Study on the size dependence of calibration parameters of the new local approach model for cleavage fracture Текст научной статьи по специальности «Физика»

УДК 539.42

Исследование размерной зависимости калибровочных параметров новой модели локального приближения для моделирования

разрушения отрывом

A.R. Shen1, P.C. Li1, Z.S. Yu1, G.A. Qian2, F. Berto3, W. Wu4

1 Шанхайский технический университет, Шанхай, 201620, Китай 2 Институт механики, Китайская академия наук, Пекин, 100190, Китай 3 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия 4 Пекинский технологический институт, Пекин, 100081, Китай

Исследовано влияние геометрических размеров образца на параметры Вейбулла новой модели локального приближения для моделирования разрушения отрывом. С использованием результатов испытаний на разрушение стали A508-C проведено численное моделирование низкотемпературного растяжения стержней из стали A508-C с круглыми надрезами разной величины при помощи программного комплекса ABAQUS и получены распределения напряжений. Параметры Вейбулла для двух стержней с надрезом определены методом линейной регрессии. Показано различие параметров Вейбулла у образцов с надрезами разной величины, обусловленное зависимостью калибровочных параметров от размера надреза.

Ключевые слова: разрушение отрывом, новая модель локального приближения, параметры Вейбулла, размер надреза

DOI 10.24411/1683-805X-2019-15007

Study on the size dependence of calibration parameters of the new local approach

model for cleavage fracture

A.R. Shen1, P.C. Li1, Z.S. Yu2, G.A. Qian3, F. Berto4, and W. Wu5

1 School of Mechanical and Automotive Engineering, Shanghai University of Engineering Science, Shanghai, 201620, China 2 School of Materials Engineering, Shanghai University of Engineering Science, Shanghai, 201620, China

3 State Key Laboratory for Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China 4 Department of Mechanical and Industrial Engineering, Norwegian University of Science and Technology, Trondheim, 7491, Norway 5 Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing, 100081, China

This paper investigates whether the Weibull parameters of the new local approach model for cleavage fractures are affected by the geometric size of the specimen. Based on the fracture test data of A508-C steel, low temperature round notched bar tensile specimens of A508-C steel with two different notch sizes are numerically simulated by using finite element analysis software ABAQUS, and the stress distributions are obtained. The Weibull parameters of two notched bars are calibrated by linear regression method. The results show that the Weibull parameters of the specimens with different notch sizes are different. This suggests that the calibration parameters are dependent on the notch size.

Keywords: cleavage fracture, new local approach model, Weibull parameters, notch size

Nomenclature

dV—differential volume, mm3; E—Young's modulus, MPa; F—external load, kN; i—the serial number of the specimen; m —Weibull modulus; N—total number of the specimen;

P—probability of cleavage fracture;

P(i)—prescribed probability for the ith specimen;

r—notch radius, mm;

V0 —mean volume occupied by each microcrack in a solid, mm3;

Vpl —plastic volume, mm3; v—Poisson's ratio;

© Shen A.R., Li P.C., Yu Z.S., Qian G.A., Berto F., Wu W., 2019

ct0 —Weibull scale parameter, MPa;

CTj —maximum tensile principal stress, MPa;

CTj o —value of ctV at initial yield of a differential volume

element dV, MPa;

avM —von Mises stress, MPa;

aW —Weibull stress, MPa;

CTWnew —newly defined Weibull stress, MPa;

tfyj, —yield stress, MPa.

1. Introduction

Cleavage fracture [1] is a kind of brittle fracture, which is the most dangerous form of fracture and often leads to disastrous failure in engineering applications [2], such as the brittle fracture of reactor pressure vessels [3]. Therefore, cleavage fracture has been the focus of research from many researchers in recent years [e.g., 4-7]. Cleavage fracture consists of three nonstop successive stages: nucleation of a crack in a second-phase particle; the second phase particle-sized crack propagates across the particle and grain boundary; grain-sized crack extends through grain boundary into contiguous grains [8, 9]. In order to properly assess the fracture of a structure, local approach models [10] of fracture have been developed in the past 30 years. There are four commonly used local approach models [11]. Among them, Beremin model [12] is the only one related to the cleavage fracture. The Beremin model is a two-parameter Weibull distribution model based on the weakest link theory. It was first proposed by the Beremin group to describe the cleavage fracture probability [13, 14]:

