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Stress triaxiality constraint and crack tip parameters
Engineering
Fracture
Mechanics
Vol. 44,No.5,pp.789-806,
1993
0013-7944/93
Printedin Great Britain.
$6.00
+ 0.00
0 1993 Pergamon Press Ltd.
STRESS TRIAXIALITY CONSTRAINT TIP PARAMETERS
AND CRACK
SUN JUNt Department
of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R.C.
Abstract-A relationship between crack tip constraints of specimens and those defined by the Hutchinson-Rice-Rosengren (HRR) solution in the plane strain case has been derived from the corrected stress-strain field presented by Anderson, by taking account of losses in triaxiality of specimens. An analytical expression has been established and verified which describes a sufficient and necessary condition for the validity of a single parameter characterizing crack tip stresses; namely the crack tip triaxiality constraints defined by the single parameter should be satisfied by those of specimens. The simple correction for crack tip stresses proposed by Anderson has been extended to situations of different specimen geometry. Furthermore, a new elastic-plastic fracture criterion, including the effect of crack tip triaxiality on the parameter characterizing fracture, has been developed and proven to be a material constant independent from specimen geometry for a given material state. It has also been demonstrated that it is impossible to obtain the plane strain fracture toughness J,c directly from laboratory test specimens because the crack tip triaxiality constraints required by the HRR solution cannot be met for various specimens.
1. INTRODUCTION THE DEVELOPMENT and confirmation of proper fracture criteria for crack growth initiation has been and still is one of the most important objectives in the study of elastic-plastic fracture mechanics. As a single parameter characterizing the crack tip field, the J-integral has played an important part in fracture mechanics and engineering applications over the past 20 years, and was considered as the most appealing criterion for crack growth initiation [l-4]. It is well known that geometry independence is a crucial test of the acceptability of a proper elastic-plastic fracture criterion for a given material state. Therefore, there have been many experimental investigations to evaluate the effect of specimen size, geometry and thickness on the Jlc or Ji values, some of which have revealed that JIG and Ji depend on the specimen geometry [5--l, crack depth [8,9], and specimen thickness [lo, 1l] in the plane strain case, even though ASTM E813-81 requirements for specimen dimension were met. Hence, it remains unknown whether or not J,, and Ji values measured from standard laboratory test specimens are suitable to evaluate the safety of the cracked structure with various geometries, thicknesses and crack depths. Thus, considerable doubts have been cast on the capability of J,c as a valid single parameter fracture criterion to characterize the material resistance against crack growth initiation. Further studies showed that the relationship between the stress triaxiality in the crack tip field of specimens and that defined by the Hutchinson-Rice-Rosengren (HRR) solution is the decisive factor which will determine whether J-dominance is valid or not [12]. In the plane strain case, the stress triaxiality constraints in crack tip regions of real specimens, which vary with the variations of specimen geometries, crack depths and load levels, are unable to meet the high triaxial constraint requirements defined by the HRR solution, which is the essential reason for losing the J-dominance and causing the specimen geometry dependence of J,c [13, 141. Having discussed the defects of J-dominance and the effects of stress triaxiality, Hutchinson proposed that two parameters, the J-integral and the measure of near-tip triaxiality, might suffice to characterize the full range of near-tip fracture environments [ 151. Another investigation has also indicated that it is very necessary to introduce a constraint parameter in addition to J to obtain a better characterization of the local stress field in crack tip regions [16]. Recently, significant developments for this problem have been made in Anderson’s research [17], in which a simple correction for crack tip stresses was derived from the quasi-deformation assumption, to take account of losses in triaxiality in the crack tip regions of specimens. However, it was limited to only a center-cracked panel that predictions of crack tip stress fields agreed well with some finite tPresent address: Department Canada K7L 3N6.
of Materials and Metallurgical Engineering, Queen’s University, Kingston, Ontario,
789
SUN JUN
790
element results because no quantitative relationship exists between the crack tip triaxiality and the ~haracte~~ng parameters. In the present article, Anderson’s results are employed, together with some finite element calculations and experiments, to discuss the effects of triaxiality on crack tip stress fields and characterizing parameters. By means of the relationship proposed here between the triaxiality of specimens and that defined by the HRR solution, the triaxiality constraints are introduced analytically into crack tip stress fields and characterizing parameters. Some interesting results have been found along this line. 2. STRESS
TRIAXIALTIY
CONSTRAINT
For the HRR solution, the stress-strain
IN CRACK TIP REGIONS
fields are given by [l, 21
where eij and cij are the stress and plastic strain tensors, c, and ce are the effective stress and strain, respectively, E is Young’s modulus, r is the radial distance from the crack tip, I,, is a dimensionless constant depending on strain hardening, and ciij, 4, 6e and Z, are dimensionless constants which depend on strain hardening and the angle (0) from the crack plane. The other parameters are defined by a power law idealization of the flow behavior, which for uniaxial deformation is given by n-I 0; cr, &=a (3) 2' 0 CO The stress cro is usually defined as 0.2% offset yield strength. The ratio of hydrostatic stress o;, to the von Mises effective stress cr, is used to characterize the stress triaxiality in crack tip regions. From (l), the stress triaxiality on the ligaments ahead of the crack tip, defined by the HRR solution in the plane strain case, can be given by
According to the HRR solution for a given material state (n-value), the stress triaxiality on the ligaments of a specimen should be a constant independent of the specimen geometry, the crack depth, the distance from the crack tip and the load level. Figure 1 depicts the stress triaxiality on
9 8
-
HRR field
7
d
1, 6-o 3 -
n-3
0
‘614-
0 3-o
u 2-a l0 0
0.1
0.2
0.3
0.4
0.5
l/n Fig. 1. Stress triaxiality constraints defined by the HRR solution along the crack tip ligaments in the plane strain case.
0
0
Ill10 11100
Cl
. 0
l
‘
I 0.01
I 0.02
I 0.03
I 0.04
1 0.05
J/ho,, Fig. 2. Global triaxiality constraints on the crack tip ligaments of a center-cracked panel in the plane strain case given by (16). The data were taken from the results in refs [IS, 191and Fig. 1.
Stress triaxiality constraint and crack tip parameters
791
the ligaments of specimens with different strain hardening, which is required by the HRR solution in the plane strain case [1,2]. It can be seen that the crack tip triaxiality defined by the HRR solution is strict and high, which is based on the infinitely large plane of a nonlinear elastic solid by obeying the pure power hardening relation, and is too high to account for any finite size specimen of real elastic-plastic materials in the plane strain case. Finite element results have shown that, in the plane strain case, the triaxiality constraints of specimens are all lower than those of the HRR solution at different degrees [12-141. In order to understand the effects of crack tip t~axiality constraints on fracture parameters, it is necessary to get a quantitative relationship between the triaxiality of specimens and that defined by the HRR solution. Anderson’s research, completed recently [17], makes it possible to discuss the problem further. Based on the quasi-deformation theory assumption, for loading in the direction perpendicular to the crack plane, Anderson gives the modified flow behavior of materials as [17] and the crack tip stress as (6) The quasi-defo~ation
theory modifies the J-CTOD
relationship:
Al00 4
J=Il’“d,p
(7)
where 6, is the CTOD and d,, is the dimensionless constant which depends on flow properties (i.e. n, a and co/E). Consequently, Anderson’s simple corrections for crack tip stress and strain can be expressed as [17]
where X is the distance of the ligament from the crack tip. Thus, a J-like parameter characterizes the stress normal to the crack plane is given by [17]
which
For material on the plane of the crack in plane strain, one has: 6, =
Substituting
(11)
(9) and (11) into (3) results in
(12) From the HRR solution and (3), one can obtain: (13) Thus, (12) becomes: (14)
792
SUN JUN
which can be considered as the correction for the effective stress. In the plane strain case, one can express the stress triaxiality as
am
_=-....--- OYY
@e cr, Introducing
j?i-
(8) and (14) into (15), the following expression is derived:
or equivalently: d,, dJ --_= co da,
%I/% + llfi @&,)tiRR + ll.Js
1
(17)
’
where o,jo, is the crack tip stress triaxiality of real specimens, which depends not only on the n-value but also on the value (d”/~o)(dJld~~), namely the specimen geometry for a given material state (n-value). When the HRR assumptions are valid, @J/d&) = (co/d,,) and the quantity in the brackets = 1.0, and (16) reduces to the HRR solution (4). For (16), the (dJ/db,) values can be obtained by
where A is the slope of the curve o. S,/J vs J/(ba,) at a fixed point of the J-CTOD relationship calculated by Shih [ 18, 191,and b is the untracked ligament length. Since J/h, is very small, dJ/d& is nearly equivalent to J/6,. As a result, the stress triaxiality constraints given by (16), on the ligaments for the deeply notched center-cracked panel (CCP; a/W = 0.75) with IZ= 3 and 10, have been computed based on the J-CTOD relationship obtained by Shih [18, 19], and compared with those defined by the HRR solution in Fig. 2. It can be seen that the triaxiality constraints of the CCP specimen decrease rapidly with increasing J and are lower than those defined by the HRR solution in the entire deformation range. However, if (16) is employed to evaluate the crack tip constraints of a deeply notched bend specimen, one would obtain the result that the crack tip constraints elevate gradually with increased J for the specimen, which is not the real case of losses in triaxiality and may be due to errors in estimating the CTOD from the finite element mesh [17]. From (17), it is known that constraint factors such as J/a0 CTOD 1161or (dJ/d~,)/~~~~d~) [17] only characterize the relative differences of the crack tip triaxiality constraint of the specimens from those defined by the HRR solution and not the constraints themselves. In addition, (a,/o,), presented in (16), characterizes only the global constraint levels in crack tip regions and does not reflect the real distribution of triaxiality along the ligaments of the specimen. In order to investigate the real situation of losses in triaxiality constraints of specimens, 4-
HRR field.
