Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 1, pp. 75-89, Warsaw 2013 THE KEY PROBLEMS OF LOCAL APPROACH TO CLEAVAGE FRACTURE Sergiy Kotrechko Institute for Metal Physics, NAS of the Ukraine, Kyiv, Ukraine e-mail: kotr@imp.kiev.ua Based on the suggested multi-scale model of Local Approach (LA) to fracture, four main problems of LA are considered, namely: (i) the effect of micro-stress fluctuations on the crack nuclei instability; (ii) intensity of micro-crack nucleation and its influence on fracture probability; (iii) theoretical and experimental assessment of the value of threshold stress; (iv) stochastic analysis of “multi-barrier” effect at micro-crack growth in polycrystalline metal. Key words: local approach to fracture, multi-scale model, local fracture stress 1. Introduction TheLocalApproach to fracture (LA) has been significantly developed for the last three decades. This approach has enabled not only to clarify the nature andmicromechanism ofmetal fracture, but also to describe the effect of loading conditions on the fracture limit of structures,which can- not be easily realized with the conventional global approach (Pineau, 2006; Pineau andBenoot, 2010; Bordet et al., 2005; Beremin 1983, Margolin et al., 1998). However, recent findings have demonstrated limitation of the conventional version of LA both in the theoretical and applied sense. This is due, first of all, to unnecessarily oversimplified description of the fracture process in Beremin’s version of LA and its further modifications. Simultaneously with this version, the multi-scale approach to brittle fracture was offered in Kotrechko (1995, 2002, 2003), Kotrechko andMeshkov (2001), Kotrechko et al. (2007). Specific feature of this approach lies in possibility to describe regularities of a pre-cracked solid on macroscopic scale based on realistic physical models of the crack nuclei (CN) creation and instability in polycrystalline metals and alloys. This version of LA ismore sophisticated for practical use; however, itmay be applied as the the- oretical basis for further development of conventional LA. This paper is aimed at consideration of key problems of LA, namely: • accounting for the effect of stochastic micro-stress field on the CN instability in polycry- stalline aggregate; • prediction of the effect of plastic strain and temperature on the CN generation; • theoretical and experimental determination of the value of threshold stress; • properties and experimental determination of the local fracture stress; • analysis of “multiple-barrier” effect at the CN propagation in polycrystal. 2. Crack nuclei instability The Griffith criterion is conventionally used for description of the beginning of macro-crack unstable growth. However, most works do not account that the CNbecomes unstable under the influence ofmicro-stresses. Onlymacroscopic stresses are usually considered.Pineau andBennot 76 S. Kotrechko (2010) noted that the effect of micro-stresses should be accounted because it is the reason for both scatter of fracture stress and the value of statistical scale effect. The statistical model of cleavage fracture of a polycrystalline metal, in which the CN instability in stochastic micro- stress field is considered, was suggested in Kotrechko (1995). Microstresses in polycrystalline aggregate are characterized by a wide spectrum of amplitudes and wave lengths. Therefore, at modelling, two components were accounted separately, namely: (1) the microstresses ξij produced by grain-to-grain elastic misfit and (2) the microstresses ξ p ij due to dislocations. In the first approximation, the microstresses ξij may be considered as homogeneous within the grain and changing from a grain to grain. The statistical distribution of these stress values may be approximated with sufficient accuracy by the normal law. The values of variances of these microstresses Dξ11, Dξ22, Dξ33 are functions of principalmacro-stresses σ1, σ2, σ3, and themean values ξ11, ξ22, ξ33 are equal to σ1, σ2, σ3, respectively Dξ11 = DIσ 2 1 +DII(σ 2 2 +σ 2 3)+2[µI(σ1σ2+σ1σ3)+µIIσ2σ3] Dξ22 = DIσ 2 2 +DII(σ 2 1 +σ 2 3)+2[µI(σ2σ1+σ2σ3)+µIIσ1σ3] Dξ33 = DIσ 2 3 +DII(σ 2 2 +σ 2 1)+2[µI(σ3σ1+σ3σ2)+µIIσ1σ2] (2.1) where DI = 1.7 · 10−2, DII = µII = 0.66 · 10−2, µI = 0.72 · 10−2 for polycrystalline iron and Fe-based alloys. From (2.1) it follows that even at uniaxial macroscopic tension (σ1 > 0, σ2 = σ3 = 0), the microscopic stress state is triaxial (ξ11 > 0, ξ22 6= 0, ξ33 6= 0). In this case, for iron, the value of microscopic stress ξ11 changes from 0.6σ1 to 1.4σ1, and the values ξ22 and ξ33 change from −1.24σ1 to +1.24σ1. This specific feature is one of the reasons for scatter of cleavage fracture onmacroscopic scale, but it does not account in most conventional models. Micro-stress fields induced by dislocations are significantly