Mechanical Behaviour of Engineering Materials - Metals, Ceramics, Polymers and Composites 2010 Part 10 pptx

352 10 Fatigue ∆K th (R) =  (1 − R) γ ∆K th | R=0 for R < R t , const for R ≥ R t . (10.5) with R t = 0.5 . . . 0.7. In low- to medium-strength ferritic steels, γ ≈ 1, in high-strength martensitic steels, γ → 0. Because the stress intensity factor K op needed to open the crack depends on the deformation near the crack tip, it also depen ds on Young’s modulus, for the crack opening in a linear-elastic material is the smaller, the higher Young’s modulus is (see equation (5.3)). Accordingly, Schwalbe [133] provides the following approximation for the fatigue-crack-growth threshold in metals: ∆K th (R) = (2.75 ±0.75) ×10 −5 E(1 − R) 0.31 √ m for R < 1 . (10.6) Equation (10.6) also shows the dependen ce of the fatigue-crack-growth thresh- old on the R ratio, which, however, is not in agreement with equation (10.5) above. Although equations like these exist, it should be kept in mind that K op and thus ∆K th depends on many other material parameters e. g., the grain size. Accordingly, large differences in the exact values can be found even within a certain material class. Nevertheless, the equations are useful in estimating the order of magnitude of ∆K th . If we take steel as an example (with E = 210 000 MPa), we find ∆K th = 5.8 MPa √ m for R = 0. This is more than one order of magnitude smaller than the static fracture toughness K Ic of ductile steels and thus illustrates how dangerous even small cracks can be under cyclic loads. Crack propagation If the cyclic stress intensity factor ∆K exceeds the fatigue-crack-growth thresh- old ∆K th (i. e., if K max > K op ), the crack grows in every cycle. The crack- growth rate da/dN is determined by those parts of ∆K that exceed K op i. e., (for the case K min < K op ) by the effective cyclic stress intensity factor ∆K eff = K max − K op . Because K op is usually unknown, da/dN cannot be plotted against ∆K eff . Instead, its dependence on ∆K and R is used. As figure 10.15 illustrates, ∆K eff increases with ∆K and with the mean stress intensity factor K m i. e., the R ratio. During crack propagation, the cyclic stress intensity factor ∆K increases due to the increase of the crack length. Therefore, the crack-growth rate da/dN also increases even if the cyclic load of the component is constant. If the max- imum stress intensity factor K max approaches the fracture toughness, the crack accelerates rapidly and eventually becomes unstable after a few more cycles. 13 Final fracture of the component ensues. Similar to the fatigue-crack- growth threshold, the transition to unstable crack propagation is determined 13 Because of the preceding cyclic crack propagation, the crack may not become unstable exactly when the stress intensity factor equals K Ic (cf., for example, 10.6 Phenomenological description of the fatigue strength 353 K t ) (¢a/¢N) 2 > (¢a/¢N) 1 ¢K 2 = ¢K 1 R 2 > R 1 ) (¢a/¢N) 3 > (¢a/¢N) 1 ¢K 3 > ¢K 1 R 3 = R 1 ¢K 1 K op 0 ¢K 2 ¢K 3 ¢K eff2 ¢K eff3 crack propagationcrack propagation no crack propagationno crack propagation ¢K eff1 Fig. 10.15. Illustration of the increase of the crack growth per cycle, da/dN, with increasing R ratio (corresponding to an increasing mean stress intensity factor K m ) or cyclic stress intensity factor ∆K. The fraction of the cycle with opened crack increases by the maximum stress intensity factor K max (K max = K Ic ). According to equation (10.4), the critical stress intensity range ∆K Ic can be determined: ∆K Ic = 2(K Ic − K m ) = (1 −R)K Ic . (10.7) Again, an increase of K m or the R ratio decreases the allowed cyclic stress intensity factor ∆K Ic . If we plot the crack-growth rate da/dN versus the cyclic stress intensity factor ∆K for a constant R ratio in a double-logarithmic plot, we get a crack- growth curve or da/dN curve (figure 10.16). A marked increase of the crack- growth rate is apparent in region III where the maximum stress intensity factor K max approaches K Ic (∆K → ∆K Ic ). The crack slows down in region I when K max approaches K op from above (∆K → ∆K th ). In between, there is a region marked ‘II’ where the dependence between log(da/dN) and log(∆K) is almost linear. Accordingly, the crack-growth rate follows the so-called Paris law da dN = C∆K n = C ∗  ∆K K Ic  n (10.8) in this region. Here, C is a constant depending on the material and the R ratio. Similar to the subcritical crack growth of ceramics in equation (7.1), the unit of the constant C depend on the exponent n, whereas C ∗ has the units of a length. In metals, the exponent n is usually in the region 2 ≤ n ≤ 7 [35], but section 5.2.5). However, this is irrelevant under cyclic loads because this effect can only alter the life time of the component by a few cycles. For simplicity, we will use K Ic in the following. 