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Effect of stress triaxiality in crack tip field on validity of J-dominance
Engineering Fracture Mechanics Vol. 34, No. 3,
pp.637-643,
0013-7944/89 $3.00 + 0.00 0 1989 Pergamon Press plc.
1989
Printed in Great Britain.
EFFECT
OF STRESS TRIAXIALITY IN CRACK ON VALIDITY OF J-DOMINANCE SUN JUN, DENG ZENGJIE,
LI ZHONGHUA
and
TIP FIELD
TU MINGJING
Department of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an, P.R.C. Abstract-The distribution and variation of stress triaxiality in crack tip regions of different specimen geometries have been calculated by the finite element method, and then the validity of J-dominance has also been analyscd. It is found that the mutual relation between the constraint intensity-stress triaxiality in a crack tip field of real specimens and that of HRR field, is a decisive factor for that whether J-dominance is valid or not.
1. INTRODUCTION As A PARAMETER characterizing crack tip field, the J-integral has played an important part in elastic-plastic fracture mechanics and engineering applications. According to the definition of J-dominance in a crack tip field, the stress-strain field at a crack tip should be uniquely determined as the J-value was given[l, 21. As long as the requirements of relevant dimensions were met for real specimens, the validity of the J-dominance of the crack tip field would be ensured [3-51, and then, .&--a fracture criterion of the J-integral would become a material constant independent from the specimen geometry[6,7]. But recent research shows that only for plain stress is the J-dominance of crack tip field valid[8], and J,c can be used as a fracture criterion which is an invariant with the variation of specimen geometries[9]; but for plain strain, the obvious difference of the stress-strain field will exist in crack tip regions of different types of specimens, and it is also different from that of the HRR field. Thus, the J-dominance of a crack tip field is seriously disturbed[lO]. Eventually, J,c is dependent on the type of specimen geometries[ll]. Consequently, considerable doubts have been cast on the capability of J,c as a valid single parameter fracture criterion to characterize the entire crack tip field. From the solution of the slip-line field of elastic-perfectly plastic materials we know that: the difference of the stress-strain fields in crack tip regions of different specimen geometries are accompanied by different constraint intensities of crack tip field-stress triaxiality[l2]. For hardening materials, the high triaxial constraint in crack tip regions was required by an HRR solution. The relation between stress triaxiality of various specimen geometries and that of the HRR field will strongly affect the validity of the J-dominance of the crack tip field. However, so far clear conclusions for the problem have not been obtained. This paper presents the distribution and variation of the stress triaxiality in the crack tip regions of different specimen geometries and their affecting factors. The essential reason for deciding the validity of the J-dominance of a crack tip field was revealed.
2. SPECIMENS,
MATERIALS
AND FINITE
ELEMENT
CALCULATIONS
The large displacement finite element program of the J-integral was used. The stress-strain response of materials obey the power hardening relation: L =olE;
o
(1)
c = cia”; a >aO.
(2)
HYSO Steel was employed: yield stress a,, = 560 MPa; Young’s modulus E = 200 GPa; and Poisson’s ratio v = 0.3; strain hardening exponent n = 9 and coefficient a = 0.544. Four types of specimens and loadings were calculated: center-cracked plain, CCP; single-edge-cracked, SEC; double-edge-cracked, DEC and three-point-bending specimen, TPB (as shown in Fig. 1). 631
638
SUN JUN et al.
I a El60
40
40
20
20
1-q
0
:
Fig. I. Schematic representation
is
z
of the specimen geometries, loadings and dimensions (in mm).
