74 2 Brittle and Ductile Fracture centrations as much as several orders of magnitude higher than that in the grain interior [160.The most dangerous impurities segregating in bcc iron and steels are phosphorus,tin and antimony.For example,the disintegra- tion of the rotor at the Hinkley Point Power Station turbine generator in 1969 was caused by 50%of phosphorus segregated at grain boundaries of the 3Crl/2Mo low-alloy steel containing a few tenths of a percent of phosphorus in the bulk 161. Brittle intercrystalline (intergranular)decohesion caused by impurity seg- regation exhibits relatively high microroughness of fracture surfaces.More- over,the secondary cracks identifying the splitting of the main crack front are often observed preferentially at triple points.Both these phenomena lead to the so-called geometrically induced shielding (GIS)of the crack tip that has a favourable effect on decreasing the local stress intensity factor,thereby increasing the fracture toughness.This kind of shielding is one of the so- called extrinsic components of fracture toughness that can be considered as a possible toughening mechanism in the research and technology of advanced materials. In the next subsections,the theory of GIS and its practical application to an improvement of fracture toughness of brittle materials is outlined. 2.1.1 Geometrically Induced Crack Tip Shielding Crack front interactions with secondary-phase particles or grain (phase) boundaries in the matrix structure cause deflections of the crack front from the straight growth direction resulting in the microscopic tortuosity of cracks. As already mentioned,such waviness combined with crack branching (split- ting)is a natural property of intergranular cracks in metals as well as ce- ramics.In general,the tortuosity induces a local mixed-mode I+II+III at the crack front even when only a pure remote mode I loading is applied. In order to describe the crack stability under mixed-mode loading,various LEFM-based criteria were proposed(see,e.g.,[162-164).Several of the most frequently used mixed-mode criteria can be found in Appendix B,where con- ditions of their validity are also briefly described.When selecting a suitable criterion one should note that an unstable brittle fracture in metallic mate- rials is usually preceded by a stable corrosion and/or fatigue crack growth to some critical crack size.During such growth the crack always turns per- pendicularly to the direction of maximal principal stress,i.e.,to the opening mode I loading.This physically corresponds to minimization of both the crack closure (see Chapter 3 for more details)and the friction so that the rough crack flanks behind the tortuous crack front do not experience any significant sliding contact.Because the crack-wake friction is responsible for somewhat higher fracture toughness values measured under remote sliding modes II and III when compared to those under mode I [164],one can consider an approx-
74 2 Brittle and Ductile Fracture centrations as much as several orders of magnitude higher than that in the grain interior [160]. The most dangerous impurities segregating in bcc iron and steels are phosphorus, tin and antimony. For example, the disintegration of the rotor at the Hinkley Point Power Station turbine generator in 1969 was caused by 50% of phosphorus segregated at grain boundaries of the 3Cr1/2Mo low-alloy steel containing a few tenths of a percent of phosphorus in the bulk [161]. Brittle intercrystalline (intergranular) decohesion caused by impurity segregation exhibits relatively high microroughness of fracture surfaces. Moreover, the secondary cracks identifying the splitting of the main crack front are often observed preferentially at triple points. Both these phenomena lead to the so-called geometrically induced shielding (GIS) of the crack tip that has a favourable effect on decreasing the local stress intensity factor, thereby increasing the fracture toughness. This kind of shielding is one of the socalled extrinsic components of fracture toughness that can be considered as a possible toughening mechanism in the research and technology of advanced materials. In the next subsections, the theory of GIS and its practical application to an improvement of fracture toughness of brittle materials is outlined. 