《纺织复合材料》课程参考文献（Micromechanisms of Fracture and Fatigue）Chapter 2 Brittle and Ductile Fracture

Chapter 2 Brittle and Ductile Fracture This chapter is devoted to damage and fracture micromechanisms operating in the case when monotonically increasing forces are applied to engineering materials and components.According to the amount of plastic deformation involved in these processes,the fracture events can be categorized as brittle, quasi-brittle or ductile. Brittle fracture is typical for ceramic materials,where plastic deformation is strongly limited across extended ranges of deformation rates and temper- atures.In polycrystalline ceramics the reasons lie in a high Peierls-Nabarro stress of dislocations due to strong and directional covalent bonds (this holds also for some ionic compounds),and in less than five independent slip systems in ionic crystals (e.g.,[149).In amorphous ceramics it is simply because of a lack of any dislocations and,simultaneously,strong covalent and ionic in- teratomic bonds.Metallic materials or polymers exhibit brittle fracture only under conditions of extremely high deformation rates,very low temperatures or extreme impurity concentrations at grain boundaries.In the case of a strong corrosion assistance,brittle fracture can also occur at very small load- ing rates or even at a constant loading (stress corrosion cracking).A typical micromechanism of brittle fracture is so-called cleavage,where the atoms are gradually separated by tearing along the fracture plane in a very fast way (comparable to the speed of sound).During the last 50 years,the resistance to unstable crack initiation and growth,i.e.,the fracture toughness,became a very efficient measure of brittleness or ductility of materials.In the case of cleavage,this quantity can be simply understood in a multiscale context. The macroscopic (continuum)linear-elastic fracture mechanics (LEFM)de- veloped by Griffith and Irwin brought to light an important relationship between the crack driving force G (the energy drop related to unit area of a new surface)and the stress intensity factor Kr as G=1- EK好 69

Chapter 2 Brittle and Ductile Fracture This chapter is devoted to damage and fracture micromechanisms operating in the case when monotonically increasing forces are applied to engineering materials and components. According to the amount of plastic deformation involved in these processes, the fracture events can be categorized as brittle, quasi-brittle or ductile. Brittle fracture is typical for ceramic materials, where plastic deformation is strongly limited across extended ranges of deformation rates and temper￾atures. In polycrystalline ceramics the reasons lie in a high Peierls–Nabarro stress of dislocations due to strong and directional covalent bonds (this holds also for some ionic compounds), and in less than five independent slip systems in ionic crystals (e.g., [149]). In amorphous ceramics it is simply because of a lack of any dislocations and, simultaneously, strong covalent and ionic in￾teratomic bonds. Metallic materials or polymers exhibit brittle fracture only under conditions of extremely high deformation rates, very low temperatures or extreme impurity concentrations at grain boundaries. In the case of a strong corrosion assistance, brittle fracture can also occur at very small load￾ing rates or even at a constant loading (stress corrosion cracking). A typical micromechanism of brittle fracture is so-called cleavage, where the atoms are gradually separated by tearing along the fracture plane in a very fast way (comparable to the speed of sound). During the last 50 years, the resistance to unstable crack initiation and growth, i.e., the fracture toughness, became a very efficient measure of brittleness or ductility of materials. In the case of cleavage, this quantity can be simply understood in a multiscale context. The macroscopic (continuum) linear–elastic fracture mechanics (LEFM) de￾veloped by Griffith and Irwin brought to light an important relationship between the crack driving force G (the energy drop related to unit area of a new surface) and the stress intensity factor KI as G = 1 − ν2 E K2 I . 69

