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 micro￾crack size, b is a micro-crack shape factor; b = p for penny shaped and b = 4/p for through thickness mi￾cro-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 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 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 pro￾vides 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, 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