7.2 Mechanisms of crack propagation 229 ceramics without significant porosity between the former powder particles. Sintering aids, like magnesium oxide (MgO) for silicon nitride (Si 3 N 4 ), can be added that pro d uc e a liquid phase at the sintering temperature and thus facilitate the sintering process. One disadvantage is that this phase, the so- called glassy phase, is amorphous and thus reduces the strength at elevated service tem peratures (creep strength, see chapter 11). 7.2 Mechanisms of crack propagation Because dislocations are completely immobile in ceramics at ro om temper- ature due to the directed atomic bonds and the complex crystal structures (see section 1.3), ceramics can in general not deform plastically. Failure can occur only by cleavage fracture, usually with initial cracks growing and prop- agating. The pores remaining after compaction are defects acting as initial cracks and thus cause failure by crack propagation. As there is no plastic deformability, it cannot unload thes e initial cracks by evening out stress con- centrations or dissipate energy during crack propagation. Therefore, the frac- ture toughness of ceramics is comparably small. This is also reflected in the toughness-strength diagram 5.11 on page 146. Because of the crack sensitivity of ceramics, even small defects can determine the strength – the pre-cracks formed during manufacturing are thu s crucial for the m echanical behaviour. The theoretical strength of a perfect ceramic without any defects is technically irrelevant. Usually, ceramics always contain cracks of different sizes with different ori- entations. The strength of the ceramic is determined by the cracks with the lowest failure strength. Under tensile loads, cracks can, depending on their orientation, be loaded in all modes, I, II, or III (cf. section 5.1.1), under com- pressive loads only in mode II or III, for the stress component perpendicular to the crack surface closes the crack. Because the fracture toughness is much smaller for mode I than for modes II or III, ceramics under tensile loads usu- ally fail in this mode and are thus more sensitive to tensile than to compressive stresses. The compressive strength of ceramics is usually 10 to 15 times larger than its tensile strength. 3 The fracture toughness is primarily determined by the strength of the chemical bonds within the ceramic because this determines the energy needed to create fresh surface. Beyond that, other effects within ceramics can occur that impede crack propagation because they require additional energy and thus increase the fracture toughness. The basic mechanisms are discussed in this section; in section 7.5, we will see how they can be utilised to strengthen ceramics. 3 This property of ceramics is exploited in the design of ferroconcrete (steel- reinforced concrete), for example, see section 9.1.1. 230 7 Mechanical behaviour of ceramics (a) Without particles (b) With particles Fig. 7.2. Deflection of a crack by appropriate particles in a ceramic 7.2.1 Crack deflection When the crack can be deflected from its straight path, the surface of the crack per advanced distance becomes larger, thus requiring additional energy for crack propagation and increasing the fracture toughness. This can be achieved in s everal ways, often by adding particles. One possible mechanism is the modulus interaction, already discussed in another context in the chapter on metals (section 6.4.3). If the particles have a larger Young’s modulus than the matrix, the matrix is p artly unloaded in the vicinity of the particles, and the stress available to propagate the crack is reduced. The crack is deflected away from the particle (see figure 7.2). If Young’s modulus of the particles is smaller than that of the matrix, the stress is raised in the vicinity of the particles, and the crack is attracted by the particle. If the crack cannot penetrate the particle, the crack must proceed along its boundary. In all these cases, the crack path becomes longer. Another way to deflect cracks are residual stresses caused by the particles. Compressive stresses reduce the force opening the crack tip and thus repel cracks. Such residual stresses can stem from differences in the coefficient of thermal expansion of the particles or from phase transformations on cooling from the sintering temperature. 7.2.2 Crack bridging When the two opposed crack surfaces interact during crack propagation, the energy dissipation during crack propagation can be increased or the crack tip can be partially relieved. This kind of interaction can occur in coarse-grained microstructures with intercrystalline crack propagation. In this case, the crack surfaces can contact and rub against each other when the crack is opened (see figure 7.3) or can even be geometrically clamped, so that the crack cannot open at all. Fibres or particles are another crack bridging mechanism. Fibres will be discussed in chapter 9. 