Center-cracked
IlllO
pm01
30 4 d
0
0
-
0
2 i
ad
... 0
1 -
q
0 0
DCI
o
badJa200 b q,/J-60 u badJ=30 l
0
t
I
I
0.5
1.0
1.5
LogIxI(Jla~)l Fig. 3. Stress triaxiality constraints on the crack tip ligament of a center-cracked panel in plane strain with a/W = 0.75 and n = 10.
793
Stress triaxiality constraint and crack tip parameters
4 -
3
HRR field. 8
Edge-cracked .
o
q
bead specimen
.
n-10
0 0
D
O0
2
2-
.
q
.
0
0
0 q
l-
o b o,/J=ZOO l b 00/J=60 0 b a,/J=30
0
,
I
I
0.5
1 .o
1.5
J-oaIW(Jla,)l Fig. 4. Stress triaxiality constraints on the crack tip ligament of an edge-cracked bend specimen in plane strain with a/W = 0.75 and n = 10.
Shih’s finite element calculations [19] have been repeated for the deeply notched CCP (a/W = 0.75) and the edge-cracked bend specimen (ECBS; a/W = 0.75) with n = 3 and 10 respectively. The properties are the same as those used by Shih and German [19]. The details of FEM calculations have been given elsewhere [ 12-14,201. As shown in Figs 3-6, the stress triaxiality constraints, along the ligaments ahead of the crack tip of two specimens with n = 10 and 3, are obviously lower than those defined by the HRR solution (the solid line in Figs 3-6). The further the distance from the crack tip on the ligaments, the lower the triaxiality. As the load increases, the stress triaxiality constraints also decrease due to progressive relaxation of constraints in the crack tip field. Up to then the ligaments are in large scale yielding, shown by FEM computations, and the decreasing tendency of triaxiality has just started to be reduced and then reaches a stable level. For a given n-value, the triaxiality of the ECBS specimen is higher than that of the CCP specimen at the same J level. The decrease of triaxiality with the increased distance from the crack tip, for the ECBS specimen is steeper than that for the CCP specimen. After the comparison of the data in Fig. 2 with those in Figs 3 and 5, it is interesting to see that the “global” triaxiality constraints, given by (16), exactly correspond to the “local” triaxiality constraints at the location of X = J/a, on the crack tip ligaments at different J levels for CCP specimens with n = 3 and 10 respectively, as shown in Fig. 7, which will be discussed further below. On the other hand, even though the (a,,,/a,) value given by (16) does not properly describe the real distribution of the crack tip triaxiality along the crack tip ligaments of specimens, (16) will play the role of a bridge for us to introduce the stress triaxiality constraints into the characterizing parameters in crack tip fields.
98 -
-HRR
field.
prnel
Centor-cracked
n=3
l-
3 2 _ 1
0
o bq,/J=POO l b a#-60 0 b adJ=30
,
I
I
0.5
1.0
1.5
LogWI(Jla,)l Fig. 5. Stress triaxiality constraints on the crack tip ligament of a center-cracked panel in plane strain with a/W = 0.75 and n = 3.
SUN JUN
794
9 8
-
HRR fisld.
O
.
6 ,-
Do
2
0
.O
’
a
2
1
_
.
a
5 43 -
Edgecracked
bend
rpocimsn
.
a
.
l
0
0
0
0
0
o
b o,,/J=200 b a,,/J=60 a bq,/J-30
a=3
l
,
-0
I 1.0
0.5
I 1.5
LosIXI(Jla,)l Fig. 6. Stress triaxiality constraints on the crack tip ligament of an edge-cracked bend specimen in plane strain with a/W = 0.75 and n = 3.
3. CRACK TIP TRIAXIALITY
CONSTRAINT
AND STRESS FIELD
In previous studies [12-141, finite element calculations have indicated that the difference in crack tip stress fields of specimens from those of the HRR solution correspond to the mutual relationship between the crack tip triaxial constraints of specimens and those defined by the HRR solution. This conclusion is to be verified below by an analytical expression together with the FEM numerical calculations. By taking (6) into account, one has: (19)
where(~JHRRis the stress defined by the HRR solution. Substituting (7) and (19) into (8) gives:
8
r o
1 0
Given by (16) for CCP from FEM rcwlts
At X-J/a,,,
I
I
I
I
I
0.01
0.02
0.03
0.04
0.05
J/b Q,, Fig. 7. Relationship between “global” triaxiality given by (16) (after Fig. 2) and “local” triaxiality at the location of X = J/a, on the crack tip ligament of a centercracked Panel (CCP) with a/W = 0.75; n = 3 and 10.
Stress triaxiality constraint and crack tip parameters
195
We can obtain:
(21) Substituting (17) into (21), the following relation holds:
(22) where ovu and a,,,/~, are the “real” stress and triaxiality constraint on the crack tip ligaments of specimens, which can be given by FEM. From (22) and (17) we know that, obviously, when and only when (%I/~,) = (%lm/cAR,
(23)
namely the “real” stress triaxiality constraints on crack tip ligaments of specimens fully satisfy the relevant constraint requirements of the HRR solution, it leads to ayv = (Gyy)HRR
(24)
and
*dJ
2-z
a,, d&
1.
(25)
’
namely, J-dominance of crack tip fields is valid and vice versa. Therefore, it can be considered that a sufficient and necessary condition, in crack tip triaxiality constraints, for the validity of J-dominance has been presented by (22) and (17). In the meantime, it is suggested that the lower the ~,,,/a, value, relative to the relevant (a,,,/~,)~~~ value, the larger the differences of crack tip
CCP
0 b adJ=ZOO
CCP
n-10
0 b odJ=60
n=3
0 b odJ=30
1.0
0 b udJ=ZOO 0 b udJ=60 0 b udJ=30
1.0
0 0 0
qr.Ia
::L *
0
1.5
1.0
0.5
0
0
’
0.5
LosIXI(J/Ql
1.0
ECBS
0 b q,/J=200
BCBS
n=lO
. b o,/J=60
n=3
b 0
$0.9
q
h 0.8
$0, b
0 b 5,/J-30
o0
. 0
'
b u,/J-60
I 0 l
00
0 b
0
0
0.6
0
0.5 0.4
1.0
.
a b uo/J=30
0
q
1.5
0 b u,/J=200
0 .
1.0
LottIJU(Jlo,N
0
0
I
I
I
0.5
1.0
1.5
LodXI(J/a,,)l
V.-n
0
0.5
1.0
1.5
LOisIXI(J/a,)l
Fig. 8. Normal stress ratio ahead of the crack for a center-cracked panel (CCP) and an edge-cracked bend specimen (ECBS) with a/W = 0.75; n = 10 and 3 respectively. After Shih and German [19].
796
SUN JLJN CCP IFlO
ab ‘s 0 Y a 0 ‘s g
0 b a,+200 0 b oo/J=60 0 b a#-30
O
0.8
0
0.7 . 0
0.6
. On
0.4
b s 0” ‘a 2
ECBS n*lO
l
a
0.7
0
0.7 0.6
0
3 a 0 B 0 Y a 0 ‘e g
0
e 0 D
0.6
1.0
0.5
1.5
0 b u,,/J=200
ECBS n=3
0
q
0.8
0
Loi~lW(J/a~)l
lO
0
0.8
1.5
0 b q,/J-200 l b q,/J=60 0 b QIJ-30
.