The finite element mesh involves 128 elements and 435 node points. Eight node isoparametric elements are employed. The blunted notch is chosen as the crack tip model (r = 0.004 mm in Fig. 2). The radial length of the elements closest to crack tip is l/2560 of the ligament size. About 200 increments are employed to cover the range from the small scale yielding to the deep general yielding. The ratio of hydrostatic stress 6, to the Von Mises effective stress ~7- o,,,/r? is used to characterize the constraint intensity of the crack tip field-stress triaxiality. For the HRR solution, the stress field in the crack tip is (q, CT)=
rJ0
EJ [ ao:Zr
1Ai
kqe, n), we, n)I
(3)
from which the stress triaxiality of HRR field is given as (4) where M = 2 for plain strain and M = 3 for plain stress. According to the HRR solution for the given material (n value), a,,,/~?in the crack tip regions will be only relative to the O-angles and independent of the specimen geometries, the distance from crack tip and the load level, keeping in mind the feature of stress triaxiality required by HRR field, which is significant for the validity of J-dominance of crack tip field in real specimens.
3. VALIDITY
OF J-DOMINANCE
FOR PLAIN STRESS
Figure 3(a) shows the distribution and variation of the stress triaxiality in the crack tip field with the variations of O-angles from the crack line, in which the solid line is that required by the HRR solution for HY80 steel (n = 9 from ref. [2]). For all types of four specimen geometries, the stress triaxiality in crack tip regions calculated by FEM coincides with each other and agree well with that of the HRR field. Along the ligaments of the four specimens, the stress triaxiality is very close to that of HRR field (as shown in Fig. 3b). At a different load level, the computations by FEM also indicate that the results are the same as above for the four specimens, in accordance with that required by HRR’s triaxial constraint independent of the types of specimen geometry, the distance from the crack-tip and the load level.
Fig. 2. Finite element mesh for four specimens. The crack-tip detail is magnified on the right.
Stress triaxiality and J-dominance (0)
639
(b) I
I
0.5
0.5
bEilb
611b 0
0
0
0
n -s-
8
0.2
0.6
0.4
06
n R(mm)
Fig. 3. The distribution of stress triaxiality in the crack-tip field for plane stress: (a) vs the O-angle from the plane ahead of the crack-tip; (b) vs the distances from the notch root on the plane ahead of the blunting crack-tip.
From eq. (3), when the J-integral as a single parameter is capable of characterizing the entire crack-tip field, the plots of crYY/oO for 8 = 0 vs the distance from the crack-tip R, normalized by (J/o,,), should fall on the same curve. Figure 4(a) shows that all of the data for the four specimen geometries fall on same curve with each other and accord with that given by the HRR solution (the solid line in Fig. 4a). The viewpoints shown in the Fig. 4(b), which is a log-log plot applying the same data of Fig. 4(a), are more obvious. From the eq. (3), for log(u&,) and log[R/(J/a,)] the stress distribution given by the HRR solution should be a straight line, the slope of which is l/(n + 1). If the J-dominance of the crack-tip field is valid, the stress distribution of all four specimen geometries should be located on the same straight line together with that of the HRR solution at different sections. The calculations by FEM have indicated that all data from the four specimen geometries are such as that mentioned above, even at different load levels. Other stress and strain distributions are also same as a,,,/a,. In the plain stress, from the results above it is concluded that, as a single parameter, the J-integral is capable of characterizing the entire crack tip field. At any given J value, the crack tip field in these four different specimen geometries is the same and agrees well with that of the HRR field. That is to say that the J-dominance of the crack tip field is valid. All of these are because a unique correspondence exists between the stress triaxiality in crack tip regions of the four specimens and that required by the HRR solution.
4. LOSING
OF J-DOMINANCE
FOR PLAIN STRAIN
Contrary to plain stress, the stress triaxialities in the crack tip regions of all four types of specimens are obviously lower than that required by the HRR field for plain strain. The larger the e-angles from the crack line, the greater the difference (as shown in Fig. 5). The further the distance 3 (0)
2
b' \ b: I
-0
152.8
5
IO R
/(J/a.)
I5
20
25
-1.5
-1
-0.5
. . . . .
ccp
----
SEC
0
log CR /J/ue
0.5 ))
Fig. 4. (a) Normal stress on the ligaments for four specimens for plane stress: (b) The log-log plot of (a).
