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Probability of brittle failure in different geometries using a simplified constraint based local criterion method
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Probability of brittle failure in different geometries using a simplified constraint based local criterion method
T
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Ahmad Mohammadi Najafabadia, Farid Reza Biglaria, , Kamran Nikbinb a b
Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Avenue, Tehran, Iran Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2Az, United Kingdom
A R T I C LE I N FO
A B S T R A C T
Keywords: Fracture toughness Crack tip constraints Local approach T-stress
This paper introduces a novel calibration method for the original Beremin model to predict cleavage fracture in various fracture mechanics geometries with differing crack tip constraints. Calibration of Weibull parameters were conducted with respect to the goodness of fitness and the probability of the fracture toughness distributions for low and high constraint specimens. By introducing a reference diagram for a specific material based on the Tstress as a crack tip constraint parameter, the applicability of calibrated parameters was investigated for the experimental data of specimens with different geometry. The findings indicated that the ratio of Weibull stresses to associated calibrated scale parameters (here denoted as σw,R) at fracture loads for specimens with different shapes were in a same range. Therefore, by calculating T-stress as crack tip constraint for a typical geometry, a suitable prediction of probability of failure would be possible with respect to the reference diagram.
1. Introduction
cleavage fracture of specimens based on the micromechanics of failure at the onset of plastic deformation of the crack tip region. It is assumed that micro-cracks could be formed as the plastic deformation begins and unstable crack growth would occur when the maximum principal stress reaches sufficiently high level. The Beremin model introduces Weibull stress which is based on the Weibull distribution to predict failure probability of components. The values of the Weibull stress for a specified geometry are to be calculated from a Finite Element Analysis (FEA) at different load levels. It should be noted that shape parameters for the probability distribution has to be determined based on the measured fracture data. Values of the shape parameter are reported from 1 to 50 [7,8]. Constraint effect is usually described as dependence of fracture toughness values on specimen configuration, size and loading conditions which influence on the characterization of stress fields in the vicinity of the crack tip. As the crack front fields are not uniquely defined by the classical fracture mechanics parameters (K or J), fracture behavior of metallic materials is influenced by higher-order non-singular elastic and elastic plastic stress distribution [9–11]. To assess the effect of specimen shape and loading mode on fracture toughness, the concept of including T-stress or Q-parameter additionally to K or J respectively, provides an essentially more realistic description of the crack-tip stress field [9–11] and thus, results in different two-parameter concepts like K-T or J-Q to failure assessment. The importance of constraint effects
Catastrophic cleavage fracture has been the subject of current researches in structural integrity assessments of structures as it is the most severe failure modes in components constructed of ferritic steels [1,2]. Cleavage fracture preferentially occurs over specific crystallographic planes. In the case of BCC metals, as ferritic steels, cleavage occurs predominantly on the [1 0 0] plane [3]. The micromechanics of cleavage fractures are based on the weakest link theory which assumes that a body of material could be fragmented to many independent volumes, linked together like a chain, and the failure of the whole specimen occurs when its weakest element link fails. As a result of the highly localized cleavage mechanism and the microstructural inhomogeneity of the material, cleavage fracture in steels is of a statistical nature and therefore the scatter in fracture toughness results will behave similarly. There exist two main approaches to assess the integrity of mechanical structures constructed of ferritic steels. The global approach assumes that fracture toughness could be determined by a single (eventually two) parameter, such as KIc or JIc. On the other hand, the local approach to fracture requires detailed metallographic evaluation of the samples (such as grain size, shape, orientation, phase structure, second-phase particle size and spacing, grain boundary composition, etc.) as well as advanced FEM calculations [4–6]. The Beremin model [4] is a statistical model for predicting scatter in
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Corresponding author. E-mail address: [email protected] (F.R. Biglari).
https://doi.org/10.1016/j.tafmec.2019.102331 Received 14 April 2019; Received in revised form 29 July 2019; Accepted 16 August 2019 Available online 17 August 2019 0167-8442/ © 2019 Elsevier Ltd. All rights reserved.
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Nomenclature A a i J JIc K KIc l m N P Pf Q
r T T V0 Vp W α β
test statistic of Komolgorov-Smirnov test crack or notch length position of a given test Contour integral critical fracture toughness in terms of the J integral stress intensity factor plane strain fracture toughness micro-crack sizes Weibull modulus total number of tests cumulative distribution function (CDF) failure probability hydrostatic parameter quantifying the level of crack-tip
ξ σij σu σw σw,R
and specifically the two-parameter fracture mechanics approaches has been extensively studied for a long time [12–14]. For example, compact tension (CT) specimen, is usually of high crack-tip constraint, while some other specimens or real cracked components have low crack-tip constraint [15]. Therefore, the constraint effect on the fracture toughness might be modified so that the fracture toughness measured in laboratory could be applied to real cracked structures. The different factors influencing the in-plane constraint are associated with crack size (a/W), geometry of specimen, type of loading and notch. The out-ofplane constraint is changed with thickness [13,16]. Some Authors have suggested that failure prediction by means of the K-T or J-Q concepts is consistent with a more general Weibull stress approach [7,17–19]. Gao et al. [7] described an approach in which a three-parameter Weibull distribution instead of the two-parameter form was employed with regard to measured fracture toughness data obtained for high constraint and low constraint configurations. The third parameter, known as threshold stress, considered as an engineering distribution which parallels that of adopted in ASTM E-1921 for use with measured KJ c-values, below which fracture cannot occur. [7,20]. Sometimes, a modified expression of the Weibull distribution is applied, through directly incorporating the threshold stress into the definition of the Weibull stress [21,22]. Despite its wide use, the three-parameter Weibull distribution has not been explicated theoretically [7]. In this paper, due to a level of ambiguity in using the three parameter Weibull distribution [7], the relationship between the original Beremin model (two parameter Weibull distribution) and constraint effects of high constraint and low constraint specimens is investigated. Due to some shortcoming in the Maximum Likelihood method and the Linear Least Squares (LLS) method [23], a novel calibration method based on Non-linear Least Squares method and Anderson-Darling test is introduced to obtain a unique parameters for the original Beremin model as material and geometry constants. Thus, the dependence of the Weibull parameters and T-stress as a constraint parameter of the material with elastic behavior are examined.
