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The critical void growth ratio and crack initiation condition in the crack tip zone of 20g steel
0013-7944/91 $3.00 + 0.00 0 1991 Pergamon Press pk.
Engineering Fracfure Mechanics Vol. 38, No. 2/3, pp. 175-183, 1991 Printed in Great Britain.
THE CRITICAL VOID GROWTH RATIO AND CRACK INITIATION CONDITION IN THE CRACK TIP ZONE OF 20g STEEL ZHU HANXING,
LI CHANGCHUN
and LI GUANGXIA
Department of Mechanics, Huazhong University of Science and Technology, Wuhan, P. R. China Abstract-In this paper, the elastic-plastic FEM method with a kinematic hardening model has been used to analyse a three-point-bend specimen. The relation between the maximum strain and the crack opening displacement has been investigated. Based on the Rice-Tracey void growth
model, the VGvalue has been calculatedwith the numericalintegrationmethod. For ductile materials,the cause of void formation and the condition of the void coalescence with the main crack have been discussed and, the results agree well with those of Li Changchun et al. [J. Huuzhong Univ. Sci. Technol. 17, 139-144 (1989)] and our experiments in this paper.
1. INTRODUCTION IN PRACTICALengineering quite a lot of destructions are ductile fracture problems, therefore, many researchers have been thoroughly studying the ductile fracture properties and the micro-mechanisms of metal materials in recent years. For ductile materials, although it has been known that the fracture processes generally are void nucleation, growth and coalescence with the principal crack, much work is still needed in search of the relationship between the macro and micro natural parameters in ductile fracture. The void critical growth ratio VGi, the natural parameter in ductile material fracture, has been extensively explored by Zheng and others[l-4] and the void critical growth ratio criterion is obtained. Vo is calculated by:
V,=t,exp
i? (
. >
(1)
The above equation is obtained by supposing a,/6 as a constant, where V, is the instantaneous void growth ratio, cp is equivalent plastic strain, 6 is Von Mises’s equivalent stress, 6, is average stress, 0,/c? is stress triaxiality parameter. For an elastic-plastic cracked body, the stress, strain and a,/6 vary markedly with the increment of the applied load. Based on the elastic-plastic FEM method and the Rice-Tracey’s void growth model[5], the VG value is calculated with a numerical integration method and the ductile crack propagation condition is investigated. 2. MATERIAL
AND SPECIMEN
The experimental material is 20g steel. The major chemical composition and the mechanical properties at room temperature are given in Table 1 and Table 2. The 64 curve of 20g steel is measured on an Instron testing machine with standard tensile specimen, as shown in Fig. l(a). The dimensions of three-point-bend specimen are shown in Fig. 2. 3. NUMERICAL
CALCULATION
3.1. Hardening model To suitably choose a hardening model, the Baushinger Effect of 20g steel is measured on an Instron testing machine. It is found that there exists a marked Baushinger Effect on 20g steel, as shown in Fig. l(b). Therefore, it is more suitable taking the kinematic hardening model for analysis. The loading surface of the kinematic hardening model can be written as f(s, - clii)- k = 0. 175
(2)
ZHU HANXING et al.
176
Table 1. The major chemical composition (wt%) C
Mn
si
s
P
0.20
0.45
0.26
0.025
0.0053
Table 2. The mechanical properties 0, (MPa) 303.8
ub @W
480.2
E
W’,)
”
1.93 x 10’
0.3
cp w-1
57.8
Here aij, the yielding center, depends on the extent and the history of plastic deformation, Shield[6] point out: c$ = cC$. For linear hardening material, c is a constant and the model becomes a linear kinematic hardening model. Thus eq. (2) can be rewritten as: f($ - CC!) - k = 0.
(3)
From Fig. 1 we known that 20g steel is non-linear hardening, so, the non-linear kinematic hardening problem is transferred into a sectionalized linear kinematic problem in present paper. In the light of von Mises’s flow rule, the yield surface of linear kinematic hardening model is expressed as J[&j
- c+>(s, - CC$)]- gs = 0.
(4)
Thus, for the sectionalized linear kinematic hardening model, the Mises’s yielding surface is as follows: J&s, here, (a,), = C;! , ck Am,
- (a,)&,
-
(5)
(qM1 - 0, = 0
n is the loading step.
3.2. Elastic-plastic FEh4 analysis For isotropic hardening material, the elastic-plastic form is as follows d(a) = where
[D],,
is elastic-plastic
matrix,
constitutive equation[7] in incremental
(6)
Plep 44 = (PI, - PI,) W [D], is
elastic matrix, [D], is plastic matrix:
(7)
PI, =
(b)
500
lb
01
5
10
15
20
1 PA)
Fig. l(a).