inhomogeneous, so, the effect of such fields may be described by the effective stress ξ (Indenbom, 1961) ξ = 2 πa a ∫ 0 ξ p 11(x) √ x a−x dx (2.2) where a is the CN length, ξ p 11(x) is the distribution function for tensile micro-stress along the path of microcrack growth. The essence of this dependence is that micro-stresses support a crack growth if they change according to the law 1/ √ x with distance x change. In Kotrechko (1995), the expression for ξ inducedby the layer of randomlydistributeddislocations near the grain boundarywhere theCN formswas obtained. The values of these stresses increase with plastic strain e growth. However, the critical value ec exists, at exceeding of which ξ decreases. As it is exhibited in Kotrechko (2002), Kotrechko et al. (1995, 2007), for typical structural steels ec ≈ 0.02. At e < ec, the dependence of ξ on strain may be approximated as ξ = kξ √ e d (2.3) where kξ is the coefficient (for α-Fe and carbon steels it is equal to ≈ 16.8MPa √ m); e is the equivalent macro-plastic strain; d is the grain size. At strain ec < e ¬ 0.2, the expression for ξ is the following1 ξ = kξ √ e d −ke ( e ec −1 ) (2.4) 1For typical ferritic steels ec ≈ 0.02. The key problems of local approach to cleavage fracture 77 where ke is the coefficient (for α-Fe and carbon steels it is approximately equal to 40MPa √ m). As it is shown in Kotrechko et al. (1995, 2007), Kotrechko (2003), the effect of dislocation micro-stresses on the CN instability results in a non-monotonic dependence of critical fracture stress σf on the value of plastic strain (Fig. 1). Fig. 1. Effect of plastic strain on the value of cleavage fracture stress σf under uniaxial tension: ec is the critical value of plastic strain corresponding to the minimum level of brittle strength of metal RMC at uniaxial tension With account of these regularities, the expression for critical stress of the CN instability on microscopic scale ξc is the following ξc = (kIc√ a −ξ ) ϕ(θ,η) (2.5) where kIc is the critical value of the stress intensity coefficient for theCN; ϕ(θ,η) is the function describing the effect of CN orientation and the micro-stress state mode on the value of ξc. In 2D-approximation ϕ(θ,η) = 1 √ cos2α+ηsin2α (2.6) where α is the angle between the normal to crack plane and ξ11 direction; η is the parameter of the micro-stress state mode (η = ξ22/ξ11). Equation (2.5) with account of (2.1) enables one to predict the value ofmacroscopic fracture stress σf based on the criterion of CN instability onmicroscopic scale ξ11 ­ ξc (2.7) As it is exhibited inKotrechko (1995, 2002),Kotrechko et al. (2001), theprobability of instability of one CN is the following P0(σf)= 1 2 ξmax c ∫ ξmin c g(ξc) [ 1− erf ( ξc−σf√ 2Dξ11 )] dξc (2.8) where g(ξc) is the density distribution function for critical micro-stresses ξc g(ξc)= 2 kIc ηmax ∫ ηmin g(η) [ θmax ∫ θmin g(θ)g(a) √ a3 ϕ(θ,η) dθ ] dη (2.9) where a = kIc [ξc/ϕ(θ,η)]+ ξ (2.10) 78 S. Kotrechko g(θ), g(η) and g(A) are thedensity distribution functions for the orientation angles θ, parameter of micro-stress state mode η and CN sizes a2, respectively. The function g(η) is determined based on the condition that ξ11 and ξ22 are distributed by a normal law with variances (2.1)1,2. According to the “weakest link” concept, cleavage fracture of a metal volume VI occurs if not less than one crack of all numbers of cracks Na (Na = ρV ) becomes unstable P(σf)= 1− [1−P0(σf)]ρV (2.11) where P(σf) is the probability of fracture of macroscopic volume V at uniform distribution of macro-stresses σf. For inhomogeneous distribution of stresses and strains ahead the crack tip or notch, the expression for probability of global fracture is P(σf)= 1− i=M ∏ i=1 [1−Pi(σf)] (2.12) where Pi(σf) is the probability of metal fracture in i-th finite element (FE) volume; M is the number of FE in the “process zone”. In conventional versions of LA, the Weibull distribution is employed instead of expression (2.11) for fracture probability Pi(σf). In Kotrechko et al. (2001) within the framework of the approach proposed, it was shown that the distribution of probability of instability of one CN, P0(σf) may be approximated by an exponential law. In this case, expression (2.11) may be presented as follows P(σf)≈ 1− exp [ −ρV (σ−σth σu )m] (2.13) where σth is the value of threshold stress; σu is the scale stress; m is the shape parameter of theWeibull distribution. 