l830 B. Strnadel, P. Byczanski Engineering Fracture Mechanics 74(2007)1825-1836 as a function of the stress intensity factor, Kl. The instance of the fracture instability, Ki= Klc and Kl rep resents 100Pf%o quantile of the experimentally assessed statistical distribution of the fracture toughness. 4. Experimental results and applications The model of brittle fracture was applied to experimental results obtained from Ni-Cr steel investigations [13]. The steel was heat treated to give the structure of tempered martensite and almost spherical carbides Transmission electron microscopy at magnification rates, 13500, was employed to study the statistical distri- bution of carbide sizes and the area density of carbides, Na= 1.45 x 10m. These relative frequencies were subjected to the statistical processing of the least squares method and the parameters of the Weibull's statis- tical distribution given by Eq(4)has been found to be, dpo=0. 263 um and d0=2.28. Good agreement obtained between the analytically determined shape of the probability density and the experimentally ascer tained distribution of relative carbide size frequencies was confirmed by a statistical coincidence test. Fo the sake of verifying the coincidence of the planar distribution of carbide particles with the Poisson's distri- bution, the number of carbides was determined on small tested areas at magnification of 3000. Through a sta- tistical coincidence test at a significance level of 0.05, it was found that the mean value of the number of carbides on the tested area is equal to variance of number of carbides in the area, which conforms to the applied precondition. Using numerical procedure published elsewhere [23] the volume density of carbides, Ny=7.4x10m', has been revealed The studied steel was mechanically tested. True stress-strain curves and the yield stress variations over a low-temperature range from 93K to 143 K, and at room temperature were assessed from uniaxial tensile tests at a strain rate of e=3x10 At room temperature of 293 K, the yield stress, oo, was 406 MPa; the Youngs modulus and the Poisson's ratio were found to be E= 207 GPa, and v=0.3, respectively. Shapes of the true stress-strain curves detected in the low-temperature range were employed for the evaluation of the strain hardening exponent, n= 5.2. Plane strain fracture toughness, Kle was evaluated in the lower bound temperature range using fatigue pre-cracked, single-edge-notched specimens 25 mm thick. These speci were tested in three points bending in accordance with standard ASTM E 399-90 at a stress intensity rate of K,=2 MPa m /2 s-l. Above the temperature of 113 K the fracture toughness values, Klc, predominately have not met the validity criteria of linear elastic fracture mechanics. In this case the values of fracture tough ness have been determined according to the concept of elastic plastic fracture mechanics through a J-integral at the onset of cleavage fracture, Je=K(1-v/E The fracture surfaces of fracture toughness specimens tested in the temperature range from 93 K to 143K were examined by scanning electron microscopy. Fractography investigation indicated that the fracture path of all specimens tested was transgranular [13]. The specimens' fracture surfaces that broke at 143 K were found to comprise a small ductile failure zone formed by dimples and directly adjacent to a narrow stretched zone. The brittle fracture zone beyond the ductile zone consisted of cleavage facets. Below 143 K no ductile zone was observed on fracture surfaces and the stretched zone extended from fatigue pre-crack tip was directly followed by a brittle fracture. As it was documented elsewhere [13] carbides at fracture surfaces of the inves tigated steel were identified to act as subsidiary sources of cleavage facets formation. Frequent star shaped cleavage facets at fracture surfaces of broken specimens were evidently initiated by local stress concentration in their centres where carbides triggering the facets were recognized. Fractography analysis also proved [13] that cleavage micro-cracks were initiated by decohesion of carbides and matrix and the initiation of a micro- void. Further propagation of initiated micro-cracks formed the cleavage facets. The direction of river mark- ings from the centre of a star shaped facet to its periphery or from microvoid clearly shows that both observed The statistical distribution of carbide sizes(Eq. (4))was employed for the numerical calculation of the total fracture probability, Pf, as a function of homogeneous acting stress, o. The effective surface energy, eff, has been taken as independent on temperature, and from previous results two alternative values have been chosen either, yeff=23 Jm[11, 24]or %ef=14 Jm[1, 7]. Calculated curves of Pf forhomogeneously stressed vol- umes,m/Ny, where on average the number of micro-cracks, m, is 1, 10, 10, 105, respectively or for volume V=10 mm, are given in Fig. 2. Homogeneously stressed volume, v, corresponding to 10 micro-cracks is a cube sized approx. 50 x 50 x 50 um asconsidered by Beremin [9]in his concept of probability, Pf, calculation.as a function of the stress intensity factor, KI. The instance of the fracture instability, KI = KIc and KIc rep￾resents 100Pf% quantile of the experimentally assessed statistical distribution of the fracture toughness. 