The tension
curve of 20g steel.
Fig. l(b). The Bauschinger
curve of 20g steel.
Critical void growth ratio and crack initiation 20
60
Y*
177
t-i * G
x ::
01 01
Y 5 Fig. 2. The dimensions of three-point-bend specimen (in mm).
Here, H’ is the slope of the material’s bicurve at present stress and strain state. In the present paper’s program, the primary yielding surface is considered as the motion of translation between loading steps. In every loading step, taking the yielding center been translated as yielding center, the primary yielding surface been translated as yielding surface, isotropic hardening method is used in calculation. While the tension curve shown in Fig. l(a) is inputted, an equivalent stress-equivalent strain point, (500 MPa, 2.5), is supplemented and that doesn’t affect the calculation precision because both the dimension of the element and the zone with large strain in the vicinity of the crack tip are very small. In the same time, load self-amendment method is taken to increase the calculation precision and decrease error. Half the three-point-bend specimen is taken to calculate because of its symmetrical characteristic and constant strain triangular element is used in the program. 312 nodes and 547 elements are partitioned and the ratio of the minimum element’s dimension at the crack tip to the crack length is 0.0013 16. The calculation is executed on Honeywell DPS/8 computer.
4. RESULTS AND DISCUSSIONS 4.1. The relationship between crack tip opening displacement and the maximum strain
To obtain the crack tip opening displacement 6,. The displacement distribution on the crack surface is calculated for every loading step, as shown in Fig. 3, and 6, is obtained with least-square fit and linear extrapolation method[8]. (L 6,,,), the dots of the crack tip opening displacement and the maximum plastic strain at the crack tip under each loading step, are shown in Fig. 4. It is clearly shown that dT is directly proportion to emax,i.e. 6, = 0.207~,,, .
(8)
Because the maximum plastic strain t,,, at the crack tip reaches a critical value cmaxas the crack tip opening displacement reaches the critical value a,, (i.e. the crack propagates forward as the maximum plastic strain at the crack tip reaches the critical value E,,,), ...-. it may be recognized that, for crack initiation, the crack critical opening displacement criterion is identical to the maximum strain criterion and both the criteria are essentially the same but different in form. 0.3
-G 0.2 E bo 0.1
0
I 0.1
0.2
I
I
I
I
I
I
I
0.3
0.4
0.6
0.6
0.7
0.6
0.9
I 1 .o
y,/a
Fig. 3. The displacement distribution on the crack surface.
178
ZHU HANXING et al.
00’
0.2
-
0.1
-
.*
/ 0’
,d 0
0.1
/
,
I
I
I
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.1
0.9
0.9
1.0
lmar
Fig. 4. The relation between crack tip opening displacement and the maximum plastic stain.
4.2. The distributions of cP a,/6
and a,, on the crack extended line
From ref.[9], we know that the fracture toughness of 20g steel is 0.22 mm, i.e. 6, = 0.22 mm. The distributions of equivalent plastic strain Q,, stress triaxiality 0,/5 and tensile stress cXXon the crack extended line as 6, = 0.205 mm are shown in Fig. 5; and it is found that, there exists a maximum point ahead of the crack for both a,,,/6 and gxx. 4.3. The calculating analysis of V, parameter and the crack initiation condition For the growth law of one spherical void in fully plastic material Rice-Tracey[S] proposed the following model:
of infinite extent,
dR/R = 0.203 exp i?
dc,. (9) 0 ) ( material, based on Von Mises’s yielding criterion, the above equation
For power-hardening can be rewritten as
dR/R =c exp i$’
dc,. ) Considering the whole process from nucleation to coalescence and taking the nucleation strain cd as very small[lO], we have: (
(11) Equation (1) is obtained from the above equation when cr,/a is a constant. For a three-point-bend specimen, although the applied load increases in proportion, the o,/d value at the crack tip zone doesn’t keep constant. Figure 6 has shown the relationship between 6, and the a,,,/5 value in the vicinity of y = 0.3 mm ahead of the crack tip. The exp(&,,/5) N Q, curve at Qx
0.6
u’
( MPa)
1600
:#ps&
_I,;
0.2
-
0.1
-
0
600
T
-2
,300 Qp
-
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
:
:
:
:
:
:
0.6
0.7
0.8
0.9
1.0
1.1
1
ytmm)
Fig. 5. The distributions of c,, oxr and ~,,,/a ahead of the crack tip when 6, = 0.205 mm.