Table 1 Critical value at significance level in the Anderson-Darling test for Weibull distribution [28]. 0.1
0.05
0.025
0.01
A2crit
1.062
1.321
1.591
1.959
θ=0
θ=± π
θ=±
T = σxx − σyy
T = σxx
T=
π 3
1 (σ 3 xx
θ=±
− σyy )
T=
π 2
1 (σ 3 xx
θ=±
− σyy )
2π 2
T = σxx − σyy
to the integral of a weighted value of the maximum principal (tensile) stress (σ1) over the plastic zone for cleavage fracture.
1 σw = ⎡ ⎢ V0 ⎣
∫V
p
1 m
σ1m dV ⎤ ⎥ ⎦
(2)
In Eq. (2), Vp denotes the volume of the cleavage fracture process zone, V0 is a reference volume and m represents the Weibull modulus or shape parameter of the probability density function for microscopic cracks in the fracture process zone, m = 2β − 2. In Eq. (1), σu represents the reference stress or scale parameter of the Weibull distribution corresponding to a failure probability of 63.2%. 2.1. A novel calibration method In order to employ the local approach model, it is necessary to determine reliable values for the Weibull distribution constants. Traditionally, this is achieved by selecting values which maximizes the agreement between the Beremin model and each experimental set of failure data. Selecting an algorithm for the fitting process can have a considerable effect on the determined model parameters. Therefore, any predictions of failure in components is affected by the fitting parameters. The variation in m and σu obtained by the fitting process can be strongly influenced by both the algorithm used to determine best values and the data used for calibration. The validity of the calibration parameters for a model can be tested by its ability to predict failure in other specimens or load cases. The calibration issue of the Beremin model parameters and the impact of the related data have been studied by many authors [24–27]. It has been shown that using common methods like matching m by maximum likelihood estimation (MLE) for original Beremin model tends to overestimate m and sometimes the supposed values do not converge for some groups of specimens [24,27]. Despite the common
Based on the assumption that fracture is nucleated at carbides or other similar inclusions, the probability density function for variation of micro-crack sizes is p(l) = ξ/lβ, where ξ is a constant and β defines the shape of the micro-crack distribution. Beremin [4] derived a twoparameter Weibull distribution known as the local approach to predict the cumulative failure probability of cleavage fracture in low-alloy steel based on the weakest link theory: m
⎜
Upper tail percentage level α
Table 2 T-stress values according to measurement direction.
2. Beremin model
σ Pf (σw ) = 1 − exp ⎡−⎛ w ⎞ ⎤ ⎢ ⎝ σu ⎠ ⎥ ⎣ ⎦
constraint distance from crack tip linear elastic T-stress temperature reference volume volume of the cleavage fracture process zone width of a specimen a parameter related to a crack configuration a parameter defines the shape of the micro-crack distribution material constant stress tensor reference stress Weibull stress reference Weibull stress
⎟
(1)
where m and σu are Weibull modulus and reference stress, σw is referred 2
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Fig. 1. Schematic illustration of (a) C(T) specimens, (b) SE(B) Specimens with different a/W of A516 steel (dimensions in mm). 900
Table 3 Measured cleavage fracture toughness values of three set of specimens, examined at the cryogenic temperatures. Specimen number
Fracture Load [kN]
KIC [MPa⋅m0.5]
C(T) (a/W = 0.5)
1 2 3 4 5 6
16.7 22.0 27.3 29.8 31.3 32.9
28.9 38.0 47.2 51.5 54.0 56.9
1 2 3 4 5 6
18.1 24.6 25.2 31.4 34.6 35.9
34.5 46.9 48.0 59.8 65.9 68.4
1 2 3 4 5 6
79.6 93.2 118.4 128.2 133.2 147.1
48.3 56.5 71.8 77.7 80.7 89.1
SE(B) (a/W = 0.5)
SE(B) (a/W = 0.1)
700 600
Stress (MPa)
Specimen Geometry
800
500 400 300 200
---
100 0 0
0.05
0.1
Strain
0.15
0.2
Fig. 3. True stress–strain curves for A516 Gr. 70 steel measured at T = −174 °C and 20 °C.
methods, fitting of data is conducted by Non-linear Least Squares method and a statistics test which is a given sample of data drawn from a given probability distribution. Among different goodness-of-fit techniques (e.g. Komolgorov-Smirnov, Anderson-Darling, Shipiro-Wilk, von Mises), the Anderson-Darling test is preferred because it is more sensitive to deviations in the tails of the distribution than is the older Komolgorov-Smirnov test. The two hypotheses for the Anderson-
Fig. 2. Schematic illustration of (a C(T)10-A533B (a/W = 0.5) and C(T)20-A533B (a/W = 0.5) specimens, (b) RNB(V45)-A533B Specimens (dimensions in mm) [48]. 3
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Darling test for the normal distribution are given below: H0: The data follows the normal distribution H1: The data do not follow the normal distribution The null hypothesis is that the data are normally distributed; the alternative hypothesis is that the data are non-normal. For the Weibull distributions, the test statistic, A2 is calculated from [28]:
1 A2 = −n − ⎛ ⎞ ⎝n⎠
n
∑ (2i − 1)[ln(Pi) + ln(1 − Pn−i +1)] i=1
(3)
where above expression follows normal distribution, but P is the CDF for the distribution under consideration. For the Weibull distribution, P is calculated form the Eq. (1) for every increment of loading. The above formula needs to be modified for small samples. A table has been listed in Ref. [28] designated as “Table 4.17” in page 146 that describes above equation for small samples. In this reference, Eq. (4) accommodates modified version of above equation for small samples.
0.3 ⎞ Am2 = A2 ⎛1 + n ⎠ ⎝
(4)
and then compared to an appropriate critical value from Table 1. The fitting process is conducted inside a Python code. The fitting routine uses the curve_fit method of the scipy.optimize library [29], where one has a parameterized model function meant to explain some phenomena and wants to adjust the numerical values for the model so that it most closely matches some data. Experimental data are arranged based on the fracture load, and assigned estimated failure probabilities using a well-known statistical ranking equation:
Fig. 4. Details of the finite element modeling and meshing of the specimens in ABAQUS software (only one quarter of the specimens was modeled).