2.1.1 Geometrically Induced Crack Tip Shielding Crack front interactions with secondary–phase particles or grain (phase) boundaries in the matrix structure cause deflections of the crack front from the straight growth direction resulting in the microscopic tortuosity of cracks. As already mentioned, such waviness combined with crack branching (splitting) is a natural property of intergranular cracks in metals as well as ceramics. In general, the tortuosity induces a local mixed-mode I+II+III at the crack front even when only a pure remote mode I loading is applied. In order to describe the crack stability under mixed-mode loading, various LEFM-based criteria were proposed (see, e.g., [162–164]). Several of the most frequently used mixed-mode criteria can be found in Appendix B, where conditions of their validity are also briefly described. When selecting a suitable criterion one should note that an unstable brittle fracture in metallic materials is usually preceded by a stable corrosion and/or fatigue crack growth to some critical crack size. During such growth the crack always turns perpendicularly to the direction of maximal principal stress, i.e., to the opening mode I loading. This physically corresponds to minimization of both the crack closure (see Chapter 3 for more details) and the friction so that the rough crack flanks behind the tortuous crack front do not experience any significant sliding contact. Because the crack-wake friction is responsible for somewhat higher fracture toughness values measured under remote sliding modes II and III when compared to those under mode I [164], one can consider an approx-
2.1 Brittle Fracture 75 imate equality KIe KIe KIle along tortuous crack fronts of remote mode I cracks.Moreover,first unstable pop-ins at these fronts follow,most probably,the local planes of already pre-cracked facets.Consequently,the simplest stability criterion Geff=GI+GII+GIII, can be accepted,where Gef is the effective crack driving force.An almost equivalent relation is often used in terms of stress intensity factors: Kg=V好+K+一k (2.7) For example,in the case of a long straight crack with an elementary kinked tip,it simply reads Kef cos2(0/2)KI, (2.8) where 6 is the kink angle.One can clearly see that Kef KI for 6>0. This inequality generally holds for any spatially complex crack front.Hence, the local stress intensity Kef at such a front is always lower than the re- mote Kr-factor applied to a straight(smooth)crack of the same macroscopic length.The geometrically induced shielding(GIS)effect belongs,according to Ritchie [165],to so-called extrinsic shielding mechanisms.The resistance to crack propagation in fracture and fatigue has,in general,many compo- nents that can be divided into two main categories:intrinsic and extrinsic toughening.The first mechanism represents the inherent matrix resistance in terms of the atomic bond strength or the global rigidity,strength and duc- tility.Appropriate modifications to both the chemical composition and the heat treatment are typical technological ways to improve the intrinsic fracture toughness.On the other hand,processes like kinking,meandering or branch- ing of the crack front,induced mostly by microstructural heterogeneities, belong typically to the extrinsic toughening mechanisms.They reduce the crack driving force and,apparently,increase the intrinsic resistance to crack growth.Thus,the measured fracture toughness can be expressed as a sum of the intrinsic toughness and extrinsic components: Ke=KIei+∑Kie (2.9) The standardized procedure for calculation of Ki-values [166 assumes a planar crack with a straight front and,therefore,does not take the extrinsic shielding effect associated with the crack microgeometry into account.Hence, surprisingly high Kre-values might be measured,particularly for materials with coarse microstructures and highly tortuous cracks.General expressions for GIS contributions in both brittle and quasi-brittle fracture were derived in [167,168]by following the approach first introduced by Faber and Evans 169.In the case of brittle fracture