640
SUN JUN er al.
. HRR field
9150
-
101l0
-
HRR
HRR
HRR
f leld
0
7r
0
B
2
TT
8
-s-
(77)
Fig. 5. The stress triaxiality in the crack-tip field vs the O-angle from the crack line for four specimens from small scale yielding to general yielding for plane strain.
from the crack tip on the ligaments, the lower the stress triaxiality (as indicated in Fig. 6). The reduction of stress triaxiality induced by crack tip blunting is limited to a very small zone close to the crack tip. As the load increases, the stress triaxiality also decreases due to progressive relaxation of constraint in the crack tip field, in which it is most remarkable for the center-cracked plain (CCP), secondly for the single-edge-cracked (SEC), and then the double-edge-cracked (DEC) and three-point-bending specimen (TPB) are slightly slow. Up to that exceeding the limit load and the ligaments in general yielding, shown by FEM computations, the decreasing tendency of stress triaxiality has just started to be reduced and then reaches a stable level. But in this situation, the stress triaxiality of all four types of specimens has already been far lower than that required by HRR theory.
_
HRR
_
field
HRR field
3’
b’/lb
0
0
L‘JO -=94 J
I
_
3
0
915
0
Lob
22.4
.
J
SEC
1154
,o
102 *
-=
A
224
.
HRRf~eld
Rtmm)
Rlmm)
Fig. 6. Variations of stress triaxiality along the crack line for the four specimen geometries. From small scale yielding to general yielding for plane strain.
Stress triaxiality and J-dominance
_
441
HRR field 1154 :
0
102
A
22.4
.
l
SEC
CCP .
-
-
field
field
6
--5:
DEC
0
2
71.6
TPB 6 4 RlfJ/obl
6
0
2
A
l
4
6
6
IO
RItJ/qt
Fig. 7. Normal stress on the plane ahead of the blunting crack-tip vs the distance R from the notch root for four specimens. From small scale yielding to general yielding for plane strain.
It is notable that stress triaxiality in the crack-tip regions varies widely from one geometry type to another at a given J value, in the order of high to low are that of TPB, DEC, SEC and CCP specimens. The J-dominance of the crack-tip field will not be universally valid in the plain strain case because the stress triaxiality of all specimens cannot meet the requirements of a high triaxial constraint in the HRR field, and depends on the specimen geometry types, the distance from crack-tip and load level. As shown in Fig. 7, the normal stress on the ligaments of all four specimen geometries are lower than that of the HRR theory (solid line in Fig. 7), and the further the distance from the crack-tip, or the higher the load level, the larger the deviations. This characteristic is relative to the distribution and the variation of stress triaxiality of the various type of specimen geometry. As far as the CCP specimen is concerned, in a small scale yielding the stress triaxiality in the crack-tip field is considerably lower than that in the HRR field, with which there is large difference between the stress field of the CCP and that given by the HRR solution. In the large general yielding, while the stress triaxiality of the CCP has already been far lower than that of HRR field, the normal stress of the CCP has been far away from that given by the HRR solution. On the other hand, for the TPB specimen with a higher stress triaxiality in the crack-tip field (still lower than that of HRR field), the stress level is also located below the HRR’s stress level in small scale yielding. As the load increases, the normal stress of the TPB has also deviated from that of the HRR theory, which is corresponding to the decreasing stress triaxiality of the TPB as the load increases. Similarly to the order of the stress triaxiality of the four specimen geometries at a given J value, the difference between the stress level of the TPB, DEC, SEC, CCP specimens and that of the HRR theory are from large to small. Namely, the higher the stress triaxiality in the crack-tip region of specimen, which is closer to that required by the HRR field, the smaller the difference between the stress iield of either and vice versa. The same conclusions are more easily obtained when the data in Fig. 6 are compared with that in Fig. 8, which are log-log plots of Fig. 7. As indicated in Fig. 8, the stress distributions of all four specimens have different sections of the straight line except for
642
SUN JUN Ed al.