0.9
0.4 1.0
1.0 0.9
1.0
lX/(J/ar,)l
‘-og
I
0 b udJ=200 . b udJ-60 0 b a,,/J=30
0.5
0.5
0
b S0
2 a 0 ?a b Y a 0 ‘s 2
0.9
0.5
3
CCP n=3
l
* b adJ=60 0 b adJ=30
1.0 0.9 0.8 0.7
*
0
a
0 0 .
0
l
0
l cl
0.6
00
v*-
0
0.5
1.0
1.5
J-o&WJ/ao)l Fig. 9. Stress triaxiality ratio ahead of crack for a center-cracked panel (CCP) and an edge-cracked bend specimen (ECBS) with a/W = 0.75; n = 10 and 3 respectively. After Figs 3-6.
stresses of specimens from those defined by the HRR solution, which agrees well with previous results reported by the present author [12-141. In order to further discuss the relationship between crack tip constraints and stresses, ~~~/(~~~)~~~and (~~/~~}/~~~/~~)~~~, along the crack tip ligaments of CCP and ECBS specimens with n = 3 and 10, are replotted in Figs 8 and 9 against X/(J/a,). The quantities o,.~and (a,,&,.) were inferred from the finite element analyses performed by Shih [19] and Figs 3-6 [20] respectively. It is evident that, for all four cases here, the distribution and the variation of u,,,/(o,,~)nRRalong the ligaments accompany those of (a,,,/~~)/(a,t/a,)HRR. For the CCP specimens, with increased JJ~u~, the deviation of the “real” stresses from those defined by the HRR solution for the specimen with n = 10 is more remarkable than that for the specimen with n = 3, which agrees better with the relevant deviating degrees of the crack tip triaxiality constraints in these specimens from those given by the HRR solution. Contrary to the situation of CCP specimens, regarding the deviations of the crack tip stresses and the triaxiality constraints of ECBS specimens from those required by the HRR solution, the smaller the n-value, the larger the difference. The ~lationship described in (22) between the stresses and the triaxiality constraints is depicted in Fig. 10, with all of the data from Figs 8 and 9. It is very clear that most of the data are located on the same straight line and other data are also close to the straight line, which defines a unique relationship between the crack tip stresses and the triaxiality constraints for a given material state. 4. CRACK TIP TRIAKIALITY
CONSTRAINT
AND CHARACTERIZING
PARAMETER
In Anderson’s research 1171,a J-like parameter characterizing the stress normal to the crack plane has been defined and shown in (10). Substituting (10) into (8) we obtain the corrected stress:
(26)
Stress
triaxiality constraint and crack tip parameters n-10
1.0 -
0.9
n=3
CCP
0
0
ECBS
.
n
/ p*
.
$
AD 2
0.8
.
0.7
797
-
/j.&& pb
. /
Y
a
1”’
8: Oy[
,
.
0.7
0.8
t&/u,+
[
,
,
zi (qJQJ”RR +111
0.6
0.5
,
0.9
l/I/3
1.0
r4
d3
Fig. 10. Relationship between normal stress ratio and stress triaxiality ratio ahead of cracks for a center-cracked panel (CCP) and an edge-cracked bend specimen (ECBS) with a/W = 0.75; n = 3 and 10. After Figs 8 and 9.
If J$ characterizing the crack tip stresses is valid, the correction for the stresses along the ligament should be suitable for different specimen geometries. However, as discussed previously, the correction of the stresses has already been limited in CCP specimens because the (dJ/dS,) in (10) is unable to properly describe the decreasing triaxiality with increased J, when the ECBS specimens are involved. Recall that the relationship between (dJ/d&) and (a,,,/~,) is given by (17). It is therefore possible to solve the problem above. Introducing (22) into (19), we derive that
I,=
%I& + l/d
[ (%/a,)“,,
By the combination
+ l/J5
n/b -1) 1
(27)
*
of (7) and (17) with (27), J$ in (10) can be rewritten as: J* = JA$l -d/n YY
d _dJ
n+’
2
( 00 d&
>
(28)
-
HRR field.
Canter-cracked
pixel
n=lO
o b q,/J=200 l b 5,/J-60 o b ~/J-30
Fig. 11. Crack tip normal stress for a center-cracked panel with a/W = 0.75 and n = 10 (from Fig. 8) normalized by .I&. J$ is given by (28).
SUN JUN
798 0.6 -
HRR field.
o
b crdJ=200
l
b udJ=60
Edge-cracked
bend specimen
n=lO
0 b ts,,/J-30
0
2
1
3
4
5
WW~,,/o,)l Fig. 12. Crack tip normal stress for an edge-cracked bend specimen with a/W = 0.75 and n = 10 (from Fig. 8) normalized by J$. .I;,, is given by (28).
where the quantity (~,/a,) is the triaxiality constraint, calculated by FEM, on the crack tip ligaments of specimens. For log[a,,,,/a,,] and log[X/(J$/a,,)], the normal stresses given by (26), on the crack tip ligaments of specimens, should be a straight line, the slope of which is l/(n + 1). If the parameter J$, is able to characterize the normal stresses, the stresses of the specimens with different geometries should be located on the same straight line at different sections for a given material state, even if the J/ha, level is different. The finite element results from Figs 8 and 9 are replotted in Figs 11-14, where X is normalized by J$/q, given by (28) and the data on the crack tip triaxiality are taken from Figs 1 and 3-6. For CCP specimens with n = 3 and 10, the stresses have been corrected and are close to those given by the HRR solution. However, it is worth noting that the normal stresses on the crack tip ligaments of ECBS specimens with n = 3 and 10 have also been corrected by using J;y; namely, the present J$ characterizing the normal stresses is valid even for different types of specimen geometry. However, it should be pointed out that the triaxiality parameter (a,,&,) in (28) has been considered as a function of the distance of the ligaments from the crack tip, i.e. X/(J/a,), as given in Figs 3-6. Thus, for Figs 1I-14, J$ given by (28) is also dependent on the distance from the crack tip, which is not right for JE as a J-like parameter characterizing the crack tip stresses. Recall that the triaxiality constraints given by (16) corresponded to those calculated by FEM at the location of X = J/0,( - 26,) on the crack tip ligaments of CCP specimens at the same J/boo (as indicated 0.9 0.8
c -
BRR fiold.
Contor-cracked
panel
n-3
0.7
2
0.6
1
0.5
$
0.4
s
0.3
o b adJ=200 -
l
badJ=60
0 b
0
adJ=30 1
2
3
LoW(l,,la,N Fig. 13. Crack tip normal stress for a center-cracked panel with a/W = 0.75 and n = 3 (from Fig. 8) normalized by J$. J$ is given by (28).
Stress triaxiality constraint and crack tip parameters 0.9 0.8 0.7 no
0.6
2fi
0.5
s2
0.4
N-l
0.3 0.2
0.1
l-
HRR field.
”
bend specimen a-3
Edge-cracked
o
b a,,/J-200 b o,,lJ-60 0 b adJ=30 l
1
0
799
I
I
I
1
2
3
LW[X/&h,)l
Fig. 14. Crack tip normal stress for an edge-cracked bend specimen with o/W = 0.75 and n = 3 (from Fig. 8) normalized by J$. .I,‘, is given by (28).
in Fig. 7). Now, we express the triaxiality constraint at the location X = J/u,, as (a&,),,,, which should be the maximum value of the triaxiality on the crack tip ligaments because there is an intense strain region within ~26, [21], where the stress triaxiality constraints have been decreased due to crack tip blunting. Therefore, we can employ (a,&,),,, as a characterizing parameter of the triaxiality constraint along the crack tip ligaments of various specimens at a given J level. Thus, (28) becomes: J* = J (Gl/~Jn+ l/d n YY [
1’ hn/~ehRR +l/J5
(2%
where the J$, value would be uniquely determined when J is given because (~,,,/a,), is only dependent on the J level for a given specimen type, which is of great interest when discussing fracture because fracture always emanates from the highly constrained zone, within which hole growth and coalescence occur intensely [22,23]. The results in Figs 11-14 are replotted in Figs 15-18, where X is normalized by J$./u,,, given by (29). It can be seen that the corrected stresses agree well with those defined by the HRR solution except for some deviation of the stresses when far from the crack tip due to a very weak triaxiality constraint. The cases of corrected stresses for the CCP specimens with n = 3 and 10 are similar to the results obtained by Anderson [17]. The results indicate clearly that the crack tip stresses of different specimen geometries could be uniquely determined by the parameter Jz for a given material state (n-value), even when the (~,,,/a,), value has been applied to characterize the
0.6
-
BRR field.
CCP
n-10
o
b q,/J-200 b udJ=60 0 b adJ-30 l
1 0
I
I
I
1
2
3
Lo8[x/(fyJa,)l
Fig. 15. Crack tip normal stress for a center-cracked panel (CCP) with o/W = 0.75 and n = 10 (from Fig. 8) normalized by J$. .I$ is given by (29).