3. Crack nucleation Prediction of a number of micro-cracks nucleating during plastic deformation is one of themost difficult and less investigated problems of LA. In classic models, determination of length of formingmicro-cracks and critical stress of their instability is accentuated. However, as it follows from(2.11) and (2.13), thenumberofCNsignificantly influences the value of fractureprobability and critical local stress σf. Besides, in most models, the main peculiarity of CN behaviour in metals is not accounted. It is the fact that only freshly nucleated micro-cracks may result in global fracture of themetal. If at themoment of crack nucleation,Griffith’s condition for it is not hold, then this crack blunts and now can not “compete” with “fresh” sharp crack nuclei, which is permanently generated during the plastic deformation. This specific feature CN behaviour in ametal was taken into account in the statistical model of fracture proposed inKotrechko (1995) and multi-scale version of LA developed based on this model (Kotrechko and Meshkov, 2001; Kotrechko, 2002, 2003). InBordet et al. (2005) it was noted that supposition of conventional LA about keeping of the CN activity over the entire loading history is invalid formetals. Therefore, the composition ρV in (2.11) and (2.13) is not the totality of CN accumulated in the metal during its loading before a certain value of plastic strain e is reached, but it is the number of CN, which arises at that strain. It means that ρ is the rate of CN generation with respect to strain. 2If fracture is initiated by carbide cracking, g(a) is the density distribution function for these particles. The key problems of local approach to cleavage fracture 79 Inhomogeneity of micro-plastic deformation, which gives rise to plastic deformation incom- patibility on grain and interphase boundaries, is a general reason for the CN formation in poly- crystalline solids. So, in Kotrechko (1995), a statistical model was offered where the formation of CN was considered as a stochastic process of reaching the critical micro-plastic strain value. Thismodel describes theCN formation onmicro-scale in terms of average strains over the grain. However, this may be described more thoroughly if one accounts that the local incompatibility of plastic deformations on the grain and interphase boundariesmay be describedwith sufficient accuracy by dislocation pile-ups. The crack nucleus arises if two conditions are held, namely: (i) relaxation in the pile-up tip is absent (pile-up blocking); (ii) formation of the pile-up of cri- tical capacity at which the value of local stresses ahead of its tip is sufficient for formation of atomically sharp flaw near the grain boundary or interphase “ferrite-carbide” boundary3. As it is exhibited in Kotrechko et al. (2011), formation of such a pile-up may be described as follows CL [ σ(kσT −M)+β √ e d ]2 ­ τc (3.1) where C is a constant depending on elastic constants of the lattice (for α-Fe C =0.0336N/m); β is constant (β ≈ 2.57MPa √ m; d is the average grain size; L is the pile-up length; σ and e are equivalent macroscopic stresses and plastic strains, respectively; kσ is a coefficient (kσ = √ Dξns/σ), where Dξns is the variance of shear microscopic stresses ξns in the slip sys- tems, (for α-Fe and slip systems {110}〈111〉kσ = 0.225); t is the dimensionless value of shear stresses ξns “applied” to the pile-up (t = ξns/ √ Dξns); M is the factor averaged over the gra- in orientation (for b.c.c. crystals M = 0.36); τc is the critical shear stress for crack nucleus formation. In dependence (3.1), the expression β √ e/d specifies the value of shearmicro-stresses caused by the interaction of grain of averaged orientation M with plastically deformed to strain e surroundingmatrix. The value of fluctuation of stresses in the slip system where a pile-up has formed ξns is specified by the expression σ(kσt−M). It should be noted that the expression β √ e/d characterizes shearmicro-stresses induced by dislocations in contrast to ξ (Eq. (2.3)), which specifies the normal component of the tensor of these stresses. Besides, while determining the value of coefficient β, a different law for averaging stresses over the pile-up length is used. Accurate to coefficients, the decrease in these stresses at a strain e greater than the critical one ec is described by the same dependence (2.4). The condition of the pile-up blocking is formulated as follows √ L r [ σ(kσt−M)+β √ e d ] ¬ mτ Y (3.2) where r is the distance from the grain boundary to the dislocation source in the neighbouring grain (r ≪ L), where the starting of a such source is possible at reaching the critical value of shear stresses The parameter m characterizes the influence of the slip system orientation of the dislocation source on the value of shear stress acting in this system. If the fluctuations of values τ Y and r are neglected, then the expression for probability of CN formation is the following Pnucl =2 tmax ∫ tc g(t) [ mmax ∫ m g(m) dm ] dt (3.3) 3At certain shape of the carbide particle, carbide cracking is more preferable than CN formation on the interphase boundary. 