2.1 Brittle Fracture 75 imate equality KIc ≈ KIIc ≈ KIIIc along tortuous crack fronts of remote mode I cracks. Moreover, first unstable pop-ins at these fronts follow, most probably, the local planes of already pre-cracked facets. Consequently, the simplest stability criterion Geff = GI + GII + GIII , can be accepted, where Geff is the effective crack driving force. An almost equivalent relation is often used in terms of stress intensity factors: Keff = K2 I + K2 II + 1 1 − ν K2 III . (2.7) For example, in the case of a long straight crack with an elementary kinked tip, it simply reads Keff = cos2(θ/2)KI , (2.8) where θ is the kink angle. One can clearly see that Keff < KI for θ > 0. This inequality generally holds for any spatially complex crack front. Hence, the local stress intensity Keff at such a front is always lower than the remote KI -factor applied to a straight (smooth) crack of the same macroscopic length. The geometrically induced shielding (GIS) effect belongs, according to Ritchie [165], to so-called extrinsic shielding mechanisms. The resistance to crack propagation in fracture and fatigue has, in general, many components that can be divided into two main categories: intrinsic and extrinsic toughening. The first mechanism represents the inherent matrix resistance in terms of the atomic bond strength or the global rigidity, strength and ductility. Appropriate modifications to both the chemical composition and the heat treatment are typical technological ways to improve the intrinsic fracture toughness. On the other hand, processes like kinking, meandering or branching of the crack front, induced mostly by microstructural heterogeneities, belong typically to the extrinsic toughening mechanisms. They reduce the crack driving force and, apparently, increase the intrinsic resistance to crack growth. Thus, the measured fracture toughness can be expressed as a sum of the intrinsic toughness and extrinsic components: KIc = KIci +KIce. (2.9) The standardized procedure for calculation of KIc-values [166] assumes a planar crack with a straight front and, therefore, does not take the extrinsic shielding effect associated with the crack microgeometry into account. Hence, surprisingly high KIc-values might be measured, particularly for materials with coarse microstructures and highly tortuous cracks. General expressions for GIS contributions in both brittle and quasi-brittle fracture were derived in [167, 168] by following the approach first introduced by Faber and Evans [169]. In the case of brittle fracture
76 2 Brittle and Ductile Fracture 1/2 geff.r KIe (2.10) RA where KIe and KIci are respectively the measured (nominal)and intrinsic values of fracture toughness,/2=is the mean effective k-factor for the tortuous crack front,normalized to the remote KI (effr=Kef/KI), and RA is the area roughness of the fracture surface.Equation 2.10 can be derived by the following simple reasoning. Let us consider a cracked body of a thickness B with an intrinsic resistance Grci against the crack growth under remote mode I loading.The coordinate system y,z is related to the crack front in the usual manner(Figure 2.1). The straight crack front with no geometrical shielding (GIS)represents a trivial case.Here,obviously,the measured fracture toughness value Gre (or KIe)is equal to its intrinsic value,i.e.,GIc GIci (or KIe KIci). detail growth direction crack front Figure 2.1 Scheme of the tortuous crack front and its segment.Reprinted with permission from John Wiley Sons,Inc.(see page 265) When the crack front is microscopically tortuous,a variable local mixed- mode 1+2+3 characterized by geff or keff values is present generally at each site along the crack front.During the external loading under increasing re- mote value GI,the proportionality gef~Gr or kef ~KI must be valid. Thus,the ratio gef.r=gef/Gr can be introduced as independent of GI but dependent on the crack front tortuosity.Let Gur be the remote crack driving force at the moment of an unstable elementary extension dz of the crack front.This value is equal to the conventionally measured(nominal)fracture toughness Gre.Then the nominal elementary energy release rate due to the creation of a new crack surface area drdz is equal to Gurdrdz.However,the actual (local)elementary energy release rate at the tortuous crack front is geffdrdz geffrGurdxdz
76 2 Brittle and Ductile Fracture KIci = g¯eff ,r RA 1/2 KIc, (2.10) where KIc and KIci are respectively the measured (nominal) and intrinsic values of fracture toughness, ¯geff ,r 1/2 = ¯ keff ,r is the mean effective k-factor for the tortuous crack front, normalized to the remote KI (keff ,r = Keff /KI ), and RA is the area roughness of the fracture surface. Equation 2.10 can be derived by the following simple reasoning. Let us consider a cracked body of a thickness B with an intrinsic resistance GIci against the crack growth under remote mode I loading. The coordinate system x, y, z is related to the crack front in the usual manner (Figure 2.1). The straight crack front