P = 1 - exp

(n Y

W

ctw =

f ^m

J ai

dV Vo

nV i

(1)

(2)

where P is the probability of cleavage fracture, denotes the Weibull stress, is the Weibull scale parameter, which represents the critical cleavage fracture stress in the Beremin model, m is the Weibull modulus representing the microscopic defect uniformity that may cause cleavage fracture, Vpl denotes the integral volume as well as the plastic zone volume, ctV is the maximum principal stress of each plastic zone volume element, V0 is a reference volume, which represents the mean volume occupied by each microcrack in a solid, d V represents the differential volume.

Before a proper application of the Beremin model, two Weibull parameters of the model (i.e., m and a0) have to be calibrated according to the existing fracture data [15]. Minami [16] and the European Structural Integrity Society (ESIS) [17] proposed a calibration procedure. The Weibull parameters m and a0 have been considered to be inherent parameters of the material completely independent from the external conditions. Accordingly, they have not been

considered to be affected by temperatures and loading conditions, as well as the geometric size of the specimens. But more recent studies have contradicted this simplistic hypothesis. For example, Akbarzadeh and Hadidimoud [18] have investigated the dependence of Weibull parameters on geometric size by using round notched bar tensile specimens on the basis of a series of fracture experiments. They have tested eight groups of specimens having a net cross-section diameters of 10 and 15 mm, and notched radius of 5, 4, 2, and 0.2 mm for each group, respectively. The have measured value of m ranging from 6 to 117. The fluctuation of m has been found so wide that it has been clearly shown how much inappropriate would have been to select a fixed value of m independent of the specimen shape and size. Cao et al. [19] have studied the temperature dependence of Beremin model parameters in ductile-to-brittle transition region with the master curve method. The experimental data of 16MnR steel have shown that in the lower transition region, as the temperature increases, the Weibull modulus m decreases. In the lower-to-mid transition region, m remains constant. Within the range of temperature under investigation in [19], the Weibull scale parameter ct0 increases with the increase of the temperature.

In order to tackle the problem that the Beremin parameters vary with temperatures and geometric constraints, several researchers have revised the initial model originally proposed by Beremin [e.g., 20-22]. However, most of these methods are empirical modifications and lack of a relevant theoretical basis. Based on the Beremin model, Lei [23] has proposed a new cumulative failure probability model of local approach, which is based on the assumption of weakest link and requires that the plastic yield is the prerequisite for the occurrence of cleavage fracture:

P = 1 - exp

i

J (CTi -tfi,o)'

dV

Vo

V m

(3)

(4)

where ctw

is the newly defined Weibull stress. Com-

pared to the Beremin model, the new model introduces the maximum principal stress at the initial yield 0 of the differential volume unit dV. As mentioned in Lei [24], in the Beremin model, the Weibull stress ctw is calculated by the instantaneous maximum principal stress in each volume element. When a specific volume element d V just reaches the initial plastic yield, there is ctV = ctV 0, according to Eq. (1), the minimum probability for the nonoccurrence of fracture is always different from zero. This is a nonsense from the physical point of view and violates the hypothesis that the plastic yielding is a necessary but insufficient condition for cleavage fracture. This is also the fundamental reason for the change of m and ct0 with temperatures and geometric sizes during the calibration process.

Conversely, in the new model, the stress increment (a1 --a1 0) after initial plastic yield of each element is introduced to calculate the failure probability, which strictly follows the physical mechanism of plastic yield and takes plastic yield as the prerequisite for cleavage fracture. Therefore, Eq. (3) is considered to be able to overcome the ambiguity of the calibration of Weibull parameters. In addition, Qian et al. [25] have recently explained that the new model can overcome the ambiguity of the parameters calibration of the Beremin model in details from the perspective of physics and mechanics. Besides, the temperature independence of parameters calibration of the new model has been verified by using circumferentially notched round tensile specimens, but the effect on the geometric size has not been studied in depth. Therefore, the objective of this work is to investigate whether the calibration of Weibull parameters (m and g0) is affected by the geometric size in the new local approach model by using two groups of fracture specimens with different notch radii.