800
SUN JUN 0.6
0
2
1
3
LWWl~~,,/O,)l Fig. 16. Crack tip normal stress
for an edge-cracked bend specimen (ECBS) with a/W = 0.75 and n = 10 (from Fig. 8) normalized by Jc.. J& is given by (29).
triaxiality constraints along the crack tip ligaments of the specimens, which is very significant for one to develop a plane strain fracture criterion independent of specimen geometry, particularly in large scale yielding. In addition, when (0,/o,) = (a,@,),,, in (28) or (a&,), = (a,/~,)~~~ in (29), (28) and (29) reduce to J;= J; (30) namely, J-dominance of crack tip fields is valid as a single parameter. In order to verify the validity of J$ characterizing crack tip stresses, it is necessary to examine the relationship between the constraints defined by Jz, and those of “real” specimens. Substituting (10) and (17) into (14) gives (31) By employing (26), one can obtain the triaxiality constraint required by the parameter J$:
where (a,,,/~,) is the triaxiality constraint presented at the crack tip ligaments of specimens. The analysis above again shows that the sufficient and necessary condition for one to ensure the validity 0.9 0.8
r
I-
-
HRR field.
Center-crrckcd
panel
n-3
Fig. 17. Crack tip normal stress for a center-cracked panel with a/W = 0.75 and n = 3 (from Fig. 8) normalized by .I$. $,, is given by (29).
Stress triaxiality constraint and crack tip parameters
0.8
no I) ‘2 s$
0.6
4
0.3
-
HRR field.
Edge-cracked
bend rpsciman
801
n=3
0.5 0.4
0.2
0
1
2
3
LoglXl(j,/o,)l Fig. 18. Crack tip normal stress for an edge-cracked bend specimen with a/W’ = 0.75 and n = 3 (from Fig. 8) normalized by Jc.. J;, is given by (29).
of a single parameter characterizing crack tip stresses is that the crack tip triaxiality constraints defined by the characterizing parameters should be satisfied by those of “real” specimens. 5. CRACK TIP TRIAXIALITY
CONSTRAINT
AND FRACTURE
CRITERIA
Studies performed recently [13,24] showed that, in the plane stress case, the triaxiality constraints in crack tip regions of real thin specimens are able to meet the very low constraint requirements defined by the HRR solution and are independent of the specimen geometry, the crack depth and the load level for a given material state, which ensures the validity of J-dominance in the plane stress case. Consequently, the plane stress Jlc and J-R curves can be considered as suitable fracture criteria to characterize the material resistance against crack initiation and stable growth based on the specimen geometry independence of J,c and J-R curves. However, “plane strain” is an immediate situation. Up to now, no substantial improvement along these lines has yet appeared to develop a plane strain fracture criterion to be a material constant independent of the specimen geometry for a given material state. Equation (29) may be helpful for one to further discuss the problem mentioned above. When crack growth initiation occurs in the crack tip regions of a specimen, from (29), we get:
J;y = (J,y 1,~.
(33)
Because the crack tip stresses on crack tip ligaments of different specimens can be uniquely determined by J$ given by (29), (Jz),, should be a material constant independent of the specimen geometry. At the same time, for some specimen we can obtain the critical values: J = Ji
(34)
and (%I/~,), = (%I/~,),,.
(35)
Thus, one has the critical form of (29) for the specimen:
namely, although the J, and (a&,), values are different for various types of specimens, the (J$>,, value is the same for a given material state (n-value), where the reason for applying (a,&,), to characterize the critical situation of the crack tip triaxiality constraints of the specimens is that the fracture is always initiated at the zone with maximum value of the stress triaxiality on the crack tip ligaments [22,23,25,26], because the separation of materials, i.e. cavity nucleation and development, occurs most easily within the zone due to maximum volume dilatation of materials EFM 44,5-K
SUN JUN
802
caused by (a,/a,), [25]. Hence, it becomes clear that this region is the most suitable for formulating a fracture criterion even in the case of small scale yielding [25], and the (~,/a,)~~ value for a specimen can be reasonably utilized to characterize the critical case of crack tip triaxiality constraint on the onset of crack growth of the specimen. If there was a nonlinear elastic cracked solid with a large enough size, on the crack tip ligament of which the triaxiality constraint defined by the HRR solution could be met, one would have (0, I, ),, = (a, /a, LRR 1 i.e. the quantity in the brackets of (36) is unity. Thus, the plane strain fracture toughness J,, could be obtained:
(J,:.),, = Jr, or equivalently
(37)
by taking account of (36) J,,
=
J,
tam/ae)mc + l/d n ’ [ (amm/a,h + l/J51
(38)
holds for some real specimen. This means that the plane strain fracture toughness J,, cannot be measured directly from real laboratory testing specimens with finite sizes because the crack tip triaxiality constraints on the crack tip ligaments of the real elastic-plastic cracked specimens are all lower than those defined by the HRR solution at different degrees [12-141, particularly in large scale yielding, as discussed previously, even though the ASTM E813-87 Standard’s requirements for measuring Jlc were fully met [27]. Similar results can be obtained in measuring the critical CTOD. Substituting (27) into (7) gives the JCTOD relationship: a0 4
J=T
[
tam/a,),+
l/Js
(a,/a,)HRR + l/$
1’
(39)
which is clearly geometry or constraint dependent. When the constraint requirements of the HRR solution are met, namely (am/a,), = (am/ee)nRRT J and CTOD are equally valid crack tip characterizing parameters and a unique relationship between them exists:
at the initiation of crack growth, and J,, = ? 6,,
(41)
n
10 -
n=lO
n-3
10 -
P2 l-7
2 x co
S-
00.4
I 0.6
I 0.8
I 1.0
Fig. 19. Predicted relationship between ratio of J value at initiation of crack growth to plane strain fracture toughness and crack tip triaxiality constraint.
n=3
n-5
II=10
S-
0 0.4
I
I
I
0.6
0.8
1 .o
Fig. 20. Predicted relationship between ratio of CTOD value at initiation of crack growth to plane strain fracture toughness and crack tip triaxiality constraint.
Stress triaxiality constraint and crack tip parameters
Fig. 21. Schematic representation
of specimen geometries and dimensions (in mm).
at the same time; for (39) we have the J-CTOD
relationship:
1Introducing
803
(42)
(41) and (42) into (38) yields:
where Jlc and 6,, are only dependent on the n-value and are material constants independent of specimen geometry, even though (a&,), , Ji and ai are different for various specimens. Figures 19 and 20 display the relationship between Ji(Si) and the corresponding (b,/o,),,,,. The results show that the Ji and ai values are very sensitive to differences of (O&J,), at the crack tip ligaments of various specimens from one geometry to another, in particular for materials with lower strain hardening (larger n-value). Therefore, generally, the plane strain fracture toughness “JIc” and a,,, measured from standard laboratory test specimens, are geometry dependent and not the “real” Jlc due to losses in the crack tip triaxiality constraints. In addition, from (38), for two types of specimens, i.e. A and B, with the same material state and different crack tip triaxiality constraints, the following relationship holds:
+ l/J5 JiA (6nl~e)mB Jis= [ (am/a,),, + l/Jr
1’ n
WI
which may be of great importance for one to predict Ji of some cracked component with a crack tip triaxiality constraint of (cr,/~,),,,, from the results measured experimentally from a laboratory test specimen for a given material state. In order to further discuss the dependence of the Ji value on (cr,,,/a,),,,, an experimental investigation along these lines has been performed, and is described below. The material used in this study was a normalized C-Mn steel for petroleum casing, in the form of a rolled plate. The chemical composition and room temperature tensile properties normal to the rolling direction have been given elsewhere [24,28]. The yield strength a0 = 400 MPa and the strain hardening exponent n = 5. The plates were machined into four types of specimen geometry: center-cracked panel (CCP), single edge-cracked (SEC), double edge-cracked (DEC) and compact cracked tension (CCT) specimens, as shown in Fig. 21, with thicknesses of B = 12.5, 25.4 and 50.8 mm. In order to discuss the problem under the condition of “pure plane strain” crack tip constraint environments and to avoid the effect of the shear lip, all of the specimens were side-grooved with 45” V sharp notches to a depth of 25%. The percentage side-groove depth is given by percentage side-groove depth = (B, - B,)/B, x 100, where Bg = gross thickness and B,, = net thickness. The fatigue pre-cracking was carried out under load control and at room temperature to give a final crack length to width ratio of about 0.5.