7.2 Mechanisms of crack propagation 231 Fig. 7.3. Coarse-grained microstructure in which the crack surfaces are in contact and dissipate energy by sliding on each other (a) Microcracks (b) Crack branching Fig. 7.4. Examples of microcracks and crack branching 7.2.3 Microcrack formation and crack branching The stress concentration near the crack tip can create microcracks at weak points in the ceramic. Examples are unfavourably oriented grain boundaries (perpendicular to the largest principal stress as drawn in figure 7.4(a)), grains with a cleavage plane perpendicular to the largest principal stress, or regions containing residual stresses. Microcrack formation raises the fracture toughness because it increases the energy dissipation. This can be understood by looking at the stress-strain diagram of a volume element that is passed by the crack tip (see figure 7.5): When the volume element approaches the crack tip, its load increases, and microcracks form. These reduce the stiffness of the volume element, causing a reduction in the slope of the stress-strain curve. On unloading (when the volume element moves away from the crack tip), the unloading curve is not the same as the loading curve. The shaded area in the diagram is the energy dissipated in this process; this additional energy has to be provided during crack propagation. 232 7 Mechanical behaviour of ceramics " ¾ unloading dissipated energy formation of micro-cracks loading Fig. 7.5. Stress-strain diagram of a volume element during microcrack formation. During the loading and unloading cycle, energy is dissipated, increasing the fracture toughness If microcracks have formed around the crack tip, further crack propagation is hampered also because Young’s modulus is locally reduced. This reduces the stress in this region and thus the driving force for crack propagation. 4 Crack branching (figure 7.4(b)) also increases the crack surface and de- creases Young’s modulus locally and can thus also impede crack propagation. 7.2.4 Stress-induced phase transformations So-called stress-induced phase transformations can produce additional com- pressive res idual stresses during crack propagation and thus increase the crack- growth resistance K IR . This is caused by particles in the matrix that can in- crease their volume by a phase transformation. Initially, the particles have to be in a metastable state which is thermodynamically unfavourable, but cannot transform to the thermodynamically stable phase because a nucleation barrier has to be overcome for this, similar to the process in precipitation hardening (see section 6.4.4). If a sufficiently large tensile stress is applied, for instance, at the crack tip, it may need less energy to transform the particles to the phase with the greater volume than to deform them elastically (figure 7.6). This case can also be understood by considering the stress-strain diagram of a volume element passed by the crack tip (figure 7.7). The phase transformation starts when the elastic energy is sufficiently large. Because the particle was metastable prior to the transformation, the transformation proceeds even when the stress be- comes smaller due to the volume increase. During the transformation, tensile 4 This argument is only valid when the microcracks are restricted to the region around the crack tip. If the whole material is cracked, the global stiffness is reduced and fracture toughness decreases, see also section 7.5.3. 7.2 Mechanisms of crack propagation 233 process zone (a) Before transformation process zone (b) After transformation Fig. 7.6. Unloading of a crack by a phase transformation of particles. In the process zone, residual compressive stresses are superimpos ed to the external tensile stress field " ¾ dissipated energy phase transformation loading unloading Fig. 7.7. Stress-strain diagram of a volume element, showing stress-induced phase transformations (after [46]). Similar to the formation of microcracks (figure 7.5), energy is dissipated during the loading and unloading cycle and thus increases the fracture toughness stresses in circumferential and compressive stresses in radial direction around the particles are superimposed to the external load. After unloading, part of the volume increase remains and compressive residual stresses are generated that redu ce the stress on the crack and thus may partly or totally close it. Because of the tensile stresses in circumferential direction around the par- ticles, microcracks can form and – as described in the previous section – cause further dissipation of energy (see also section 7.5.3). To be more precise, the stress-induced transformation is based on a reduction of the free enthalpy, defined in equation (C.4). The phase with the larger volume is thermodynamically stable, but, as already stated, a nucleation barrier has to be overcome to transform the particle. If hydrostatic tensile stress is added, the free enthalpy of the phase with 234 7 Mechanical behaviour of ceramics the larger volume decreases more strongly, according to equation (C.4), thus increasing the driving force for the transformation that enables the particle to overcome the nucleation barrier. In some metals, an analogous behaviour is observed: The stainless steel X 5 CrNi 18-10, which is austenitic (face-centred cubic) at room temp erature, is only metastable. Thus, the ferritic phase is thermody- namically stable, but the transformation does not occur because the driving force is too small. Under mechanical load, for instance during forming, a martensitic transformation can take place in parts of the comp onent, easily detectable by the component becoming ferromag- netic lo cally. 