148 5 Fracture mechanics 0 K Ic K I 10σ 1 5σ 1 σ c1 3σ 1 2σ 1 1σ 1 K IR 0 σ c1 0 a 1 a σ stationary unstable stationary unstable (a) During crack propagation, the stress de- creases and crack propagation is unstable from the start at σ c Fig. 5.12. Plot of the crack-growth resistance K IR and the stress σ versus the crack length a for the case of constant K IR during crack propagation (1/2) In contrast, figure 5.12(c) shows a configuration with the same crack- growth resistance curve, but a smaller initial crack length a 3 . In this case, the stress needed for the crack to propagate decreases from the start, although the crack-growth resistance increases. Hence, crack propagation is unstable as soon as K Ic is reached. The critical stress σ c3 is nevertheless larger than that for the longer crack (σ c2 ). As the figures 5.12(b) and 5.12(c) illustrate, an increasing crack-growth resistance K IR does not guarantee stable crack propagation because the stress can still decrease. If the load propagating the crack is displacement-controlled, a decreasing load does not necessarily cause f ailure of the component. As the compliance of the component increases with increasing crack length, the stress may become small enough to stabilise the crack, a phenomenon called crack arresting. We now want to estimate the stress intensity factor K I that marks the onset of unstable crack propagation. This happens when the re- quired external stress σ decreases or at least stops to increase on crack propagation by a distance da. During crack propagation, the current stress intensity factor K I , which depends on the stress σ and the crack length a, must equal the crack-growth resistance K IR : 5.2 Linear-elastic fracture mechanics 149 0 K Ic K ∗ 2 K I 10σ 1 5σ 1 σ ∗ 2 σ c2 3σ 1 2σ 1 1σ 1 K IR (∆a) 0 σ c2 σ ∗ 2 0 a 2 a ∗ 2 a σ stationary stable unstable stationary stable unstable (b) For a crack of length a 2 , stable crack propagation begins at a stress σ c . At a stress of σ ∗ , the crack becomes unstable 0 K Ic K I 10σ 1 5σ 1 σ c3 3σ 1 2σ 1 1σ 1 K IR (∆a) 0 σ c3 0 a 3 a σ stationary unstable stationary unstable (c) For a short crack of length a 3 , crack propagation is unstable from the start at σ c Fig. 5.12. Plot of the crack-growth resistance K IR and the stress σ versus the crack length a for the case of increasing K IR during crack propagation. In both figures, the same crack-growth resistance curve K IR (∆a) is plotted (after [104]) (2/2) K I (σ, a) = K IR (a) . In order to meet this condition at any arbitrary time, the derivatives must also be equal: dK I (σ, a) da = dK IR (a) da . If we separate the left-hand side into its constituent terms, we get ∂K I (σ, a) ∂σ ∂σ ∂a + ∂K I (σ, a) ∂a = dK IR (a) da . As long as the load increases during crack propagation, ∂σ/∂a > 0 holds, and the crack propagation is stable. If the external load does not increase anymore, ∂σ/∂a ≤ 0 holds. The condition for the transition b etween stable and unstable crack propagation is thus ∂σ/∂a = 0. K ∗ I can b e calculated by ∂K I (σ, a) ∂a ˛ ˛ ˛ ˛ K I =K ∗ I ≥ dK IR (a) da ˛ ˛ ˛ ˛ K I =K ∗ I . (5.29) 150 5 Fracture mechanics If the crack-growth resi stance curve K IR (a) is known, this criterion together with the condition K I (σ, a) = K IR (a) allows to calculate the b egi nning of unstable crack propagation. ∗ 5.2.6 Subcritical crack propagation So far, we have assumed that a crack is stationary if the stress intensity factor is smaller than the fracture toughness K Ic . This, however, is not always the case. If, for instance, a corrosive medium penetrates the material along the crack surface, it might weaken the material near the crack tip. This decreases the fracture toughness locally and the c rack can propagate, but only until it reaches undamaged material. If this happens, a crack can slowly propagate at loads be low the critical load, a phenomenon called subcritical crack growth. The crack propagates subcritically until the stress intensity factor K I equals the fracture toughness K Ic and the crack propagation becomes unstable. 14 As the subcritical crack growth is time-dependent, it can b e described by the crack-growth rate da/dt, specifying the increment da by which the crack propagates during an infinitesimal time dt. Generally, subcritical crack growth occurs when the material near the crack tip is weakened by time-dependent processes. Different physical phenomena may be responsible for this. In many metals, corrosive media like electrolytes can cause stress corrosion cracking. Two different kinds of stress corrosion cracking can be distinguished. In anodic stress corrosion cracking, loading accelerates corrosion at the crack tip, resulting in crack propagation. This may happen in metals whose surface is usually protected by a passivating layer. The crack tip can be acti- vated, for instanc e by local plastic deformation, and the crack can propagate. If a new passivating layer forms immediately on the freshly formed surface, corrosion can only proceed at the crack tip, causing the crack to remain sharp- edged. Whether anodic stress corrosion cracking can occur depends on the surrounding me dia, the material and its state, the temperature, and the me- chanical stress. In unfavourable circumstances, the required stress level may be very small and even residual stresses may be sufficient to cause failure by a delayed fracture (see section 3.5.3). Anodic stress corrosion cracking can, for example, occur in the presence of chloride ions (e. g., in saline air) in aluminium alloys and austenitic chrome nickel steels. 