179
Critical void growth ratio and crack initiation
0.02
0.06
0.10
0.14
0.16
8Thm)
Fig. 6. The a,,,/5 - 6, curve at y = 0.3 mm ahead of the crack tip.
y = 0.177 mm ahead of the crack tip is shown in Fig. 7. At cP= 0.03, from (11) the Vo value is the area of the shaded portion OACD which can be obtained from the numerical integration method, but from eq. (1) the Vo value is the area of the square OBCD. To further compare eq. (11) with eq. (l), the vicinity of y = 0.56 mm ahead of the crack tip is investigated and the exp($r,/o) - cP curve of the vicinity is shown in Fig. 8. The Vo values of the three points: A, B and C in Fig. 8 are calculated individually using eqs (11) and (1) and the results are given in Table 3. From Table 3, it is found that, Vo(,i) increases monotonically with the increment of cp, but Vo(i)(the results of eq. (l)), does not. In the physical sense of Vo, Vo(,)clearly doesn’t correspond to the experimental result, because there exists a triaxial stress state in front of the three-point-bend specimen’s crack and the void can not become smaller but can enlarge. Therefore, it is better to calculate the Vo value with eq. (11) than with eq. (1). When 6, = 0.205 mm, the Vo value on the crack extended line is obtained from eq. (1 l), as shown in Fig. 9. It is found that, there exists a maximum value of Vo at y = 0.3 mm and that can be explained by Fig. 5 and eq. (11) because there exists a maximum point of a,/6 in the vicinity of y = 0.3 mm and Vo is approximately in direct proportion to exp&,/a) and those determine that there also exists a maximum point of Vo at the very place although Vo is approximately in direct proportion to cP too, but at y = 0.3 mm, the variation of cP is quite plain and that hasn’t much effect on the maximum Vo position of y = 0.3 mm. From ref. [l 11,we know that the maximum gXX position is directly proportion to 6, and that is quite reasonable. For example, as 6, tends to 0, the three-point-bend specimen reaches an elastic state and the maximum cr, position certainly tends to the crack tip point. For plane strain three-point-bend specimen’s, the maximum o,, position generally is the maximum value position of a,,,/6 (see Fig. 5) and also approximately is the maximum Vo position for high ductile material, therefore, the maximum Vo position moves forward with the increased BT (that has been proved by our calculation results). For 20g steel, assuming the distance between the maximum Vo position and the crack tip as yc , we approximately have Y,z 1.58r.
(12)
100
I+
-
a E
60 ___---_
40 10
-
20
0
0.01
0.02
0.03
0.04
0
0.02
Fig. 7. The exp(&,/oZ) -Q, curve at y = 0.177 mm ahead of the crack tip.
0.06
0.10
0.14
0.18
lP
‘P
Fig. 8. The exp($,/@
- cP curve at y = 0.56 mm ahead of the crack tip.
180
ZHU
HANXING
et al.
Table 3. 6,
VW) VoW,
0.0736
0.135
0.196
1.0628 2.06
0.9489 2.69
0.752 3.02
Figure 10 is the crack tip photogram of 20g steel at the crack initiation and it is found that the distance between the crack and the void ahead of the crack tip is approximately 0.34 mm. In the light of ref.[9], the crack critical opening displacement of 20g steel is 0.22mm, i.e. 6,, = 0.22mm. Those agree quite well with eq. (12) and show that the void formation at Y, ahead of the crack tip is owing to the existence of the maximum Vo at the very place and the crack propagates forward when the maximum Vo reaches a critical value VGc
.
The experimental results of 08F steel in ref.[l2] also show that the void ahead of the crack tip is formed before the principal crack propagates. Owing to that the experiment is in the plane stress state, the ratio of the distance Y, between the void and the crack tip to the crack opening displacement 6, is smaller than that in plane strain state because the stress triaxiality parameter a,,,/~?in plane stress state is much smaller than that in plane strain state. The weighting of equivalent plastic stain cP in V, value of plane stress state is larger than in that of plane strain state. As shown in Fig. 5, the cP value on the crack extended line decreases promptly with the increased Y, so, the maximum VG position ahead of the crack tip in plane stress state is certainly closer to the crack tip than in the plane strain state. Therefore, the experimental results of ref.[l2] also agree with this paper. In general, for the void ahead of the crack, its nucleation, growth and coalescence with the principal crack are caused by the increment of VG. When the maximum V, value ahead of the crack tip reaches a critical value VGc, the void coalesces with the principal crack, i.e. the crack propagates forward, that is the crack initiation characteristic of ductile material. The larger VG is the larger &, and cm,,, are, and vice versa. All of them can be taken for the crack propagation condition, but, for ductile material, V,, is the real cause of the crack initiation and, 6,, and L,,,, are only the phenomena existing in the fracture process. Taking account of the convenience in practical application, it is more suitable to choose dTCas the crack initiation criterion for ductile material. On the other hand, the value and the position Y, of the maximum V, ahead of the crack tip increase with the increased crack opening displacement &, as to where a large void is formed in front of the crack tip, that is closely related to the material’s property. For low strength steel, a void is usually formed by the separation of the interface between the ferrite matrix and non-metallic inclusions, or by the inclusion cracking[l3]. These steels are mainly manganese sulphide[l4]. Generally, for high ductile materials, V,, 6,, and cmaxcare quite large because it is difficult for a void to form and grow in the vicinity of the crack tip.