7.2.5 Stable crack growth In the previous sections (7.2.2 to 7.2.4), we encountered several mechanisms (crack bridging, microcracking, stress-induced phase transformations) that may increase the crack-growth resistance of a material. They all have in com- mon that the resistance initially grows during crack growth because a process zone forms near the crack tip and are thus examples for the mechanism dis- cussed in section 5.2.5. If conditions are appropriate, stable crack growth may thus occur in a certain stress range. The crack-growth resistance increases during crack propagation as long as the process zone grows. For example, if friction of the crack surfaces (figure 7.3) occurs, the crack-growth resistance initially increases because the contact area of the surfaces grows. If the crack propagates further, a stationary state is reached because parts of the surface far away from the crack tip will not touch anymore when the crack has opened too much. Then, the crack-growth resistance stays constant because for every newly formed region of fracture an equally large region is lost further away from the crack tip. When energy is dissipated within the material, as it happens in microcracking or stress- induced phase transformations, the process zone initially grows, for initially there are no microcracks or transformed particles near the crack tip. Only after a stable equilibrium is reached does the crack-growth res istance remain constant. ∗ 7.2.6 Subcritical crack growth in ceramics In section 5.2.6, it was explained that ceramics may, under certain conditions, exhibit subcritical crack propagation which can be quantified using the crack- growth rate da/dt. Figure 7.8 shows crack growth curves for a glass in different environments. Frequently, the crack growth curve is a line when a log-log scale is used (region 1 ○). In some cases, a plateau follows (region 2 ○), and, finally, the crack-growth rate rapidly increases shortly before reaching the fracture toughness K Ic (region 3 ○). 7.2 Mechanisms of crack propagation 235 10 −11 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 K I /MPa √ m da/dt m/s 1  1  2  3  in water in toluol K Ic K I0 Fig. 7.8. Dependence of the crack-growth rate on the stress intensity factor of soda-lime glass in water and toluene (phenylmethane) at 25℃ [104] Often, a power law is used to describe the crack-growth rate in region 1 ○: da dt = AK n I = A ∗  K I K Ic  n (7.1) where n, A, and A ∗ are temperature-dependent material parameters [103]. 5 The unit of A depends on the exponent n and can thus contain fractional exponents. To avoid this, A ∗ can be used which has always the unit of a velocity because K I /K Ic is dimensionless. As already mentioned in section 5.2.6, the dependence of the crack-growth rate on the stress intensity factor is sometimes very strong, resulting in a large value of the exponent n in equation (7.1). If a time-dependent stress σ(t) is applied, the life time of the component can be calculated by integration of equation (7.1). If the load is static, it is t f = B ∗ σ −n = Bσ n−2 c σ −n , (7.2) where B ∗ and B depend on the material and, via the geometry factor Y , also on the geometry. σ c is the so-called inert strength, the load required to break the specimen in a chemically inert environment where it would fail not by subcritical crack growth, but by fracture at K Ic . Because the stress exponent n is large, there is a strong dependence of the time to failure on the applied stress. In table 7.1, some examples for the stress exponent n and the prefactor are summarised. The effect of time-dependent loads on subcritical crack growth will be discussed in section 10.3. In exercise 24, an example for designing ceramic components against subcritical crack growth is given. 5 This shape of the crack growth curve da/dt is very similar to the crack-growth rate da/dN in cyclic loading of metals, to be discussed in section 10.6.1. 236 7 Mechanical behaviour of ceramics Table 7.1. Exemplary parameters of subcritical crack propagation [104]. For the specimen geometry investigated, Y = 1 holds true; the parameter B ∗ thus contains only material parameters ceramic medium T/℃ n lg ` B ∗ /(MPa n h) ´ Al 2 O 3 water 20 52.2 . . . 67.6 121.1 . . . 