P fri =
i − 0.5 N
(5)
where i is the position of a given test in the ranked data set and N is the total number of tests. Determining m and associated σu by the goodness of fitness technique requires a grid search approach. Grid search is a method to perform hyper-parameter optimization. It is a method to find the best combination of hyper-parameters. In this situation, there are different combination of hyper-parameters which correspond to a single model, that can be said to lie on a point of a “grid”. Here, fitting the data of experiments with the Beremin model gives different values for m which in turn gives different values for σw. The σw is a parameter of the Beremin model. The goal is then to train each of these combinations of m and σw and evaluate them using cross-validation to select best fitted values for Beremin model. Values of σw are calculated for a range of m. Then a corresponding σu is found from the fits.
Fig. 5. Calibration of the shape parameter of Weibull distribution with Anderson-Darling test for A516 specimens.
3. T-stress as crack tip constraint Williams [30] argues that the stress distribution as r → 0 depends on the polar angle θ. However, for some particular θ angles (Table 2), the T-stress is obtained by the method proposed by Yang et al., [31]. The method, the Stress Difference Method (SDM) incorporated the iterative single dual-boundary element method and a tip impose zero displacement jump at the crack tip [32]. Particularly for θ = 0, the T-stress is given by:
T = (σxx − σyy )θ = 0
(6)
where σyy is the opening stress and σxx refers to the stress parallel to the crack. The T-stress represents a stress parallel to the crack faces. Although the T-stress is calculated from the linear elastic material properties, but also it is a useful quantity in elastic-plastic fracture studies. Larsson et al. [9] demonstrated that the T-stress play significant roles in fracture
Fig. 6. Calibration of the shape parameter of Weibull distribution with Anderson-Darling test for A533B specimens.
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Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Fig. 7. Comparison of (σw/σu-calibrated) and (T/σy) over different m values for (a) C(T)25-A516 (a/W = 0.5), (b) SE(B)25-A516 (a/W = 0.5) and (c) SE(B)25-A516 (a/ W = 0.1) specimens.
4. Experiments
mechanics and can have a significant effect on the plastic zone shape and size and that the small plastic zones in various sizes of specimens can be predicted adequately by including the T-stress as a second cracktip parameter. Crack tip constraint which is a result of triaxial stress state at the crack tip affects the fracture toughness of a material. Some investigations [10,33–37] showed that the T-stress can correlate well with the tensile stress triaxiality of elastic-plastic crack-tip fields. The following studies [18,38–44] demonstrated that the apparent fracture toughness of the material varied with specimen geometry or the constraint level, and nonsingular T-stress term needs to be accounted for to accurately measure the fracture resistance in FGM. A key feature of using Weibull stress in Beremin model is that under increased remote loading, differences in evolution of the Weibull stress incorporates both the effects of stressed volume (the fracture process zone) and the potentially strong variation in crack front stress fields due to constraint loss which thus provides the necessary framework to correlate fracture toughness for varying crack depth, section thickness, specimen size, crack geometry and loading configuration. Previous research efforts to overcome the transferability issue is attempted by developing a model to elastic-plastic fracture toughness values depend on the Weibull stress as a crack-tip driving force [18,20,45–47].
The practicality of the proposed parameter calibration method for predicting failure in the specimens with different levels of constraints were studied in this work. The relation between calibrated parameters with elastic T-stress as a constraint effects of a crack tip were also considered for testing materials. Three different specimen geometries of A516 ferritic steel were chosen to cover different crack tip constraints. All the experiments were carried out in −174 °C, which was achieved by sinking the specimens in the liquid nitrogen. To ensure constant temperature during loading to fracture, two thermocouples was used. First one was fixed to the specimen and the other used to monitor and control the environmental temperature of the nitrogen inside the container. The thermocouple voltage was used to determine the current specimen temperature. All of the experiments were conducted when a steady state specimen temperature had been reached. The reason to choose this temperature is that the materials show near elastic behavior in their failures when cooled down to very low temperatures and −174 °C was the lowest possible temperature that could be reached in our Lab. Experimental standard tensile response of material obtained from −174 °C tests will be used in FE simulations. All test specimens 5
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Fig. 8. Comparison of (σw/σu-calibrated) and (T/σy) over different m values for (a) C(T)20-A533B (a/W = 0.5), (b) C(T)10-A533B (a/W = 0.5),. and (c) RNB(V45)A533B specimens.
Fig. 9. Prediction and measured fracture toughness of the high and low constraint specimens with the calibrated parameters for A516 specimens.
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Theoretical and Applied Fracture Mechanics 104 (2019) 102331
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Fig. 10. Prediction and measured fracture toughness of the high and low constraint specimens with the calibrated parameters for A533B specimens.
Fig. 11. σw,R vs. T/σy for the three sets of A516 specimens with different levels of constraints.
Fig. 12. σw,R vs. T/σy for the three sets of A533B specimens with different levels of constraints.
specimens, and six SE(B) specimens with a/W = 0.1 as low level constraint specimens were examined at cryogenic temperatures (Fig. 1). Measured fracture toughness data of tested specimens are presented in Table 3. In addition to the above fracture tests, three other sets of A533B
were chosen with large enough dimensions to give predominantly plane strain conditions at the crack tip. To control the initial crack length, a notch in all specimens was introduced by EDM wire of 0.1 mm in radius. Three experimental data sets divided into six C(T) specimens with a/W = 0.5, six SE(B) specimens with a/W = 0.5 as high level constraint 7
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Fig. 14. Prediction and measured fracture toughness of (a) C(T)20-A533B (a/ W = 0.5), (b) C(T)10-A533B (a/W = 0.5) and (c) RNB(v45)-A533B specimens with respect to reference diagram of A533B material.
Fig. 13. Prediction and measured fracture toughness of (a) C(T)25-A516 (a/ W = 0.5), (b) SE(B)25-A516 (a/W = 0.5) and (c) SE(B)25-A516 (a/W = 0.1) specimens with respect to reference diagram of A516 material.