SUN JUN
804 0 l
CCP SEC
I
0 DEC . CCT 200
200
180 P 2E
150 I
$,
100
e;-
P
1
1 50
0
I
10
P
ii
20
30
Thickness
40
50
B(mm)
g
140
d
120
z i-
100
0 12.5 . 25.4
O
I-
160
T
i
r
0
0
0
50.8
i3
80
2.6
2.4
60
2.8
3.0
3.2
3.4
B(mm)
Fig. 22. Effect of thickness for side-grooved specimens with four types of geometry on J values at the onset of crack
Fig. 23. Dependence of J values at crack growth initiation of four specimen geometries with different thicknesses on crack tip triaxiality constraints.
The J, values, at the onset of crack growth initiation, were determined from J-R-Au curves, using the procedure described in ASTM E8 13-87 [27] (which assumes a 2a, Au blunting line) and the methods presented in Gibson et aZ.‘s research [29] for four types of specimen geometry. The (am/ae)mc values, the triaxiality constraints at the location of X = J/a, on the crack tip ligaments of the four specimens, were calculated by the FEM [12-14,201, related to the corresponding Ji value of each specimen. Figure 22 shows the experimental results of J, values for the four types of specimen geometries with different thicknesses. In all cases, Ji shows a small decrease with increasing specimen thickness due to side-grooving. However, Ji evidently depends on the specimen geometry even for different specimen thickness, which is easy to understand from the viewpuint of the crack tip triaxiality constraints. As shown in Fig. 23, the J, of the specimen with lower crack tip constraint is much larger than that with higher triaxiality, the mechanical mechanisms of which have been revealed in previous work [ 13,141. As a proper fracture criterion, the parameter should be a material constant independent from the specimen geometry for a given material state. Obviously, the present J, (or “JIc”) is unable to meet this requirement. As analyzed earlier, (JG),, = Jlc should become geometry independent, where J,, is “real” and not that measured directly from the laboratory test specimens. Figure 24 illustrates the case of (JTy),c = J,c as the fracture criterion at crack growth initiation, which was calculated by (37) and (38) by taking the data from Fig. 23. It can be seen that (J$),c and Jlc are indeed material constants independent of specimen geometry with different crack tip triaxiality constraints. On the other hand, the “real” plane strain fracture toughness JIc value is only about one-third of the Ji value,
B(mm)
26
$j 2 7 .I! -x *a.
0 12.5 l 25.4 0 50.8
0
22’
0
0
l
-VA.
18
- ‘2.4
0
0
•I
2.6
2.8
3.0
3.2
3.4
Rl’=,L
Fig. 24. Specimen geometry independence of plane strain fracture toughness for a given material state.
Stress triaxiality constraint and crack tip parameters
805
or “JIc” as is usually considered, measured from compact tensile specimens. Commonly, the compact tensile specimens were suggested to measure the Jlc value of materials only because they have higher triaxiality constraints in crack tip regions than those of other types of specimens and closer to that required by HRR singularity, namely such “JIc” measured from CCT specimens are just lower than that from other specimens but are still not the “real” J,c of the materials. In other words, the plane strain fracture toughness has been overestimated by “J,c” measured from the standard CCT specimens [l 11, or at least it can be thought that the ASTM E813-87 size requirements are not restrictive enough [30]. 6. CONCLUSIONS (1) A quantitative relationship between the crack tip triaxiality constraints on the ligaments of specimens and those defined by the HRR solution in the plane strain case was derived from the simple correction of the crack tip stress-strain field presented by Anderson. It was found that the crack tip triaxiality given by the relationship corresponds well with that of the location X = J/cq,( - 26,) on the crack tip ligaments of a center cracked panel. (2) An analytical expression was established and verified, which describes a sufficient and necessary condition for the validity of a single parameter characterizing crack tip stresses. The crack tip triaxial constraints required by the single parameter should be fully satisfied by those presented at the crack tip ligaments of specimens. (3) A characterizing parameter of crack tip constraints was introduced into the crack tip parameter proposed by Anderson, and then the validity of the single parameter characterizing crack tip stresses was extended to situations of different specimen geometry. A corrected relationship between J and CTOD was given which clearly shows its dependence (4) on the specimen geometry and load level by taking account of different losses in triaxiality constraint for various specimens with increased load level. relationship between Ji(Si) and the corresponding (rr,/e,),,,, was (5) The interdependent presented, which showed that Ji(Si) is very sensitive to differences of crack tip triaxiality constraints for specimens from those defined by the HRR solution, particularly in the case of low strain hardening (large n-value). Based on the validity of Jz, uniquely characterizing the crack tip stresses of different specimen (6) geometries, a new elastic-plastic fracture criterion was developed and proven experimentally to be a material constant independent of specimen geometry for a given material in the plane strain case, which is the “real” plane strain fracture toughness Jlc. (7) From the viewpoint of crack tip triaxiality constraints, it is impossible to obtain the plane strain Jlc value directly from the laboratory test specimens because the crack tip triaxiality constraints of various specimens are unable to meet those required by the HRR solution, even in small scale yielding. Acknowledgement-The acknowledged.
financial support of Fok Ying Tung Education
Foundation
for this research is gratefully
REFERENCES [l] [2] [3] [4] [5] [6]
[7] [8] [9]
[lo] [ll] [12] [13]
[14]
J. R. Rice and G. F. Rosengren, J. Mech. Phys. Solids 16, 1-12 (1968). J. W. Hutchinson, J. Mech. Phys. Solids 16, 13-31 (1968). R. A. Begley and J. D. Landes, ASTM STP 514, 1-23 (1972). C. F. Shih, H. G. de Lorenzi and W. R. Andrews, ASTM STP 668,65-120 (1979). E. Roos, V. Eisele, H. Silcher and F. Speath, J. Nuclear Engng Design 102, 439-449 (1987). J. W. Hancock and M. J. Cowling, Metal Sci. 14, 293-304 (1980). G. Rousselier, J. Nuclear Engng Design 105, 97-l 11 (1987). 0. L. Towers and S. J. Garwood, ASTM STP 905.454-484 (1986). P. M. S. T. de Castro, Engng Fracture Mech. 19, 341-357 (1984). R. Hobe, R. Twickler and W. Dahl, in Adounces in Fracture Research, Proc. Seuenth Inr. Conf. on Fracture (ICF7) (Edited by K. Salama et al.), pp. 2509-2517. Pergamon Press, Oxford (1989). P. De Roo, B. Marander, G. Phelippeau and G. Rousselier, ASTM STP 833, 606-621 (1984). Sun Jun, Deng Zengjie, Li Zhonghua and Tu Mingjing, Engng Fracrure Mech. 34, 637-643 (1989). Sun Jun, Deng Zengjie, Li Zhonghua and Tu Minding, Engng Fracture Mech. 34, 413-418 (1989). Sun Jun, Deng Zengjie, Li Zhonghua and Tu Minding, Engng Fracture Mech. 36, 321-326 (1990).
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SUN JUN
[15] J. W. Hutchinson, in Advances in Fracture Research, Proc. Fifth Int. Conf. on Fracture (ICFS) (Edited by D. Francois er al.), pp. 2669-2684. Pergamon Press, Oxford (1981). [16] S. Dedovic, A. Bakker and D. G. H. Latxko, Fatigue Fracture Engng Mater. Structures 11, 251-266 (1988). [17] T. L. Anderson, Inr. J. Fracture 41, 79-104 (1989). (181 C. F. Shih, J. Mech. Phys. Solids 29, 305-326 (1981). [19] C. F. Shih and M. D. German, Znr. J. Fracture 17, 27-43 (1981). [20] Sun Jun, Ph.D. Dissertation, Xi’an Jiaotong University, Xi’an (July 1989). [21] R. M. McMeeking and D. M. Parks, ASTM STP 668, 175-194 (1979). [22] J. T. Bamby, Y. W. Shi and A. S. Nakarni, Int. J. Fracture 25, 271-284 (1984). [23] J. F. Knott, Metal Sci. 14, 327-336 (1980). [24] Sun Jun, Deng Zengjie and Tu Mingjing, Engng Fracture Mech. 37, 675-680 (1990). [25] P. S. Theocoris and T. P. Philippidis, Inr. J. Fracture 35 21-37 (1987). [26] G. A. Papadopoulos, Engng Fracture Mech. 27, 643-652 (1987). [27] E813-87, Standard test method for .I,=, a measure of fracture toughness. American Society of Testing and Materials, Philadelphia, PA (1987). [28] Sun Jun, Deng Zengjie, Li Zhonghua and Tu Mingjing, Inf. J. Fracture 42, R39-42 (1990). [29] G. P. Gibson, S. G. Druce and C. E. Turner, Int. J. Fracture 32, 219-240 (1987). [30] T. L. Anderson and R. H. Dodds Jr., J. Test. Eval. 19, 123-134 (1991). (Received 3 March 1992)
Fracture
Mechanics
Vol. 44,No.5,pp.789-806,
1993
0013-7944/93
Printedin Great Britain.