7r
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0
0.9
1.0
1.1
1.2
ytmm)
Fig. 9. V, value ahead
of the crack
tip when 6, = 0.205 mm.
Critical void growth ratio and crack initiation
e
Fig. 10. The crack tip photogram of 20g steel 50x
181
Critical void growth ratio and crack initiation
183
5. CONCLUSIONS (1) For 20g steel, the crack opening displacement S, is directly proportional to the maximum plastic strain, i.e. 6, = 0.207~,,, . Therefore, it may be recognized that, the crack begins to propagate as the maximum plastic strain at the crack tip reaches a critical value cmaxC,6, and 6maxecan all be taken as crack initiation criteria and both the criteria are not different in essentially but in form. (2) The CJ,/C values of the crack tip zone do not remain constant. The Vo value calculated from eq. (11) differs remarkably from that calculated from eq. (1) and VouIj increases monotonically with the increased cP, but Vo(,) does not keep this property, but decreases gradually as cP increases. Therefore, to calculate the Vo value with eq. (11) is more suitable than with eq. (1). (3) There exists a maximum Vo value ahead of the crack tip that is the cause of the void formation at the very place, i.e. at the position of Y, = 1.56r. (4) For ductile materials, when the maximum Vo value ahead of the crack tip reaches a critical value VoC,the void coalesces with the principal crack, i.e. the crack propagates forward. Therefore, VoCis the real cause of the crack initiation and may be taken as the crack initiation condition. Although 6, and cmaxcare the crack initiation conditions they are only the phenomena existing in the crack initiation process. Acknowledgement-Project
supported by National Science Foundation of China.
REFERENCES [1] C. Q. Zheng and J. C. Radon, Proc. ICF Inr. Symp. Fracture Mech. (pp. 1052-1056), Beijing (1983). [2] C. Q. Zheng and J. C. Radon, Proc. ICF Inr. Symp. Fracture Mech. (pp. 1057-1062), Beijing (1983). [3] C. Q. Zheng, L. Zhou and J. M. Liu, Proc. 9rh Congress Mater. Testing and 3rd Danubia-Adria Symp. (pp. 8691) Budapest (1986). [4] C. Q. Zheng, L. Zhou and J. M. Liu, Proc. KM5 (pp. 213-218), Beijing (1987). [5] J. R. Rice and P. M. Tracey, J. Mech. Phys. Solids 17, 201-217 (1969). [6] R. Shield and M. Ziegler, ZAMP, 9a, 260 (1958). [7] Yiquan Xie and Fubao He, FEM Method in Elastic and Plastic Mechanics, pp. 206-207. Machinery Industry Press, China (1983). [8] Xiaobin Lin, Guoyou Sun and Jiliang Xue, J. Zhejiang Uniuersiry, China 21(3), 75-84 (1987). [9] Hanxing Zhu, Master’s degree thesis. Zhejing University, HangZhou, China (1986). [lo] J. W. Hancock and A. C. Mackengie, J. Mech. Phys. Solids, 24, 147-169 (1976). [11] Hanxing Zhu, Guogou Sun and Jiliang Xue, Acrn Mechanica Solida sinica, (China), 1990. [12] Changchun Li, Xiping Li and Guangxia Li, J. Huazhong Univ. Sci. Technol. (Wuhan, China), 17(3) 139-144 (1989). [13] Y. W. Shi and J. T. Batnby, Inr. J. Fracture 25, 143-151 (1984). [14] D. A. Shockey, L. Seaman, K. C. Dao and R. Curran, Paper No78-PVP-92. Presented at the ASME/CSME Pressure Vessels and Piping Conference, Montreal, Canada, June 25-30, 1978. (Received 17 Ocrober 1989)
Engineering Fracfure Mechanics Vol. 38, No. 2/3, pp. 175-183, 1991 Printed in Great Britain.