Appropriate boundary conditions were applied in symmetry faces. The mesh along the crack path should be very fine to provide adequate resolution of the stress fields. For all of the specimens, a regular mesh configuration has been adopted initially. A focused ring of elements in the radial direction surrounding the crack tip was used. A set of trial mesh with higher densities were employed during mesh sensitivity analysis for the Weibull stress in different load levels. The focused element configuration was radially identical around the crack tip to ensure accurate calculation of opening stresses. The finite element models and its refined mesh are presented in Fig. 4.
specimens were also analyzed. The first two sets of fracture tests are C(T) specimens of 10 mm and 20 mm thickness (Fig. 2a). The sample size is N = 15 for the C(T)10-A533B (a/W = 0.5) specimens and N = 24 for the C(T)20-A533B (a/W = 0.5) specimens [48]. The third data set is related to round notched bar specimens (the sample number is N = 12) with a V-notch of 45 degree groove (RNB (V45)-A533B), depicted in Fig. 2b. 5. FE modeling
6. Results and discussion
Nonlinear analyses of different geometries were conducted using 3D finite element modelling in ABAQUS V6.14. The material properties of A516 steel followed true stress–strain data measured at T = −174 °C, with the isotropic hardening behavior shown in Fig. 3. In the case of A533B specimens, the material properties are given in Ref. [49]. Due to geometric and loading symmetry, only one quarter of C(T) and SE(B) specimens and one eighth of RNB specimens were modeled.
Calibration of Weibull parameters were conducted by the novel method described in Section 2.1. A python script was developed for post-processing of the output data obtained from FE analyses in ABAQUS to calibrate Weibull parameters of examined data. Results of A2 parameter calculation for A516 and A533B specimens are shown in 8
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Fig. 5 and Fig. 6, respectively. A2 parameter results are used to select shape parameter of the Weibull distribution. A 95% confidence has been reported in Fig. 5 for Anderson-Darling test with a range of m varying from 1 to 30. However, A2 parameter results illustrated in Fig. 6 shows a 99% confidence. For A516 specimens all of (m, σu) data pairs can predict failure probability of 95% and 99% confidence. The use of minimum values of A2 parameters in Fig. 5 and Fig. 6 is not recommended. This is due to sensitivity of the Beremin model to m values and inevitable uncertainty in experimental test results. As it is discussed in Section 3, T-stress can affect both plastic zone size and crack tip constraint effects which are used in calculation of Weibull stress. So, there should be a relationship between these two parameters. σu is calibrated for each set of specimen configuration by a given m value and is stored into σu-calibrated. Figs. 7 and 8 compare the ratio of σw/σu-calibrated which is denoted as σw,R and named as reference Weibull stress versus T/σy for each set of data in fracture loads for different pairs of (m, σu-calibrated). As the figures show, there are a linear relationship between the reference Weibull stress and T-stress. Assuming this relationship, best pair of (m, σu-calibrated) is obtained where the intercept of the linear regression line tend to pass from zero. This is due to the fact that the value of both Weibull stresses and calculated T-stresses are directly related to the loading values. Before applying any load on the specimen, since there is no plastic zone at the crack tip, therefore the Weibull stress equals zero. In similar circumstances, the value of calculated T-stress is also equals zero. According to the figures, the intercept of the regression lines of m = 4 for all three specimen geometries of the A516 material are zero. Therefore, m = 4 can be selected as a material constant, regardless of its crack configuration for this material. This is in consistent with the previous reports in the literature [48]. The σu-calibrated values of 2089, 3199, and 4074 MPa are obtained for C(T)25-A516 (a/W = 0.5), SE(B)25-A516 (a/W = 0.5), and SE(B)25A516 (a/W = 0.1) specimens respectively. In the case of A533B material, both sets of C(T)20-A533B (a/ W = 0.5) and C(T)10-A533B (a/W = 0.5) specimens the plots of σw/σucalibrated intercept zero for m = 4. In contrast, the best regression line of m = 7 intercepts zero for RNB (V45)-A533B specimens. In this paper, it is assumed that m = 4 can be selected as the material constant and the case of m = 7 is ignored due to high constraint specimens. The results of failure probability estimation by this parameter will be discussed further. Figs. 9 and 10 show the probability of failures of different data sets according to the best calibrated m (equal to 4) and σu obtained by the calibration method presented in this paper. Figs. 11 and 12 show σw,R with respect to T/σy for A516 and A533B specimens, respectively. These figures have been generated using m = 4 as calibrated shape parameter. As they show, for all specimens, σw,R is between a range of minimum and maximum values which have a linear relationship with T-stress for each specimen configuration. As a result, a line with a slope of α which is drawn from the origin of this reference diagram represent each crack configuration. Thus, σw,R could be determined by knowing a given α which is slope of the line related to a crack configuration, and calculating T-stress. Then σu-calibrated can be obtained from σw,R/αT. As previously stated [7], the value of σw can reflect the potentially strong variations in crack-front stress fields due to the effects of constraint loss and volume sampling. The term σw,R/αT gives the scale parameter which can be interpreted as a fraction of Weibull stress related to a crack configuration which is participate to failure. The minimum and maximum values of the σw,R for A516 material including all different constraint specimens are between a range of 0.55 and 1.17, respectively. The 0.54 and 1.23 values are obtained for A533B material, respectively. In this article, mid-ranges of the data sets were used for even distribution of data. The mid-ranges of 0.86 for the A516 and 0.88 for A533B specimens are obtained for σw,R respectively.
Figs. 13 and 14 show prediction of failure probability based on the reference diagram for each data sets of A516 and A533B material. Each diagram contains two plots which show prediction just by lowest and highest fracture load of each data sets. Thus, it is concluded that failure predictions will be in the range based on those specimens with their fracture loads bounded by the minimum and maximum of the data sets. Predictions based on the mid-ranges are also shown in Figs. 9 and 10 which are almost similar to the prediction. Therefore, it is assumed that the failure probability occurs between minimum and maximum range of failure prediction by calculating T-stress for a given crack configuration and midrange of σw,R of reference diagram. Assessing the method for different material and different crack configuration shows a reasonable agreement between the fracture tests and this method. 7. Conclusions In this study, a novel calibration method for the original Beremin model was introduced, tested, and verified through experimental tests carried out in different geometries and sample sizes. Summary of the main results are as follows:
• Introducing a novel robust method for calibrating unique Weibull • • • •
parameters, so as the original model assumed, the shape parameter are to be a material constant. There is a direct relation between the scale parameter of Weibull distribution and T-stress as constraint effects. Thus, in the real components the constraint of the cracked components should be considered. By introducing a reference diagram, a reasonable failure prediction is obtained for different shape and different crack configurations. It is concluded that the failure probability occurs between a minimum and maximum range of failure prediction by calculating T-stress for a given crack configuration and midrange of σw,R of reference diagram. Assessing the method for different material and different crack configuration shows a reasonable agreement between the fracture tests and this method.