Ei
0.e
1010 . . . . .
HRR f ie(d
-
_
1154....* LG --_=
102
-
-
22 4 ----
b’ \ b’ G s
SEC
CCP 0 HRR field
-
9!5
l
...*
HRR field
-
BOB
0 a-
71.6 26
0.4
-
0.2
DEC
0 -15
-I
. . . . . -
-
_-__
TPB -0.5
0
0.5
I
15
-I
-05
0
0.5
I
I.5
log(R/fJ/a,)l
Fig. 8. The log--log plot of the distribution
of normal stress along the crack line. Data are taken from Fig. 7.
the stress deviation in the region near to crack-tip and remote field with the weak stress triaxiality constraint. However, the positions and slopes of these straight-line sections vary with the variations of specimen geometries from one to another, and load level, which corresponds to the cases of the stress triaxiality in Fig. 6. In other words, the different distribution feature of the stress field for various types of specimen is respectively accompanied with the fact that the stress triaxialities of specimens differ from each other, and then the lower stress triaxiality in the crack-tip regions of real specimens is not able to meet the higher triaxial constraint required by the HRR field according to the stress level lower than that given by the HRR solution at the same J value, and the higher the load level, the larger the difference. Therefore, as a single parameter chracterizing the crack-tip field, the J-integral has not been able to define the stress field in the crack-tip region effectively. From the viewpoint of constraint intensity of the crack-tip field-stress triaxiality to say strictly-the validity of J-dominance of the crack-tip field has not existed for the plain strain case. As a result, different J,c values have been measured from specimens with different geometric types, a difference of a factor of four for HY80 steel [l 11. The effect of the stress triaxiality on the fracture criterion of the J-integral will be discussed in the author’s other articles.
5. CONCLUSIONS (1) Under the condition of plain stress, the validity of the J-dominance has been ensured for the crack-tip field, and the constraint intensity in crack-tip regions-stress triaxiality-has met the triaxial constraint required by the HRR field, independent of the specimen geometries, the distance from the crack-tip and the load level. (2) For plain strain, the different stress triaxiality in the crack-tip regions for various specimen geometries, accompanied by the different stress fields at any given J value, has not met the requirements of the high triaxial constraint required by the HRR field. So J-dominance of the crack tip field is not valid again. (3) The mutual relations between the constraint intensity-stress triaxiality in the crack-tip field of real specimens and that required by the HRR solution, are the decisive factors which will determine whether J-dominance is valid or not. Consequently, it is necessary to use two parameters, J and a measure of near-tip triaxiality, to characterize the full range of near-tip fracture environments.
Stress triaxiality and J-dominance
REFERENCES [1] J. R. Rice and C. F. Rosengren, J. Mech. P&s. Solids 16, 1 (1968). [2] J. W. Hutchinson, J. Mech. Phys. Solids 16, 13 (1968). [3] R. M. McMeeking and D. M. Parks, ASTM STP 668, 175 (1979). [4] C. F. Shih and M. D. German, Int. J. Fracture 17, 27 (1981). [5] P. C. Paris, ASTM STP 514, 21 (1972). [6] J. A. Begley and J. D. Landes, Inr. J. Fracrure 12, 764 (1976). [7] C. F. Shih et al., ASTM STP 668, 65 (1979). [8] H. W. Liu er al., Proc. 6rh. Inr. Co@ Frucrure. India, Vol. 2, 777 (1984). [9] Y. W. Mai et al., Engng Fracture Mech. 21, 123 (1985). ‘IO] Zhuang Tao, Acru. Mech. Sinicu (in Chinese), 19, Sup, 154 (1987). fll] J. W. Hancock and M. J. Cowling, Metal Science 293 Aug.-Sep. 1980. [12] K. Z. Huang and S. W. Yu, Elusric-Plastic Fracture Mechanics, 115 (1986). (Received 23 November 1988)
643
pp.637-643,
0013-7944/89 $3.00 + 0.00 0 1989 Pergamon Press plc.