2. Experimental data

In order to study the effect of geometric size on the two Weibull parameters, two kinds of round notched bar tensile specimens with different notch sizes are used in this work. The notch radii are 3 and 6 mm, respectively. The specimens under loading are shown in Fig. 1. The experimental material is A508-C steel and the experimental data are available from Ref. [26]. Table 1 shows the chemical composition of A508-C steel. The yield stress and Young's modulus of A508-C steel at -196°C are Gys =898 MPa and E = 2.14 • 105 MPa. Poisson's ratio v = 0.3. Figure 2 represents the true stress-strain curve of A508-C steel at -196 °C. In [26], Hui and Li made low-temperature tensile tests on the rods with notch radius of 3 and 6 mm, respectively. Among them, 13 groups of specimens with notch radius of 3 mm and 5 groups of specimens with notch radius of 6 mm were tested at the low temperature. The tensile fracture loads of two kinds of specimens are summarized in Table 2.

3. Finite element analysis and Weibull parameters calibration

3.1. Finite element analysis and calibration method

In this work, the 3D elastic-plastic finite element analysis is used to numerically analyze the specimens. Considering the geometric symmetry and the loading symmetry of the specimens, we only select 1/8 of the structure for modeling so as to reduce the computational time. The ele-

6

5" L ------ Z/ -----

30 80 30

140 3 r

------ — -2-1 -----

30 80 30

140

Fig. 1. Round notched bar tensile specimens with notch radius of 6 and 3 mm. All sizes are in mm

ment employed in this work is C3D20R. It is worth noting that the grids in the notch region need to be refined. The number of elements are 546624 for notched specimen with r =6 mm and 510692 for notched specimen with r = 3 mm, respectively. The calibration procedure developed by Qian et al. [25] for the new local approach model is adopted in this work. In the procedure, the process of finite element analysis includes two steps. In the first step, the distribution of the maximum principal stress G1 of each volume element is obtained by using the true elastic-plastic hardening constitutive law for the material. As stated by Qian et al. [27], due to the uneven stress distribution of the notch, plastic yield may occur at different time for each volume element, and the maximum principal stress of each unit may be different at the initial yield. In order to obtain the maximum principal stress a1 0 of differential volume element dV introduced in Eq. (4), we assume the material as a perfect elastic-plastic material in the second step of the finite element simulation. Then the maximum principal stress of each element in the plastic zone is taken as a1 0. The distributions of the maximum principal stress of the two specimens at -196°C are shown in Figs. 3 and 4.

According to the result file of ABAQUS, we can obtain the number, volume, and corresponding values of g1 and a1 ,0 of each plastic zone element. The volume of the plastic zone here is defined as the volume of the unit of a1 > 1.0 a ys [28]. Then the above results can be substituted into Eq. (4) to calculate the Weibull stress. Here we take V0 = 0.001mm3. Since the value of m in Eq. (4) is unknown, we assume that the initial value of m is 10, and then the linear regression method is used to calibrate the two Weibull parameters. Equation (3) can be rewritten as follows [29]:

ln ln(1/(1- P)) = m lna w - m lna 0. (5)

Table 1

Chemical composition of A508-C steel (wt %)

C Si Mn P S Ni Gr Mo V Cu Co

0.20 0.25 1.22 0.01 0.007 0.72 0.09 0.52 0.003 0.066 0.014

a>

£ 400 -1

§ I

^ 200 -1

0 "T-1-1-1-1-1-1-

0.00 0.02 0.04 0.06 True strain 8

Fig. 2. True stress-strain curve of A508-C steel at -196°C

In the linear regression method, the cumulative failure probability P is calculated according to the fracture loads of the specimens in ascending order. Assuming that there are N specimens, the fracture probability of the ith specimen is calculated by the formula P (i) = (i - 0.5)/N. The value of m can be calibrated from Eq. (5), and the value of G0 can be calculated via the intercept term, where the intercept term is -mln a0.