15 14 In section 5.2.5, we saw that even above K Ic a crack may still be stable if the crack-growth resistance increases. If subcritical crack growth occurs, this effect is usually negligible because the crack grows fast at stress intensity values close to K Ic . 15 This latter case is especially problematic because these steels are usually called ‘stainless’, implying a high level of corrosion resistance. However, in saline atmo- sphere, the corrosion resistance of these steels is reduced. 5.2 Linear-elastic fracture mechanics 151 A different case is hydrogen-induced stress corrosion cracking. It occurs in materials showing hydrogen embrittlement (see section 3.5.3, page 117). The stress concentration near the crack tip dilates the crystal lattice. Hydrogen, generated during the corrosion process, is therefore preferentially stored in this region near the crack tip, reducing the fracture toughness in the most stressed region. The crack propagates through the weakened material, and, subsequently, hydrogen diffuses to the new crack tip. Hydrogen-induced stress corrosion cracking is observed particularly in high-strength steels because here elastic strains near the crack tip can be large and large amounts of hydrogen can thus be stored. Polymers show a similar effect in the presence of solvents. Solvents pre- ferredly enter the material near the crack tip because the distance between the molecules is increased there by the large tensile stresses. If, for instance, a rod made of polymethylmethacrylate (Plexiglas) is bent and the tensile side is wetted with acetone or alcohol, brittle fracture can occur after a short ex- posure time. In this case, the cleavage strength is reduced because the dipole bonds between the molecules are replaced by bonds formed with the s olvent (see also section 8.8). Ceramics can also fail by subcritical crack growth due to stresse s and localised chemical reactions. In glasses, for instance, water can enter surface defects and can attack the bonds between the silicon and the oxygen atoms if these are strained by an external stress [9]: Si−O−Si + H 2 O → Si−O−H + H−O−Si . This reaction can cause a seemingly sudden failure of glasses. Subcritical crack growth also occurs in crystalline ceramics, especially if they have a glassy phase (see section 7.1) on the grain boundaries. Among the most sensitive ceramics are silicate ceramics like porcelain or mullite, for they usually contain a large amount of more than 20% glassy phases [19, 142]. Subcritical crack growth can also be present in engineering ce ramics, for instance in aluminium oxide (Al 2 O 3 ) in humid atmosphere or saline solution [104]. At elevated temperatures, metals and ceramics exhibit time-dependent plastic deformation, called creep, a phenomenon to be discussed in detail in chapter 11. If a pre-cracked material is loaded at high temperatures, the crack can grow. In metals and ceramics, pores are frequently responsible for this because they form and coalesce in the highly-stressed region in front of the crack tip, often on grain boundaries [119] (see section 11.3). The crack thus frequently propagates between the grains (intercrystalline fracture). This pro- cess is called creep crack growth (ccg). In polymers, time-dependent plastic deformation occurs already at ambient temperature (see section 8), and they are thus also susceptible to creep crack growth. Subcritical crack growth is determined by the temperature, the material, and, for the low-temperature processes, the environment. In a given system, the crack-growth rate frequently depends on the stress intensity factor K I only. Below a certain, temperature-dependent limiting value K I0 , crack growth 152 5 Fracture mechanics ¾(t) ¾(t) constructive notch fatigue initial crack ¾ t Fig. 5.13. Creating an initial crack by cyclic loading vanishes completely or almost completely [104]. If K I exceeds K I0 , the crack- growth rate rapidly increases with increasing stress intensity factor. In many cases (for example, in stress corrosion cracking), a plateau region follows i. e., the crack-growth rate is nearly constant for a