$6.00
+ 0.00
0 1993 Pergamon Press Ltd.
STRESS TRIAXIALITY CONSTRAINT TIP PARAMETERS
AND CRACK
SUN JUNt Department
of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R.C.
Abstract-A relationship between crack tip constraints of specimens and those defined by the Hutchinson-Rice-Rosengren (HRR) solution in the plane strain case has been derived from the corrected stress-strain field presented by Anderson, by taking account of losses in triaxiality of specimens. An analytical expression has been established and verified which describes a sufficient and necessary condition for the validity of a single parameter characterizing crack tip stresses; namely the crack tip triaxiality constraints defined by the single parameter should be satisfied by those of specimens. The simple correction for crack tip stresses proposed by Anderson has been extended to situations of different specimen geometry. Furthermore, a new elastic-plastic fracture criterion, including the effect of crack tip triaxiality on the parameter characterizing fracture, has been developed and proven to be a material constant independent from specimen geometry for a given material state. It has also been demonstrated that it is impossible to obtain the plane strain fracture toughness J,c directly from laboratory test specimens because the crack tip triaxiality constraints required by the HRR solution cannot be met for various specimens.
1. INTRODUCTION THE DEVELOPMENT and confirmation of proper fracture criteria for crack growth initiation has been and still is one of the most important objectives in the study of elastic-plastic fracture mechanics. As a single parameter characterizing the crack tip field, the J-integral has played an important part in fracture mechanics and engineering applications over the past 20 years, and was considered as the most appealing criterion for crack growth initiation [l-4]. It is well known that geometry independence is a crucial test of the acceptability of a proper elastic-plastic fracture criterion for a given material state. Therefore, there have been many experimental investigations to evaluate the effect of specimen size, geometry and thickness on the Jlc or Ji values, some of which have revealed that JIG and Ji depend on the specimen geometry [5--l, crack depth [8,9], and specimen thickness [lo, 1l] in the plane strain case, even though ASTM E813-81 requirements for specimen dimension were met. Hence, it remains unknown whether or not J,, and Ji values measured from standard laboratory test specimens are suitable to evaluate the safety of the cracked structure with various geometries, thicknesses and crack depths. Thus, considerable doubts have been cast on the capability of J,c as a valid single parameter fracture criterion to characterize the material resistance against crack growth initiation. Further studies showed that the relationship between the stress triaxiality in the crack tip field of specimens and that defined by the Hutchinson-Rice-Rosengren (HRR) solution is the decisive factor which will determine whether J-dominance is valid or not [12]. In the plane strain case, the stress triaxiality constraints in crack tip regions of real specimens, which vary with the variations of specimen geometries, crack depths and load levels, are unable to meet the high triaxial constraint requirements defined by the HRR solution, which is the essential reason for losing the J-dominance and causing the specimen geometry dependence of J,c [13, 141. Having discussed the defects of J-dominance and the effects of stress triaxiality, Hutchinson proposed that two parameters, the J-integral and the measure of near-tip triaxiality, might suffice to characterize the full range of near-tip fracture environments [ 151. Another investigation has also indicated that it is very necessary to introduce a constraint parameter in addition to J to obtain a better characterization of the local stress field in crack tip regions [16]. Recently, significant developments for this problem have been made in Anderson’s research [17], in which a simple correction for crack tip stresses was derived from the quasi-deformation assumption, to take account of losses in triaxiality in the crack tip regions of specimens. However, it was limited to only a center-cracked panel that predictions of crack tip stress fields agreed well with some finite tPresent address: Department Canada K7L 3N6.
of Materials and Metallurgical Engineering, Queen’s University, Kingston, Ontario,
789
SUN JUN
790
element results because no quantitative relationship exists between the crack tip triaxiality and the ~haracte~~ng parameters. In the present article, Anderson’s results are employed, together with some finite element calculations and experiments, to discuss the effects of triaxiality on crack tip stress fields and characterizing parameters. By means of the relationship proposed here between the triaxiality of specimens and that defined by the HRR solution, the triaxiality constraints are introduced analytically into crack tip stress fields and characterizing parameters. Some interesting results have been found along this line. 2. STRESS
TRIAXIALTIY
CONSTRAINT
For the HRR solution, the stress-strain
IN CRACK TIP REGIONS
fields are given by [l, 21
where eij and cij are the stress and plastic strain tensors, c, and ce are the effective stress and strain, respectively, E is Young’s modulus, r is the radial distance from the crack tip, I,, is a dimensionless constant depending on strain hardening, and ciij, 4, 6e and Z, are dimensionless constants which depend on strain hardening and the angle (0) from the crack plane. The other parameters are defined by a power law idealization of the flow behavior, which for uniaxial deformation is given by n-I 0; cr, &=a (3) 2' 0 CO The stress cro is usually defined as 0.2% offset yield strength. The ratio of hydrostatic stress o;, to the von Mises effective stress cr, is used to characterize the stress triaxiality in crack tip regions. From (l), the stress triaxiality on the ligaments ahead of the crack tip, defined by the HRR solution in the plane strain case, can be given by
According to the HRR solution for a given material state (n-value), the stress triaxiality on the ligaments of a specimen should be a constant independent of the specimen geometry, the crack depth, the distance from the crack tip and the load level. Figure 1 depicts the stress triaxiality on
9 8
-
HRR field
7
d
1, 6-o 3 -
n-3
0
‘614-
0 3-o
u 2-a l0 0
0.1
0.2
0.3
0.4
0.5
l/n Fig. 1. Stress triaxiality constraints defined by the HRR solution along the crack tip ligaments in the plane strain case.
0
0
Ill10 11100
Cl
. 0
l
‘
I 0.01
I 0.02
I 0.03
I 0.04
1 0.05
J/ho,, Fig. 2. Global triaxiality constraints on the crack tip ligaments of a center-cracked panel in the plane strain case given by (16). The data were taken from the results in refs [IS, 191and Fig. 1.
Stress triaxiality constraint and crack tip parameters
791
the ligaments of specimens with different strain hardening, which is required by the HRR solution in the plane strain case [1,2]. It can be seen that the crack tip triaxiality defined by the HRR solution is strict and high, which is based on the infinitely large plane of a nonlinear elastic solid by obeying the pure power hardening relation, and is too high to account for any finite size specimen of real elastic-plastic materials in the plane strain case. Finite element results have shown that, in the plane strain case, the triaxiality constraints of specimens are all lower than those of the HRR solution at different degrees [12-141. In order to understand the effects of crack tip t~axiality constraints on fracture parameters, it is necessary to get a quantitative relationship between the triaxiality of specimens and that defined by the HRR solution. Anderson’s research, completed recently [17], makes it possible to discuss the problem further. Based on the quasi-deformation theory assumption, for loading in the direction perpendicular to the crack plane, Anderson gives the modified flow behavior of materials as [17] and the crack tip stress as (6) The quasi-defo~ation
theory modifies the J-CTOD
relationship:
Al00 4
J=Il’“d,p
(7)
where 6, is the CTOD and d,, is the dimensionless constant which depends on flow properties (i.e. n, a and co/E). Consequently, Anderson’s simple corrections for crack tip stress and strain can be expressed as [17]
where X is the distance of the ligament from the crack tip. Thus, a J-like parameter characterizes the stress normal to the crack plane is given by [17]
which
For material on the plane of the crack in plane strain, one has: 6, =
Substituting
(11)
(9) and (11) into (3) results in
(12) From the HRR solution and (3), one can obtain: (13) Thus, (12) becomes: (14)
792
SUN JUN
which can be considered as the correction for the effective stress. In the plane strain case, one can express the stress triaxiality as
am
_=-....--- OYY
@e cr, Introducing
j?i-
(8) and (14) into (15), the following expression is derived:
or equivalently: d,, dJ --_= co da,
%I/% + llfi @&,)tiRR + ll.Js
1
(17)
’
where o,jo, is the crack tip stress triaxiality of real specimens, which depends not only on the n-value but also on the value (d”/~o)(dJld~~), namely the specimen geometry for a given material state (n-value). When the HRR assumptions are valid, @J/d&) = (co/d,,) and the quantity in the brackets = 1.0, and (16) reduces to the HRR solution (4). For (16), the (dJ/db,) values can be obtained by
where A is the slope of the curve o. S,/J vs J/(ba,) at a fixed point of the J-CTOD relationship calculated by Shih [ 18, 191,and b is the untracked ligament length. Since J/h, is very small, dJ/d& is nearly equivalent to J/6,. As a result, the stress triaxiality constraints given by (16), on the ligaments for the deeply notched center-cracked panel (CCP; a/W = 0.75) with IZ= 3 and 10, have been computed based on the J-CTOD relationship obtained by Shih [18, 19], and compared with those defined by the HRR solution in Fig. 2. It can be seen that the triaxiality constraints of the CCP specimen decrease rapidly with increasing J and are lower than those defined by the HRR solution in the entire deformation range. However, if (16) is employed to evaluate the crack tip constraints of a deeply notched bend specimen, one would obtain the result that the crack tip constraints elevate gradually with increased J for the specimen, which is not the real case of losses in triaxiality and may be due to errors in estimating the CTOD from the finite element mesh [17]. From (17), it is known that constraint factors such as J/a0 CTOD 1161or (dJ/d~,)/~~~~d~) [17] only characterize the relative differences of the crack tip triaxiality constraint of the specimens from those defined by the HRR solution and not the constraints themselves. In addition, (a,/o,), presented in (16), characterizes only the global constraint levels in crack tip regions and does not reflect the real distribution of triaxiality along the ligaments of the specimen. In order to investigate the real situation of losses in triaxiality constraints of specimens, 4-
HRR field.