THE CRITICAL VOID GROWTH RATIO AND CRACK INITIATION CONDITION IN THE CRACK TIP ZONE OF 20g STEEL ZHU HANXING,
LI CHANGCHUN
and LI GUANGXIA
Department of Mechanics, Huazhong University of Science and Technology, Wuhan, P. R. China Abstract-In this paper, the elastic-plastic FEM method with a kinematic hardening model has been used to analyse a three-point-bend specimen. The relation between the maximum strain and the crack opening displacement has been investigated. Based on the Rice-Tracey void growth
model, the VGvalue has been calculatedwith the numericalintegrationmethod. For ductile materials,the cause of void formation and the condition of the void coalescence with the main crack have been discussed and, the results agree well with those of Li Changchun et al. [J. Huuzhong Univ. Sci. Technol. 17, 139-144 (1989)] and our experiments in this paper.
1. INTRODUCTION IN PRACTICALengineering quite a lot of destructions are ductile fracture problems, therefore, many researchers have been thoroughly studying the ductile fracture properties and the micro-mechanisms of metal materials in recent years. For ductile materials, although it has been known that the fracture processes generally are void nucleation, growth and coalescence with the principal crack, much work is still needed in search of the relationship between the macro and micro natural parameters in ductile fracture. The void critical growth ratio VGi, the natural parameter in ductile material fracture, has been extensively explored by Zheng and others[l-4] and the void critical growth ratio criterion is obtained. Vo is calculated by:
V,=t,exp
i? (
. >
(1)
The above equation is obtained by supposing a,/6 as a constant, where V, is the instantaneous void growth ratio, cp is equivalent plastic strain, 6 is Von Mises’s equivalent stress, 6, is average stress, 0,/c? is stress triaxiality parameter. For an elastic-plastic cracked body, the stress, strain and a,/6 vary markedly with the increment of the applied load. Based on the elastic-plastic FEM method and the Rice-Tracey’s void growth model[5], the VG value is calculated with a numerical integration method and the ductile crack propagation condition is investigated. 2. MATERIAL
AND SPECIMEN
The experimental material is 20g steel. The major chemical composition and the mechanical properties at room temperature are given in Table 1 and Table 2. The 64 curve of 20g steel is measured on an Instron testing machine with standard tensile specimen, as shown in Fig. l(a). The dimensions of three-point-bend specimen are shown in Fig. 2. 3. NUMERICAL
CALCULATION
3.1. Hardening model To suitably choose a hardening model, the Baushinger Effect of 20g steel is measured on an Instron testing machine. It is found that there exists a marked Baushinger Effect on 20g steel, as shown in Fig. l(b). Therefore, it is more suitable taking the kinematic hardening model for analysis. The loading surface of the kinematic hardening model can be written as f(s, - clii)- k = 0. 175
(2)
ZHU HANXING et al.
176
Table 1. The major chemical composition (wt%) C
Mn
si
s
P
0.20
0.45
0.26
0.025
0.0053
Table 2. The mechanical properties 0, (MPa) 303.8
ub @W
480.2
E
W’,)
”
1.93 x 10’
0.3
cp w-1
57.8
Here aij, the yielding center, depends on the extent and the history of plastic deformation, Shield[6] point out: c$ = cC$. For linear hardening material, c is a constant and the model becomes a linear kinematic hardening model. Thus eq. (2) can be rewritten as: f($ - CC!) - k = 0.
(3)
From Fig. 1 we known that 20g steel is non-linear hardening, so, the non-linear kinematic hardening problem is transferred into a sectionalized linear kinematic problem in present paper. In the light of von Mises’s flow rule, the yield surface of linear kinematic hardening model is expressed as J[&j
- c+>(s, - CC$)]- gs = 0.
(4)
Thus, for the sectionalized linear kinematic hardening model, the Mises’s yielding surface is as follows: J&s, here, (a,), = C;! , ck Am,
- (a,)&,
-
(5)
(qM1 - 0, = 0
n is the loading step.
3.2. Elastic-plastic FEh4 analysis For isotropic hardening material, the elastic-plastic form is as follows d(a) = where
[D],,
is elastic-plastic
matrix,
constitutive equation[7] in incremental
(6)
Plep 44 = (PI, - PI,) W [D], is
elastic matrix, [D], is plastic matrix:
(7)
PI, =
(b)
500
lb
01
5
10
15
20
1 PA)
Fig. l(a).
