Availableonlineatwww.sciencedirect.com ScienceDirect Engineering Fracture Mechanics ELSEVIER Engineering Fracture Mechanics 74 (2007)1825-1836 www.elsevier.com/locate/engfracmech Micro-Structural reliability design of brittle materials B.Strnadel a,, P. Byczanski b Technical University of Ostrava, 17. listopadu 15. 708 33 Ostrava, Czech repu Received 24 January 2005; received in revised form 12 May 2006: accepted 18 August 2006 ailable online 2 November 2006 Abstract The paper analyses the effects of statistical distribution of micro-structural defect sizes concerning a scatter of brittle material fracture toughness. The results can be utilized for reliability assessment of selected engineering components oper ating under conditions of imminent brittle fracture. The reliability, taken as a complementary probability of brittle fracture initiation, is discussed, taking into account the character of the defect size statistical distribution, material mechanical properties, and varying loading and stress conditions of the component. Application of this method on Ni-Cr steel has demonstrated that there is very good agreement of the fracture behaviour predicted scatter with experimental results. This probability approach is compared with a deterministic reliability method originating from computations of safety factors Its rational evaluation, as a function of the acceptable probability of fracture instability, provides a highly effective tool for designing of engineering components. 2006 Elsevier Ltd. All rights reserved Keywords: Cleavage strength; Brittle fracture; Fracture toughness; Fracture probability; Reliability; Safety factor 1. Introduction Engineering components made from brittle materials such a inter-metallics, glasses or carbon steels at low-temperatures must be designed with regard to flaws d inclusions in structure. The load applied to the component causes the local stress concentrations these defects which initiates micro- cracking. If these micro-cracks extend further and interact with each other, fracture instability occurs and macroscopic failure may arise. The usual combination of high strength and low fracture toughness of brittle materials leads to a relatively small critical crack size, detected with great difficulty by current non-destructive evaluation methods. As a result, service reliability of components made from brittle materials is very sensitive to micro-structural parameters such as micro-crack size distribution, micro-crack shape, their orientation and spatial allocations in the component stress field. E-mail address: bohumir. strnadelavsb cz(B. Strnadel) 0013-7944S. see front matter 2006 Elsevier Ltd. All rights reserved doi: 10. 1016/j-engfracmech 2006.08.027
Micro-structural reliability design of brittle materials B. Strnadel a,*, P. Byczanski b a Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic b Institute of Geonics, Academy of Sciences, Studentska´ 1768, 708 00 Ostrava, Czech Republic Received 24 January 2005; received in revised form 12 May 2006; accepted 18 August 2006 Available online 2 November 2006 Abstract The paper analyses the effects of statistical distribution of micro-structural defect sizes concerning a scatter of brittle material fracture toughness. The results can be utilized for reliability assessment of selected engineering components operating under conditions of imminent brittle fracture. The reliability, taken as a complementary probability of brittle fracture initiation, is discussed, taking into account the character of the defect size statistical distribution, material mechanical properties, and varying loading and stress conditions of the component. Application of this method on Ni–Cr steel has demonstrated that there is very good agreement of the fracture behaviour predicted scatter with experimental results. This probability approach is compared with a deterministic reliability method originating from computations of safety factors. Its rational evaluation, as a function of the acceptable probability of fracture instability, provides a highly effective tool for designing of engineering components. 2006 Elsevier Ltd. All rights reserved. Keywords: Cleavage strength; Brittle fracture; Fracture toughness; Fracture probability; Reliability; Safety factor 1. Introduction Engineering components made from brittle materials such as ceramics, inter-metallics, glasses or carbon steels at low-temperatures must be designed with regard to flaws, holes, and inclusions in structure. The load applied to the component causes the local stress concentrations around these defects which initiates microcracking. If these micro-cracks extend further and interact with each other, fracture instability occurs and macroscopic failure may arise. The usual combination of high strength and low fracture toughness of brittle materials leads to a relatively small critical crack size, detected with great difficulty by current non-destructive evaluation methods. As a result, service reliability of components made from brittle materials is very sensitive to micro-structural parameters such as micro-crack size distribution, micro-crack shape, their orientation and spatial allocations in the component stress field. 0013-7944/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.08.027 * Corresponding author. E-mail address: bohumir.strnadel@vsb.cz (B. Strnadel). Engineering Fracture Mechanics 74 (2007) 1825–1836 www.elsevier.com/locate/engfracmech