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Probability of brittle failure in different geometries using a simplified constraint based local criterion method
T
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Ahmad Mohammadi Najafabadia, Farid Reza Biglaria, , Kamran Nikbinb a b
Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Avenue, Tehran, Iran Department of Mechanical Engineering, Imperial College London, Exhibition Road, London SW7 2Az, United Kingdom
A R T I C LE I N FO
A B S T R A C T
Keywords: Fracture toughness Crack tip constraints Local approach T-stress
This paper introduces a novel calibration method for the original Beremin model to predict cleavage fracture in various fracture mechanics geometries with differing crack tip constraints. Calibration of Weibull parameters were conducted with respect to the goodness of fitness and the probability of the fracture toughness distributions for low and high constraint specimens. By introducing a reference diagram for a specific material based on the Tstress as a crack tip constraint parameter, the applicability of calibrated parameters was investigated for the experimental data of specimens with different geometry. The findings indicated that the ratio of Weibull stresses to associated calibrated scale parameters (here denoted as σw,R) at fracture loads for specimens with different shapes were in a same range. Therefore, by calculating T-stress as crack tip constraint for a typical geometry, a suitable prediction of probability of failure would be possible with respect to the reference diagram.
1. Introduction
cleavage fracture of specimens based on the micromechanics of failure at the onset of plastic deformation of the crack tip region. It is assumed that micro-cracks could be formed as the plastic deformation begins and unstable crack growth would occur when the maximum principal stress reaches sufficiently high level. The Beremin model introduces Weibull stress which is based on the Weibull distribution to predict failure probability of components. The values of the Weibull stress for a specified geometry are to be calculated from a Finite Element Analysis (FEA) at different load levels. It should be noted that shape parameters for the probability distribution has to be determined based on the measured fracture data. Values of the shape parameter are reported from 1 to 50 [7,8]. Constraint effect is usually described as dependence of fracture toughness values on specimen configuration, size and loading conditions which influence on the characterization of stress fields in the vicinity of the crack tip. As the crack front fields are not uniquely defined by the classical fracture mechanics parameters (K or J), fracture behavior of metallic materials is influenced by higher-order non-singular elastic and elastic plastic stress distribution [9–11]. To assess the effect of specimen shape and loading mode on fracture toughness, the concept of including T-stress or Q-parameter additionally to K or J respectively, provides an essentially more realistic description of the crack-tip stress field [9–11] and thus, results in different two-parameter concepts like K-T or J-Q to failure assessment. The importance of constraint effects
Catastrophic cleavage fracture has been the subject of current researches in structural integrity assessments of structures as it is the most severe failure modes in components constructed of ferritic steels [1,2]. Cleavage fracture preferentially occurs over specific crystallographic planes. In the case of BCC metals, as ferritic steels, cleavage occurs predominantly on the [1 0 0] plane [3]. The micromechanics of cleavage fractures are based on the weakest link theory which assumes that a body of material could be fragmented to many independent volumes, linked together like a chain, and the failure of the whole specimen occurs when its weakest element link fails. As a result of the highly localized cleavage mechanism and the microstructural inhomogeneity of the material, cleavage fracture in steels is of a statistical nature and therefore the scatter in fracture toughness results will behave similarly. There exist two main approaches to assess the integrity of mechanical structures constructed of ferritic steels. The global approach assumes that fracture toughness could be determined by a single (eventually two) parameter, such as KIc or JIc. On the other hand, the local approach to fracture requires detailed metallographic evaluation of the samples (such as grain size, shape, orientation, phase structure, second-phase particle size and spacing, grain boundary composition, etc.) as well as advanced FEM calculations [4–6]. The Beremin model [4] is a statistical model for predicting scatter in
⁎
Corresponding author. E-mail address: [email protected] (F.R. Biglari).
https://doi.org/10.1016/j.tafmec.2019.102331 Received 14 April 2019; Received in revised form 29 July 2019; Accepted 16 August 2019 Available online 17 August 2019 0167-8442/ © 2019 Elsevier Ltd. All rights reserved.
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Nomenclature A a i J JIc K KIc l m N P Pf Q
r T T V0 Vp W α β
test statistic of Komolgorov-Smirnov test crack or notch length position of a given test Contour integral critical fracture toughness in terms of the J integral stress intensity factor plane strain fracture toughness micro-crack sizes Weibull modulus total number of tests cumulative distribution function (CDF) failure probability hydrostatic parameter quantifying the level of crack-tip
ξ σij σu σw σw,R
and specifically the two-parameter fracture mechanics approaches has been extensively studied for a long time [12–14]. For example, compact tension (CT) specimen, is usually of high crack-tip constraint, while some other specimens or real cracked components have low crack-tip constraint [15]. Therefore, the constraint effect on the fracture toughness might be modified so that the fracture toughness measured in laboratory could be applied to real cracked structures. The different factors influencing the in-plane constraint are associated with crack size (a/W), geometry of specimen, type of loading and notch. The out-ofplane constraint is changed with thickness [13,16]. Some Authors have suggested that failure prediction by means of the K-T or J-Q concepts is consistent with a more general Weibull stress approach [7,17–19]. Gao et al. [7] described an approach in which a three-parameter Weibull distribution instead of the two-parameter form was employed with regard to measured fracture toughness data obtained for high constraint and low constraint configurations. The third parameter, known as threshold stress, considered as an engineering distribution which parallels that of adopted in ASTM E-1921 for use with measured KJ c-values, below which fracture cannot occur. [7,20]. Sometimes, a modified expression of the Weibull distribution is applied, through directly incorporating the threshold stress into the definition of the Weibull stress [21,22]. Despite its wide use, the three-parameter Weibull distribution has not been explicated theoretically [7]. In this paper, due to a level of ambiguity in using the three parameter Weibull distribution [7], the relationship between the original Beremin model (two parameter Weibull distribution) and constraint effects of high constraint and low constraint specimens is investigated. Due to some shortcoming in the Maximum Likelihood method and the Linear Least Squares (LLS) method [23], a novel calibration method based on Non-linear Least Squares method and Anderson-Darling test is introduced to obtain a unique parameters for the original Beremin model as material and geometry constants. Thus, the dependence of the Weibull parameters and T-stress as a constraint parameter of the material with elastic behavior are examined.