1989
Printed in Great Britain.
EFFECT
OF STRESS TRIAXIALITY IN CRACK ON VALIDITY OF J-DOMINANCE SUN JUN, DENG ZENGJIE,
LI ZHONGHUA
and
TIP FIELD
TU MINGJING
Department of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an, P.R.C. Abstract-The distribution and variation of stress triaxiality in crack tip regions of different specimen geometries have been calculated by the finite element method, and then the validity of J-dominance has also been analyscd. It is found that the mutual relation between the constraint intensity-stress triaxiality in a crack tip field of real specimens and that of HRR field, is a decisive factor for that whether J-dominance is valid or not.
1. INTRODUCTION As A PARAMETER characterizing crack tip field, the J-integral has played an important part in elastic-plastic fracture mechanics and engineering applications. According to the definition of J-dominance in a crack tip field, the stress-strain field at a crack tip should be uniquely determined as the J-value was given[l, 21. As long as the requirements of relevant dimensions were met for real specimens, the validity of the J-dominance of the crack tip field would be ensured [3-51, and then, .&--a fracture criterion of the J-integral would become a material constant independent from the specimen geometry[6,7]. But recent research shows that only for plain stress is the J-dominance of crack tip field valid[8], and J,c can be used as a fracture criterion which is an invariant with the variation of specimen geometries[9]; but for plain strain, the obvious difference of the stress-strain field will exist in crack tip regions of different types of specimens, and it is also different from that of the HRR field. Thus, the J-dominance of a crack tip field is seriously disturbed[lO]. Eventually, J,c is dependent on the type of specimen geometries[ll]. Consequently, considerable doubts have been cast on the capability of J,c as a valid single parameter fracture criterion to characterize the entire crack tip field. From the solution of the slip-line field of elastic-perfectly plastic materials we know that: the difference of the stress-strain fields in crack tip regions of different specimen geometries are accompanied by different constraint intensities of crack tip field-stress triaxiality[l2]. For hardening materials, the high triaxial constraint in crack tip regions was required by an HRR solution. The relation between stress triaxiality of various specimen geometries and that of the HRR field will strongly affect the validity of the J-dominance of the crack tip field. However, so far clear conclusions for the problem have not been obtained. This paper presents the distribution and variation of the stress triaxiality in the crack tip regions of different specimen geometries and their affecting factors. The essential reason for deciding the validity of the J-dominance of a crack tip field was revealed.
2. SPECIMENS,
MATERIALS
AND FINITE
ELEMENT
CALCULATIONS
The large displacement finite element program of the J-integral was used. The stress-strain response of materials obey the power hardening relation: L =olE;
o
(1)
c = cia”; a >aO.
(2)
HYSO Steel was employed: yield stress a,, = 560 MPa; Young’s modulus E = 200 GPa; and Poisson’s ratio v = 0.3; strain hardening exponent n = 9 and coefficient a = 0.544. Four types of specimens and loadings were calculated: center-cracked plain, CCP; single-edge-cracked, SEC; double-edge-cracked, DEC and three-point-bending specimen, TPB (as shown in Fig. 1). 631
638
SUN JUN et al.
I a El60
40
40
20
20
1-q
0
:
Fig. I. Schematic representation
is
z
of the specimen geometries, loadings and dimensions (in mm).