B. Strnadel, P. Byczanski Engineering Fracture Mechanics 74(2007)1825-1836 Ao material constant characteristic width of the crack front micro-crack size critical size of micro-crack dpmax the largest carbide dpmin the smallest carbide dpo size parameter of function v(dp) E Young's modulus I, dimensionless parameter in HRR stress fiel o 0) dimensionless function of 0 in elastic stress fie path independent J-integral J J-integral at the onset of cleavage fracture the safety factor corresponding to survival probability for loading kla micro-crack arrest toughnes Mode I stress intensity factor Klc fracture toughness KJe elastic-plastic fracture toughness number of micro-cracks work hardening exponent Na area density of carbides volume density of carbides probability of micro-crack initiation in a carbide P total fracture probability radial coordinate of the polar system, centred at crack tip volume deviation between applied stress direction and perpendicularity to the cleavage plane zo size parameter of o,(emax) function B micro-crack shape factor Bo shape parameter of I(emax)function effective surface energy crack tip opening displacement do shape parameter of function y(dpi s(8V, i Poissons distribution function 6 angular coordinate of the polar system, centred at crack tip K(kp) statistical distribution of the safety factor Poissons ratio S(a) probability density function of disorientation angle a local maximum effective stress Ofmax the highest local strength Omin the lowest local strength ar local cleavage strength oir, e)stress field around the crack tip yield stress
Nomenclature A0 material constant Aa constant b characteristic width of the crack front dp micro-crack size dpf critical size of micro-crack dpmax the largest carbide dpmin the smallest carbide dp0 size parameter of function w(dp) E Young’s modulus hij(h) dimensionless function of h in elastic stress field In dimensionless parameter in HRR stress field J path independent J-integral Jc J-integral at the onset of cleavage fracture kp safety factor kp0 the safety factor corresponding to survival probability for loading kIa micro-crack arrest toughness KI Mode I stress intensity factor KIc fracture toughness KJc elastic–plastic fracture toughness m number of micro-cracks n work hardening exponent NA area density of carbides NV volume density of carbides pf probability of micro-crack initiation in a carbide Pf total fracture probability r radial coordinate of the polar system, centred at crack tip dV volume element V volume a deviation between applied stress direction and perpendicularity to the cleavage plane a0 size parameter of /1(remax) function b micro-crack shape factor b0 shape parameter of /1(remax) function ceff effective surface energy d crack tip opening displacement d0 shape parameter of function w(dp) e0 yield strain f(dV,i) Poisson’s distribution function h angular coordinate of the polar system, centred at crack tip j(kp) statistical distribution of the safety factor m Poisson’s ratio n(a) probability density function of disorientation angle a r stress remax local maximum effective stress rfmax the highest local strength rfmin the lowest local strength rf local cleavage strength rij(r,h) stress field around the crack tip r0 yield stress 1826 B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836
B. Strnadel, P. Byezanski/ Engineering Fracture Mechanics 74(2007)1825-1836 1827 dii(r, 0)angular function of n and Ki in HRR stress field p angle of the wedge active region ahead of the crack tip pI(Emax) statistical distribution of local maximum effective stress p(or) statistical distribution of cleavage strength v(dp) probability density function of carbide sizes Low-temperature transgranular cleavage of carbon structural steels has been experimentally proved to be initiated by a slip induced micro-cracking of carbides [1-3]. Some other works [4, 5] prove that depending on the temperature there are other micro-mechanisms initiating cleavage of steels other than those caused by micro-cracking of carbides. There are other micro-structural barriers, such as packet boundaries in bainitic teels or grain boundaries in ferritic steels that controll the size of initiated micro-cracks. In this paper, only the propagation of micro-cracks, which initiate within carbides or inclusions, is considered as the critical stage in brittle fracture process Local heterogeneity in deformation may result in the initiation of micro-cracks and their propagation into the matrix whenever the applied stress, o, exceeds the local cleavage strength, ar [6] ≥m=(0y where kla=[2Eye/(1-v]is the micro-crack arrest toughness introduced by Hahn [6], dp is