80 S. Kotrechko where m = τ Y √ r L σ(kσt−M)+β √ e d (3.4) The distribution density function g(m) is determined based on the distribution of a scalar angle of misorientation of grain boundaries (Lindley et al., 1970)4. According to (3.1), an expression for the critical value tc is described by the dependence tc = 1 kσ [ M + 1 σ ( √ τc CL −β √ e d )] (3.5) The density distribution function g(t) is determined as g(t) = 1√ 2π exp (t2 2 ) (3.6) In some cases, during calculations, it is reasonably to use an approximate expression for Pnucl. It may be obtained if m fluctuation is neglected. In this case Pnucl ≈ P(tc < t < tr)=2 tr ∫ tc g(t) dt (3.7) Accounting for (3.6), Pnucl is the following Pnucl ≈ 2[Φ(tr)−Φ(tc)] (3.8) where Φ(tr) and Φ(tc) are values of the Laplace function at the corresponding value of the parameter t. The expression for the parameter tr that characterises relaxation conditions is the following tr = 1 kσ [ M + 1 σ ( τ Y m √ r L −β √ e d )] (3.9) In a general case, the rate of CN generation in the metal volume unit may be specified as ρ = kρPnucl (3.10) where kρ is the coefficient dependingon the density of carbide particles and grain size. Thevalue of this coefficient may be estimated using an experimental evidence by a calibration procedure. The approach proposed enables one to model the effect of many factors on the rate of CN generation, such as metallurgical factors (average grain size d and maximum grain size L ≈ (0.5-1.0)dmax), loading condition (temperature and loading rate (parameters τY and σ), crystallographic texture (function g(m)), the value of plastic strain e. Figure 2 presents the dependence of ρ on the value of plastic strain at different test temperatures for the reactor pressure vessel steel 2Cr-Ni-Mo-V. The specific feature of these dependences is a non-monotonic change of ρ with e growth. This agrees well with data of the work by Lindley et al. (1970) demonstrating that the number of cleaved carbides grows up to a certain level, after which the nucleation rate decreasesmonotonically. It should be noted that comparison of the experimental evidencewith the obtained results requires to account that ρ is not a cumulativeCNdensity, but is the rate of its nucleation with respect to strain. Therefore, description of the CN formation on submicroscales enables one to solve adequately one of the key problems of LA related to the prediction of crack formation under the action of plastic flow. 4Expression for g(m) is somewhat intricate, so it is not presented in an explicit form. The key problems of local approach to cleavage fracture 81 Fig. 2. Dependence of the CN density in RPV steel ρ on the value of plastic strain and temperature: e and ec are the equivalent plastic strain and its critical value, respectively 4. Threshold fracture stress As it is known, the assessment of the threshold stress value is one of the important problems of LA. In most cases, to simplify the calibration procedure, two-parameter Weibull’s distribution is employed (Pineau, 2006; Pineau and Benoot, 2010; Beremin, 1983), i.e. it is supposed that σth =0.However, for steels, thevalueof σth is ratherhigh, andmayamount to σth ≈ (0.4-0.6)σf , where σf is the local fracture stress. So, neglecting the σth value, gives rise to essential errors at the estimation of theWeibull modulus m by experimental evidence. This results in errors of the prediction of scatter and temperaturedependenceof the fracture toughness.Attempts are known to estimate the value σth by the yield stress at low temperatures. However, it contradicts the physical essence of the threshold fracture stress. According to themodel proposed, the value σth may be specified as σth = ξminc 1+3Iξ11 (4.1) where Iξ11 is the coefficient of variation of principal tensile micro-stresses ξ11 (for ferritic steels under uniaxial tension Iξ11 ≈ 0.13); coefficient “3” means that σth is estimated with probabili- ty 0.997; ξminc is the minimum value of the critical stress of the CN instability. According to (2.5) ξminc = ( kIc√ amax − ξmax ) ϕmin(θ,η) (4.2) where amax, ξmax and ϕmin(θ,η) are the maximum and minimum values of corresponding parameters in dependence (2.5)5. Figure3 illustrates the ideaof experimentaldeterminationof thevalueof threshold stress σth. According to these data, in the case of uniformdistribution of stresses, the average value of frac- ture stress tends to σth very fastly with an increase in the specimen volume. It enables one to estimate values of σth by the minimum value of brittle fracture stress RMC for standard (V = 1000mm3) tensile specimens over the ductile-to-brittle transition temperature range (Fig. 4) (Kotrechko andMeshkov, 2001) σth = λRMC (4.3) where λ is the coefficient whose value depends on the rate of CN generation under the action of plastic deformation. For typical structural steels λ ≈ 0.75, . . . ,0.95. 5In the first approximation ϕmin(θ,η)≈ 1. 