Center-cracked
IlllO
pm01
30 4 d
0
0
-
0
2 i
ad
... 0
1 -
q
0 0
DCI
o
badJa200 b q,/J-60 u badJ=30 l
0
t
I
I
0.5
1.0
1.5
LogIxI(Jla~)l Fig. 3. Stress triaxiality constraints on the crack tip ligament of a center-cracked panel in plane strain with a/W = 0.75 and n = 10.
793
Stress triaxiality constraint and crack tip parameters
4 -
3
HRR field. 8
Edge-cracked .
o
q
bead specimen
.
n-10
0 0
D
O0
2
2-
.
q
.
0
0
0 q
l-
o b o,/J=ZOO l b 00/J=60 0 b a,/J=30
0
,
I
I
0.5
1 .o
1.5
J-oaIW(Jla,)l Fig. 4. Stress triaxiality constraints on the crack tip ligament of an edge-cracked bend specimen in plane strain with a/W = 0.75 and n = 10.
Shih’s finite element calculations [19] have been repeated for the deeply notched CCP (a/W = 0.75) and the edge-cracked bend specimen (ECBS; a/W = 0.75) with n = 3 and 10 respectively. The properties are the same as those used by Shih and German [19]. The details of FEM calculations have been given elsewhere [ 12-14,201. As shown in Figs 3-6, the stress triaxiality constraints, along the ligaments ahead of the crack tip of two specimens with n = 10 and 3, are obviously lower than those defined by the HRR solution (the solid line in Figs 3-6). The further the distance from the crack tip on the ligaments, the lower the triaxiality. As the load increases, the stress triaxiality constraints also decrease due to progressive relaxation of constraints in the crack tip field. Up to then the ligaments are in large scale yielding, shown by FEM computations, and the decreasing tendency of triaxiality has just started to be reduced and then reaches a stable level. For a given n-value, the triaxiality of the ECBS specimen is higher than that of the CCP specimen at the same J level. The decrease of triaxiality with the increased distance from the crack tip, for the ECBS specimen is steeper than that for the CCP specimen. After the comparison of the data in Fig. 2 with those in Figs 3 and 5, it is interesting to see that the “global” triaxiality constraints, given by (16), exactly correspond to the “local” triaxiality constraints at the location of X = J/a, on the crack tip ligaments at different J levels for CCP specimens with n = 3 and 10 respectively, as shown in Fig. 7, which will be discussed further below. On the other hand, even though the (a,,,/a,) value given by (16) does not properly describe the real distribution of the crack tip triaxiality along the crack tip ligaments of specimens, (16) will play the role of a bridge for us to introduce the stress triaxiality constraints into the characterizing parameters in crack tip fields.
98 -
-HRR
field.
prnel
Centor-cracked
n=3
l-
3 2 _ 1
0
o bq,/J=POO l b a#-60 0 b adJ=30
,
I
I
0.5
1.0
1.5
LogWI(Jla,)l Fig. 5. Stress triaxiality constraints on the crack tip ligament of a center-cracked panel in plane strain with a/W = 0.75 and n = 3.
SUN JUN
794
9 8
-
HRR fisld.
O
.
6 ,-
Do
2
0
.O
’
a
2
1
_
.
a
5 43 -
Edgecracked
bend
rpocimsn
.
a
.
l
0
0
0
0
0
o
b o,,/J=200 b a,,/J=60 a bq,/J-30
a=3
l
,
-0
I 1.0
0.5
I 1.5
LosIXI(Jla,)l Fig. 6. Stress triaxiality constraints on the crack tip ligament of an edge-cracked bend specimen in plane strain with a/W = 0.75 and n = 3.
3. CRACK TIP TRIAXIALITY
CONSTRAINT
AND STRESS FIELD
In previous studies [12-141, finite element calculations have indicated that the difference in crack tip stress fields of specimens from those of the HRR solution correspond to the mutual relationship between the crack tip triaxial constraints of specimens and those defined by the HRR solution. This conclusion is to be verified below by an analytical expression together with the FEM numerical calculations. By taking (6) into account, one has: (19)
where(~JHRRis the stress defined by the HRR solution. Substituting (7) and (19) into (8) gives:
8
r o
1 0
Given by (16) for CCP from FEM rcwlts
At X-J/a,,,
I
I
I
I
I
0.01
0.02
0.03
0.04
0.05
J/b Q,, Fig. 7. Relationship between “global” triaxiality given by (16) (after Fig. 2) and “local” triaxiality at the location of X = J/a, on the crack tip ligament of a centercracked Panel (CCP) with a/W = 0.75; n = 3 and 10.
Stress triaxiality constraint and crack tip parameters
195
We can obtain:
(21) Substituting (17) into (21), the following relation holds:
(22) where ovu and a,,,/~, are the “real” stress and triaxiality constraint on the crack tip ligaments of specimens, which can be given by FEM. From (22) and (17) we know that, obviously, when and only when (%I/~,) = (%lm/cAR,
(23)
namely the “real” stress triaxiality constraints on crack tip ligaments of specimens fully satisfy the relevant constraint requirements of the HRR solution, it leads to ayv = (Gyy)HRR
(24)
and
*dJ
2-z
a,, d&
1.
(25)
’
namely, J-dominance of crack tip fields is valid and vice versa. Therefore, it can be considered that a sufficient and necessary condition, in crack tip triaxiality constraints, for the validity of J-dominance has been presented by (22) and (17). In the meantime, it is suggested that the lower the ~,,,/a, value, relative to the relevant (a,,,/~,)~~~ value, the larger the differences of crack tip
CCP
0 b adJ=ZOO
CCP
n-10
0 b odJ=60
n=3
0 b odJ=30
1.0
0 b udJ=ZOO 0 b udJ=60 0 b udJ=30
1.0
0 0 0
qr.Ia
::L *
0
1.5
1.0
0.5
0
0
’
0.5
LosIXI(J/Ql
1.0
ECBS
0 b q,/J=200
BCBS
n=lO
. b o,/J=60
n=3
b 0
$0.9
q
h 0.8
$0, b
0 b 5,/J-30
o0
. 0
'
b u,/J-60
I 0 l
00
0 b
0
0
0.6
0
0.5 0.4
1.0
.
a b uo/J=30
0
q
1.5
0 b u,/J=200
0 .
1.0
LottIJU(Jlo,N
0
0
I
I
I
0.5
1.0
1.5
LodXI(J/a,,)l
V.-n
0
0.5
1.0
1.5
LOisIXI(J/a,)l
Fig. 8. Normal stress ratio ahead of the crack for a center-cracked panel (CCP) and an edge-cracked bend specimen (ECBS) with a/W = 0.75; n = 10 and 3 respectively. After Shih and German [19].
796
SUN JLJN CCP IFlO
ab ‘s 0 Y a 0 ‘s g
0 b a,+200 0 b oo/J=60 0 b a#-30
O
0.8
0
0.7 . 0
0.6
. On
0.4
b s 0” ‘a 2
ECBS n*lO
l
a
0.7
0
0.7 0.6
0
3 a 0 B 0 Y a 0 ‘e g
0
e 0 D
0.6
1.0
0.5
1.5
0 b u,,/J=200
ECBS n=3
0
q
0.8
0
Loi~lW(J/a~)l
lO
0
0.8
1.5
0 b q,/J-200 l b q,/J=60 0 b QIJ-30
.