Table 1 Critical value at significance level in the Anderson-Darling test for Weibull distribution [28]. 0.1
0.05
0.025
0.01
A2crit
1.062
1.321
1.591
1.959
θ=0
θ=± π
θ=±
T = σxx − σyy
T = σxx
T=
π 3
1 (σ 3 xx
θ=±
− σyy )
T=
π 2
1 (σ 3 xx
θ=±
− σyy )
2π 2
T = σxx − σyy
to the integral of a weighted value of the maximum principal (tensile) stress (σ1) over the plastic zone for cleavage fracture.
1 σw = ⎡ ⎢ V0 ⎣
∫V
p
1 m
σ1m dV ⎤ ⎥ ⎦
(2)
In Eq. (2), Vp denotes the volume of the cleavage fracture process zone, V0 is a reference volume and m represents the Weibull modulus or shape parameter of the probability density function for microscopic cracks in the fracture process zone, m = 2β − 2. In Eq. (1), σu represents the reference stress or scale parameter of the Weibull distribution corresponding to a failure probability of 63.2%. 2.1. A novel calibration method In order to employ the local approach model, it is necessary to determine reliable values for the Weibull distribution constants. Traditionally, this is achieved by selecting values which maximizes the agreement between the Beremin model and each experimental set of failure data. Selecting an algorithm for the fitting process can have a considerable effect on the determined model parameters. Therefore, any predictions of failure in components is affected by the fitting parameters. The variation in m and σu obtained by the fitting process can be strongly influenced by both the algorithm used to determine best values and the data used for calibration. The validity of the calibration parameters for a model can be tested by its ability to predict failure in other specimens or load cases. The calibration issue of the Beremin model parameters and the impact of the related data have been studied by many authors [24–27]. It has been shown that using common methods like matching m by maximum likelihood estimation (MLE) for original Beremin model tends to overestimate m and sometimes the supposed values do not converge for some groups of specimens [24,27]. Despite the common
Based on the assumption that fracture is nucleated at carbides or other similar inclusions, the probability density function for variation of micro-crack sizes is p(l) = ξ/lβ, where ξ is a constant and β defines the shape of the micro-crack distribution. Beremin [4] derived a twoparameter Weibull distribution known as the local approach to predict the cumulative failure probability of cleavage fracture in low-alloy steel based on the weakest link theory: m
⎜
Upper tail percentage level α
Table 2 T-stress values according to measurement direction.
2. Beremin model
σ Pf (σw ) = 1 − exp ⎡−⎛ w ⎞ ⎤ ⎢ ⎝ σu ⎠ ⎥ ⎣ ⎦
constraint distance from crack tip linear elastic T-stress temperature reference volume volume of the cleavage fracture process zone width of a specimen a parameter related to a crack configuration a parameter defines the shape of the micro-crack distribution material constant stress tensor reference stress Weibull stress reference Weibull stress
⎟
(1)
where m and σu are Weibull modulus and reference stress, σw is referred 2
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
A. Mohammadi Najafabadi, et al.
Fig. 1. Schematic illustration of (a) C(T) specimens, (b) SE(B) Specimens with different a/W of A516 steel (dimensions in mm). 900
Table 3 Measured cleavage fracture toughness values of three set of specimens, examined at the cryogenic temperatures. Specimen number
Fracture Load [kN]
KIC [MPa⋅m0.5]
C(T) (a/W = 0.5)
1 2 3 4 5 6
16.7 22.0 27.3 29.8 31.3 32.9
28.9 38.0 47.2 51.5 54.0 56.9
1 2 3 4 5 6
18.1 24.6 25.2 31.4 34.6 35.9
34.5 46.9 48.0 59.8 65.9 68.4
1 2 3 4 5 6
79.6 93.2 118.4 128.2 133.2 147.1
48.3 56.5 71.8 77.7 80.7 89.1
SE(B) (a/W = 0.5)
SE(B) (a/W = 0.1)
700 600
Stress (MPa)
Specimen Geometry
800
500 400 300 200
---
100 0 0
0.05
0.1
Strain
0.15
0.2
Fig. 3. True stress–strain curves for A516 Gr. 70 steel measured at T = −174 °C and 20 °C.
methods, fitting of data is conducted by Non-linear Least Squares method and a statistics test which is a given sample of data drawn from a given probability distribution. Among different goodness-of-fit techniques (e.g. Komolgorov-Smirnov, Anderson-Darling, Shipiro-Wilk, von Mises), the Anderson-Darling test is preferred because it is more sensitive to deviations in the tails of the distribution than is the older Komolgorov-Smirnov test. The two hypotheses for the Anderson-
Fig. 2. Schematic illustration of (a C(T)10-A533B (a/W = 0.5) and C(T)20-A533B (a/W = 0.5) specimens, (b) RNB(V45)-A533B Specimens (dimensions in mm) [48]. 3
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
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Darling test for the normal distribution are given below: H0: The data follows the normal distribution H1: The data do not follow the normal distribution The null hypothesis is that the data are normally distributed; the alternative hypothesis is that the data are non-normal. For the Weibull distributions, the test statistic, A2 is calculated from [28]:
1 A2 = −n − ⎛ ⎞ ⎝n⎠
n
∑ (2i − 1)[ln(Pi) + ln(1 − Pn−i +1)] i=1
(3)
where above expression follows normal distribution, but P is the CDF for the distribution under consideration. For the Weibull distribution, P is calculated form the Eq. (1) for every increment of loading. The above formula needs to be modified for small samples. A table has been listed in Ref. [28] designated as “Table 4.17” in page 146 that describes above equation for small samples. In this reference, Eq. (4) accommodates modified version of above equation for small samples.
0.3 ⎞ Am2 = A2 ⎛1 + n ⎠ ⎝
(4)
and then compared to an appropriate critical value from Table 1. The fitting process is conducted inside a Python code. The fitting routine uses the curve_fit method of the scipy.optimize library [29], where one has a parameterized model function meant to explain some phenomena and wants to adjust the numerical values for the model so that it most closely matches some data. Experimental data are arranged based on the fracture load, and assigned estimated failure probabilities using a well-known statistical ranking equation:
Fig. 4. Details of the finite element modeling and meshing of the specimens in ABAQUS software (only one quarter of the specimens was modeled).