354 10 Fatigue ¢K Ic ¢K tr K max = K Ic K max = K op ¢K (log) ¢K th R 2 > R 1 R 2 R 1 da/dN (log) I II III Fig. 10.16. Crack-growth curve, plotting da/dN versus the cyclic stress intensity factor ∆K in a double-logarithmic plot. There are three characteristic regions as shown for the curve with the R ratio R 1 in brittle materials it can be as large as 50 [120]. For ferritic-pearlitic steels, Landgraf [87] states the following upper limit for the da/dN curve at R = 0: da dN = 6.9 ×10 −9 mm cycle ×  ∆K MPa √ m  3 . If we load a crack with a constant stress range ∆σ with ∆K > ∆K th , the crack grows. According to equation (10.3), ∆K increases, and the loading point in the da/dN curve in figure 10.16 moves to the right. The crack-growth rate increases in each cycle until ∆K Ic is reached, and final fracture destroys the component. If we consider a specific material and increase the mean stress intensity factor K m (thus usually also increasing the R ratio, see section 10.1), K max increases as well. The cyclic stress intensity factor ∆K that the component can bear decreases, shifting the curve to the left, as shown by the dashed line in figure 10.16. This is a direct consequence of what we discussed above concerning the mean-stress dependence of ∆K th , ∆K Ic , and da/dN . In equa- tion (10.8), this shift of the curve is accounted for by the R-dependence of the factor C. There are a large number of, sometimes contradictory, approaches to describe the dependence of C on the R ratio and the da/dN curve in all three regions (see, for example, Broek [23], Radaj [113], and Schott [130]). Some exemplary da/dN curves are shown in figure 10.17. Only in the case of the steel was the range of the cyclic stress intensity factor sufficiently large to capture all three regions of the cu rve. The slope of the curve is much larger for ceramics than in the Paris region of metals, resulting in a cyclic stress intensity factor that is almost the same for negligible and rapid crack growth. The reason for this is that the strength of ceramics is at most only slightly 10.6 Phenomenological description of the fatigue strength 355 10 −11 10 −10 10 −9 10 −8 10 −7 10 −6 10 −5 2 5 10 20 50 ∆K MPa √ m da/dN m/cycle Al SiC steel TiAl6V4 Ni base psz Fig. 10.17. Crack-growth curves of several materials: An aluminium alloy (AlZn 6 CuMgZr), a steel (20 MnMoNi 4-5), a nickel-base superalloy (Waspaloy), a titanium alloy (TiAl 6 V 4), silicon carbide, and a psz ceramic [6,38,53,57,87] reduced by cyclic loading. The fatigue-crack-growth threshold ∆K th is almost the same in ceramics and metals. Nevertheless, metals do have one advantage if loaded cyclically: Their fatigue strength is larger if they are designed against a small number of cycles since the Paris region can be exploited in the design. Because of the extended Paris region, significant crack growth must take place before the component fails so that regular in spection cycles may detect the growing crack before failure. In polymers, the da/dN curves are similar to those of metals. Below a certain threshold value ∆K th , there is no crack growth, at larger values, three regions can be distinguished, with region II being described by a Paris law. The exponent n takes a value of about 4 in many polymers [97]. In composites, the fatigue behaviour can frequently not be described ade- quately by da/dN curves because the material usually fails by accumulating local damage, not by propagation of a single crack. Measuring da/dN curves is thus a rather involved procedure [29]. If a single crack determines the failure behaviour, the da/dN curves can be described with a Paris law. Compared to the matrix material alone, K Ic is often reduced in polymer and metal matrix composites (see section 9.3.4), but ∆K th is increased. Despite the reduced fracture toughness, the fatigue life of a composite may thus be larger than that of the matrix material. Assessing life times Using equation (10.3), we can calculate the critical crack length a f at which unstable or accelerated crack growth occurs (transition between regions II and III in figure 10.16). If we require that this crack length must not be exceeded, 356 10 Fatigue we can calculate the number of cycles to failure for a given initial crack length a 0 < a f . To do so, we exploit the equality ∆K = ∆K Ic or ∆K = ∆K tr (for the transition between region II and III), respectively. The number of cycles until the critical crack length is reached can be estimated for the initial crack length a 0 by [8,40]: N f (a 0 ) = Z N f 0 dN = Z a f a 0 1 C „ 1 ∆K « n da . (10.9) Here we assume that we are already in region I I at the initial crack length. Inserting ∆K from equation (10.3) and assuming a constant stress range ∆σ, we find N f (a 0 ) = 1 C „ 1 ∆σ √ π « n Z a f a 0 1 ` Y √ a ´ n da . (10.10) If the geometry factor Y is independent of the crack length – an as- sumption unfortunately not true in most cases –, we can take Y out of the integral and solve the integral. 