162.7 Al 2 O 3 conc. saline solution 70 20 45.5 Si 3 N 4 + 5.5% Y 2 O 3 1 100 37 106.5 1 200 30 84.2 Si 3 N 4 + 2.5% MgO 1 000 26 69.8 1 100 22.6 61.8 7.3 Statistical fracture mechanics Because ceramics cannot compensate for inner defects by plastic deformation, the statistical scatter of defect sizes causes a large scatter in the mechanical properties, different from metals and polymers. Therefore, it is usually not sufficient to simply state a failure load. Because it is not feasible to measure the size and position of every single defect within a component and thus to predict its strength exactly (deterministically), the statistics of the defect distribution is cons idered, and, using the methods of statistical fracture mechanics, a failure or survival probability is calculated. The objective of this section is to describe the probability of failure of a ceramic component analytically, using statistical fracture mechanics. Sim- plifyingly, we ass ume that defects with a certain defect size are distributed homogeneously in the material and that crack propagation at only one of them will cause complete failure. Initially, we will also assume a c onstant stress σ within the component. The probability of failure P f (σ) states the probability of the component failing when the stress σ is applied. If, for instance, in a batch of (macroscop- ically) identical specimens, the probability of failure is P f (200 MPa) = 0.3, 30% of the specimens will fracture when we try to apply a load of 200 MPa. The value is not be understood in such a way that 30% of the specimens will fail exactly at this stress value, but at stresses lower or equal to it. If defects were not statistically distributed, the behaviour of the material would be deterministic: It would fail at a c ritical stress σ 0 and the probability of failure would discontinuously change from 0 to 1. In reality, there is always a probability that the material will bear larger loads or will fail at smaller ones, and the ‘edge’ at σ 0 is round ed off. 7.3.1 Weibull statistics Usually, ceramics fail as soon as a crack starts to propagate. Therefore, their strength is determined by the stress value at which the first, and thus critical, 7.3 Statistical fracture mechanics 237 crack starts to grow. The probability of failure is thus given by the probability that the critical crack has a certain failure stress. If loads are tensile, one of the cracks that are at least partially loaded in mode I will govern the strength. To describe the probability of failure, a statistical approach is needed that takes into account the statistical distribution of the density and size of the cracks [104]. If we consider a component with homogeneous stress distribution and known number of defects, the size distribution of the defects can b e used to determine the failure probability. This is equal to the prob- ability that at least one crack has the critical crack length. As the critical crack determines failure, only the largest defects are relevant. The probability of finding a large defect eventually becomes smaller with increasing defect size; therefore, different defect size distributions will look similar in the relevant region. The details of the defect size distribution are thus not important. Because the number of defects differs between different components, the defect density distribution must also be taken into account by using it to accumulate the probability of failure for all possible numbers of defects. From these statistical considerations, it can b e shown that the failure stress σ of the critical defect is distributed according to the so-called Weibull distribu- tion. From this, the probability of failure can be calculated as 6 P f (σ) = 1 − exp  −  σ σ 0  m  . (7.3) The parameter m, called the Weibull modulus, quantifies the scatter of the strength values and is thus a measure of how strongly the edge in a plot of the probability of failure is rounded off as shown in figure 7.9 for some examples. As can be seen, a larger Weibull modulus reduces the scatter of the failure stress. For m → ∞, there is no scatter anymore, and σ 0 is equal to the fracture stress. The Weibull modulus is a material parameter; the reference stress σ 0 depends on the material and the specimen volume. Equation (7.3) is only valid when a cons tant, homogeneously loaded specimen volume is considered. The influence of the volume will be discussed from the following page onwards. Table 7.2 gives a synopsis of some values for the Weibull modulus in different materials. Frequently, a linearised representation of the probability of failure is use d. For this purpose, equation (7.3) is re-written as follows: 1 − P f = exp  −  σ σ 0  m  , 6 To correctly describe the probability of failure, the volume of the component must also b e taken into account, see equation (7.6) below. 238 7 Mechanical behaviour of ceramics 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 σ/σ 0 P f 1−e −1 m = 5 m = 10 m = 20 m = 50 Fig. 7.9. Dependence of the failure probability on the stress for several Weibull moduli Table 7.2. Weibull modulus m and reference stress σ 0 of some ceramics (after [58, 64,104]). V 0 is the reference volume in equation (7.6). For comparison, the Weibull modulus of cast iron and steel is specified material m σ 0 /MPa V 0 /mm 3 SiC 8 . . . 27 250 . . . 600 1 Al 2 O 3 8 . . . 20 100 . . . 600 1 Si 3 N 4 8 . . . 9 750 . . . 1 350 1 ZrO 2 10 . . . 15 200 . . . 