certain range of K I -values. When the stress intensity factor approaches K Ic , the crack-growth rate increase s rapidly again. Examples for crack-growth rate curves and a mathematical description of the crack-growth rate are discussed in section 7.2.6. ∗ 5.2.7 Measuring fracture parameters In the previous sections, we introduced several important material parame- ters: The fracture toughness K Ic , the critical energy release rate G Ic , and the crack-growth resistance curve. We now want to see how these quantities are measured. A common feature of all experiments is that the test specimens are p re- cracked. To achieve th is, a notched sample is used and a crack is propagated from this notch by cyclic loading (see chapter 10) as shown in figure 5.13. Creating the initial crack in this way is necessary because the notch tip is usually not sharp enough to behave like a true crack. Cyclic loading allows to produce the initial crack at a load that is much smaller than that needed for static experiments (see chapter 10). Several specimen geometries are standardised, and the corresponding ge- ometry factors are given in tabular form or by approximation functions [21,133, 138]. Among the most common specimen geometries are the three-point bend- ing specimen (figure 5.14(a)) and the compact tension specimen, or ct speci- men for short (figure 5.14(b)). For the compact tension specimen, the stress intensity factor can be calcu- lated from the external load F by K I = F B √ W f( a /W ) (5.30) 5.2 Linear-elastic fracture mechanics 153 (a) Three-point bending specimen B s G W H a F F (b) ct specimen Fig. 5.14. Standardised specimen geometries for fracture mechanical experiments where f( a /W ) = 2 + a W  1 − a W  3 /2 × (5.31)  0.886 + 4.64 a W − 13.32  a W  2 + 14.72  a W  3 − 5.6  a W  4  is the geometry factor. In calculating K I , it is important to note that the crack length a is measured from the point of loading, not from the beginning of the initial crack. This is easily understood because it is completely irrelevant for the stress state near the crack tip how ‘wide’ the crack is some distance away. It is only important that the initial crack is sharp-edged with a small radius of curvature, and this is ensured by creating it through cyclic loading. The quantities G, H, B, W, and s from figure 5.14(b) must, according to the standard astm e 399, be related in a certain way: As an example, B = W/2, s = 0.55W , H = 1.2W , and G = 1.25W are used for a standard ct specimen. The initial crack length should be limited by 0.45W ≤ a ≤ 0.55W . 154 5 Fracture mechanics 0 2 4 6 8 10 12 14 0.0 0.5 1.0 1.5 2.0 ∆s/mm F kN Fig. 5.15. Load-displacement curve of an AlCuMg 2 ct specimen ¢s F F Q =F max linear elastic 95 % line F 5 (a) F 5 right of F max ¢s F F max linear elastic 95 % line F Q =F 5 (b) F 5 left of F max ¢s F F Q linear elastic 95 % line F max F 5 (c) F 5 left of F max , but right of the first maximum Fig. 5.16. Determination of the critical force F Q . The 95% lines are drawn with a slope of 90% only ∗ Measuring the fracture toughness The procedure to measure K Ic or G Ic is independent of the specimen geometry. The loading points are displaced with constant speed and the required force is measured. If the force is plotted against the displacement of the loading points, a load-displacement curve results as shown in figure 5.15. The onset of unstable crack propagation can be seen from this curve because the force reaches a maximum and drops, resulting in a larger compliance of the specimen. As the experiment is displacement-controlled, the crack usually stabilises again due to the unloading and propagates only when the crack is opened further. If the size of the plastic zone near the crack tip is small comp ared to the volume of the specimen, crack propagation starts without any noticeable deviation of the loading curve from linear-elastic behaviour (see figure 5.16(a)). 5.2 Linear-elastic fracture mechanics 155 In contrast, the material behaviour shown in figure 5.16(b) indicates signif- icant plastic deformation. The crack propagation is stable at first and b ecomes unstable at a force F max . This shape of the curve is typical for ductile materi- als. In this case, it has to be ensured that linear-elastic fracture mechanics is still valid. Furthermore, it is not possible to determine from the curve alone at which force crack propagation has started because stable crack propagation and plastic deformation both reduce the slope of the curve. To determine the facture toughness K Ic , a pragmatic app roach is taken, similar to the definition of the yield strength R p0.2 . This will be described below. A special case is shown in figure 5.16(c). On reaching a load F Q , the crack propagates unstably for a certain distance and then becomes arrested. This is called pop-in. To determine the fracture toughness K Ic , the following procedure has been agreed upon (see e. g., standards astm e 399 and iso 12737): We start by drawing a line with a slope of 95% of that of the elastic line from the experiment (figure 5.16). The intersection of this line with the load- displacement curve determines the force F 5 . 