The tension
curve of 20g steel.
Fig. l(b). The Bauschinger
curve of 20g steel.
Critical void growth ratio and crack initiation 20
60
Y*
177
t-i * G
x ::
01 01
Y 5 Fig. 2. The dimensions of three-point-bend specimen (in mm).
Here, H’ is the slope of the material’s bicurve at present stress and strain state. In the present paper’s program, the primary yielding surface is considered as the motion of translation between loading steps. In every loading step, taking the yielding center been translated as yielding center, the primary yielding surface been translated as yielding surface, isotropic hardening method is used in calculation. While the tension curve shown in Fig. l(a) is inputted, an equivalent stress-equivalent strain point, (500 MPa, 2.5), is supplemented and that doesn’t affect the calculation precision because both the dimension of the element and the zone with large strain in the vicinity of the crack tip are very small. In the same time, load self-amendment method is taken to increase the calculation precision and decrease error. Half the three-point-bend specimen is taken to calculate because of its symmetrical characteristic and constant strain triangular element is used in the program. 312 nodes and 547 elements are partitioned and the ratio of the minimum element’s dimension at the crack tip to the crack length is 0.0013 16. The calculation is executed on Honeywell DPS/8 computer.
4. RESULTS AND DISCUSSIONS 4.1. The relationship between crack tip opening displacement and the maximum strain
To obtain the crack tip opening displacement 6,. The displacement distribution on the crack surface is calculated for every loading step, as shown in Fig. 3, and 6, is obtained with least-square fit and linear extrapolation method[8]. (L 6,,,), the dots of the crack tip opening displacement and the maximum plastic strain at the crack tip under each loading step, are shown in Fig. 4. It is clearly shown that dT is directly proportion to emax,i.e. 6, = 0.207~,,, .
(8)
Because the maximum plastic strain t,,, at the crack tip reaches a critical value cmaxas the crack tip opening displacement reaches the critical value a,, (i.e. the crack propagates forward as the maximum plastic strain at the crack tip reaches the critical value E,,,), ...-. it may be recognized that, for crack initiation, the crack critical opening displacement criterion is identical to the maximum strain criterion and both the criteria are essentially the same but different in form. 0.3
-G 0.2 E bo 0.1
0
I 0.1
0.2
I
I
I
I
I
I
I
0.3
0.4
0.6
0.6
0.7
0.6
0.9
I 1 .o
y,/a
Fig. 3. The displacement distribution on the crack surface.
178
ZHU HANXING et al.
00’
0.2
-
0.1
-
.*
/ 0’
,d 0
0.1
/
,
I
I
I
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.1
0.9
0.9
1.0
lmar
Fig. 4. The relation between crack tip opening displacement and the maximum plastic stain.
4.2. The distributions of cP a,/6
and a,, on the crack extended line
From ref.[9], we know that the fracture toughness of 20g steel is 0.22 mm, i.e. 6, = 0.22 mm. The distributions of equivalent plastic strain Q,, stress triaxiality 0,/5 and tensile stress cXXon the crack extended line as 6, = 0.205 mm are shown in Fig. 5; and it is found that, there exists a maximum point ahead of the crack for both a,,,/6 and gxx. 4.3. The calculating analysis of V, parameter and the crack initiation condition For the growth law of one spherical void in fully plastic material Rice-Tracey[S] proposed the following model:
of infinite extent,
dR/R = 0.203 exp i?
dc,. (9) 0 ) ( material, based on Von Mises’s yielding criterion, the above equation
For power-hardening can be rewritten as
dR/R =c exp i$’
dc,. ) Considering the whole process from nucleation to coalescence and taking the nucleation strain cd as very small[lO], we have: (
(11) Equation (1) is obtained from the above equation when cr,/a is a constant. For a three-point-bend specimen, although the applied load increases in proportion, the o,/d value at the crack tip zone doesn’t keep constant. Figure 6 has shown the relationship between 6, and the a,,,/5 value in the vicinity of y = 0.3 mm ahead of the crack tip. The exp(&,,/5) N Q, curve at Qx
0.6
u’
( MPa)
1600
:#ps&
_I,;
0.2
-
0.1
-
0
600
T
-2
,300 Qp
-
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
:
:
:
:
:
:
0.6
0.7
0.8
0.9
1.0
1.1
1
ytmm)
Fig. 5. The distributions of c,, oxr and ~,,,/a ahead of the crack tip when 6, = 0.205 mm.