P fri =
i − 0.5 N
(5)
where i is the position of a given test in the ranked data set and N is the total number of tests. Determining m and associated σu by the goodness of fitness technique requires a grid search approach. Grid search is a method to perform hyper-parameter optimization. It is a method to find the best combination of hyper-parameters. In this situation, there are different combination of hyper-parameters which correspond to a single model, that can be said to lie on a point of a “grid”. Here, fitting the data of experiments with the Beremin model gives different values for m which in turn gives different values for σw. The σw is a parameter of the Beremin model. The goal is then to train each of these combinations of m and σw and evaluate them using cross-validation to select best fitted values for Beremin model. Values of σw are calculated for a range of m. Then a corresponding σu is found from the fits.
Fig. 5. Calibration of the shape parameter of Weibull distribution with Anderson-Darling test for A516 specimens.
3. T-stress as crack tip constraint Williams [30] argues that the stress distribution as r → 0 depends on the polar angle θ. However, for some particular θ angles (Table 2), the T-stress is obtained by the method proposed by Yang et al., [31]. The method, the Stress Difference Method (SDM) incorporated the iterative single dual-boundary element method and a tip impose zero displacement jump at the crack tip [32]. Particularly for θ = 0, the T-stress is given by:
T = (σxx − σyy )θ = 0
(6)
where σyy is the opening stress and σxx refers to the stress parallel to the crack. The T-stress represents a stress parallel to the crack faces. Although the T-stress is calculated from the linear elastic material properties, but also it is a useful quantity in elastic-plastic fracture studies. Larsson et al. [9] demonstrated that the T-stress play significant roles in fracture
Fig. 6. Calibration of the shape parameter of Weibull distribution with Anderson-Darling test for A533B specimens.
4
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
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Fig. 7. Comparison of (σw/σu-calibrated) and (T/σy) over different m values for (a) C(T)25-A516 (a/W = 0.5), (b) SE(B)25-A516 (a/W = 0.5) and (c) SE(B)25-A516 (a/ W = 0.1) specimens.
4. Experiments
mechanics and can have a significant effect on the plastic zone shape and size and that the small plastic zones in various sizes of specimens can be predicted adequately by including the T-stress as a second cracktip parameter. Crack tip constraint which is a result of triaxial stress state at the crack tip affects the fracture toughness of a material. Some investigations [10,33–37] showed that the T-stress can correlate well with the tensile stress triaxiality of elastic-plastic crack-tip fields. The following studies [18,38–44] demonstrated that the apparent fracture toughness of the material varied with specimen geometry or the constraint level, and nonsingular T-stress term needs to be accounted for to accurately measure the fracture resistance in FGM. A key feature of using Weibull stress in Beremin model is that under increased remote loading, differences in evolution of the Weibull stress incorporates both the effects of stressed volume (the fracture process zone) and the potentially strong variation in crack front stress fields due to constraint loss which thus provides the necessary framework to correlate fracture toughness for varying crack depth, section thickness, specimen size, crack geometry and loading configuration. Previous research efforts to overcome the transferability issue is attempted by developing a model to elastic-plastic fracture toughness values depend on the Weibull stress as a crack-tip driving force [18,20,45–47].
The practicality of the proposed parameter calibration method for predicting failure in the specimens with different levels of constraints were studied in this work. The relation between calibrated parameters with elastic T-stress as a constraint effects of a crack tip were also considered for testing materials. Three different specimen geometries of A516 ferritic steel were chosen to cover different crack tip constraints. All the experiments were carried out in −174 °C, which was achieved by sinking the specimens in the liquid nitrogen. To ensure constant temperature during loading to fracture, two thermocouples was used. First one was fixed to the specimen and the other used to monitor and control the environmental temperature of the nitrogen inside the container. The thermocouple voltage was used to determine the current specimen temperature. All of the experiments were conducted when a steady state specimen temperature had been reached. The reason to choose this temperature is that the materials show near elastic behavior in their failures when cooled down to very low temperatures and −174 °C was the lowest possible temperature that could be reached in our Lab. Experimental standard tensile response of material obtained from −174 °C tests will be used in FE simulations. All test specimens 5
Theoretical and Applied Fracture Mechanics 104 (2019) 102331
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Fig. 8. Comparison of (σw/σu-calibrated) and (T/σy) over different m values for (a) C(T)20-A533B (a/W = 0.5), (b) C(T)10-A533B (a/W = 0.5),. and (c) RNB(V45)A533B specimens.
Fig. 9. Prediction and measured fracture toughness of the high and low constraint specimens with the calibrated parameters for A516 specimens.
6
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Fig. 10. Prediction and measured fracture toughness of the high and low constraint specimens with the calibrated parameters for A533B specimens.
Fig. 11. σw,R vs. T/σy for the three sets of A516 specimens with different levels of constraints.
Fig. 12. σw,R vs. T/σy for the three sets of A533B specimens with different levels of constraints.
specimens, and six SE(B) specimens with a/W = 0.1 as low level constraint specimens were examined at cryogenic temperatures (Fig. 1). Measured fracture toughness data of tested specimens are presented in Table 3. In addition to the above fracture tests, three other sets of A533B
were chosen with large enough dimensions to give predominantly plane strain conditions at the crack tip. To control the initial crack length, a notch in all specimens was introduced by EDM wire of 0.1 mm in radius. Three experimental data sets divided into six C(T) specimens with a/W = 0.5, six SE(B) specimens with a/W = 0.5 as high level constraint 7
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Fig. 14. Prediction and measured fracture toughness of (a) C(T)20-A533B (a/ W = 0.5), (b) C(T)10-A533B (a/W = 0.5) and (c) RNB(v45)-A533B specimens with respect to reference diagram of A533B material.
Fig. 13. Prediction and measured fracture toughness of (a) C(T)25-A516 (a/ W = 0.5), (b) SE(B)25-A516 (a/W = 0.5) and (c) SE(B)25-A516 (a/W = 0.1) specimens with respect to reference diagram of A516 material.