70 2 Brittle and Ductile Fracture This relation holds for a straight front of an ideally flat crack under con- ditions of both the remote mode I loading and the plane strain.The energy necessary for creation of new fracture surfaces can be supplied from the elastic energy drop of the cracked solid and/or from the work done by external forces (or the drop in the associated potential energy).Thus,at the moment of un- stable fracture,the Griffith criterion gives Ge2y,where y is the surface (or fracture)energy that represents a resistance to cleavage.Consequently 71~2 2E Kie (2.1) However,the surface energy can be expressed also in terms of the cohesive (bonding)energy needed to break down an ideal crystal or an amorphous solid into individual atoms.The bonding energy of a surface atom is a half of that associated with an internal atom 150 and,because of two fracture surfaces,one can simply write U Y=4S' (2.2) where U is the cohesive energy assigned to one atom and S is the area per atom on the fracture surface.With regard to Equations 2.1 and 2.2 it reads EU 1/2 KIc≈ 2S (2.3) Values of U can be calculated either ab initio or by using semi-empirical interatomic potentials(see the previous chapter),and they can also be exper- imentally determined as twice the sublimation energy.For most metallic and ceramic crystals,values of U and S are in units of ev/atom and 10-19 m2. respectively.Thus,according to Equation 2.3,values of fracture toughness in the case of an ideal brittle fracture are as low as KIeE(0.5,1)MPam1/2.This range represents a lower-bound physical benchmark for the fracture tough- ness of engineering materials,and it corresponds well to experimental results achieved in tests with classical ceramic materials such as glasses or porcelain. Similar considerations can also be applied to classical ceramic materials that do not contain macroscopic pre-cracks.Indeed,some pores or microcracks are always present in such materials. In advanced ceramic materials for engineering applications,however,the level of fracture toughness is substantially enhanced.This can be achieved by microstructurally induced crack tortuosity combined with the presence of many small particles (or even microcracks)around the crack front.In this way the crack tip becomes shielded from the external stress supply and the stress intensity factor at the crack tip reduces.Both the theoretical background and the practical example of that technology are discussed in Section 2.1 in more details.Another method,commonly utilized for an additional improvement of fracture toughness of ceramics,is the distribution of supplied energy to

70 2 Brittle and Ductile Fracture This relation holds for a straight front of an ideally flat crack under con￾ditions of both the remote mode I loading and the plane strain. The energy necessary for creation of new fracture surfaces can be supplied from the elastic energy drop of the cracked solid and/or from the work done by external forces (or the drop in the associated potential energy). Thus, at the moment of un￾stable fracture, the Griffith criterion gives Gc ≈ 2γ, where γ is the surface (or fracture) energy that represents a resistance to cleavage. Consequently γ ≈ 1 − ν2 2E K2 Ic. (2.1) However, the surface energy can be expressed also in terms of the cohesive (bonding) energy needed to break down an ideal crystal or an amorphous solid into individual atoms. The bonding energy of a surface atom is a half of that associated with an internal atom [150] and, because of two fracture surfaces, one can simply write γ = U 4S , (2.2) where U is the cohesive energy assigned to one atom and S is the area per atom on the fracture surface. With regard to Equations 2.1 and 2.2 it reads KIc ≈ EU 2S 1/2 . (2.3) Values of U can be calculated either ab initio or by using semi-empirical interatomic potentials (see the previous chapter), and they can also be exper￾imentally determined as twice the sublimation energy. For most metallic and ceramic crystals, values of U and S are in units of eV/atom and 10−19 m2, respectively. Thus, according to Equation 2.3, values of fracture toughness in the case of an ideal brittle fracture are as low as KIc ∈ (0.5, 1) MPa m1/2. This range represents a lower-bound physical benchmark for the fracture tough￾ness of engineering materials, and it corresponds well to experimental results achieved in tests with classical ceramic materials such as glasses or porcelain. Similar considerations can also be applied to classical ceramic materials that do not contain macroscopic pre-cracks. Indeed, some pores or microcracks are always present in such materials. In advanced ceramic materials for engineering applications, however, the level of fracture toughness is substantially enhanced. This can be achieved by microstructurally induced crack tortuosity combined with the presence of many small particles (or even microcracks) around the crack front. In this way the crack tip becomes shielded from the external stress supply and the stress intensity factor at the crack tip reduces. Both the theoretical background and the practical example of that technology are discussed in Section 2.1 in more details. Another method, commonly utilized for an additional improvement of fracture toughness of ceramics, is the distribution of supplied energy to

