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 temperatures. 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 interatomic 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 loading 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) developed 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 conditions 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 unstable 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 experimentally 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 toughness 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 microscopically 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 unstable 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 preexisting 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 dislocation 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 (temperature 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 quasibrittle) 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 simple 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 occurring in multi-phase engineering materials is outlined. This two-level concept can be used to give quantitative interpretation of some rather surprising results obtained when measuring the fracture toughness and the absorbed impact 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 deformation. 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 Equation 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 preferentially 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 induced by a difference in thermal dilatations of the matrix and inclusions. Nucleated voids experience their stable growth controlled by the plastic deformation. In the tensile test, for example, the voids become cylindrically prolonged 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 engineering practice, the research of that process is important for forging technologies. 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 analytical 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 temperature (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 appears 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 example, 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 material processing or segregation of deleterious impurities at grain boundaries. Grain boundary segregation can result in a local enrichment of thin but continuous interfacial layers throughout the polycrystalline material with con-