B. Strnadel, P. Byezanski/ Engineering Fracture Mechanics 74(2007)1825-1836 1833 5. Discussion The proposed theory of brittle fracture applied to Ni-Cr steel at very low-temperatures indicates how nicro-cracking originated in carbides bear on the fracture instability of the main crack in a body. Not only the size of the nucleated micro-cracks, but also the way the orientation of cleavage planes in the matrix affect ing the total probability of the fracture, is taken into account. It is shown in Eq (3)that this factor can sub- stantially increase the critical size of a micro-crack, and therefore randomly orientated micro-cracks diminish brittle fracture probability. This micro-crack orientation effect is stronger in a homogeneously loaded body than in a non-homogenous stress field like that around the macro-crack tip. Increasing effective surface energy lowering yield stress, and localized plasticity in the vicinity of the macro-crack tip reduce the risk of brittle fracture occurrence. The weakest link theory applied on stress field described by the stress intensity factor enables the prediction of the statistical distribution of fracture toughness. Elastic stress field singularity ahead of the crack tip implies a more pronounced brittleness of steel than a small scale yielding HRR stress field Carbide through thickness micro-cracks, when B=4/, are more prone to create brittle fracture instability than penny shaped micro-cracks nucleated in spherical carbides with B=T. All these general conclusions can be directly utilized in the micro-structural design of steels operating at low-temperatures or at conditions promoting embrittlement. By varying temperature and time of spheroidization in heat treatment of steel, the resulting space and size distributions of carbides alter and that affects the relation between strength and tough ness of the steel As it has been analyzed earlier, the actual strength or fracture toughness of the material could vary depend ing on micro-structural parameters. In addition to this fact, it is usually difficult to precisely predict the exter nal loads acting on the component made under actual service condition. The risk of the brittle fracture can be expressed in terms of statistical distributions of the local maximum effective stress, emax, acting in the com- ponent volume, V,(1(emax)and of the local cleavage strength, 2(or), as given in [25]: Here the probability density of the local cleavage strength, 2(ar), corresponds to the given volume, v, where the total probability of the brittle fracture, P(Fig. 2)is dP G=of perp B=3 V= 100 mm Fig. 6. Probability of brittle fracture of Ni-Cr steel as a function of loading spectrum parameter, o, and for different values of the effective5. Discussion The proposed theory of brittle fracture applied to Ni–Cr steel at very low-temperatures indicates how micro-cracking originated in carbides bear on the fracture instability of the main crack in a body. Not only the size of the nucleated micro-cracks, but also the way the orientation of cleavage planes in the matrix affect￾ing the total probability of the fracture, is taken into account. It is shown in Eq. (3) that this factor can sub￾stantially increase the critical size of a micro-crack, and therefore randomly orientated micro-cracks diminish brittle fracture probability. This micro-crack orientation effect is stronger in a homogeneously loaded body than in a non-homogenous stress field like that around the macro-crack tip. Increasing effective surface energy, lowering yield stress, and localized plasticity in the vicinity of the macro-crack tip reduce the risk of brittle fracture occurrence. The weakest link theory applied on stress field described by the stress intensity factor enables the prediction of the statistical distribution of fracture toughness. Elastic stress field singularity ahead of the crack tip implies a more pronounced brittleness of steel than a small scale yielding HRR stress field. Carbide through thickness micro-cracks, when b = 4/p, are more prone to create brittle fracture instability than penny shaped micro-cracks nucleated in spherical carbides with b = p. All these general conclusions can be directly utilized in the micro-structural design of steels operating at low-temperatures or at conditions promoting embrittlement. By varying temperature and time of spheroidization in heat treatment of steel, the resulting space and size distributions of carbides alter and that affects the relation between strength and tough￾ness of the steel. As it has been analyzed earlier, the actual strength or fracture toughness of the material could vary depend￾ing on micro-structural parameters. In addition to this fact, it is usually difficult to precisely predict the exter￾nal loads acting on the component made under actual service condition. The risk of the brittle fracture can be expressed in terms of statistical distributions of the local maximum effective stress, remax, acting in the com￾ponent volume, V, u1(remax) and of the local cleavage strength, u2(rf), as given in [25]: Pf ¼ Prðremax P rfÞ ¼ Z þ1 0 u1ðremaxÞ Z remax 0 u2ðrfÞdrf dremax: ð10Þ Here the probability density of the local cleavage strength, u2(rf), corresponds to the given volume, V, where the total probability of the brittle fracture, Pf (Fig. 2) is u2ðrfÞ ¼ dPf dr  r¼rf ð11Þ 1.2 1.4 1.6 1.8 2 2.2 –15 –10 –5 log10 Pr( σemax ≥ σf ) α0 [GPa] β0 = 3 V = 100 mm3 γ eff = 23 J/m2 23 14 rand. perp. perp. Fig. 6. Probability of brittle fracture of Ni–Cr steel as a function of loading spectrum parameter, a0, and for different values of the effective surface energy. B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836 1833