The finite element mesh involves 128 elements and 435 node points. Eight node isoparametric elements are employed. The blunted notch is chosen as the crack tip model (r = 0.004 mm in Fig. 2). The radial length of the elements closest to crack tip is l/2560 of the ligament size. About 200 increments are employed to cover the range from the small scale yielding to the deep general yielding. The ratio of hydrostatic stress 6, to the Von Mises effective stress ~7- o,,,/r? is used to characterize the constraint intensity of the crack tip field-stress triaxiality. For the HRR solution, the stress field in the crack tip is (q, CT)=
rJ0
EJ [ ao:Zr
1Ai
kqe, n), we, n)I
(3)
from which the stress triaxiality of HRR field is given as (4) where M = 2 for plain strain and M = 3 for plain stress. According to the HRR solution for the given material (n value), a,,,/~?in the crack tip regions will be only relative to the O-angles and independent of the specimen geometries, the distance from crack tip and the load level, keeping in mind the feature of stress triaxiality required by HRR field, which is significant for the validity of J-dominance of crack tip field in real specimens.
3. VALIDITY
OF J-DOMINANCE
FOR PLAIN STRESS
Figure 3(a) shows the distribution and variation of the stress triaxiality in the crack tip field with the variations of O-angles from the crack line, in which the solid line is that required by the HRR solution for HY80 steel (n = 9 from ref. [2]). For all types of four specimen geometries, the stress triaxiality in crack tip regions calculated by FEM coincides with each other and agree well with that of the HRR field. Along the ligaments of the four specimens, the stress triaxiality is very close to that of HRR field (as shown in Fig. 3b). At a different load level, the computations by FEM also indicate that the results are the same as above for the four specimens, in accordance with that required by HRR’s triaxial constraint independent of the types of specimen geometry, the distance from the crack-tip and the load level.
Fig. 2. Finite element mesh for four specimens. The crack-tip detail is magnified on the right.
Stress triaxiality and J-dominance (0)
639
(b) I
I
0.5
0.5
bEilb
611b 0
0
0
0
n -s-
8
0.2
0.6
0.4
06
n R(mm)
Fig. 3. The distribution of stress triaxiality in the crack-tip field for plane stress: (a) vs the O-angle from the plane ahead of the crack-tip; (b) vs the distances from the notch root on the plane ahead of the blunting crack-tip.
From eq. (3), when the J-integral as a single parameter is capable of characterizing the entire crack-tip field, the plots of crYY/oO for 8 = 0 vs the distance from the crack-tip R, normalized by (J/o,,), should fall on the same curve. Figure 4(a) shows that all of the data for the four specimen geometries fall on same curve with each other and accord with that given by the HRR solution (the solid line in Fig. 4a). The viewpoints shown in the Fig. 4(b), which is a log-log plot applying the same data of Fig. 4(a), are more obvious. From the eq. (3), for log(u&,) and log[R/(J/a,)] the stress distribution given by the HRR solution should be a straight line, the slope of which is l/(n + 1). If the J-dominance of the crack-tip field is valid, the stress distribution of all four specimen geometries should be located on the same straight line together with that of the HRR solution at different sections. The calculations by FEM have indicated that all data from the four specimen geometries are such as that mentioned above, even at different load levels. Other stress and strain distributions are also same as a,,,/a,. In the plain stress, from the results above it is concluded that, as a single parameter, the J-integral is capable of characterizing the entire crack tip field. At any given J value, the crack tip field in these four different specimen geometries is the same and agrees well with that of the HRR field. That is to say that the J-dominance of the crack tip field is valid. All of these are because a unique correspondence exists between the stress triaxiality in crack tip regions of the four specimens and that required by the HRR solution.
4. LOSING
OF J-DOMINANCE
FOR PLAIN STRAIN
Contrary to plain stress, the stress triaxialities in the crack tip regions of all four types of specimens are obviously lower than that required by the HRR field for plain strain. The larger the e-angles from the crack line, the greater the difference (as shown in Fig. 5). The further the distance 3 (0)
2
b' \ b: I
-0
152.8
5
IO R
/(J/a.)
I5
20
25
-1.5
-1
-0.5
. . . . .
ccp
----
SEC
0
log CR /J/ue
0.5 ))
Fig. 4. (a) Normal stress on the ligaments for four specimens for plane stress: (b) The log-log plot of (a).