2 Brittle and Ductile Fracture 71 damage mechanisms other than pure cleavage.This can be succeeded,for example,by an enforcement of phase transformations in the vicinity of the advancing crack front [149]. In cracked metallic solids,however,the measured values of Kic are at least an order of magnitude higher than the lower-bound benchmark.This holds even for ferrite (bcc Fe)at very low temperatures,where almost mi- croscopically smooth cleavage fractures along {001}planes appear(note that the(001)direction in Fe is associated with the lowest ideal tensile strength). The value of related fracture energy was experimentally found to be about 14Jm-2[149].This means that the energy supplied for the unstable fracture is also considered here for the development of localized plastic deformation around the crack tip.Hence,the general thermodynamic criterion for unsta- ble crack growth [19]can be written in the Griffith-Orowan form 1- EK2≥2+，(K,, (2.4) where wp(K,Y)is the plastic work needed for building the plastic zone at the crack tip.While this work can be neglected in the case of brittle fracture,it is of the same order of magnitude as 2 in the case of quasi-brittle fracture in metals.Note that the crack tip emission of dislocations in metals already occurs at very low K values in units of MPam1/2(see Section 3.2 for more details).The dislocations emitted from the crack tip generate an opposite stress intensity factor so that the crack tip becomes shielded from increasing external(remote)loading.The plastic work consumption proceeds until the moment when the sum of external and internal stress intensity factors at the crack tip (the local K-factor)exceeds the critical value necessary for separating atoms to produce new surfaces in an unstable(cleavage)manner (151,152.This is mathematically expressed in Equation 2.4 so that the plastic work wp(K,is written as a function of both y and K.Thus,the moment of cleavage fracture is somewhat delayed and,as reported by many authors [153-155],a short stage of stable crack growth often precedes the unstable propagation.The microstructurally induced heterogeneity in the resistance to both the unstable crack growth (y)and the dislocation emission can, sometimes,produce a series of elementary advances and arrests of the crack tip. Many quasi-brittle fractures in practice occur as a consequence of pre- existing corrosion dimples,large inclusions or fatigue cracks.However,the localized plastic deformation at favourable sites in the bulk also enables the creation of microcracks as nucleators of the quasi-brittle fracture in solids which do not contain any preliminary defects.At phase or grain boundaries it can be accomplished by many different and well known micromechanisms conditioned by the existence of high stress concentrations in front of dislo- cation pile-ups.Let us briefly mention another mechanism of crack initiation in bcc metals first introduced by Cottrell 156.When two edge dislocation pile-ups are driven by the applied stress o and meet on different {110}glide