14 Otherwise, equation (10.10) must b e integrated numerically. The result is (for n = 2 and geometry factor Y independent of the crack length) N f (a 0 ) = 1 C  1 ∆σ √ π Y  n · 2 2 − n  a 2−n 2 f − a 2−n 2 0  . (10.11) This equation can be used to estimate the number of cycles to failure for a known length of the largest crack (see exercise 30). Growth of short cracks As explained above, the crack-growth rate da/dN depends on the cyclic stress intensity factor ∆K = ∆σ √ πa Y and on the R ratio. According to this, a short crack loaded with a large stress range will propagate with the same rate as a long crack loaded with a small stress range provided the cyclic stress intensity factor ∆K is the same. In many cases, this simple picture is correct. However, the statement of the previous paragraph only holds for macro- cracks. Microcracks may grow faster than expected from the da/dN curve (figure 10.16), and they may even grow at a cyclic stress intensity factor below ∆K th [21,113]. On the one hand, this is due to the fact that the crack growth resistance of the material varies on the microscopic scale. A microcrack that is, for example, surrounded by favourably oriented grains may grow rather 14 For the case n = 2, we have to integrate 1/a, leading to ln a. This case is dealt with in exercise 30. 10.6 Phenomenological description of the fatigue strength 357 σ E R m N E N f (log) σ A (log) lcf hcf low-cycle fatigue strength high-cycle fatigue strength endurance limit Fig. 10.18. Example of an S-N diagram with data points quickly, whereas another crack is stopped at a grain boundary because neigh- bouring grains are less favourably oriented. On the other hand, short cracks may remain open even under compressive loads because they are embedded in a plastic deformation field [113]. This explains why crack propagation may take place even below ∆K th . If a component contains only micro cracks or is not cracked at all, da/dN curves cannot be used to assess the life time. In this case, other methods are required that are the subject of the next section. 10.6.2 Stress-cycle diagrams (S-N diagrams) At the beginning of the chapter, we already saw that the complex load-time curves occurring in real life are usually replaced by simplified curves in the laboratory e. g., using sinusoidal loading. Frequently, smooth specimens are used, similar to the tensile specimens discussed in section 3.2. They are loaded cyclically with a fixed period, prescribing the stress amplitude σ a or the strain amplitude ε a , and also the R ratio (R or R ε , respectively). The advantages and disadvantages of these two experimental procedures will be discussed at the end of this section; in the following, we will consider stress-controlled experiments only. For each fatigue experiment, the number of cycles to failure 15 is measured. If several fatigue experiments are performed and the number of cycles to failure N f is plotted versus the stress amplitude σ A or the stress range ∆σ, the resulting diagram is called a stress-cycle (or S-N) diagram (sometimes also stress-life or Wöhler diagram, see figure 10.18). We denote the stress values in the S-N diagram with capitalised subscripts. For example, we denote the stress amplitude that causes failure after N f cycles as σ A instead of σ a . The number of cycles can also be specified in the subscript, as in σ AN f , stating, for example, 15 Failure can be defined as fracture of the specimen or occurrence of a crack. 