82 S. Kotrechko Fig. 3. Dependence of the fracture probability on stress at different volumes V of specimens for RPV steel: Na is the number of forming CN; σth is the threshold stress; RMC is the minimum level of brittle strength of the standard (V =1000mm3) tensile specimen Fig. 4. Temperature dependence of mechanical properties of RPV steel at uniaxial tension: σ0.2 is the proof stress; Rf is the true fracture stress; RMC is the brittle strength; ψ is the reduction in area Theductile-to-brittle transition temperature range of high-ductile structural steels for tensile specimens is located below the temperature of boiling of liquid nitrogen (T =−196◦C). In this case, RMC value may be determined by the results of tests of cylindrical specimens with notch radius 2mm at T = −196◦C. The use of this technique for σth determination enables one to employ three-parameterWeibull distribution inLAand to improve predictive capabilities of LA. 5. Local fracture stress Initially, LAwas aimed at the prediction of a temperature dependence of fracture toughness of steel. However, further development of this approach on the ground of multi-scale models has enabled not only improvement of predictive capabilities of the approach, but also clarificatiob of specific features of the mechanism of cleavage fracture initiation in a highly inhomogeneous stress-strain field ahead the crack tip. Specifically, it appears in the possibility of differentia- ting the effects both of metallurgical factors and loading conditions (temperature, constraint The key problems of local approach to cleavage fracture 83 lost effect etc.) on the fracture limit of structures. The CN instability is the reason for brittle fracture, so the effect of these defects on the local fracture stress is determined through both properties of separate CN (CN length and orientation) and the rate of CN generation during plastic deformation. In terms of the “weakest link” concept, properties of one separate CN pre- determine the type and parameters of the function P0(σf) in (2.11), and, respectively, the values of Weibull distribution parameters σth, σu and m in dependence (2.13). The effect of the rate of CN generation with respect to strain is characterised by the term ρ in expressions (2.11) and (2.13). Dependences of theWeibull distribution parameters on themost probable value of the grain size at different magnitudes of grain structure inhomogeneity were obtained in Kotrechko et al. (2001). It was exhibited that the values of σth and σu rise linearly with growth of 1/ √ dmpv (dmpv is themost probable grain size). The value of shape parameter m is virtually independent of dmpv, however, it decreases with an increase in the variance of the grain size logarithm Dlnd m = a1− b1 √ Dlnd (5.1) where for iron: a1 =3.35±0.27, b1 =1.84±0.11. At the same time, the absolute value of parameter m decreases from 3.0 to 2.1 for typical range of √ Dlnd ( √ Dlnd =0.2-0.7). The normalised value of scaling stress σu/σth does not depend on the absolute grain size either, and it is a linear function of the variance of grain size logarithms Dlnd σu σth = a2+ b2Dlnd (5.2) where for iron: a2 =3.81±0.07, b2 =11.97±0.02 at e =0.02. Their sense is that the parameters of distribution of grain sizes or carbide particles pre- determine the distribution of CN lengths. The values of coefficients a2 and b2 depend on the magnitudes of equivalent strain e. This is due to the effect of dislocation stresses on the value of critical stress of the CN instability ξc (Eqs. (2.3)-(2.5)). These dependences are obtained for polycrystalline iron; however, they are correct also for the case of fracture initiation by carbide cracking. As it was mentioned above, the number N of CN forming at the given value of plastic strain (term ρV in (2.11) and (2.13)) is the second important factor affecting the local fracture stress σf. As it is shown in Fig. 3, an increase in N gives rise to a decrease in both the average fracture stress and its scatter. This is just the physical nature of the statistical scale effect at cleavage fracture of metals and alloys. It should be remarked that in the conventional version of LA, theCNdensity is characterised by the expression 1/V0 (where V0 is the reference volume). In the calibration procedure, this value is supposed to be constant. It is one of the reasons for σu and m dependence on test temperature and notch parameters. Figure 5 presents the dependence σf(Na) for uniform stress distribution in an explicit form. According to this evidence, an essential excess of σf over the brittle strength RMC of standard (V = 1000mm 3, e = 0.02) tensile specimens is observed at N ­ 50000. At these N values, scatter limits of fracture stress increase significantly. The magnitude of local fracture stress of a metal also depends on the plastic strain value (Fig. 