640
SUN JUN er al.
. HRR field
9150
-
101l0
-
HRR
HRR
HRR
f leld
0
7r
0
B
2
TT
8
-s-
(77)
Fig. 5. The stress triaxiality in the crack-tip field vs the O-angle from the crack line for four specimens from small scale yielding to general yielding for plane strain.
from the crack tip on the ligaments, the lower the stress triaxiality (as indicated in Fig. 6). The reduction of stress triaxiality induced by crack tip blunting is limited to a very small zone close to the crack tip. As the load increases, the stress triaxiality also decreases due to progressive relaxation of constraint in the crack tip field, in which it is most remarkable for the center-cracked plain (CCP), secondly for the single-edge-cracked (SEC), and then the double-edge-cracked (DEC) and three-point-bending specimen (TPB) are slightly slow. Up to that exceeding the limit load and the ligaments in general yielding, shown by FEM computations, the decreasing tendency of stress triaxiality has just started to be reduced and then reaches a stable level. But in this situation, the stress triaxiality of all four types of specimens has already been far lower than that required by HRR theory.
_
HRR
_
field
HRR field
3’
b’/lb
0
0
L‘JO -=94 J
I
_
3
0
915
0
Lob
22.4
.
J
SEC
1154
,o
102 *
-=
A
224
.
HRRf~eld
Rtmm)
Rlmm)
Fig. 6. Variations of stress triaxiality along the crack line for the four specimen geometries. From small scale yielding to general yielding for plane strain.
Stress triaxiality and J-dominance
_
441
HRR field 1154 :
0
102
A
22.4
.
l
SEC
CCP .
-
-
field
field
6
--5:
DEC
0
2
71.6
TPB 6 4 RlfJ/obl
6
0
2
A
l
4
6
6
IO
RItJ/qt
Fig. 7. Normal stress on the plane ahead of the blunting crack-tip vs the distance R from the notch root for four specimens. From small scale yielding to general yielding for plane strain.
It is notable that stress triaxiality in the crack-tip regions varies widely from one geometry type to another at a given J value, in the order of high to low are that of TPB, DEC, SEC and CCP specimens. The J-dominance of the crack-tip field will not be universally valid in the plain strain case because the stress triaxiality of all specimens cannot meet the requirements of a high triaxial constraint in the HRR field, and depends on the specimen geometry types, the distance from crack-tip and load level. As shown in Fig. 7, the normal stress on the ligaments of all four specimen geometries are lower than that of the HRR theory (solid line in Fig. 7), and the further the distance from the crack-tip, or the higher the load level, the larger the deviations. This characteristic is relative to the distribution and the variation of stress triaxiality of the various type of specimen geometry. As far as the CCP specimen is concerned, in a small scale yielding the stress triaxiality in the crack-tip field is considerably lower than that in the HRR field, with which there is large difference between the stress field of the CCP and that given by the HRR solution. In the large general yielding, while the stress triaxiality of the CCP has already been far lower than that of HRR field, the normal stress of the CCP has been far away from that given by the HRR solution. On the other hand, for the TPB specimen with a higher stress triaxiality in the crack-tip field (still lower than that of HRR field), the stress level is also located below the HRR’s stress level in small scale yielding. As the load increases, the normal stress of the TPB has also deviated from that of the HRR theory, which is corresponding to the decreasing stress triaxiality of the TPB as the load increases. Similarly to the order of the stress triaxiality of the four specimen geometries at a given J value, the difference between the stress level of the TPB, DEC, SEC, CCP specimens and that of the HRR theory are from large to small. Namely, the higher the stress triaxiality in the crack-tip region of specimen, which is closer to that required by the HRR field, the smaller the difference between the stress iield of either and vice versa. The same conclusions are more easily obtained when the data in Fig. 6 are compared with that in Fig. 8, which are log-log plots of Fig. 7. As indicated in Fig. 8, the stress distributions of all four specimens have different sections of the straight line except for
642
SUN JUN Ed al.