358 10 Fatigue σ E N E N f (log) σ A (log) lcf hcf endurance limit (a) Typ e I σ E N E N f (log) σ A (log) lcf hcf endurance limit (b) Typ e II Fig. 10.19. The characteristic types of S-N curves σ A(1.5×10 4 ) = 130 MPa. The number of cycles to failure is always plotted logarithmically in the S-N diagram; the stress can be plotted logarithmically or linearly. Some materials exhibit a true fatigue limit (sometimes also called the endurance limit). In this case, there exists an limiting number of cycles N E , with the S-N curve being almost horizontal at a larger number of cycles. In this case, the S-N diagram is of type I (figure 10.19(a)). A specimen that has survived N E cycles never fails. The experiment can be stopped and the specimen can be marked accordingly, usually with an arrow in the diagram (sometimes denoted as ‘run out’, see figure 10.18). Frequently, N E takes values between 2 × 10 6 and 10 7 , depending on the material. The stress level that corresponds to N E in the S-N curve is called the fatigue strength, endurance limit, or fatigue limit σ E . In many materials, there is no horizontal part of the S-N curve (typ e II, figure 10.19(b)). Although the slope of the S-N curve becomes smaller beyond a certain number of cycles, failure can still occur. These materials thus have no true fatigue limit. To ensure safety of the component, a limiting number of cycles of 10 8 is often used, ten times larger than the usual value for materials with a true fatigue limit. To state explicitly that a fatigue strength corresponds only to a certain number of cycles, not to a true fatigue limit, the number of cycles can be added to the subscript, as in σ E(10 8 ) . So far, we have only looked at large numbers of cycles, the so-called high- cycle fatigue (hcf) regime. As we already saw in the introduction of the chapter for the example of the car engine (section 10.1), it is sometimes nec- essary to design against a rather limited number of cycles. If this number is smaller than about 10 4 , the term low-cycle fatigue (lcf) is used. However, the number of cycles that characterises the transition from low- to high-cycle 10.6 Phenomenological description of the fatigue strength 359 fatigue is not well-defined [130]. A stress amplitude that causes failure in the lcf regime is called low-cycle fatigue strength, an amplitude causing failure in the hcf regime is called high-cycle fatigue strength. As can be seen from figure 10.18, the slope of the S-N cu rve is usually much smaller in the lcf than in the hcf regime so that a small change in the stress amplitude has a large effect on the number of cycles. This phenomenon is restricted to metals and polymers and will be discussed for the case of metals in the next section. If the maximum stress σ max reaches the strength of the monotonous exper- iment in the first cycle (the tensile strength R m for the case of axial loading), the specimen fractures during this cycle. Often, the number of cycles to failure is then taken to be N f = 0.5. The left end of the S-N curve is thus determined by σ A(0.5) = 0.5(1 −R)R m . Independent of the material tested, the scatter of the cycles to failure is usually rather large, for even small defects in the material or on the surface can have a strong effect on the life time. Different specimens thus are never identical. For this reason, several experiments have to be performed at each stress level (usually 6 to 10) to allow ascertaining the width of the scatter band. Using statistical methods, limiting curves can be constructed that represent a certain probability of failure (for example, 95%). This is elaborated on in Forrest [50], Radaj [113] or Schott [130]. As the introductory example of a car engine (see section 10.1) shows, real- life fatigue loads can be stress- or strain-controlled. Stress-controlled loads occur if the loads are determined by external forces, strain-controlled loads, for example, if there are temperature changes causing thermal strains. In many cases, lcf loads are strain-controlled and hcf loads stress-controlled. This, however, cannot be used as a rule. For example, loads in a rotating disc are determined by centrifugal forces. Since these are constant during rota- tion, switching the device on and off corresponds to a single cycle. The load is thus stress-controlled, but the number of cycles is low (lcf). Usually, stress- or force-controlled experiments are easier to perform than strain-controlled exp er- iments and are thus often preferred. This is especially true in the hcf regime. If we look at an S-N curve (fi gure 10.18), we can see that the number of cycles to failure strongly depends on the stress in the lcf regime. Small scatter in the stress-strain properties of different specimens (due to scatter in the material properties, for example) would caus e large changes in the number of cycles to failure measured in the experiment. The scatter band would thus be rather wide. In this regime, strain-controlled experiments are more useful since, with a prescribed strain amplitude, the scatter of the stress