6). This is due to: (i) the effect of dislocation micro-stresses ξ on the instability of separate CN, and (ii) dependence of their rate creation on the value of plastic strain (Fig. 2). It should be noticed that in the existing versions of LA, the attempts were made to account for this effect by the introduction of phenomenological dependences of the Weibull stress σW on the strain value (Bordet et al., 2005; Beremin, 1983). Dependences in Fig. 5 are obtained for the case of uniform stress and strain distribution (solid and dash lines). In the vicinity of a macro-crack or notch, these distributions are essentially non-uniform. This gives rise to difficulties in the 84 S. Kotrechko Fig. 5. Dependence of the normalized value of fracture stress σf on the CN number N for fracture probabilities 5%, 50% and 95% at T =−196◦C under uniform uniaxial tension; (N, � are values of the local fracture stress σf ahead the macro-crack or notch of R =0.25mm at the corresponding values of fracture probability (the arrows indicate the effect of local plastic strain gradient) Fig. 6. The effect of plastic strain on the value local fracture stress σf for different volumes of the plastically deformedmetal determination of the local fracture stress value. In the conventional Beremin version of LA, the Weibull stress σW is used as ameasure of local stress. This stress is an integral characteristic of brittle strength for the region subjected to local yielding. Another approach to this problemwas offered in Lin et al. (1986) and developed in Kotrechko andMeshkov (2001), Kotrechko (2002). In this case, the local fracture stress σf is determined as the value of tensile stress σ11 in the locus where the probability of fracture initiation reaches it maximum value (Fig. 7). Such an approach enables one to compare directly the calculated magnitude σf with the experimental evidence determined by the value of tensile stresses at the cleavage initiation site ahead of the macro-crack tip. Besides, it permits one to ascertain the regionwhere fracture initiates (“process zone”) (Fig. 7). As it is exhibited inKotrechko andMeshkov (2001, Kotrechko (2003), this area is much less than the whole region of local plastic yielding. According to the computer simulation findings on fracture of the reactor pressure vessel steel, in addition to the CN number, the value of stress-strain field inhomogeneity ahead the macro-crack influences the value of local fracture stress σf (Fig. 5). This effect depends on CN density. The smaller the density of forming CN, the stronger the effect of strain gradient on the value of σf. The higher the stress value σf, the higher the fracture toughness KIc, so The key problems of local approach to cleavage fracture 85 Fig. 7. Distribution of local tensile stresses σ11, equivalent local plastic strain e, and local probability Pni of fracture initiation ahead the crack tip in the pre-crackedCharpy surveillance specimen at temperature−120◦, KJc =59.4MPa √ m and the probability of global fracture PΣ =0.63; XPZ is the size of the “process zone” in the minimum cross-section of the specimen manufacturing of steels with a low rate of CNgenerationwill significantly increase their fracture toughness and decrease their sensibility to crack-like defects. The latter is especially important for high-strength steels. 6. Multiple-barrier effect Ascertainment of the critical event of micro-crack growth, which governs the global fracture, is one of urgent problems of LA (Pineau and Benoot, 2010). The simplest model that enables one to estimate the critical size of carbidemicro-crack, which instability gives rise to global fracture, was offered by Martin-Meizoso et al. (1994) a d ¬ ( k c/f Ia k f/f Ia )2 (6.1) where d is the ferrite grain size; k c/f Ia and k f/f Ia are the critical values of stress intensity coef- ficient corresponding to overcoming the interphase boundary “carbide-ferrite” and the ferrite grain boundary, respectively. The value of k c/f Ia /k f/f Ia must be less than ∼ 1/5. It means that unstable propagation of the carbide crack will give rise to global fracture if its size is at least 25 times less than the grain size. For typical steels, this condition is usually held. However, a great number of arrested cleavage micro-cracks is observed in steels, see for example Lambert- Perlade et al. (2004). This is due to threemain factors: (i) random size of carbide particles and grains; (ii) statistic distribution of ferrite grain boundary misorientations; (iii) fluctuation of tensile micro-stresses which changes from grain to grain. In Kotrechko (1995) an approach was formulated that enables modelling of micro-crack propagation in the polycrystalline aggregate accounting for these factors. Within the framework of such an approach, an expression for the 86 S. Kotrechko critical