16 Two cases can be distinguished: • If F 5 lies to the right of the force value at which the first reduction in the load occu rs, the force F Q is d etermined by this maximum (F Q = F max in figure 5.16(a), F Q in figure 5.16(c)). • If F 5 is left to this maximum, F Q = F 5 is used as a critical value (fig- ure 5.16(b)). In this case, it has to be ensured that plastic deformation was small enough to allow using linear-elastic fracture mechanics. A nec- essary condition for this is F max F Q ≤ 1.1 . (5.32) If this condition does not hold, the experiment has to be e valuated ac- cording to the rules of elastic-plastic fracture mechanics, discussed in sec- tion 5.3. For a ct specimen, the critical stress intensity factor K Q is calculated from F Q by using equation (5.30), K Q = F Q B √ W f( a /W ) . For this, it is necessary to know the initial crack length. This can be measured optically after the spec imen has been fractured because the fracture surface of the initial crack produced by cyclic loading can easily be discerned from the statically produced crack surface (figure 5.17). The value of K Q determined in this way does not depend on the material only. Instead, the stress state near the crack tip, that in itself depe nd s on 16 The subscript ‘5’ denotes the reduction of the slope by 5% used in constructing the line. 156 5 Fracture mechanics Fig. 5.17. Measuring the initial crack length plain stress plain stress plain strain plastic zone Fig. 5.18. Size and shap e of the plastic zone near the crack tip (dog bone, after [58]) the specimen geometry, may influence this value. This can be illustrated by inspecting the plastic zone near the crack tip (see figure 5.18). At the surface, the specimen is in a state of plane stress because no nor- mal forces can be transmitted here. The smallest principal stress is thus zero. Within the specimen, the stress state is a state of nearly plane strain because the transversal contraction near the crack tip is constrained by the surround- ing material. The stress state is thus a state of triaxial tension. Therefore, the equivalent stresses and thus the plastic deformations are larger near the sur- face th an within the specimen. As plastic deformation dissipates energy, the crack-growth resistance increases with decreasing thickness of the specimen (see figure 5.19). For sufficiently thick specimen s, the influence of the surface zone with its state of plane stress can be neglected and the crack-growth resistance K Q approaches a constant value, the fracture toughness K Ic . To ensure independence of the geometry and to determine the fracture toughness as lower (and therefore safe) limiting value for the crack-growth resistance of a material, a state of plane strain is required. Only if this can be guaranteed, the measured value K Q is called fracture toughness K Ic . Accord- ing to the standards astm e 399 and iso 12737, this requires 5.2 Linear-elastic fracture mechanics 157 K Q B K Ic plain strain mixed Fig. 5.19. Dependence of the measured K Q -values on the specimen thickness B. For very thin specimens, the state of plane stress is dominant, for thick specimens the behaviour is determined by the state of plane strain within the specimen. In the region denoted as ‘mixed’, both states contribute    B a W − a    ≥ 2.5  K Q R p  2 . (5.33) This is a reasonable requirement as can be seen by estimating the size r of the plastic zone from equation (5.1). If we insert K I = K Q in this equation, we can estimate this size by setting ˜σ 22 (˜x 2 = 0) = R p and solving for r = ˜x 1 : r ≈ 1 2π  K Q R p  2 . (5.34) Thus, equation (5.33) ensures that the plastic zone is small compared to the specimen size. ∗ Measuring the crack-growth resistance curve As explained in section 5.2.5, the crack-growth resistance curve is a plot of the stress intensity factor versus the crack length a. Experiments are usu- ally displaceme nt-controlled to enable measurement of the load-displacement curve after the maximum force has been exceeded. To measure the crack-growth resistance curve according to astm e 561, the load is applied step-wise, and the crack length is measured for each load value after the crack has stabilised. For a complete c urve, 10 to 15 measurement values are needed. To avoid using several specimens for a single curve, the crack length is measured during the experiment. This can be done in several ways [21, 43]. Optical methods measure the crack length directly on the polished surface of the specimen. This method is rather simple, but its main disadvantage is that the crack is only measured on the surface; the crack length within the specimen is unknown. [...]