4. Experimental results and applications The model of brittle fracture was applied to experimental results obtained from Ni–Cr steel investigations [13]. The steel was heat treated to give the structure of tempered martensite and almost spherical carbides. Transmission electron microscopy at magnification rates, 13 500, was employed to study the statistical distri￾bution of carbide sizes and the area density of carbides, NA = 1.45 · 1012 m2 . These relative frequencies were subjected to the statistical processing of the least squares method and the parameters of the Weibull’s statis￾tical distribution given by Eq. (4) has been found to be, dp0 = 0.263 lm and d0 = 2.28. Good agreement obtained between the analytically determined shape of the probability density and the experimentally ascer￾tained distribution of relative carbide size frequencies was confirmed by a statistical coincidence test. For the sake of verifying the coincidence of the planar distribution of carbide particles with the Poisson’s distri￾bution, the number of carbides was determined on small tested areas at magnification of 3000. Through a sta￾tistical coincidence test at a significance level of 0.05, it was found that the mean value of the number of carbides on the tested area is equal to variance of number of carbides in the area, which conforms to the applied precondition. Using numerical procedure published elsewhere [23] the volume density of carbides, NV = 7.4 · 1018 m3 , has been revealed. The studied steel was mechanically tested. True stress–strain curves and the yield stress variations over a low-temperature range from 93 K to 143 K, and at room temperature were assessed from uniaxial tensile tests at a strain rate of e_ ¼ 3  104 s1. At room temperature of 293 K, the yield stress, r0, was 406 MPa; the Young’s modulus and the Poisson’s ratio were found to be E = 207 GPa, and m = 0.3, respectively. Shapes of the true stress–strain curves detected in the low-temperature range were employed for the evaluation of the strain hardening exponent, n = 5.2. Plane strain fracture toughness, KIc was evaluated in the lower bound temperature range using fatigue pre-cracked, single-edge-notched specimens 25 mm thick. These specimens were tested in three points bending in accordance with standard ASTM E 399-90 at a stress intensity rate of K_ I ¼ 2 MPa m1=2 s1. Above the temperature of 113 K the fracture toughness values, KIc, predominately have not met the validity criteria of linear elastic fracture mechanics. In this case the values of fracture tough￾ness have been determined according to the concept of elastic plastic fracture mechanics through a J-integral at the onset of cleavage fracture, Jc ¼ K2 Jcð1 m2Þ=E. The fracture surfaces of fracture toughness specimens tested in the temperature range from 93 K to 143 K were examined by scanning electron microscopy. Fractography investigation indicated that the fracture path of all specimens tested was transgranular [13]. The specimens’ fracture surfaces that broke at 143 K were found to comprise a small ductile failure zone formed by dimples and directly adjacent to a narrow stretched zone. The brittle fracture zone beyond the ductile zone consisted of cleavage facets. Below 143 K no ductile zone was observed on fracture surfaces and the stretched zone extended from fatigue pre-crack tip was directly followed by a brittle fracture. As it was documented elsewhere [13] carbides at fracture surfaces of the inves￾tigated steel were identified to act as subsidiary sources of cleavage facets formation. Frequent star shaped cleavage facets at fracture surfaces of broken specimens were evidently initiated by local stress concentration in their centres where carbides triggering the facets were recognized. Fractography analysis also proved [13] that cleavage micro-cracks were initiated by decohesion of carbides and matrix and the initiation of a micro￾void. Further propagation of initiated micro-cracks formed the cleavage facets. The direction of river mark￾ings from the centre of a star shaped facet to its periphery or from microvoid clearly shows that both observed micro-mechanisms of the initiation of micro-cracks are controlled by local stress in the area around carbides. The statistical distribution of carbide sizes (Eq. (4)) was employed for the numerical calculation of the total fracture probability, Pf, as a function of homogeneous acting stress, r. The effective surface energy, ceff, has been taken as independent on temperature, and from previous results two alternative values have been chosen either, ceff = 23 Jm2 [11,24] or ceff = 14 Jm2 [1,7]. Calculated curves of Pf forhomogeneously stressed vol￾umes, m/NV, where on average the number of micro-cracks, m, is 1, 10, 102 , 103 , respectively or for volume, V = 10 mm3 , are given in Fig. 2. Homogeneously stressed volume, V, corresponding to 106 micro-cracks is a cube sized approx. 50 · 50 · 50 lm asconsidered by Beremin [9] in his concept of probability, Pf, calculation. 1830 B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836