2 Brittle and Ductile Fracture 71 damage mechanisms other than pure cleavage. This can be succeeded, for example, by an enforcement of phase transformations in the vicinity of the advancing crack front [149]. In cracked metallic solids, however, the measured values of KIc are at least an order of magnitude higher than the lower-bound benchmark. This holds even for ferrite (bcc Fe) at very low temperatures, where almost mi￾croscopically smooth cleavage fractures along {001} planes appear (note that the 001 direction in Fe is associated with the lowest ideal tensile strength). The value of related fracture energy was experimentally found to be about 14 Jm−2 [149]. This means that the energy supplied for the unstable fracture is also considered here for the development of localized plastic deformation around the crack tip. Hence, the general thermodynamic criterion for unsta￾ble crack growth [19] can be written in the Griffith–Orowan form 1 − ν2 E K2 ≥ 2γ + wp(K, γ), (2.4) where wp(K, γ) is the plastic work needed for building the plastic zone at the crack tip. While this work can be neglected in the case of brittle fracture, it is of the same order of magnitude as 2γ in the case of quasi-brittle fracture in metals. Note that the crack tip emission of dislocations in metals already occurs at very low K values in units of MPa m1/2 (see Section 3.2 for more details). The dislocations emitted from the crack tip generate an opposite stress intensity factor so that the crack tip becomes shielded from increasing external (remote) loading. The plastic work consumption proceeds until the moment when the sum of external and internal stress intensity factors at the crack tip (the local K-factor) exceeds the critical value necessary for separating atoms to produce new surfaces in an unstable (cleavage) manner [151,152]. This is mathematically expressed in Equation 2.4 so that the plastic work wp(K, γ) is written as a function of both γ and K. Thus, the moment of cleavage fracture is somewhat delayed and, as reported by many authors [153–155], a short stage of stable crack growth often precedes the unstable propagation. The microstructurally induced heterogeneity in the resistance to both the unstable crack growth (γ) and the dislocation emission can, sometimes, produce a series of elementary advances and arrests of the crack tip. Many quasi-brittle fractures in practice occur as a consequence of pre￾existing corrosion dimples, large inclusions or fatigue cracks. However, the localized plastic deformation at favourable sites in the bulk also enables the creation of microcracks as nucleators of the quasi-brittle fracture in solids which do not contain any preliminary defects. At phase or grain boundaries it can be accomplished by many different and well known micromechanisms conditioned by the existence of high stress concentrations in front of dislo￾cation pile-ups. Let us briefly mention another mechanism of crack initiation in bcc metals first introduced by Cottrell [156]. When two edge dislocation pile-ups are driven by the applied stress σ and meet on different {110} glide

72 2 Brittle and Ductile Fracture planes in the grain interior,their interaction results in the nucleation of a 001]sessile dislocation.This dislocation can be considered to be a wedge in the (001 cleavage plane.Interaction of n dislocations of Burgers vector b then creates a microcrack with flank opening nb.The work W=on262 done by the force onb acting at the front of n dislocations along the distance nb must be equal to the energy 2ynb for the creation of new crack surfaces.This gives the microscopic criterion for quasi-brittle fracture as Ocnb =2Ys; (2.5) where oc is the critical (fracture)stress.Assuming the relation connecting the number of dislocations with the grain size d in terms of the Hall-Petch relation,Equation 2.5 can be rearranged to (ova+ku ky BGT (2.6) where oo is the yield stress,ky constant in the Hall-Petch relation (tem- perature dependent),B the temperature independent constant and G the shear modulus (weakly temperature dependent).Thus,the right-hand side of Equation 2.6 is practically independent of temperature.If the left-hand side is equal to or higher than the right-hand side,the brittle (or quasi- brittle)fracture initiates just at the moment of reaching the yield stress.In an opposite case,the ductile failure occurs after some deformation hardening period.Both the high deformation rate and the low temperature enhance oo as well as ky,thereby giving rise to quasi-brittle fracture.The same is caused by a large grain size.Thus,the criterion at Equation 2.6 correctly predicts the experimentally observed fracture behaviour.Note that this sim- ple model for single-phase bcc metals is of a two-level type,since the Hall- Petch relation can be easily interpreted by combined atomistic-dislocation considerations 149. In Section 2.2 a statistical approach to geometrical shielding effects occur- ring in multi-phase engineering materials is outlined.This two-level concept can be used to give quantitative interpretation of some rather surprising re- sults obtained when measuring the fracture toughness and the absorbed im- pact energy (notch toughness)of some metallic materials.Examples of such interpretation are documented for ultra-high-strength low-alloyed(UHSLA) steels and Fe-V-P alloys. Unlike brittle or quasi-brittle fracture,the ductile fracture starts with a rather long period of stable crack or void growth due to the bulk plastic de- formation.In the case of pre-cracked solids this means that the surface energy 2y becomes negligible when compared to the plastic term wp(K,Y)in Equa- tion 2.4,and this criterion loses its sense.Therefore,instead of stress-based criteria(fracture stress,critical stress intensity factor)the deformation-based criteria are more appropriate for a quantitative description of ductile fracture. In the first stage of ductile fracture,microvoids (micropores)nucleate pref- erentially at the interface between the matrix and secondary phase particles