... value of T depends on the type of dislocation i e., on the orientation of b and t, and on the curvature of the dislocation line For the considerations that follow, the estimate given here is sufficient 6.2 Dislocations 171 σ11 / MPa 100 50 0 50 −100 −100 50 0 x1 / nm 50 50 100 50 x 0/ nm 100 −100 2 σ22 / MPa 100 50 0 50 −100 −100 50 0 x1 / nm 50 50 100 50 x 0/ nm 100 −100 2 τ12 / MPa 100 50 0 50 ... from equation (5. 35) So, finally, we have shown the equivalence between equation (5. 10) and (5. 35) as shown in equation (5. 36) ∗ 5. 3.3 Material behaviour during crack propagation We saw in section 5. 2 .5 that the crack-growth resistance KIR in linear-elastic fracture mechanics depends on the crack length increment ∆a Similarly, the ¿m ¿m ¿m ax ax 5 Fracture mechanics ax 162 1 2 3 x ax plastic 5 ¿m ax ax... regions Equations (5. 10) and (5. 35) are equivalent Therefore, for linear-elastic materials and a state of plane stress, the equation GI = J = 2 KI E (5. 36) holds As for G, there is a critical value Jc , denoting the onset of crack propagation This will be discussed below in section 5. 3.3 We now want to show the equivalence of the J integral and the energy release rate (equation (5. 10)) by calculating... −100 50 0 x1 / nm 50 50 100 50 x 0/ nm 100 −100 2 σm / MPa 100 50 0 50 −100 −100 50 0 x1 / nm 50 50 100 50 x 0/ nm 100 −100 2 Fig 6.6 Stress distribution near an edge dislocation oriented in the x3 direction in aluminium Values are cut at ±100 MPa Compressive stresses are printed in dark colour 172 6 Mechanical behaviour of metals III x2 II IV I x1 V VIII VI VII Fig 6.7 Qualitative illustration of. .. face-centred cubic crystal is close-packed (see section 1.2.2) Planes of type {111} and directions of the type 110 are close-packed and thus form the slip systems (figure 6.14) If we consider planes as identical differing only 176 6 Mechanical behaviour of metals (a) Close-packed plane or direction Slip is relatively easy, as the upper array of spheres has to be lifted only slightly (b) Non-close-packed... 1 and ○ During formation of the stretch zone and of cavities (subfigures ○ 4 to ○), the relation between ∆a and JR is almost linear With increasing load, the cavities coalesce with each other and the crack, 5 causing a ‘true’ growth of the crack (subfigure ○) This is called crack initiation and occurs when J reaches Jc [58 ].19 The slope of the crack-growth resistance curve now decreases (see figure 5. 22)... tip is blunted by the 5. 3 Elastic-plastic fracture mechanics 159 Fig 5. 20 One possible definition of the crack tip opening displacement δt Fig 5. 21 Coordinate system and integration contour for the J integral plastic deformation and that both crack surfaces are almost parallel, see figure 5. 20 One possibility to determine δt is to draw two lines at an angle of 45 to the crack line and measure the distance... with the crack surface ∗ 5. 3.2 J integral In section 5. 2.2, we calculated the energy balance of a propagating plane crack and predicted crack growth if the energy release rate reaches a critical value GIc The definition of GI in equation (5. 10) was independent of the material behaviour Linear-elastic behaviour was only assumed to calculate the terms in equation (5. 12) The so-called J integral can quantify... metallic single crystals is thus between 1 GPa and 25 GPa If we measure the strength of single crystals of pure metals, the values found are several orders of magnitudes below this theoretical value and even lie below that of engineering alloys Typical values are in the range of a few 1 For simplicity, we usually use a simple cubic lattice in the sketches of crystals shown in this chapter, although this... a) d(∆s) , (5. 40) 0 with ∆s denoting the displacement of the loading points From this, the J integral for a ct specimen is given by [21] J= with WF ·η B(W − a) (5. 41) 164 5 Fracture mechanics Fig 5. 23 Load-displacement curve to determine the JR crack-growth resistance η = 2 + 0 .52 2 1 − a W As can be expected because of the definition of the J integral as an energy release rate, the value of the J integral . equation (5. 10) and (5. 35) as shown in equa- tion (5. 36). ∗ 5. 3.3 Material behaviour during crack propagation We saw in section 5. 2 .5 that the crack-growth resistance K IR in linear-elastic fracture. 1 .5 2.0 ∆s/mm F kN Fig. 5. 15. Load-displacement curve of an AlCuMg 2 ct specimen ¢s F F Q =F max linear elastic 95 % line F 5 (a) F 5 right of F max ¢s F F max linear elastic 95 % line F Q =F 5 (b). = 0 .55 W , H = 1.2W , and G = 1.25W are used for a standard ct specimen. The initial crack length should be limited by 0.45W ≤ a ≤ 0 .55 W . 154 5 Fracture mechanics 0 2 4 6 8 10 12 14 0.0 0 .5 1.0