amplitude is small. Furthermore, stress-controlled experiments would also cause more rapid failure due to the reduction in the cross section of the specimen caused by crack propagation [113]. To assert the influence of notches and inhomogeneous stress distributions on fatigue life, experiments can also be performed with notched components 360 10 Fatigue or specimens, resulting in specific S-N curves. The influence of notches on fatigue life is discussed in more detail in section 10.7. S-N curves of metals In a double-logarithmic plot, the S-N curve of many me tals is a straight line for a wide range of the number of cycles (see figure 10.18). This line can be described by the Basquin equation [14] σ A = σ  f (2N f ) −a . (10.12) The fatigue strength coefficient σ  f is related to the tensile strength. In plain carbon and low-alloy steels, a rule of thumb states σ  f = 1.5R m ; in aluminium and titanium alloys, σ  f = 1.67R m holds approximately [113]. The fatigue strength exponent a depends on the material and the specimen geometry; in many materials, it takes values between 0.05 and 0.12 if smooth specimens are used [8, 113]. In p lain carbon steels and titanium alloys with body-centred cubic lattice, there is a true fatigue limit with a horizontal S-N curve at a number of cycles beyond 2 × 10 6 to 10 7 [130] (type I, figure 10.19(a)). This, however, is not true for notched specimens (and thus also for components) or if corrosion or oxidation occur during the experiment. Face-centred cubic metals and hardened steels do not have a true fatigue limit (S-N curve of type II, figure 10.19(b)). At a number of cycles beyond 10 7 , the slope of the S-N curve is rather small and a limiting number of cycles of N E = 10 7 to 10 8 can be used to design safely against fatigue [130]. Recently, it has been found even in body-centred cubic metals that a specimen can fail in fatigue even beyond the limiting number of cycles (10 7 ). At a very large number of cycles (more than 10 10 ), the S-N curve may drop again [93,135]. This is called ultra-high-cycle fatigue (uhcf) or very-high-cycle fatigue. In contrast to failure at smaller numbers of cycles, which usu- ally start from the surface, failure in the uhcf regime is caused by microcracks being initiated at microscopic inclusions slightly below the surface of the specimen, visible as so-called fish eyes at the sur- face [135, 139]. S-N curves of metals have a small slope at low numbers (N f  10 3 ) of cy- cles as well as in the regime N f > N E . In this region, the yield strength of the material is exceeded, and the strain amplitude inc reases rapidly with the stress amplitude. A slight increase of the stress causes much larger plastic deformations and thus strongly reduces the life time. If we plot the strain amplitude ε A (N f ) versus N f in a double-logarithmic plot, we get a strain-cycle diagram as shown in figure 10.20. Two linear regimes, with a smooth transition between them, can be discerned. As we will see soon, 10.6 Phenomenological description of the fatigue strength 361 10 −4 10 −3 10 −2 10 −1 10 1 10 2 10 3 10 4 10 5 10 6 10 7 N f ε A ε (el) A ε (pl) A ε A Fig. 10.20. Strain-cycle diagram with σ  f = 470 MPa, E = 210 000 MPa, a = 0.1, ε  f = 0.1, and b = 0.5 these linear regimes are related to the elastic and the plastic part of the total strain. The total strain amplitude can be decomposed as ε A = ε (el) A + ε (pl) A . (10.13) Large numbers of cycles (hcf) can only be reached with a small stress amplitude so that the amount of plastic deformation is small. The total strain thus corresponds mainly to the elastic part of the strain. The line can be described using the Basquin equation, re-written with the help of Hooke’s law: ε (el) A = σ  f E (2N f ) −a . (10.14) At a small number of cycles (lcf), the stresses are large and the total strain is mainly determined by plastic deformation. In the lcf regime, a good approximation for the relation between plastic strain amplitude and cyclic life is given by the Coffin-Manson equation [32, 94, 95]. ε (pl) A = ε  f · (2N f ) −b . (10.15) For the fatigue ductility coefficient ε  f , the true fracture strain in tensile loading can be us ed as a good approximation. The fatigue ductility exponent b depends on the hardening of the material. Typical values for b are in the range of 0.4 to 0.73 [113, 130]. Adding both parts of the strain (equation (10.13)) yields ε A = σ  f E (2N f ) −a + ε  f · (2N f ) −b , (10.16) [...]... after [41]) σA MPa 200 100 40 % C 50 40 % glass 20 % glass 10 % glass unreinforced 10 100 101 102 103 104 105 106 107 Nf Fig 10. 