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
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 represents 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 distribution 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 statistical 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 ascertained 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 distribution, the number of carbides was determined on small tested areas at magnification of 3000. Through a statistical 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 toughness 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 investigated 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 microvoid. Further propagation of initiated micro-cracks formed the cleavage facets. The direction of river markings 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 volumes, 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
B. Strnadel, P. Byezanski/ Engineering Fracture Mechanics 74(2007)1825-1836 l831 T=143K 02 B= 10 GPa Fig. 2. Total probability of brittle fracture dependence on local stress in volumes corresponding to various number, m, of micro-cracks T=143K andom [GPa Fig. 3. Dependence of brittle fracture probability on local stress in the volume encompassing one micro-crack for perpendicular and random orientation of cleavage planes with respect to local acting stress, as well as two various shapes of micro-cracks, (penny shaped Increasing volume, m/Nv, lowers the strength, or of the body and the curve is getting straight so that the tran- sition from state, Pr=0 to Pf= I has a character of a sudden jump. The influence of the micro-crack shape factor, B, and the way of space array on the total fracture probability, Pf, is illustrated by Fig. 3. The prob- ability, Ps, is calculated by Eq.(6)for the volume corresponding to one cracked carbide V=1/Nv; then Pr=l-expl-Pdo) Fig. 6 illustrates how the probability of fracture, Pr, depends on the stress field singularity around a sharp crack tip. Experimentally assessed statistical distribution of fracture toughness, Klc, of the investigated steel at temperatures of 113 K and 143 K respectively are predicted using the integral form of Eq.(6)for HRR (Eq (7))or the elastic(Eq. 8)stress field. A comparison of PAa) curves for elastic and HRR stress fields at 113 K(Fig. 4)demonstrates the affirmative effect of a small scale yielding on the growth of fracture tough- ness. The angle of the wedge of the active region ahead of the crack tip, affects the course of fracture prob- bility only a little at low-temperatures
Increasing volume, m/NV, lowers the strength, rf of the body and the curve is getting straight so that the transition from state, Pf = 0 to Pf = 1 has a character of a sudden jump. The influence of the micro-crack shape factor, b, and the way of space array on the total fracture probability, Pf, is illustrated by Fig. 3. The probability, Pf, is calculated by Eq. (6) for the volume corresponding to one cracked carbide, V = 1/NV; then Pf = 1 exp[pf(r)]. Fig. 6 illustrates how the probability of fracture, Pf, depends on the stress field singularity around a sharp crack tip. Experimentally assessed statistical distribution of fracture toughness, KIc, of the investigated steel at temperatures of 113 K and 143 K respectively are predicted using the integral form of Eq. (6) for HRR (Eq. (7)) or the elastic (Eq. (8)) stress field. A comparison of Pf(r) curves for elastic and HRR stress fields at 113 K (Fig. 4) demonstrates the affirmative effect of a small scale yielding on the growth of fracture toughness. The angle of the wedge of the active region ahead of the crack tip, u, affects the course of fracture probability only a little at low-temperatures. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Pf σ [GPa] T = 143 K β = π V = m / NV V = 10 mm3 m = 1 10 102 103 106 Fig. 2. Total probability of brittle fracture dependence on local stress in volumes corresponding to various number, m, of micro-cracks in Ni–Cr steel. 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 Pf σ [GPa] T = 143 K V = 1 NV perpendicular random 4/π π 4/π β = π Fig. 3. Dependence of brittle fracture probability on local stress in the volume encompassing one micro-crack for perpendicular and random orientation of cleavage planes with respect to local acting stress, as well as two various shapes of micro-cracks, (penny shaped, b = p, and through thickness, b = 4/p). B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836 1831
l832 B. Strnadel, P. Byczanski Engineering Fracture Mechanics 74(2007)1825-1836 HRR 113K 113K 90 02 143K MPa m"I Fig. 4. The total probability of brittle fracture predicted as a function of fracture toughness for two different temperatures considering elastic and small scale yielding HRR stress fields in investigated Ni-Cr steel Fig. 5 shows the influence of some parameters on the total probability of fracture, Pf. The fracture probability curve predicted for the temperature, T=143 K, corresponding to yield stress, oo= 582 MPa and eff= 23 Jm- when initiated micro-cracks are penny shaped, and B=T is complemented by other three curves. The first curve is calculated when the brittle fracture is initiated by the through-thickness micro-cracks of grain boundaries carbides, and the shape parameter, B=4/. In the second curve, the yield stress is higher by 100 MPa and the third one was made out when the effective surface energy in comparison to the original value was by 10 Jm- larger. It is obvious from the curves that both the lowering yield stress, Go, and increasing effective surface energy, 7ef, suppress the brittle fracture When the brittle fracture is initiated by through-thickness micro-cracks, the main crack propagation is 0.6 T=143K Weft=23 J/m2 Co=582 MPa K。DPam Fig. 5. Effects of increasing temperature, effective surface energy, and changes of micro-crack shapes as regards predicted statistical distribution of fracture toughness of steel tested at temperature of 143K