Ei
0.e
1010 . . . . .
HRR f ie(d
-
_
1154....* LG --_=
102
-
-
22 4 ----
b’ \ b’ G s
SEC
CCP 0 HRR field
-
9!5
l
...*
HRR field
-
BOB
0 a-
71.6 26
0.4
-
0.2
DEC
0 -15
-I
. . . . . -
-
_-__
TPB -0.5
0
0.5
I
15
-I
-05
0
0.5
I
I.5
log(R/fJ/a,)l
Fig. 8. The log--log plot of the distribution
of normal stress along the crack line. Data are taken from Fig. 7.
the stress deviation in the region near to crack-tip and remote field with the weak stress triaxiality constraint. However, the positions and slopes of these straight-line sections vary with the variations of specimen geometries from one to another, and load level, which corresponds to the cases of the stress triaxiality in Fig. 6. In other words, the different distribution feature of the stress field for various types of specimen is respectively accompanied with the fact that the stress triaxialities of specimens differ from each other, and then the lower stress triaxiality in the crack-tip regions of real specimens is not able to meet the higher triaxial constraint required by the HRR field according to the stress level lower than that given by the HRR solution at the same J value, and the higher the load level, the larger the difference. Therefore, as a single parameter chracterizing the crack-tip field, the J-integral has not been able to define the stress field in the crack-tip region effectively. From the viewpoint of constraint intensity of the crack-tip field-stress triaxiality to say strictly-the validity of J-dominance of the crack-tip field has not existed for the plain strain case. As a result, different J,c values have been measured from specimens with different geometric types, a difference of a factor of four for HY80 steel [l 11. The effect of the stress triaxiality on the fracture criterion of the J-integral will be discussed in the author’s other articles.
5. CONCLUSIONS (1) Under the condition of plain stress, the validity of the J-dominance has been ensured for the crack-tip field, and the constraint intensity in crack-tip regions-stress triaxiality-has met the triaxial constraint required by the HRR field, independent of the specimen geometries, the distance from the crack-tip and the load level. (2) For plain strain, the different stress triaxiality in the crack-tip regions for various specimen geometries, accompanied by the different stress fields at any given J value, has not met the requirements of the high triaxial constraint required by the HRR field. So J-dominance of the crack tip field is not valid again. (3) The mutual relations between the constraint intensity-stress triaxiality in the crack-tip field of real specimens and that required by the HRR solution, are the decisive factors which will determine whether J-dominance is valid or not. Consequently, it is necessary to use two parameters, J and a measure of near-tip triaxiality, to characterize the full range of near-tip fracture environments.
Stress triaxiality and J-dominance
REFERENCES [1] J. R. Rice and C. F. Rosengren, J. Mech. P&s. Solids 16, 1 (1968). [2] J. W. Hutchinson, J. Mech. Phys. Solids 16, 13 (1968). [3] R. M. McMeeking and D. M. Parks, ASTM STP 668, 175 (1979). [4] C. F. Shih and M. D. German, Int. J. Fracture 17, 27 (1981). [5] P. C. Paris, ASTM STP 514, 21 (1972). [6] J. A. Begley and J. D. Landes, Inr. J. Fracrure 12, 764 (1976). [7] C. F. Shih et al., ASTM STP 668, 65 (1979). [8] H. W. Liu er al., Proc. 6rh. Inr. Co@ Frucrure. India, Vol. 2, 777 (1984). [9] Y. W. Mai et al., Engng Fracture Mech. 21, 123 (1985). ‘IO] Zhuang Tao, Acru. Mech. Sinicu (in Chinese), 19, Sup, 154 (1987). fll] J. W. Hancock and M. J. Cowling, Metal Science 293 Aug.-Sep. 1980. [12] K. Z. Huang and S. W. Yu, Elusric-Plastic Fracture Mechanics, 115 (1986). (Received 23 November 1988)
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