72 2 Brittle and Ductile Fracture planes in the grain interior, their interaction results in the nucleation of a [001] sessile dislocation. This dislocation can be considered to be a wedge in the {001} cleavage plane. Interaction of n dislocations of Burgers vector b then creates a microcrack with flank opening nb. The work W = σn2b2 done by the force σnb acting at the front of n dislocations along the distance nb must be equal to the energy 2γnb for the creation of new crack surfaces. This gives the microscopic criterion for quasi-brittle fracture as σcnb = 2γs, (2.5) where σc is the critical (fracture) stress. Assuming the relation connecting the number of dislocations with the grain size d in terms of the Hall–Petch relation, Equation 2.5 can be rearranged to  σ0 √ d + ky  ky = βGγs, (2.6) where σ0 is the yield stress, ky constant in the Hall–Petch relation (tem￾perature dependent), β the temperature independent constant and G the shear modulus (weakly temperature dependent). Thus, the right-hand side of Equation 2.6 is practically independent of temperature. If the left-hand side is equal to or higher than the right-hand side, the brittle (or quasi￾brittle) fracture initiates just at the moment of reaching the yield stress. In an opposite case, the ductile failure occurs after some deformation hardening period. Both the high deformation rate and the low temperature enhance σ0 as well as ky, thereby giving rise to quasi-brittle fracture. The same is caused by a large grain size. Thus, the criterion at Equation 2.6 correctly predicts the experimentally observed fracture behaviour. Note that this sim￾ple model for single-phase bcc metals is of a two-level type, since the Hall– Petch relation can be easily interpreted by combined atomistic-dislocation considerations [149]. In Section 2.2 a statistical approach to geometrical shielding effects occur￾ring in multi-phase engineering materials is outlined. This two-level concept can be used to give quantitative interpretation of some rather surprising re￾sults obtained when measuring the fracture toughness and the absorbed im￾pact energy (notch toughness) of some metallic materials. Examples of such interpretation are documented for ultra-high-strength low-alloyed (UHSLA) steels and Fe-V-P alloys. Unlike brittle or quasi-brittle fracture, the ductile fracture starts with a rather long period of stable crack or void growth due to the bulk plastic de￾formation. In the case of pre-cracked solids this means that the surface energy 2γ becomes negligible when compared to the plastic term wp(K, γ) in Equa￾tion 2.4, and this criterion loses its sense. Therefore, instead of stress-based criteria (fracture stress, critical stress intensity factor) the deformation-based criteria are more appropriate for a quantitative description of ductile fracture. In the first stage of ductile fracture, microvoids (micropores) nucleate pref￾erentially at the interface between the matrix and secondary phase particles