Appropriate boundary conditions were applied in symmetry faces. The mesh along the crack path should be very fine to provide adequate resolution of the stress fields. For all of the specimens, a regular mesh configuration has been adopted initially. A focused ring of elements in the radial direction surrounding the crack tip was used. A set of trial mesh with higher densities were employed during mesh sensitivity analysis for the Weibull stress in different load levels. The focused element configuration was radially identical around the crack tip to ensure accurate calculation of opening stresses. The finite element models and its refined mesh are presented in Fig. 4.
specimens were also analyzed. The first two sets of fracture tests are C(T) specimens of 10 mm and 20 mm thickness (Fig. 2a). The sample size is N = 15 for the C(T)10-A533B (a/W = 0.5) specimens and N = 24 for the C(T)20-A533B (a/W = 0.5) specimens [48]. The third data set is related to round notched bar specimens (the sample number is N = 12) with a V-notch of 45 degree groove (RNB (V45)-A533B), depicted in Fig. 2b. 5. FE modeling
6. Results and discussion
Nonlinear analyses of different geometries were conducted using 3D finite element modelling in ABAQUS V6.14. The material properties of A516 steel followed true stress–strain data measured at T = −174 °C, with the isotropic hardening behavior shown in Fig. 3. In the case of A533B specimens, the material properties are given in Ref. [49]. Due to geometric and loading symmetry, only one quarter of C(T) and SE(B) specimens and one eighth of RNB specimens were modeled.
Calibration of Weibull parameters were conducted by the novel method described in Section 2.1. A python script was developed for post-processing of the output data obtained from FE analyses in ABAQUS to calibrate Weibull parameters of examined data. Results of A2 parameter calculation for A516 and A533B specimens are shown in 8
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Fig. 5 and Fig. 6, respectively. A2 parameter results are used to select shape parameter of the Weibull distribution. A 95% confidence has been reported in Fig. 5 for Anderson-Darling test with a range of m varying from 1 to 30. However, A2 parameter results illustrated in Fig. 6 shows a 99% confidence. For A516 specimens all of (m, σu) data pairs can predict failure probability of 95% and 99% confidence. The use of minimum values of A2 parameters in Fig. 5 and Fig. 6 is not recommended. This is due to sensitivity of the Beremin model to m values and inevitable uncertainty in experimental test results. As it is discussed in Section 3, T-stress can affect both plastic zone size and crack tip constraint effects which are used in calculation of Weibull stress. So, there should be a relationship between these two parameters. σu is calibrated for each set of specimen configuration by a given m value and is stored into σu-calibrated. Figs. 7 and 8 compare the ratio of σw/σu-calibrated which is denoted as σw,R and named as reference Weibull stress versus T/σy for each set of data in fracture loads for different pairs of (m, σu-calibrated). As the figures show, there are a linear relationship between the reference Weibull stress and T-stress. Assuming this relationship, best pair of (m, σu-calibrated) is obtained where the intercept of the linear regression line tend to pass from zero. This is due to the fact that the value of both Weibull stresses and calculated T-stresses are directly related to the loading values. Before applying any load on the specimen, since there is no plastic zone at the crack tip, therefore the Weibull stress equals zero. In similar circumstances, the value of calculated T-stress is also equals zero. According to the figures, the intercept of the regression lines of m = 4 for all three specimen geometries of the A516 material are zero. Therefore, m = 4 can be selected as a material constant, regardless of its crack configuration for this material. This is in consistent with the previous reports in the literature [48]. The σu-calibrated values of 2089, 3199, and 4074 MPa are obtained for C(T)25-A516 (a/W = 0.5), SE(B)25-A516 (a/W = 0.5), and SE(B)25A516 (a/W = 0.1) specimens respectively. In the case of A533B material, both sets of C(T)20-A533B (a/ W = 0.5) and C(T)10-A533B (a/W = 0.5) specimens the plots of σw/σucalibrated intercept zero for m = 4. In contrast, the best regression line of m = 7 intercepts zero for RNB (V45)-A533B specimens. In this paper, it is assumed that m = 4 can be selected as the material constant and the case of m = 7 is ignored due to high constraint specimens. The results of failure probability estimation by this parameter will be discussed further. Figs. 9 and 10 show the probability of failures of different data sets according to the best calibrated m (equal to 4) and σu obtained by the calibration method presented in this paper. Figs. 11 and 12 show σw,R with respect to T/σy for A516 and A533B specimens, respectively. These figures have been generated using m = 4 as calibrated shape parameter. As they show, for all specimens, σw,R is between a range of minimum and maximum values which have a linear relationship with T-stress for each specimen configuration. As a result, a line with a slope of α which is drawn from the origin of this reference diagram represent each crack configuration. Thus, σw,R could be determined by knowing a given α which is slope of the line related to a crack configuration, and calculating T-stress. Then σu-calibrated can be obtained from σw,R/αT. As previously stated [7], the value of σw can reflect the potentially strong variations in crack-front stress fields due to the effects of constraint loss and volume sampling. The term σw,R/αT gives the scale parameter which can be interpreted as a fraction of Weibull stress related to a crack configuration which is participate to failure. The minimum and maximum values of the σw,R for A516 material including all different constraint specimens are between a range of 0.55 and 1.17, respectively. The 0.54 and 1.23 values are obtained for A533B material, respectively. In this article, mid-ranges of the data sets were used for even distribution of data. The mid-ranges of 0.86 for the A516 and 0.88 for A533B specimens are obtained for σw,R respectively.
Figs. 13 and 14 show prediction of failure probability based on the reference diagram for each data sets of A516 and A533B material. Each diagram contains two plots which show prediction just by lowest and highest fracture load of each data sets. Thus, it is concluded that failure predictions will be in the range based on those specimens with their fracture loads bounded by the minimum and maximum of the data sets. Predictions based on the mid-ranges are also shown in Figs. 9 and 10 which are almost similar to the prediction. Therefore, it is assumed that the failure probability occurs between minimum and maximum range of failure prediction by calculating T-stress for a given crack configuration and midrange of σw,R of reference diagram. Assessing the method for different material and different crack configuration shows a reasonable agreement between the fracture tests and this method. 7. Conclusions In this study, a novel calibration method for the original Beremin model was introduced, tested, and verified through experimental tests carried out in different geometries and sample sizes. Summary of the main results are as follows:
• Introducing a novel robust method for calibrating unique Weibull • • • •
parameters, so as the original model assumed, the shape parameter are to be a material constant. There is a direct relation between the scale parameter of Weibull distribution and T-stress as constraint effects. Thus, in the real components the constraint of the cracked components should be considered. By introducing a reference diagram, a reasonable failure prediction is obtained for different shape and different crack configurations. It is concluded that the failure probability occurs between a minimum and maximum range of failure prediction by calculating T-stress for a given crack configuration and midrange of σw,R of reference diagram. Assessing the method for different material and different crack configuration shows a reasonable agreement between the fracture tests and this method.
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