value of micro-stresses ξLc required to support the unstable propagation of a crack of length L, is the following ξLc = 2 πL ( dc ∫ 0 ξdc √ x L−x dx+ 2d ∫ d ξ2dc √ x L−x dx ) (6.2) where ξdc is the critical level of tensile stresses in the first grain, which guarantees its cleavage (Fig. 9); ξ2dc is the critical cleavage stress for the second grain. The value of critical stress required for cleavage of the first grain and overcoming the grain boundarymay be specified as ξdc = αξc (6.3) where α = √ a d k f/f Ia k c/f Ia (6.4) The probability of cleavage of grain #1 (Fig. 8) resulting in formation of a disk-like crack of diameter L = d at the given level of macro-stresses σf is described by the dependence P2(σf)= ξd cmax ∫ ξd cmin g(ξdc)P1(σf|ξdc) dξDc (6.5) where g(ξdc) is the distribution density function determined by (2.9) accounting for conditions (6.3); P1(σf|ξdc) is the conditional probability P1(σ1|ξdc)= 1 √ 2πDξ11 ξmax 11 ∫ ξd c exp (ξ11−σf √ 2Dξ11 ) dξ11 (6.6) Fig. 8. Scheme of micro-crack growth in a polycrystalline metal: a is the initial CN size; d is the grain size; 1, 2, 3 are numbers of disk-like cracks at different steps of their extension As it is shown in Fig. 8a, the further stage of formed disk-like crack #1 is the transition to not less than one of m neighbouring grains (at this step of the crack growth m =6). This gives rise to formation of disk-like crack #2. For this event, the value of probability is following P4(σf)= ξd cmax ∫ ξd cmin ξ2d cmax ∫ ξ2d cmin g(ξdc ,ξ 2d c )P3(σf|ξdc ,ξ2dc ) dξdc dξ2dc (6.7) The key problems of local approach to cleavage fracture 87 Disk-like crack #3 and the further ones are formed similarly. If the fluctuation of the coefficient α in (6.4) is neglected, then dependence (6.7)will simplify to P4(σf)= ξd cmax ∫ ξd cmin g(ξdc)P3(σf|ξdc ,ξ2dc ) dξdc (6.8) where P3(σf|ξdc ,ξ2dc )= 1− [1−P(ξ11 > ξDc ,ξ2D11 > ξ2Dc )]m (6.9) Therefore, the crack growth in a polycrystalline metal consists in realisation of two sequentially repeating events related to cleavage of not less than one grain neighbouring to the disk-like crack with consequent formation of the disk-like crack with a greater diameter. Such a mechanism is similar to the dislocation kink moving. The difference is that the dislocation kink is created by thermal fluctuations, and in the crack movement it is due to fluctuation of tensile micro- stresses ξ11 and stochastic misorientation of grain boundaries. The fanlike type of the cleavage initiation site is one of the consequences of such a micromechanism (Fig. 8b). Figure 9 shows computer simulation findings on crack growth in polycrystalline iron with a average grain size 97µm and the variance of grain logarithms Dlnd = 0.19. This simulation was executed for an extremely unfavourable condition α = 1 for overcoming the grain boundary by the crack. According to the data obtained, in the case α =16, the level of average critical macro- stress of unstable propagation of carbide crack within the grain σf is sufficient for further growth of this crack onmacroscopic scale. It should be noted that this is a dependence for averagemacroscopic fracture stress. As it follows from the suggested model, some cracks are arrested due to micro- stress ξ11 fluctuations and grain boundaries. This gives rise to presence of arrested cracks with a size equal to 1-2 grain sizes in the fractured specimens. Fig. 9. Change in the average value of stress of micro-crack instability σf on its diameter a: σ ∗ f is the critical stress of overcoming the grain boundary (of grain cleavage); dg is the average grain size 7. Conclusions • Themulti-scale approach to fracture enables one to overcome several essential challenges of the conventional Local Approach: – to offer a statistica criterion of cleavage fracture initiation accounting for fluctuation of micro-stresses; 6It means that at d/a=100, the ratio k f/f Ia and k c/f Ia reaches an extremely high level equal to 10. 88 S. Kotrechko – to propose a dependence describing the effect of plastic strain, temperature and lo- ading rate on the crack nuclei generation rate; – to differentiate contribution of properties of a separate crack nucleus and the rate of crack nuclei generation to change in the value of local fracture stress σf and fracture toughness. • The value of shape parameter of the Weibull distribution for one crack nucleus instabili- ty mdoes notdependon the grain/carbide particle size; however, it is a linearly decreasing function of square root from logarithm of this size variance √ Dlnd. • The normalized value of the scaling stress σu/σth is a linear function of √ Dlnd. • The dependence of the value of critical cleavage stress on the number of forming crack nuclei is the reason for the statistical scale effect at cleavage fracture of metals. For α-Fe and steels, this effect