500 1 cast iron ≈ 40 steel ≈ 100 1 1 − P f = exp  σ σ 0  m  , ln 1 1 − P f =  σ σ 0  m , ln  ln 1 1 − P f  = m ln σ σ 0 . (7.4) If we now plot ln  ln[1/(1 − P f )]  as a measure of the probability of failure versus ln(σ/σ 0 ) as a measure for the applied stress, a linear equation with slope m through the origin results as sketched in figure 7.10. The probability of failure in equation (7.3) depends on the material volume. This is plausible if we assume that a single defect of critical size will cause the component to fail, for, if the volume and thus the number of defects is increased, the probability of a critical defect being present increases as well. This will now be shown using an example. It is useful to consider the probability of survival P s instead of the probability of failure, defined as [...]... amorphous regions 262 8 Mechanical behaviour of polymers fluid T viscous rubbery Tm ductile Tg crystalline 101 102 103 semi-crystalline 104 105 106 1 07 degree of polymerisation Fig 8.5 Dependence of the glass and melting temperature as well as the mechanical behaviour of a crystalline or semi-crystalline polymer on the degree of polymerisation (after [ 27] ) At very low degrees of polymerisation, complete... therefore time-dependent, and it is not always easy to distinguish elastic and plastic deformations The mechanical properties of polymers are the subject of sections 8.2 to 8.4 Methods to improve the mechanical properties of polymers are discussed subsequently The chapter closes with a brief discussion of the sensibility of polymers against environmental influences 8.1 Physical properties of polymers 8.1.1... Pf,i 0.6 1 234.0 0.042 0.4 2 2 57. 4 0.125 3 273 .0 0.208 σ0 0.2 4 273 .8 0.292 5 275 .3 0. 375 0.0 6 276 .9 0.458 200 250 300 350 7 280.8 0.542 σ / MPa 8 288.6 0.625 9 290.2 0 .70 8 Fig 7. 16 Plot of the failure probabilities 10 296.4 0 .79 2 for the experimental data in table 7. 3 Addi11 312.0 0. 875 tionally, the curve calibrated to the experi12 335.4 0.958 mental data using equation (7. 3) is included measured fracture... discussed in section 7. 2 Frequently, particles or fibres are added to the ceramic Ceramics strengthened by particles are called dispersion-strengthened ceramics and will be discussed here; fibre-reinforced ceramics are one topic of chapter 9 Table 7. 4 summarises the mechanical properties of some technically important ceramics 7. 5 Strengthening ceramics 249 Table 7. 4 Mechanical properties of some ceramics... model describes the behaviour of a purely viscoelastic material 1 The name ‘creep modulus’ is somewhat misleading because it denotes viscoelastic behaviour, although the term ‘creep’ is usually used for viscoplastic behaviour only 8.2 Time-dependent deformation of polymers (a) Kelvin model 265 (b) Four-parameter model Fig 8 .7 Mechanical models of viscoelastic and viscoplastic materials, built as systems... from the probability of failure Pf If we increase the stress σ by ∆σ, a fraction of Pf (σ + ∆σ) − Pf (σ) of all specimens 244 7 Mechanical behaviour of ceramics i j=0 nj/N, Pf (σ) 1.0 60 i j=0 0.8 nj i= 0 1 2 3 4 5 6 7 8 9 10 11 12 50 40 0.6 30 0.4 20 0.2 10 0.0 0 ∆σ 0 100 200 300 400 500 600 σ / MPa P Fig 7. 15 Plot of the accumulated number of specimens i nj failed until the j=0 end of the current stress... order of magnitude Therefore, polymers can exhibit much larger elastic strains without deforming plastically When components made of polymers are designed, this large elastic deformation has to be taken into account Both the elastic and the plastic behaviour of polymers are time-dependent even at room temperature; polymers are thus viscoelastic and viscoplastic In this section, we discuss the time-dependent... positions, and the intermolecular bonds are strained under mechanical loads At elevated temperatures, the behaviour of polymers is much more complex because thermally activated rearrangements and movements within and between the chains can occur, which are frequently reversible These processes are mainly responsible for the physical and mechanical properties of polymers They are called relaxation processes and. .. probability of failure for components tested with a proof stress σp It is only valid for σ > σp ; otherwise, the probability of failure is zero Figure 7. 19 shows the probability of failure before (Pf (σ)) and after (Gf (σ)) the proof test with a linear scale and in the linearised form Below σp , the probability of failure is zero But even above the proof stress, it is smaller than before the proof test... near particles may, as explained in section 7. 2.1, also deflect a crack One example is silicon nitride reinforced with titanium nitride [20] Due to the difference in thermal expansion of the two materials, the matrix is stressed compressively near the particles, impeding crack propa- 7. 5 Strengthening ceramics 251 Fig 7. 20 Microcracks around a particle (after [1 37] ) gation The fracture toughness KIc of . design of ferroconcrete (steel- reinforced concrete), for example, see section 9.1.1. 230 7 Mechanical behaviour of ceramics (a) Without particles (b) With particles Fig. 7. 2. Deflection of a crack. the particle (see figure 7. 2). If Young’s modulus of the particles is smaller than that of the matrix, the stress is raised in the vicinity of the particles, and the crack is attracted by the particle during crack propagation. 232 7 Mechanical behaviour of ceramics " ¾ unloading dissipated energy formation of micro-cracks loading Fig. 7. 5. Stress-strain diagram of a volume element during