2.1 Brittle Fracture 73 The physical reasons are clear:high interfacial energy (low fracture energy), the incompatibility strains(dislocation pile-ups)and the mosaic stresses in- duced by a difference in thermal dilatations of the matrix and inclusions. Nucleated voids experience their stable growth controlled by the plastic de- formation.In the tensile test,for example,the voids become cylindrically pro- longed by uniaxial deformation up to the moment when the ultimate strength is reached.Beyond that limit they also expand in transverse directions under the triaxial state of stress inside the volume of developing macroscopic neck. Although the bulk ductile fracture occurs only very exceptionally in engi- neering practice,the research of that process is important for forging tech- nologies.Besides the two-scale analysis of plastic deformation,some models of void coalescence during the tensile test are outlined in the last section of this chapter.It should be emphasized that the damage process inside the crack-tip plastic zone of many metallic materials can also be described in terms of the ductile fracture mechanism (e.g.,[157).Therefore,an analyti- cal model that enables a prediction of fracture toughness values by means of more easily measurable ductile characteristics is also presented. 2.1 Brittle Fracture From the historical point of view,brittle fracture proved to be one of the most frequent and dangerous failures occurring in engineering practice.Besides the well known brittleness of utility ceramics and glasses,metallic materials may also exhibit intrinsically brittle properties dependent on temperature;there exists a critical temperature,the so-called ductile-brittle transition tempera- ture(DBTT)under which the material is brittle,while it is ductile above that temperature.This holds particularly for bcc metals,in which cores of screw dislocation are split into sessile configurations [4,158.They remain immobile at low temperatures so that,under such conditions,cleavage is a dominant fracture mechanism.However,a steep exponential increase of ductility ap- pears when approaching the DBTT owing to thermal activation helping to increase the mobility of screw segments.Improper application of a material below this temperature can have catastrophic consequences,such as,for ex- ample,the sinking of the RMS Titanic nearly one hundred years ago.The material of Titanic,although representing the best-grade steel at that time, was characterized by coarsed grain and high level of inclusions so that DBTT was higher than 32C.No wonder this ship was catastrophically destroyed by brittle fracture during its impact with the iceberg at the water temperature of-2°C[159]. However,brittleness is often induced by other effects such as flawed ma- terial processing or segregation of deleterious impurities at grain boundaries. Grain boundary segregation can result in a local enrichment of thin but con- tinuous interfacial layers throughout the polycrystalline material with con-

2.1 Brittle Fracture 73 The physical reasons are clear: high interfacial energy (low fracture energy), the incompatibility strains (dislocation pile-ups) and the mosaic stresses in￾duced by a difference in thermal dilatations of the matrix and inclusions. Nucleated voids experience their stable growth controlled by the plastic de￾formation. In the tensile test, for example, the voids become cylindrically pro￾longed by uniaxial deformation up to the moment when the ultimate strength is reached. Beyond that limit they also expand in transverse directions under the triaxial state of stress inside the volume of developing macroscopic neck. Although the bulk ductile fracture occurs only very exceptionally in engi￾neering practice, the research of that process is important for forging tech￾nologies. Besides the two-scale analysis of plastic deformation, some models of void coalescence during the tensile test are outlined in the last section of this chapter. It should be emphasized that the damage process inside the crack-tip plastic zone of many metallic materials can also be described in terms of the ductile fracture mechanism (e.g., [157]). Therefore, an analyti￾cal model that enables a prediction of fracture toughness values by means of more easily measurable ductile characteristics is also presented. 2.1 Brittle Fracture From the historical point of view, brittle fracture proved to be one of the most frequent and dangerous failures occurring in engineering practice. Besides the well known brittleness of utility ceramics and glasses, metallic materials may also exhibit intrinsically brittle properties dependent on temperature; there exists a critical temperature, the so-called ductile-brittle transition tempera￾ture (DBTT) under which the material is brittle, while it is ductile above that temperature. This holds particularly for bcc metals, in which cores of screw dislocation are split into sessile configurations [4,158]. They remain immobile at low temperatures so that,under such conditions, cleavage is a dominant fracture mechanism. However, a steep exponential increase of ductility ap￾pears when approaching the DBTT owing to thermal activation helping to increase the mobility of screw segments. Improper application of a material below this temperature can have catastrophic consequences, such as, for ex￾ample, the sinking of the RMS Titanic nearly one hundred years ago. The material of Titanic, although representing the best-grade steel at that time, was characterized by coarsed grain and high level of inclusions so that DBTT was higher than 32◦C. No wonder this ship was catastrophically destroyed by brittle fracture during its impact with the iceberg at the water temperature of −2◦C [159]. However, brittleness is often induced by other effects such as flawed ma￾terial processing or segregation of deleterious impurities at grain boundaries. Grain boundary segregation can result in a local enrichment of thin but con￾tinuous interfacial layers throughout the polycrystalline material with con-

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 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 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 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-

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 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 re￾mote 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 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: 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 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 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 in￾trinsic (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 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 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 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−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 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 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