Table 2

Tensile fracture loads of specimens with notch radius of 3 and 6 mm at -196°C

Fracture specimens Fracture loads of specimens, kN

r = 3 mm r = 6 mm

1 95.79 92.08

2 94.04 92.11

3 96.57 92.50

4 97.73 92.51

5 98.55 93.32

6 99.21

7 99.38

8 99.40

9 99.53

10 99.54

11 99.55

12 99.65

13 100.58

Fig. 3. Distribution of av in the notched specimen with r = 3 mm at -196°C. The external load is 95.79 (a) and 100.58 kN (b) (color online)

Fig. 4. Distribution of av in the notched specimen with r = 6 mm at -196°C. The external load is 92.08 (a) and 93.32 kN (b) (color online)

lnln(l/(l-P))

4-1-,-1-,-

6.6 6.7 6.8 6.9 lna,

lnln(l/(l-P))

Fig. 5. Calibration result of m of round notched bar tensile specimens with r = 3 (a) and 6 mm (b)

3.2. Results of Weibull parameter calibration

Figure 5 illustrates the calibration results of Weibull modulus m for round notched bar tensile specimens with notch radius of 3 and 6 mm. The calibration results of Weibull parameters for two specimens at -196°C are provided in Table 3.

Table 3 reveals that both the values of m and a0 of the two kinds of specimens are significantly different. That is to say, when the notch sizes of the specimens are different at a given temperature, the corresponding Weibull parameters make a difference.

4. Analysis and discussion

As stated above, we can see that the calibration results of the new model parameters are influenced by the geometric size. Specimens with different notch sizes correspond to different Weibull parameters. Akbarzadeh and Hadidi-moud [ 18] have studied the dependence of Weibull parameters on geometry in the Beremin model. They have pointed out that the dependence of parameters on geometric size can be attributed to the stress concentration, but they did not explain it in details. In view of this, we investigate the stress concentration of two kinds of round notched bar tensile specimens with different notch radii. By looking up the practical stress concentration manual [30], we know that the stress concentration factor of round notched bar tensile specimen with notch radius of 3 mm is about 1.57, while it

Table 3

Calibration results of Weibull parameters for specimens with different notch radii

r, mm m a0, MPa

3 11.888 984.360

6 25.833 664.255

is about 1.32 in the case of notch radius of 6 mm. Obviously, when the notch radius is different, the stress concentration factor of the specimen is different. In addition, we examine the influence of notch depth on the maximum principal stress of the two types of specimens. The curves of the maximum principal stress with the distance from the notch root under different loads are displayed in Fig. 6.

It can be seen from Fig. 6, a that the maximum principal stress with notch radius of 3 mm increases first and

ab MPa

1000-1-1-1-1-1-

0 1 2 3 4 5

Distance from the notch root X, mm

ab MPa

Distance from the notch root X, mm

Fig. 6. Variations of av of specimens with the distance from notch root, r = 3 (a) and 6 mm (b) (color online)

Fig. 7. Variations of a1 of the two specimens with the distance from the notch root, F = 94.2 kN, r = 3 (1) and 6 mm (2)

Fig. 8. Variations of a10 of the two specimens with the distance from the notch root, r = 3 (1) and 6 mm (2)

then decreases with the increase of the distance from the notch root under different loads, and reaches maximum at the distance around 2.3 mm from the notch root. As depicted in Fig. 6, b, for the tensile specimens with notch radius of 6 mm, as the distance from the notch root increases, the maximum principal stress under different loads shows an increasing trend, and the maximum principal stress reaches maximum around the center of the narrowest surface of the notched tensile bar. Combined with Figs. 6, a and 6, b, we can find that the variation trends of the maximum principal stress of the two specimens with different notch radii are different, and the regions where the maximum principal stress reaches maximum are also distinguishing. The main reason for the local high stress state in discontinuous specimens is that the orientation of crystals compared to the direction of external loading is random. Actually, if the existence of macrodiscontinuity (such as notch) is the cause of the local high stress field in the specimen, the fracture volume of the material will be merely affected by the local high strength stress field namely the process zone [31]. This is probably the main reason why the calibration results of cleavage fracture parameters of two kinds of notch sizes are inconsistent. In order to further compare the difference between the distributions of the maximum principal stress of the two specimens, we select the same load of 94.2 kN, which is an intermediate value between the fracture loads of the two specimens. The distributions of the maximum principal stress under this load are plotted in Fig. 7.