becomes quantitatively essential only for extremely small volumes V ¬ 0.1mm3, andmay amount to double increase for V ¬ 0.001mm3. Such small volumes limit the “process zone” ahead the sharp cracks. • The gradient of local plastic strain is one of the factors affecting the value of local fracture stress σf.Reduction in the cracknuclei density gives rise to an increase in the susceptibility of σf to the magnitude of the gradient value. • A relation exists between the value of threshold stress of cleavage fracture σth and themi- nimum fracture stress of standard tensile specimens RMC, namely σth =(0.75-0.95)RMC . This enables one to suggest a simple procedureof σth determination in theLocalApproach to fracture. • Themechanismofmicro-crack propagation in polycrystallinemetals consists in realization of two sequentially repeating steps related to cleavage of not less than one grain neighbo- uring to the disk-like crack with further crack growth in the tangential direction resulting in formation of the disk-like crack with a greater diameter. References 1. Beremin F., 1983, A local criterion for cleavage fracture of a nuclear pressure vessel steel,Metal- lurgical Transactions,A 14, 2277-2287 2. BordetS.R.,KarstensenA.D.,KnowlesD.M.,WiesnerC.S., 2005,Anew statistical local criterion for cleavage fracture in steel. Part I:Model presentation,Engineering FractureMechanics, 72, 435-452 3. Indenbom V., 1961, Criteria of fracture in dislocation theories of strength,Physica Status Solidi, 3, 7, 2071-2079 [in Russian] 4. Kotrechko S., 1995, Statistic model of brittle fracture of polycrystalline metals,Physic Metals, 14, 1099-1120 5. Kotrechko S., 2002, Physical Fundamentals of Local Approach to analysis of cleavage fracture, [In:]Transferability of FractureMechanical Characteristic, I. Dlouhy (Edit.), NATOScience Series, Series II, 78, 135-150 6. Kotrechko S., 2003, A local approach to brittle fracture analysis and it physical interpretation, Strength of Materials, 35, 4, 334-345 7. Kotrechko S., Meshkov Yu., 2001, Physical fundamentals of a local approach to analysis of brittle fracture of metals and alloys,Materials Science, 37, 4, 583-597 8. Kotrechko S., Meshkov Yu., Dlouhy I., 2001, Computer simulation of effect of grain size distribution onWeibull parameters,Theoretical and Applied Fracture Mechanics, 35, 255-260 The key problems of local approach to cleavage fracture 89 9. Kotrechko S.,MeshkovYu.,MettusG., 1995, Physical nature of strength of polycrystalline metal near the embrittlement point,Metal Physics and Advanced Technologies, 14, 1205-1210 10. KotrechkoS., StrnadelB.,Dlouhy I., 2007,Fracture toughness of cast ferritic steel applying local approach,Theoretical and Applied Fracture Mechanics, 47, 171-181 11. Kotrechko S., et al., 2011, Local scale effect at cleavage fracture of metals,Metal Physics and Advanced Technologies, 33, 687-705 12. Lambert-PerladeA.,GourguesA.F.,Besson J., SturelT.,PineauA., 2004,Mechanisms andmodeling of cleavage fracture in simulatedheat-affected zonemicrostructures of ahigh-strength low alloy steel,Metallurgical and Materials Transactions, A35, 1039-1053 13. Lin T., Evans A.G., Ritchie R.0., 1986, Statistical analysis of cleavage fracture ahead of sharp cracks and rounded,Acta Metallurgica, 34, II, 2205-2216 14. Lindley T.C., Oates G., Richards C.E., 1970, A critical appraisal of carbide crackingmecha- nisms in ferride/carbide aggregates,Acta Metallurgica, 18, 1127-1136 15. Margolin B., Gulenko A., Shvetsova V., 1998, Improved probabilistic model for fracture toughness prediction for nuclear pressure vessel steels, International Journal of Pressure Vessels and Piping, 75, 843-855 16. Martin-Meizoso A., Ocana-Arizcorreta I., Gil-Sevillano J. and Fuentes-Perez M., 1994,Modelling cleavage fracture of bainitic steels,ActaMetallurgica et Materialia, 42, 2057-2068 17. Pineau A., 2006, Development of the local approach to fracture over the past 25 years: theory and applications, International Journal of Fracture, 138, 139-166 18. PineauA,BenootT., 2010,Advances in cleavage fracturemodelling in steels:Micromechanical, numerical andmultiscale aspects,Comptes Rendus Physique, 11, 316-325 Kluczowe zagadnienia w lokalnym ujęciu kruchego pękania Streszczenie W oparciu o zaproponowany wieloskalowy model lokalnego sformułowania procesu pękania (Local Approach – LA) wyróżniono cztery podstawowe problemy do rozważenia: (i) efekt fluktuacji mikrona- prężeń na niestabilność jądra pęknięcia, (ii) intensywność zawiązywania się mikropęknięcia i jego wpływ na prawdopodobieństwo powstania przełomu, (iii) teoretyczne i eksperymentalne oszacowanie wartości naprężenia krytycznego, (iv) stochastyczna analiza efektu „wieloprogowego” na wzrost mikropęknięcia wmetalu polikrystalicznym. Manuscript received November 9, 2011; accepted for print March 15, 2012