the micro- crack size B is a micro-crack shape factor; B=t for penny shaped and =4/ for through thickness mi cro-crack [1-7]. E is Youngs modulus, v is the Poissons ratio, and ?efr is the effective surface energy given by the sum of the true surface energy of the matrix and its plastic work. Over the past years, experimental investigations of the low-temperature brittle fracture in steels have been completed by attempts to model the fracture process by statistical methods [7-13] using local criteria for the initiation of micro-cracks. These approaches can reveal the relationship between the micro-structural param- eters and macroscopic mechanical properties From the size distribution of carbides, using Weibull's weakest link statistical theory, the cumulative probability of cleavage failure and the temperature dependence of frac- ture toughness scatter were computed [7-13] Even though the original Beremin's model [9]considers the statistical distribution of carbides as originators of micro-cleavage, other random parameters controlling the fracture process were not taken into consider- ation in the model. Except of rather accidental effects of carbides, this paper has also taken into account the influences of micro-cracks'accidental orientations inclusive their spatial distributions. Nevertheless, also temperature variations, as well as characteristics of the stress-strain field adjacent to a sharp crack, plasticity properties, yield stress, and effective surface energy related to brittle fracture risk implications: they all deserve closer inspection and investigation. This paper is concerned with the probability of brittle fracture in steels loaded under conditions of homogenous and non-homogenous elastic and elasto-plastic stress fields and pro- ides a method how to calculate these parameters effects on the fracture probability. The acquired results are capable of being applied on the micro-structural reliability design concerning not only brittle steels but also other brittle materials 2. Statistical analysis of micro-cracking The initiated micro-crack as obeying the criterion given in Eq. (1) crosses the particle-matrix interface more easily if the cleavage planes in the matrix are favourably orientated relative to the cleavage plane in the carbide particle. Substantial misalignment between these planes, or when particles are too small to satisfy the propa- gation criterion, they both cause the initiation of stable micro-cracks. Similarly, deviation between applied stress direction and perpendicularity to the cleavage plane a(Fig. 1)makes micro-crack propagation into the matrix difficult, and the local cleavage strength, of, specified by Eq (1)is 1/cos"a times higher [14]. Then, for every magnitude of local stress, o, there is a certain critical size of the initiated micro-crack at which point this micro-crack could spread from the carbide particle into the matrix:
Low-temperature transgranular cleavage of carbon structural steels has been experimentally proved to be initiated by a slip induced micro-cracking of carbides [1–3]. Some other works [4,5] prove that depending on the temperature there are other micro-mechanisms initiating cleavage of steels other than those caused by micro-cracking of carbides. There are other micro-structural barriers, such as packet boundaries in bainitic steels or grain boundaries in ferritic steels that controll the size of initiated micro-cracks. In this paper, only the propagation of micro-cracks, which initiate within carbides or inclusions, is considered as the critical stage in brittle fracture process. Local heterogeneity in deformation may result in the initiation of micro-cracks and their propagation into the matrix whenever the applied stress, r, exceeds the local cleavage strength, rf [6]: r P rf ¼ ðb=2Þ 1=2 kIa ffiffiffiffiffi dp p ð1Þ where kIa = [2Eceff/(1 m 2 )]1/2 is the micro-crack arrest toughness introduced by Hahn [6], dp is the microcrack size, b is a micro-crack shape factor; b = p for penny shaped and b = 4/p for through thickness micro-crack [1–7], E is Young’s modulus, m is the Poisson‘s ratio, and ceff is the effective surface energy given by the sum of the true surface energy of the matrix and its plastic work. Over the past years, experimental investigations of the low-temperature brittle fracture in steels have been completed by attempts to model the fracture process by statistical methods [7–13], using local criteria for the initiation of micro-cracks. These approaches can reveal the relationship between the micro-structural parameters and macroscopic mechanical properties. From the size distribution of carbides, using Weibull’s weakest link statistical theory, the cumulative probability of cleavage failure and the temperature dependence of fracture toughness scatter were computed [7–13]. Even though the original Beremin’s model [9] considers the statistical distribution of carbides as originators of micro-cleavage, other random parameters controlling the fracture process were not taken into consideration in the