25 Comparison of the S-N curves of unreinforced as well as glass- and carbon-reinforced polysulfone (simplified plot after [29]) S-N curves of fibre composites According to section 10. 5, the fatigue strength of fibre composites is usually higher than that of the matrix material... alloys = (0.45 0.65) × Rm ≈ 620 MPa for Rm < 1 100 MPa for Rm ≥ 1 100 MPa σA 700 MPa 600 500 400 300 20 ℃ 1 000 ℃ 1 200 ℃ 200 100 101 102 103 104 105 106 107 108 109 Nf Fig 10. 22 S-N diagram of Si3 N4 at different temperatures (measured in bending at R = −1) [113] The dashed line is a fit according to the Basquin equation, common to temperatures of 20℃ and 100 0℃, whereas the dotted line is valid at 1200℃... production and dissipation plays a crucial role To design components, experiments should be as close to real service conditions as possible 10. 6 Phenomenological description of the fatigue strength σA 50 MPa 365 pa 40 acetal copolymerisate pp 30 hdpe 20 10 pvc 0 104 105 106 107 108 Nf Fig 10. 24 S-N curves for fully reversed bending of several polymers at a temperature of 20℃ and a loading frequency of 10. .. phase of the grain boundaries (see section 7.5.2), resulting in crack bridging effects as explained in section 10. 3 However, the effect is rather small The number of cycles to failure is almost identical for 20℃ and 100 0℃ so that the same fit curve can be used to describe both Raising the tempera- 364 10 Fatigue σA MPa 35 0.5 Hz 30 1.67 Hz 25 5 Hz 0.167 Hz 20 10 Hz 15 104 105 106 107 Nf Fig 10. 23 S-N diagram... of creep depends on the homologous temperature T /Tm : High-melting materials have a large value of the binding energy and thus need a large amount of energy to create and move vacancies The activation energy for self-diffusion is thus large, and the exponential term in equation (11.2) can reach the size of that in a low-melting material only at higher temperatures As shown in table 11.1, a rule -of- thumb... formation and propagation, similar to a metal, at higher frequencies, thermal fatigue occurs (section 10. 4.1), and the fatigue strength strongly decreases The load frequency is for this reason usually limited to 10 Hz S-N curves of different polymers are depicted in figure 10. 24 In many polymers (e g., pvc, pp, pa), the S-N curve is horizontal at a large number of cycles, corresponding to a curve of type... times, the material behaviour does not change anymore and a stationary state is arrived at If the stress is measured at each of the strain maxima, the 372 10 Fatigue ε σ σ t ε t (a) Cyclic relaxation (strain-controlled experiment) σ σ ε t t ε (b) Cyclic ratchetting (stress-controlled experiment) Fig 10. 31 Cyclic stress-strain behaviour at the beginning of fatigue experiments with non-zero mean strain... ∗ Cyclic relaxation and ratchetting If a strain-controlled fatigue experiment is performed at a non-zero mean strain, cyclic relaxation may occur in addition to cyclic hardening or softening, with the mean stress decreasing over time (figure 10. 31(a)) If, on the other hand, the experiment is stress-controlled at a non-zero mean stress, the hys- 10. 6 Phenomenological description of the fatigue strength... 1 dσa σa,max dx , (10. 25) x=X where x and X are defined in figure 10. 35, and where the maximum stress amplitude at the notch root is σa,max The unit of the relative stress gradient is mm−1 The relative stress gradient corresponds to the inverse of the distance between the notch root and the intersection of the tangent at σa (x)|x=X with the coordinate axis as in figures 10. 34 and 10. 35 It only depends... also typical of ceramics S-N curves of polymers We already saw in section 10. 4 that the fatigue behaviour of polymers strongly depends on the load frequency because of their viscoelastic properties If the frequency is sufficiently large, the polymer can fail by thermal fatigue due to the heat generated during deformation This is shown for the example of a thermoplastic polymer in figure 10. 23 At low frequencies, . of 10 Hz (polyamide: 15 Hz, after [41]) 50 200 10 100 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 N f σ A MPa unreinforced 10 % glass 20 % glass 40 % glass 40 % C Fig. 10. 25. Comparison of the S-N. R m < 1 100 MPa ≈ 620 MPa for R m ≥ 1 100 MPa 200 300 400 500 600 700 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 N f σ A MPa 20 ℃ 1 000 ℃ 1 200 ℃ Fig. 10. 22. S-N diagram of Si 3 N 4 at. description of the fatigue strength 361 10 −4 10 −3 10 −2 10 −1 10 1 10 2 10 3 10 4 10 5 10 6 10 7 N f ε A ε (el) A ε (pl) A ε A Fig. 10. 20. Strain-cycle diagram with σ  f = 470 MPa, E = 210 000