and observed stress that is especially important under cyclic loading (see section 10.4). The matrix Although most of the mechanical load is borne by the fib res, there are still several requirements for the mechanical properties of the matrix. Its fracture strain 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 compressive loads and to avoid buckling or kinking of the fibres. Finally, its mechanical behaviour should remain unchanged under different environmental conditions (humidity, temperature, irradiation). Unfortunately, these requirements are partially 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 by using thermoplastic matrices which are considered for aerospace applications for this reason. However, th ey are less temperature-resistant than duromers and are more difficult to manufacture because they cannot be produced by curing a resin and thus have to be processed at higher temperatures. Depending on the application, different matrix materials are used. Among the duromers, most common are polyester and epoxy resins. Thermoplastic matrix materials are p olyethylene (pe) and polypropylene (pp), but the use of thermoplastics with aromatic rings on the chain and thus with increased temperature 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 be carefully adjusted to obtain optimal properties of the component under me- chanical loads. Under tensile loads, the fracture strain of the matrix has to be 320 9 Mechanical behaviour of fibre reinforced composites Table 9.2. Increase of Young’s modulus and tensile strength of a duromer matrix (p olyester resin) by addition of glass fibres with a volume fraction of 65% to 70% [77] typ e of fibre E/GPa R m /MPa none 3.5 90 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 not reduce the strength of the component significantly, they can cause consequen- tial damage by penetration of water or other media. In applications with high safety requirements, for example in aerospace industry, the permitted total strain of the composite is limited to a value well below the fracture strain of the fibres for this reason. Because duromer matrix composites are viscoelastic and have no plastic regime, this reduces the permitted stress accordingly. If, for example, the permitted strain is limited to half of the fracture strain, only 50% of the f racture strength can be exploited. This limitation is a crucial reason for the high interest in matrix materials with large fracture strain and temperature stability. Humidity also has a strong influence on the composite’s mechanical be- haviour because it changes the properties of the matrix as already discussed in section 8.8. The strength of the matrix decreases whereas its failure strain increases 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 stresses can be generated. This can also happen in polymer fibres. Aramid fibres, for example, do absorb water, but due to their anisotropic microstructure, they swell 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 poly- mer matrix composites are well below that of long-fibre reinforced materials. Depending on the chosen processing route, the fibres can be oriented in loading direction or irregularly (see section 9.1.1). The influence of the fibre direction on the mechanical properties can be seen from table 9.2 for the example of a glass-fibre reinforced duromer matrix. Young’s modu lus is strongly increased even when irregularly oriented fibres are added. Directing the fibres further increases the stiffness. Using continuous instead of directed short fibres has no significant effect. Relations are different concerning the tensile strength: Although irregu- larly oriented short fibres significantly increase the tensile strength, their effect is much smaller than that of directed fibres. The strength further increases [...]... polymer matrix composites Short-fibre reinforced polymers are useful in many applications where unreinforced polymers are not sufficient The design of injection moulded components made of short-fibre reinforced polymers is complicated by the fact that the orientation of the fibre is determined by the fluid flow (see section 9. 1.1) and can be irregular within the material 9. 4.2 Metal matrix composites Metals... matrix composites are also suitable for push rods in motorcycle engines and for electrically conductive and mechanically loaded connectors on power poles [1] Short-fibre reinforced metal matrix composites are significantly less expensive than long-fibre reinforced materials and can thus be used in automotive engineering or in sports equipment For example, short-fibre reinforced aluminium-silicon carbide composites. .. so-called nacre or mother -of- pearl structure, found, for example, in the pearl oyster In nacre, the ceramic takes the shape of polygonal aragonite platelets with a diameter of approximately 5 µm (see figure 9. 15) The thickness of the aragonite platelets is only 400 nm The matrix in between the platelets is organic and is very thin, with a typical thickness of only 20 nm The mechanical properties of. .. bridging effects [ 69] Fatigue is also observed in transformation-toughened ceramics, like partially stabilised zirconium oxide (see sections 7.2.4 and 7.5.4), where phase transformations occur near the crack tip This effect is attributed to the formation of microcracks in the vicinity of the crack tip [66] 346 10 Fatigue 10.4 Fatigue of polymers Similar to metals, polymers can deform plastically and thus can... strength of semi-crystalline polymers is in most cases superior to that of amorphous polymers [ 19] Whether a polymer fails by mechanical or thermal fatigue is determined by many factors e g., the load frequency, the stress level, the temperature, and the geometry of the component In general, polymers with weak viscoelastic effects, which produce only a small amount of heat in each cycle, fail by mechanical. .. tungsten carbide reinforced hard metals, their wear and temperature resistance is larger In ma- 9. 4 Examples of composites 325 chining steels, one problem is that carbon may diffuse from the silicon carbide into the steel, causing eventual failure of the tool ∗ 9. 4.4 Biological composites Composites are frequently used by organisms in nature to meet the requirements of the environment In this section,... residual (compressive or tensile) stresses that tend to straighten the tree [96 ] Even in a straightly grown tree, the wood is pre-stressed: In the centre of the tree, stresses are compressive, near the bark, they are 9. 4 Examples of composites 5 µm 327 aragonite platelet organic matrix 400nm 20nm Fig 9. 15 Structure of mother -of- pearl (nacre) Flat aragonite platelets are stacked in a staggered way The... engines at elevated temperatures [ 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 and favourable aspect ratio 9. 4 Examples of composites 323 The stiffness and strength of metals can be increased not only by adding fibres, but also using particles In contrast to fibres,... the layered structure If Young’s modulus of a shell is measured in the plane of the platelets using a three-point bending test, the result is about 50 GPa Of much √ higher 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 328 9 Mechanical behaviour of fibre reinforced composites crack propagation is perpendicular... tropocollagen molecules (see figure 9. 16) 16 These fluorine ions reduce the solubility of hydroxyapatite in acidic media To protect our teeth, which have a microstructure very similar to that of bone, tooth paste contains fluorides that improve the acid resistance of the tooth enamel 330 9 Mechanical behaviour of fibre reinforced composites This composite of tropocollagen and hydroxapatite forms fibres that . 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. 9. 3 Plasticity and fracture of comp os ites 311 F x l/2 d ¿ i ¿ i Fig. 9. 9. Pull-Out of a fibre from the ma- trix fracture of matrix fracture of fibre fibre pull-out ¾ ¾ 0 " Fig. 9. 10 application of ceramic matrix composites are cutting tools made of SiC-whisker reinforced aluminium oxide for cutting of hard-to-machine materials, especially nickel- base superalloys and hardened