179
Critical void growth ratio and crack initiation
0.02
0.06
0.10
0.14
0.16
8Thm)
Fig. 6. The a,,,/5 - 6, curve at y = 0.3 mm ahead of the crack tip.
y = 0.177 mm ahead of the crack tip is shown in Fig. 7. At cP= 0.03, from (11) the Vo value is the area of the shaded portion OACD which can be obtained from the numerical integration method, but from eq. (1) the Vo value is the area of the square OBCD. To further compare eq. (11) with eq. (l), the vicinity of y = 0.56 mm ahead of the crack tip is investigated and the exp($r,/o) - cP curve of the vicinity is shown in Fig. 8. The Vo values of the three points: A, B and C in Fig. 8 are calculated individually using eqs (11) and (1) and the results are given in Table 3. From Table 3, it is found that, Vo(,i) increases monotonically with the increment of cp, but Vo(i)(the results of eq. (l)), does not. In the physical sense of Vo, Vo(,)clearly doesn’t correspond to the experimental result, because there exists a triaxial stress state in front of the three-point-bend specimen’s crack and the void can not become smaller but can enlarge. Therefore, it is better to calculate the Vo value with eq. (11) than with eq. (1). When 6, = 0.205 mm, the Vo value on the crack extended line is obtained from eq. (1 l), as shown in Fig. 9. It is found that, there exists a maximum value of Vo at y = 0.3 mm and that can be explained by Fig. 5 and eq. (11) because there exists a maximum point of a,/6 in the vicinity of y = 0.3 mm and Vo is approximately in direct proportion to exp&,/a) and those determine that there also exists a maximum point of Vo at the very place although Vo is approximately in direct proportion to cP too, but at y = 0.3 mm, the variation of cP is quite plain and that hasn’t much effect on the maximum Vo position of y = 0.3 mm. From ref. [l 11,we know that the maximum gXX position is directly proportion to 6, and that is quite reasonable. For example, as 6, tends to 0, the three-point-bend specimen reaches an elastic state and the maximum cr, position certainly tends to the crack tip point. For plane strain three-point-bend specimen’s, the maximum o,, position generally is the maximum value position of a,,,/6 (see Fig. 5) and also approximately is the maximum Vo position for high ductile material, therefore, the maximum Vo position moves forward with the increased BT (that has been proved by our calculation results). For 20g steel, assuming the distance between the maximum Vo position and the crack tip as yc , we approximately have Y,z 1.58r.
(12)
100
I+
-
a E
60 ___---_
40 10
-
20
0
0.01
0.02
0.03
0.04
0
0.02
Fig. 7. The exp(&,/oZ) -Q, curve at y = 0.177 mm ahead of the crack tip.
0.06
0.10
0.14
0.18
lP
‘P
Fig. 8. The exp($,/@
- cP curve at y = 0.56 mm ahead of the crack tip.
180
ZHU
HANXING
et al.
Table 3. 6,
VW) VoW,
0.0736
0.135
0.196
1.0628 2.06
0.9489 2.69
0.752 3.02
Figure 10 is the crack tip photogram of 20g steel at the crack initiation and it is found that the distance between the crack and the void ahead of the crack tip is approximately 0.34 mm. In the light of ref.[9], the crack critical opening displacement of 20g steel is 0.22mm, i.e. 6,, = 0.22mm. Those agree quite well with eq. (12) and show that the void formation at Y, ahead of the crack tip is owing to the existence of the maximum Vo at the very place and the crack propagates forward when the maximum Vo reaches a critical value VGc
.
The experimental results of 08F steel in ref.[l2] also show that the void ahead of the crack tip is formed before the principal crack propagates. Owing to that the experiment is in the plane stress state, the ratio of the distance Y, between the void and the crack tip to the crack opening displacement 6, is smaller than that in plane strain state because the stress triaxiality parameter a,,,/~?in plane stress state is much smaller than that in plane strain state. The weighting of equivalent plastic stain cP in V, value of plane stress state is larger than in that of plane strain state. As shown in Fig. 5, the cP value on the crack extended line decreases promptly with the increased Y, so, the maximum VG position ahead of the crack tip in plane stress state is certainly closer to the crack tip than in the plane strain state. Therefore, the experimental results of ref.[l2] also agree with this paper. In general, for the void ahead of the crack, its nucleation, growth and coalescence with the principal crack are caused by the increment of VG. When the maximum V, value ahead of the crack tip reaches a critical value VGc, the void coalesces with the principal crack, i.e. the crack propagates forward, that is the crack initiation characteristic of ductile material. The larger VG is the larger &, and cm,,, are, and vice versa. All of them can be taken for the crack propagation condition, but, for ductile material, V,, is the real cause of the crack initiation and, 6,, and L,,,, are only the phenomena existing in the fracture process. Taking account of the convenience in practical application, it is more suitable to choose dTCas the crack initiation criterion for ductile material. On the other hand, the value and the position Y, of the maximum V, ahead of the crack tip increase with the increased crack opening displacement &, as to where a large void is formed in front of the crack tip, that is closely related to the material’s property. For low strength steel, a void is usually formed by the separation of the interface between the ferrite matrix and non-metallic inclusions, or by the inclusion cracking[l3]. These steels are mainly manganese sulphide[l4]. Generally, for high ductile materials, V,, 6,, and cmaxcare quite large because it is difficult for a void to form and grow in the vicinity of the crack tip.