with no geometrical shielding (GIS) represents a trivial case. Here, obviously, the measured fracture toughness value GIc (or KIc) is equal to its intrinsic value, i.e., GIc ≡ GIci (or KIc ≡ KIci). dz dx y x detail growth direction crack front Figure 2.1 Scheme of the tortuous crack front and its segment. Reprinted with permission from John Wiley & Sons, Inc. (see page 265) When the crack front is microscopically tortuous, a variable local mixedmode 1+2+3 characterized by geff or keff values is present generally at each site along the crack front. During the external loading under increasing remote value GI , the proportionality geff ∼ GI or keff ∼ KI must be valid. Thus, the ratio geff ,r = geff /GI can be introduced as independent of GI but dependent on the crack front tortuosity. Let GuI be the remote crack driving force at the moment of an unstable elementary extension dx of the crack front. This value is equal to the conventionally measured (nominal) fracture toughness GIc. Then the nominal elementary energy release rate due to the creation of a new crack surface area dxdz is equal to GuIdxdz. However, the actual (local) elementary energy release rate at the tortuous crack front is geff dxdz = geff ,rGuIdxdz
2.1 Brittle Fracture 77 Consequently,the total energy available for the creation of a new surface area Bdx along the crack front can be written as B dW Gurdz 9eff,rdz. (2.11) 0 As follows from Figure 2.1,however,the real new elementary surface area dS=RABdz is greater than Bdr since dz RA B (2.12) coso(z)cos(z) In Equation 2.12,RA is the roughness of the fracture surface and drdz/(cos ocos) is the area of the hatched rectangle in Figure 2.1.Because GIci is the intrinsic resistance to crack growth,the total fracture energy must be dW GIcidS GIci RA B dt. (2.13) Combining Equations 2.11 and 2.13 and denoting ⊙ gef.r= B gef,rdz, 0 one obtains GuI≡GIc= RA GIei. (2.14) geff.r In general,,Gie≥GIei since geff,r≤1 and RA≥1.Therefore,.the nominally measured fracture toughness Gic is usually higher than the in- trinsic (real)matrix resistance GIci.According to the relation Gre/GIci (KIe/KIci)2,Equation 2.14 can be eventually rewritten to obtain Equation 2.10. Values of gefr and RA must be estimated by using numerical (or ap- proximate analytical)models of the real tortuous crack front combined with appropriate experimental methods for fracture surface roughness determina- tion.In Sections 2.1.2 and 2.1.3,the so-called pyramidal-and particle-induced models are presented.In the context of 2D crack models,the tortuosity is usu- ally described by a double-or even single-kink geometry and RA =1/cos is assumed.In the 2D single kink approximation at Equation 2.8,the crack front is assumed to be straight (RA =1).Consequently,Equation 2.10 takes the following form: KIci=cos2(0/2)KIc
2.1 Brittle Fracture 77 Consequently, the total energy available for the creation of a new surface area Bdx along the crack front can be written as dW = GuIdx B 0 geff ,rdz. (2.11) As follows from Figure 2.1, however, the real new elementary surface area dS = RABdx is greater than Bdx since RA = 1 B B 0 dz cos φ(z) cos ϑ(z) . (2.12) In Equation 2.12, RA is the roughness of the fracture surface and dxdz/(cos φ cos ϑ) is the area of the hatched rectangle in Figure 2.1. Because GIci is the intrinsic resistance to crack growth, the total fracture energy must be dW = GIcidS = GIci RA B dx. (2.13) Combining Equations 2.11 and 2.13 and denoting g¯eff ,r = 1 B B 0 geff ,rdz, one obtains GuI ≡ GIc = RA g¯eff ,r GIci. (2.14) In general, GIc ≥ GIci since ¯geff ,r ≤ 1 and RA ≥ 1. Therefore, the nominally measured fracture toughness GIc is usually higher than the intrinsic (real) matrix resistance GIci. According to the relation GIc/GIci = (KIc/KIci)2, Equation 2.14 can be eventually rewritten to obtain Equation 2.10. Values of ¯geff ,r and RA must be estimated by using numerical (or approximate analytical) models of the real tortuous crack front combined with appropriate experimental methods for fracture surface roughness determination. In Sections 2.1.2 and 2.1.3, the so-called pyramidal- and particle-induced models are presented. In the context of 2D crack models, the tortuosity is usually described by a double- or even single-kink geometry and RA = 1/ cos θ is assumed. In the 2D single kink approximation at Equation 2.8, the crack front is assumed to be straight (RA = 1). Consequently, Equation 2.10 takes the following form: KIci = cos2(θ/2)KIc.