model. Except of rather accidental effects of carbides, this paper has also taken into account the influences of micro-cracks’ accidental orientations inclusive their spatial distributions. Nevertheless, also temperature variations, as well as characteristics of the stress–strain field adjacent to a sharp crack, plasticity properties, yield stress, and effective surface energy related to brittle fracture risk implications; they all deserve a closer inspection and investigation. This paper is concerned with the probability of brittle fracture in steels loaded under conditions of homogenous and non-homogenous elastic and elasto-plastic stress fields and provides a method how to calculate these parameters effects on the fracture probability. The acquired results are capable of being applied on the micro-structural reliability design concerning not only brittle steels but also other brittle materials. 2. Statistical analysis of micro-cracking The initiated micro-crack as obeying the criterion given in Eq. (1) crosses the particle-matrix interface more easily if the cleavage planes in the matrix are favourably orientated relative to the cleavage plane in the carbide particle. Substantial misalignment between these planes, or when particles are too small to satisfy the propagation criterion, they both cause the initiation of stable micro-cracks. Similarly, deviation between applied stress direction and perpendicularity to the cleavage plane a (Fig. 1) makes micro-crack propagation into the matrix difficult, and the local cleavage strength, rf, specified by Eq. (1) is 1/cos2 a times higher [14]. Then, for every magnitude of local stress, r, there is a certain critical size of the initiated micro-crack at which point this micro-crack could spread from the carbide particle into the matrix: r~ijðr; hÞ angular function of n and KI in HRR stress field / angle of the wedge active region ahead of the crack tip /1(remax) statistical distribution of local maximum effective stress /2(rf) statistical distribution of cleavage strength w(dp) probability density function of carbide sizes B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836 1827
l828 B. Strnadel, P. Byczanski Engineering Fracture Mechanics 74(2007)1825-1836 o = const main crack plane Fig. I. Schematic illustration of isostressed volume element and wedge active zone ahead of macro-crack tip. dpr(a)=202-cos+a The probability of the micro-crack propagation from the carbide into the matrix equals the probability that carbide sized micro-crack, dn, exceeds its critical size P(o)=Pr(dp≥dpr(a)= s(a)v(dp)dadd p where p(dp)=sodo()exp[-(dp/dpo) o1 is the Weibull's probability density function of carbide sizes with size, dpo, and shape, So, parameters estab- lished by the least square method from the experimental data set. The two parameter Weibull distribution is sufficiently flexible to describe experimental data, however their agreement with the analytical shape of probability density function given by Eq (4)has to be corroborated by a statistical coincidence test at the selected significance level. Lee et al. [2]have shown that carbide size follows a cut-off domain of an exponential distribution. This finding has been used by Tanguy et al. [15] to model the fracture toughness scattering of 22NiMoCr3-7 steel The distribution of micro-cracks orientation, a, (Fig. 1)will depend on the availability of cleavage planes the matrix, as well as on the physical parameters that determine the mechanisms of the cracking process. In steels, cleavage occurs on mutually perpendicular (1001 planes. If grains'spatial orientations are uniformly distributed, the distribution function of these cleavage planes can be represented by the equation S(a=A, sin a where the constant Ax follows from the normalization condition of the probability Po)(Eq (3) published elsewhere [14]. Since the upper limit of the second integral in Eq (3)actually equals to the maximum carbide particle size from carbides population, the upper limit of the micro-crack orientation, amax, in the first integral d from Eq (2)follows that amax=arccos(od dpmax)/o] /2.Then the carbides ulation permits a larger upper limit of the micro-crack orientation %fmax. In any case with decreasing local stress o, e.g. as the distance from the pre-crack tip increases, the maximum angle, %fmax, diminishes. Uniformly distributed carbide micro-cracks orientations with the upper angle limit, as those given by eq. (5), have been proven experimentally in previous works [16-18] The elementary probability that a micro-crack will be initiated in at least a single carbide particle within the stressed volume element, SVo), ahead of the macro-crack tip(Fig. 1)can be calculated from the equation:
dpfðrÞ ¼ bk2 Ia 2r2 cos4 a : ð2Þ The probability of the micro-crack propagation from the carbide into the matrix equals the probability that carbide sized micro-crack, dp, exceeds its critical size, pfðrÞ ¼ Prðdp P dpfðrÞÞ ¼ Z p=2 0 Z 1 dpfðrÞ nðaÞwðdpÞdaddp ð3Þ where wðdpÞ ¼ d0dd0 p0 dðd01Þ p exp½ðdp=dp0Þ d0 ð4Þ is the Weibull’s probability density function of carbide sizes with size, dp0, and shape, d0, parameters established by the least square method from the experimental data set. The two parameter Weibull distribution is sufficiently flexible to describe experimental data, however their agreement with the analytical shape of probability density function given by Eq. (4) has to be corroborated by a statistical coincidence test at the selected significance level. Lee et al. [2] have shown that carbide size follows a cut-off domain of an exponential distribution. This finding has been used by Tanguy et al. [15] to model the fracture toughness scattering of 22NiMoCr3-7 steel. The distribution of micro-cracks orientation, a, (Fig. 1) will depend on the availability of cleavage planes in the matrix, as well as on the physical parameters that determine the mechanisms of the cracking process. In steels, cleavage occurs on mutually perpendicular {1 0 0} planes. If grains’ spatial orientations are uniformly distributed, the distribution function of these cleavage planes can be represented by the equation: nðaÞ ¼ Aa sin a ð5Þ where the constant Aa follows from the normalization condition of the probability pf(r) (Eq. (3)) published elsewhere [14]. Since the upper limit of the second integral in Eq. (3) actually equals to the maximum carbide particle size, dpmax, from carbides population, the upper limit of the micro-crack orientation, amax, in the first integral depends on dpmax, and from Eq. (2) follows that amax = arccos[rf(dpmax)/r] 1/2. Then the coarser carbides population permits a larger upper limit of the micro-crack orientation amax. In any case with decreasing local stress r, e.g. as the distance from the pre-crack tip increases, the maximum angle, amax, diminishes. Uniformly distributed carbide micro-cracks orientations with the upper angle limit, as those given by Eq. (5), have been proven experimentally in previous works [16–18]. The elementary probability that a micro-crack will be initiated in at least a single carbide particle within the isostressed volume element, dV(r), ahead of the macro-crack tip (Fig. 1) can be calculated from the equation: 2ϕ δ σ V( ) σe = const σ r crack tip θ main crack plane σ α microcrack Fig. 1. Schematic illustration of isostressed volume element and wedge active zone ahead of macro-crack tip. 1828 B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836
B. Strnadel, P. Byezanski/ Engineering Fracture Mechanics 74(2007)1825-1836 6P(o)=1-∑(6V,川[1-(o)=1-expl-Nvo(o where C(SV, i=1/iI(NvSn'exp(-Nv8n) is the probability that i micro-cracks initiate in the volume element oVa) corresponding to Poisson's distribution and Ny is the volume density of micro-cracks. The validity of this precondition must be proved by a statistical coincidence test. For homogenously stressed volume, V, the final brittle fracture probability, Pf, is easy to calculate from Eq. (6) by replacing, 8V, with V, and conse- quently dPa)=Pf 3. Cleavage in non-homogenous stress field A non-homogenous stress field around the sharp macro-crack tip on small scale yielding condition satisfies the HRR singular solution [ 19, 20 l/(n+1) o(r,0)=0o ou (n, 0), i,j=r, 8 where go is the yield stress; n is the work hardening exponent following from the constitutive law given by Eo= Ao(o/oo)" Eo is the yield strain; Ao is a material constant of order unity; In is a dimensionless parameter slightly dependent on the work hardening exponent, n; au(n, 0) are angular functions of n and J is the path independent J-integral. If the HRR stress field described above is truncated directly by crack tip blunting [21], the bracketed term on the right side of Eq (7)can be replaced by(1-v)Ki/oor) within the near crack tip region. Further away from the crack tip, typically at r>108, where 8 is the crack tip displacement, the stress field differs from HRR solution and at the elastic-plastic interface the stress field approximates the linear lastic asymptotic solution by Williams [22] iy(r, 0) K v2r hy(0) i,j=r, 0 where hi e) are dimensionless functions of 0 and KI is Mode I stress intensity factor. Since at low-temperatures the plastic zone in proximity of the main crack tip is small, initiation of micro-cracking is localized with the greatest probability in the vicinity of the elastic-plastic interface. With the increasing temperature, the size of the plastic zone increases and the place with the greatest probability of micro-cracking is therefore found in its ange. Then in the investigated temperature interval of the micro-cleavage initiation, it is appropriate to carry out the statistical analysis of micro-cleavage for limited, idealized cases of stress distribution ahead of the crack tip. The fracture character of steel at very low-temperatures is rather controlled by linear, elastic stress field ahead of the crack(Eq. 8)and probability analysis of cracking results can be useful in estimating the lowest values of fracture toughness, Klc. At higher temperatures the fracture behaviour of steel is controlled by the HrR stress-strain field(Eq (7)). The effective stress field around the crack tip, de(r, 0), has been considered as the maximum eigenvalue calculated from HRR and elastic stress tensors given by eqs. (7)and (8) The isostressed volume element, dV, is given by the integral of the area element in polar coordinates, ordo (Fig. 1). The bounds of integral, -o and p, delimitate the wedge active region where propagation of the main crack is the most probable. Since the stress field of de is symmetric, the volume element is expressed by the equation 8v(oe)=2b/ rode where b is the characteristic width of the crack front [8, 111. The total probability of brittle fracture initiation can now be established by integrating Eq. (6) within the limits of the lowest, omin, and the highest, Ofmax, local strengths. These extreme values of the local cleavage strength were calculated using Eq (1)from the values of the largest, and smallest, domin, carbides as given by their statistical distribution(Eq.(4)). The Eq.(6) integrated within the active region in the non-homogenous stress field around the crack tip enables calculating of the total fracture probability, Pt
dPfðrÞ ¼ 1 X1 i¼0 fðdV ; iÞ½1 pfðrÞi ¼ 1 exp½N VdVpfðrÞ ð6Þ where f(dV,i) = 1/i!(NVdV) i exp(NVdV) is the probability that i micro-cracks initiate in the volume element dV(r) corresponding to Poisson’s distribution and NV is the volume density of micro-cracks. The validity of this precondition must be proved by a statistical coincidence test. For homogenously stressed volume, V, the final brittle fracture probability, Pf, is easy to calculate from Eq. (6) by replacing, dV, with V, and consequently dPf(r) = Pf. 3. Cleavage in non-homogenous stress field A non-homogenous stress field around the sharp macro-crack tip on small scale yielding condition satisfies the HRR singular solution [19,20]: rijðr; hÞ ¼ r0 J A0e0r0I nr 1=ðnþ1Þ r~ijðn; hÞ; i;j ¼ r; h ð7Þ where r0 is the yield stress; n is the work hardening exponent following from the constitutive law given by e/e0 = A0 (r/r0) n ; e0 is the yield strain; A0 is a material constant of order unity; In is a dimensionless parameter slightly dependent on the work hardening exponent, n; r~ijðn; hÞ are angular functions of n and J is the path independent J-integral. If the HRR stress field described above is truncated directly by crack tip blunting [21], the bracketed term on the right side of Eq. (7) can be replaced by ð1 m2ÞK2 I =ðr2 0rÞ within the near crack tip region. Further away from the crack tip, typically at r > 10d, where d is the crack tip displacement, the stress field differs from HRR solution and at the elastic–plastic interface the stress field approximates the linear elastic asymptotic solution by Williams [22]: rijðr; hÞ ¼ KI ffiffiffiffiffiffiffi 2pr p hijðhÞ; i;j ¼ r; h ð8Þ where hij(h) are dimensionless functions of h and KI is Mode I stress intensity factor. Since at low-temperatures the plastic zone in proximity of the main crack tip is small, initiation of micro-cracking is localized with the greatest probability in the vicinity of the elastic–plastic interface. With the increasing temperature, the size of the plastic zone increases and the place with the greatest probability of micro-cracking is therefore found in its range. Then in the investigated temperature interval of the micro-cleavage initiation, it is appropriate to carry out the statistical analysis of micro-cleavage for limited, idealized cases of stress distribution ahead of the crack tip. The fracture character of steel at very low-temperatures is rather controlled by linear, elastic stress field ahead of the crack (Eq. (8)) and probability analysis of cracking results can be useful in estimating the lowest values of fracture toughness, KIc. At higher temperatures the fracture behaviour of steel is controlled by the HRR stress–strain field (Eq. (7)). The effective stress field around the crack tip, re(r,h), has been considered as the maximum eigenvalue calculated from HRR and elastic stress tensors given by Eqs. (7) and (8). The isostressed volume element, dV, is given by the integral of the area element in polar coordinates, rdrdh (Fig. 1). The bounds of integral, –/ and /, delimitate the wedge active region where propagation of the main crack is the most probable. Since the stress field of re is symmetric, the volume element is expressed by the equation dV ðreÞ ¼ 2b Z u 0 rdr dh ð9Þ where b is the characteristic width of the crack front [8,11]. The total probability of brittle fracture initiation can now be established by integrating Eq. (6) within the limits of the lowest, rfmin, and the highest, rfmax, local strengths. These extreme values of the local cleavage strength were calculated using Eq. (1) from the values of the largest, dpmax, and smallest, dpmin, carbides as given by their statistical distribution (Eq. (4)). The Eq. (6) integrated within the active region in the non-homogenous stress field around the crack tip enables calculating of the total fracture probability, Pf, B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836 1829