7r
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0
0.9
1.0
1.1
1.2
ytmm)
Fig. 9. V, value ahead
of the crack
tip when 6, = 0.205 mm.
Critical void growth ratio and crack initiation
e
Fig. 10. The crack tip photogram of 20g steel 50x
181
Critical void growth ratio and crack initiation
183
5. CONCLUSIONS (1) For 20g steel, the crack opening displacement S, is directly proportional to the maximum plastic strain, i.e. 6, = 0.207~,,, . Therefore, it may be recognized that, the crack begins to propagate as the maximum plastic strain at the crack tip reaches a critical value cmaxC,6, and 6maxecan all be taken as crack initiation criteria and both the criteria are not different in essentially but in form. (2) The CJ,/C values of the crack tip zone do not remain constant. The Vo value calculated from eq. (11) differs remarkably from that calculated from eq. (1) and VouIj increases monotonically with the increased cP, but Vo(,) does not keep this property, but decreases gradually as cP increases. Therefore, to calculate the Vo value with eq. (11) is more suitable than with eq. (1). (3) There exists a maximum Vo value ahead of the crack tip that is the cause of the void formation at the very place, i.e. at the position of Y, = 1.56r. (4) For ductile materials, when the maximum Vo value ahead of the crack tip reaches a critical value VoC,the void coalesces with the principal crack, i.e. the crack propagates forward. Therefore, VoCis the real cause of the crack initiation and may be taken as the crack initiation condition. Although 6, and cmaxcare the crack initiation conditions they are only the phenomena existing in the crack initiation process. Acknowledgement-Project
supported by National Science Foundation of China.
REFERENCES [1] C. Q. Zheng and J. C. Radon, Proc. ICF Inr. Symp. Fracture Mech. (pp. 1052-1056), Beijing (1983). [2] C. Q. Zheng and J. C. Radon, Proc. ICF Inr. Symp. Fracture Mech. (pp. 1057-1062), Beijing (1983). [3] C. Q. Zheng, L. Zhou and J. M. Liu, Proc. 9rh Congress Mater. Testing and 3rd Danubia-Adria Symp. (pp. 8691) Budapest (1986). [4] C. Q. Zheng, L. Zhou and J. M. Liu, Proc. KM5 (pp. 213-218), Beijing (1987). [5] J. R. Rice and P. M. Tracey, J. Mech. Phys. Solids 17, 201-217 (1969). [6] R. Shield and M. Ziegler, ZAMP, 9a, 260 (1958). [7] Yiquan Xie and Fubao He, FEM Method in Elastic and Plastic Mechanics, pp. 206-207. Machinery Industry Press, China (1983). [8] Xiaobin Lin, Guoyou Sun and Jiliang Xue, J. Zhejiang Uniuersiry, China 21(3), 75-84 (1987). [9] Hanxing Zhu, Master’s degree thesis. Zhejing University, HangZhou, China (1986). [lo] J. W. Hancock and A. C. Mackengie, J. Mech. Phys. Solids, 24, 147-169 (1976). [11] Hanxing Zhu, Guogou Sun and Jiliang Xue, Acrn Mechanica Solida sinica, (China), 1990. [12] Changchun Li, Xiping Li and Guangxia Li, J. Huazhong Univ. Sci. Technol. (Wuhan, China), 17(3) 139-144 (1989). [13] Y. W. Shi and J. T. Batnby, Inr. J. Fracture 25, 143-151 (1984). [14] D. A. Shockey, L. Seaman, K. C. Dao and R. Curran, Paper No78-PVP-92. Presented at the ASME/CSME Pressure Vessels and Piping Conference, Montreal, Canada, June 25-30, 1978. (Received 17 Ocrober 1989)