Figure 7 shows that even under the same external load (F = 94.2 kN), the variation trends of the maximum principal stress of the specimen with notch radius of 3 mm and the specimen with notch radius of 6 mm differ. Besides, the locations where the maximum principal stress reaches maximum in the two specimens are also different. That is, the areas where the fracture occurs in the two specimens are different. Moreover, the maximum value of the maximum prin-

cipal stress of the specimen with notch radius of 3 mm is less than that of the specimen with notch radius of 6 mm, which indicates that the specimen with notch radius of 6 mm is more inclined to fracture than that with notch radius of 3 mm. This is consistent with the experimental results in Ref. [28].

In addition, Qian et al. [27] have claimed that the decisive factor that the new local approach model can overcome the ambiguity of parameters calibration of Beremin model is the introduction of a10. For this reason, we investigate the distributions of a10 of the above two kinds of specimens. Figure 8 shows the variations of a10 with the distance from the notch root in the above mentioned specimens.

As can be seen from Fig. 8, in the case of the same temperature, material and external loading, the regions where a10 reaches maximum in the two notched specimens are different. Meanwhile, the maximum value of a10 of the specimen with notch radius of 3 mm is greater than that of the specimen with notch radius of 6 mm. In other words, a10 also differs greatly with the notch size changing.

In summary, the effect of notch radius on Weibull parameters may be mainly due to the different stress concentration of specimens with different notch radii, and the different distributions of a1 and a10. Considering a1 and a10 are the most important influencing factors when calculating Weibull stress in the new local approach model, their values will directly affect the results of Weibull stress. Then it will further influence the calibration results of Weibull parameters.

5. Conclusions

By using 3D finite element analysis and the new local approach model, the Weibull parameters of two different sized round notched bar tensile specimens of A508-C steel at -196°C have been calibrated. The calibration results of

the specimens with notch radius of 3 mm are m = 11.888 and a0 = 984.360 MPa. As for the specimen with notch radius of 6 mm, the results shows that m = 25.833 and a0 = = 664.255 MPa.

For the round notched bar tensile specimens with a fixed material, the calibration results of Weibull parameters are different when the notch size is different. That is to say, the calibration results of Weibull parameters of the new model are affected by the geometric size of the specimen.

The variation of Weibull parameters of the new model with the geometric size of the specimen is mainly due to the different stress concentration factors of specimens with different notch sizes, the different distribution trends of the maximum principal stress, and the different distributions of ai 0 introduced by the new model.

In this paper, we employ the experimental data of two sets of round notched bar tensile specimens with different notch radii and finite element analysis to approve the size-dependence of parameters calibration of the new local approach model. Note that the present work is just a preliminary study on the issue of size-dependence of calibrated parameters. The underlying microscopic mechanism concerning the issue remains unclear. So more sufficient experimental and numerical data and theoretical analysis are required to support the obtained conclusions in this study. Note that the present work is only based on the data of A508-C steel under tensile conditions, the other types of ferritic steels and loading conditions shall be considered in the future work. Meanwhile, the present study indicates that probably the new local approach model also needs to be improved with respect to the determination of av 0.

Funding

This work is funded by the Natural Science Foundation of Shanghai (Grant No. 19ZR1421400), the National Natural Science Foundation of China (Grant No. 11872364), and the CAS Pioneer Hundred Talents Program.

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Received July 22, 2019, revised July 22, 2019, accepted July 31, 2019

Ceedeuun 06 aemopax

Airu Shen, Master Student, Shanghai University of Engineering Science, China, M015117119@sues.edu.cn Peichao Li, PhD, Shanghai University of Engineering Science, China, wiselee18@163.com Zhishui Yu, PhD, Shanghai University of Engineering Science, China, Yu_Zhishui@163.com Guian Qian, PhD, Institute of Mechanics, Chinese Academy of Sciences, China, qianguian@imech.ac.cn Filippo Berto, PhD, Prof., Norwegian University of Scienceand Technology, Norway, berto@gest.unipd.it Wenwang Wu, PhD, Beijing Institute of Technology, China, Wuwenwang2014@163.com