Fig. 5 shows the influence of some parameters on the total probability of fracture, Pf. The fracture probability curve predicted for the temperature, T = 143 K, corresponding to yield stress, r0 = 582 MPa and ceff = 23 Jm2 when initiated micro-cracks are penny shaped, and b = p is complemented by other three curves. The first curve is calculated when the brittle fracture is initiated by the through-thickness micro-cracks of grain boundaries carbides, and the shape parameter, b = 4/p. In the second curve, the yield stress is higher by 100 MPa and the third one was made out when the effective surface energy in comparison to the original value was by 10 Jm2 larger. It is obvious from the curves that both the lowering yield stress, r0, and increasing effective surface energy, ceff, suppress the brittle fracture. When the brittle fracture is initiated by through-thickness micro-cracks, the main crack propagation is easier. 0 100 200 300 0 0.2 0.4 0.6 0.8 1 Pf KIc [MPa m1/2] elast. 113 K 1o HRR 113 K 90o HRR 143 K 60o exp. 113 K exp. 143 K Fig. 4. The total probability of brittle fracture predicted as a function of fracture toughness for two different temperatures considering elastic and small scale yielding HRR stress fields in investigated Ni–Cr steel. 0 100 200 300 400 0 0.2 0.4 0.6 0.8 1 Pf KIc [MPa m1/2] σ0 = 582 MPa γ eff = 23 J/m2 T = 143 K β = π exp. pred. β = 4/π σ0 +← 100 MPa γ eff +← 10 J/m2 Fig. 5. Effects of increasing temperature, effective surface energy, and changes of micro-crack shapes as regards predicted statistical distribution of fracture toughness of steel tested at temperature of 143 K. 1832 B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836
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 effective
5. 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 affecting the total probability of the fracture, is taken into account. It is shown in Eq. (3) that this factor can substantially 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 toughness of the steel. As it has been analyzed earlier, the actual strength or fracture toughness of the material could vary depending on micro-structural parameters. In addition to this fact, it is usually difficult to precisely predict the external 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 component 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
l834 B. Strnadel, P. Byczanski Engineering Fracture Mechanics 74(2007)1825-1836 The local maximum effective stress distribution, l(emax), can be expressed in its easiest form as Pi(Emax)=Bo-oooEomal exp[-(demax/ao)Fo] where ao and Bo are size, and/or shape parameters depending on the loading conditions. In Figs. 6-8 graphic examples of the probability dependence of brittle fracture, Pr= Pr(emax> or) Eq (10)as a function of hom- ogenously stressed volume, V, and parameters, o, Po, are given. In these three graphics, there are always for two selected values, ao= 1.5 GPa, Bo=3, and/or V= 100 mm, a third value is taken as an independent var iable Size parameter, a0= 1.5 GPa, has been chosen so that upper limits of the acting stress, emax, of prob ability density, (p(Emax), would intervene with distributions of the lowest values of local cleavage strength, P2(or The first low option for the parameter, 00= 1.5 GPa, corresponds to relatively high value(4.4 um) of the micro-crack initiated (Eq(1)). Nevertheless, the initial shape parameter low value, Bo= 3, extends satisfactorily the value range, Emax, so that a measurable intersection for the intervals, emax and af, arises perp 3J/m2 夏=1.5GPa V= 100 mm -20 B Fig. 7. Probability of brittle fracture of Ni-Cr steel as a function of loading spectrum parameter Po, and for different values of the effective surface energy, ieff 14 o Yeff =23 J/m2 rand V(mml Fig. 8. Probability of brittle fracture of Ni-Cr steel as a function of loaded volume, and for different values of the effective surface
The local maximum effective stress distribution, u1(remax), can be expressed in its easiest form as: u1ðremaxÞ ¼ b0ab0 0 rb01 emax exp½ðremax=a0Þ b0 ð12Þ where a0 and b0 are size, and/or shape parameters depending on the loading conditions. In Figs. 6–8 graphic examples of the probability dependence of brittle fracture, Pf = Pr(remax P rf) Eq. (10) as a function of homogenously stressed volume, V, and parameters, a0, b0, are given. In these three graphics, there are always for two selected values, a0 = 1.5 GPa, b0 = 3, and/or V = 100 mm3 , a third value is taken as an independent variable. Size parameter, a0 = 1.5 GPa, has been chosen so that upper limits of the acting stress, remax, of probability density, u1(remax), would intervene with distributions of the lowest values of local cleavage strength, u2(rf). The first low option for the parameter, a0 = 1.5 GPa, corresponds to relatively high value (4.4 lm) of the micro-crack initiated (Eq. (1)). Nevertheless, the initial shape parameter low value, b0 = 3, extends satisfactorily the value range, remax, so that a measurable intersection for the intervals, remax and rf, arises. 0 200 400 600 800 1000 –9 –8 –7 –6 –5 –4 –3 log10 Pr( σ emax ≥ σf ) V [mm3 ] α0 = 1.5 GPa β0 = 3 γ eff = 23 J/m2 23 14 rand. perp. perp. Fig. 8. Probability of brittle fracture of Ni–Cr steel as a function of loaded volume, and for different values of the effective surface energy, ceff. 2 2.5 3 3.5 4 –20 –15 –10 –5 log10 Pr( σ emax ≥ σf ) β0 α0 = 1.5 GPa V = 100 mm3 γ eff = 23 J/m2 23 14 rand. perp. perp. Fig. 7. Probability of brittle fracture of Ni–Cr steel as a function of loading spectrum parameter b0, and for different values of the effective surface energy, ceff. 1834 B. Strnadel, P. Byczanski / Engineering Fracture Mechanics 74 (2007) 1825–1836