78 2 Brittle and Ductile Fracture Besides both the kinking and the meandering,the crack branching can also take place especially in the case of intergranular fracture.This process causes further reduction of SIF ahead of the crack tip and,therefore,Equation 2.10 is to be further modified.According to [170],the crack branching reduces the local SIF approximately to one half of its original magnitude.Let us denote A the area fraction of the fracture surface influenced by crack branching. When accepting a linear mixed rule,Equation 2.10 can be then modified as 1/2 KIci geff.r (1-A6)+0.546 (2.15) RA The area A can be determined by measuring the number of secondary cracks(branches)occurring on fracture profiles prepared by polishing met- allographical samples perpendicular to the fracture surface [171](see also Section 3.2).Twice the sum of projected lengths of branches into the main crack path divided by the true crack length yields a plausible estimate of Ab. When omitting the crack branching and considering Equations 2.9 and 2.10,the extrinsic GIS component of fracture toughness can be simply ex- pressed as KIce=(1-./RA)KIe.Brittle fracture in metallic materials occurs only when a pure cleavage or intergranular decohesion takes place.In these cases the extrinsic components other than geometrical (such as zone shielding or bridging)can be neglected.In the particular case of cleavage fracture (bcc metals at very low temperatures)one usually observes that RA 1.2 and gef.r>0.9.This means that GIS is rather insignificant.On the other hand,the extrinsic component Kice might be very high when the intergranular fracture cannot be avoided(strong corrosion or hydrogen assis- tance,grain-boundary segregation of impurities and tempering embrittlement of high-strength steels).In that case,however,the favourable effect of the ex- trinsic component is usually totally destroyed by an extreme reduction of the intrinsic component KIci.Nevertheless,one can still improve the fracture toughness of both metals and ceramics by increasing the extrinsic(shielding) component without the loss of general quality in mechanical properties(see Sections2.1.2,2.2.2and3.2.6). 2.1.2 Pyramidal Model of Tortuous Crack Front A plausible assessment of the GIS effect is possible only when the following steps can be realized: 1.building of a realistic model of the crack front based on a 3D determination of fracture surface roughness; 2.calculation of local normalized stress intensity factors kir,k2r and k3r along the crack front; 3.calculation of the effective stress intensity factor Keff.r
78 2 Brittle and Ductile Fracture Besides both the kinking and the meandering, the crack branching can also take place especially in the case of intergranular fracture. This process causes further reduction of SIF ahead of the crack tip and, therefore, Equation 2.10 is to be further modified. According to [170], the crack branching reduces the local SIF approximately to one half of its original magnitude. Let us denote Ab the area fraction of the fracture surface influenced by crack branching. When accepting a linear mixed rule, Equation 2.10 can be then modified as KIci = g¯eff ,r RA 1/2 (1 − Ab)+0.5Ab KIc. (2.15) The area Ab can be determined by measuring the number of secondary cracks (branches) occurring on fracture profiles prepared by polishing metallographical samples perpendicular to the fracture surface [171] (see also Section 3.2). Twice the sum of projected lengths of branches into the main crack path divided by the true crack length yields a plausible estimate of Ab. When omitting the crack branching and considering Equations 2.9 and 2.10, the extrinsic GIS component of fracture toughness can be simply expressed as KIce = (1−g¯eff ,r/RA)KIc. Brittle fracture in metallic materials occurs only when a pure cleavage or intergranular decohesion takes place. In these cases the extrinsic components other than geometrical (such as zone shielding or bridging) can be neglected. In the particular case of cleavage fracture (bcc metals at very low temperatures) one usually observes that RA < 1.2 and ¯geff ,r > 0.9. This means that GIS is rather insignificant. On the other hand, the extrinsic component KIce might be very high when the intergranular fracture cannot be avoided (strong corrosion or hydrogen assistance, grain-boundary segregation of impurities and tempering embrittlement of high-strength steels). In that case, however, the favourable effect of the extrinsic component is usually totally destroyed by an extreme reduction of the intrinsic component KIci. Nevertheless, one can still improve the fracture toughness of both metals and ceramics by increasing the extrinsic (shielding) component without the loss of general quality in mechanical properties (see Sections 2.1.2, 2.2.2 and 3.2.6). 2.1.2 Pyramidal Model of Tortuous Crack Front A plausible assessment of the GIS effect is possible only when the following steps can be realized: 1. building of a realistic model of the crack front based on a 3D determination of fracture surface roughness; 2. calculation of local normalized stress intensity factors k1r, k2r and k3r along the crack front; 3. calculation of the effective stress intensity factor keff ,r