0.9
0.4 1.0
1.0 0.9
1.0
lX/(J/ar,)l
‘-og
I
0 b udJ=200 . b udJ-60 0 b a,,/J=30
0.5
0.5
0
b S0
2 a 0 ?a b Y a 0 ‘s 2
0.9
0.5
3
CCP n=3
l
* b adJ=60 0 b adJ=30
1.0 0.9 0.8 0.7
*
0
a
0 0 .
0
l
0
l cl
0.6
00
v*-
0
0.5
1.0
1.5
J-o&WJ/ao)l Fig. 9. Stress triaxiality ratio ahead of crack for a center-cracked panel (CCP) and an edge-cracked bend specimen (ECBS) with a/W = 0.75; n = 10 and 3 respectively. After Figs 3-6.
stresses of specimens from those defined by the HRR solution, which agrees well with previous results reported by the present author [12-141. In order to further discuss the relationship between crack tip constraints and stresses, ~~~/(~~~)~~~and (~~/~~}/~~~/~~)~~~, along the crack tip ligaments of CCP and ECBS specimens with n = 3 and 10, are replotted in Figs 8 and 9 against X/(J/a,). The quantities o,.~and (a,,&,.) were inferred from the finite element analyses performed by Shih [19] and Figs 3-6 [20] respectively. It is evident that, for all four cases here, the distribution and the variation of u,,,/(o,,~)nRRalong the ligaments accompany those of (a,,,/~~)/(a,t/a,)HRR. For the CCP specimens, with increased JJ~u~, the deviation of the “real” stresses from those defined by the HRR solution for the specimen with n = 10 is more remarkable than that for the specimen with n = 3, which agrees better with the relevant deviating degrees of the crack tip triaxiality constraints in these specimens from those given by the HRR solution. Contrary to the situation of CCP specimens, regarding the deviations of the crack tip stresses and the triaxiality constraints of ECBS specimens from those required by the HRR solution, the smaller the n-value, the larger the difference. The ~lationship described in (22) between the stresses and the triaxiality constraints is depicted in Fig. 10, with all of the data from Figs 8 and 9. It is very clear that most of the data are located on the same straight line and other data are also close to the straight line, which defines a unique relationship between the crack tip stresses and the triaxiality constraints for a given material state. 4. CRACK TIP TRIAKIALITY
CONSTRAINT
AND CHARACTERIZING
PARAMETER
In Anderson’s research 1171,a J-like parameter characterizing the stress normal to the crack plane has been defined and shown in (10). Substituting (10) into (8) we obtain the corrected stress:
(26)
Stress
triaxiality constraint and crack tip parameters n-10
1.0 -
0.9
n=3
CCP
0
0
ECBS
.
n
/ p*
.
$
AD 2
0.8
.
0.7
797
-
/j.&& pb
. /
Y
a
1”’
8: Oy[
,
.
0.7
0.8
t&/u,+
[
,
,
zi (qJQJ”RR +111
0.6
0.5
,
0.9
l/I/3
1.0
r4
d3
Fig. 10. Relationship between normal stress ratio and stress triaxiality ratio ahead of cracks for a center-cracked panel (CCP) and an edge-cracked bend specimen (ECBS) with a/W = 0.75; n = 3 and 10. After Figs 8 and 9.
If J$ characterizing the crack tip stresses is valid, the correction for the stresses along the ligament should be suitable for different specimen geometries. However, as discussed previously, the correction of the stresses has already been limited in CCP specimens because the (dJ/dS,) in (10) is unable to properly describe the decreasing triaxiality with increased J, when the ECBS specimens are involved. Recall that the relationship between (dJ/d&) and (a,,,/~,) is given by (17). It is therefore possible to solve the problem above. Introducing (22) into (19), we derive that
I,=
%I& + l/d
[ (%/a,)“,,
By the combination
+ l/J5
n/b -1) 1
(27)
*
of (7) and (17) with (27), J$ in (10) can be rewritten as: J* = JA$l -d/n YY
d _dJ
n+’
2
( 00 d&
>
(28)
-
HRR field.
Canter-cracked
pixel
n=lO
o b q,/J=200 l b 5,/J-60 o b ~/J-30
Fig. 11. Crack tip normal stress for a center-cracked panel with a/W = 0.75 and n = 10 (from Fig. 8) normalized by .I&. J$ is given by (28).
SUN JUN
798 0.6 -
HRR field.
o
b crdJ=200
l
b udJ=60
Edge-cracked
bend specimen
n=lO
0 b ts,,/J-30
0
2
1
3
4
5
WW~,,/o,)l Fig. 12. Crack tip normal stress for an edge-cracked bend specimen with a/W = 0.75 and n = 10 (from Fig. 8) normalized by J$. .I;,, is given by (28).
where the quantity (~,/a,) is the triaxiality constraint, calculated by FEM, on the crack tip ligaments of specimens. For log[a,,,,/a,,] and log[X/(J$/a,,)], the normal stresses given by (26), on the crack tip ligaments of specimens, should be a straight line, the slope of which is l/(n + 1). If the parameter J$, is able to characterize the normal stresses, the stresses of the specimens with different geometries should be located on the same straight line at different sections for a given material state, even if the J/ha, level is different. The finite element results from Figs 8 and 9 are replotted in Figs 11-14, where X is normalized by J$/q, given by (28) and the data on the crack tip triaxiality are taken from Figs 1 and 3-6. For CCP specimens with n = 3 and 10, the stresses have been corrected and are close to those given by the HRR solution. However, it is worth noting that the normal stresses on the crack tip ligaments of ECBS specimens with n = 3 and 10 have also been corrected by using J;y; namely, the present J$ characterizing the normal stresses is valid even for different types of specimen geometry. However, it should be pointed out that the triaxiality parameter (a,,&,) in (28) has been considered as a function of the distance of the ligaments from the crack tip, i.e. X/(J/a,), as given in Figs 3-6. Thus, for Figs 1I-14, J$ given by (28) is also dependent on the distance from the crack tip, which is not right for JE as a J-like parameter characterizing the crack tip stresses. Recall that the triaxiality constraints given by (16) corresponded to those calculated by FEM at the location of X = J/0,( - 26,) on the crack tip ligaments of CCP specimens at the same J/boo (as indicated 0.9 0.8
c -
BRR fiold.
Contor-cracked
panel
n-3
0.7
2
0.6
1
0.5
$
0.4
s
0.3
o b adJ=200 -
l
badJ=60
0 b
0
adJ=30 1
2
3
LoW(l,,la,N Fig. 13. Crack tip normal stress for a center-cracked panel with a/W = 0.75 and n = 3 (from Fig. 8) normalized by J$. J$ is given by (28).
Stress triaxiality constraint and crack tip parameters 0.9 0.8 0.7 no
0.6
2fi
0.5
s2
0.4
N-l
0.3 0.2
0.1
l-
HRR field.
”
bend specimen a-3
Edge-cracked
o
b a,,/J-200 b o,,lJ-60 0 b adJ=30 l
1
0
799
I
I
I
1
2
3
LW[X/&h,)l
Fig. 14. Crack tip normal stress for an edge-cracked bend specimen with o/W = 0.75 and n = 3 (from Fig. 8) normalized by J$. .I,‘, is given by (28).
in Fig. 7). Now, we express the triaxiality constraint at the location X = J/u,, as (a&,),,,, which should be the maximum value of the triaxiality on the crack tip ligaments because there is an intense strain region within ~26, [21], where the stress triaxiality constraints have been decreased due to crack tip blunting. Therefore, we can employ (a,&,),,, as a characterizing parameter of the triaxiality constraint along the crack tip ligaments of various specimens at a given J level. Thus, (28) becomes: J* = J (Gl/~Jn+ l/d n YY [
1’ hn/~ehRR +l/J5
(2%
where the J$, value would be uniquely determined when J is given because (~,,,/a,), is only dependent on the J level for a given specimen type, which is of great interest when discussing fracture because fracture always emanates from the highly constrained zone, within which hole growth and coalescence occur intensely [22,23]. The results in Figs 11-14 are replotted in Figs 15-18, where X is normalized by J$./u,,, given by (29). It can be seen that the corrected stresses agree well with those defined by the HRR solution except for some deviation of the stresses when far from the crack tip due to a very weak triaxiality constraint. The cases of corrected stresses for the CCP specimens with n = 3 and 10 are similar to the results obtained by Anderson [17]. The results indicate clearly that the crack tip stresses of different specimen geometries could be uniquely determined by the parameter Jz for a given material state (n-value), even when the (~,,,/a,), value has been applied to characterize the
0.6
-
BRR field.
CCP
n-10
o
b q,/J-200 b udJ=60 0 b adJ-30 l
1 0
I
I
I
1
2
3
Lo8[x/(fyJa,)l
Fig. 15. Crack tip normal stress for a center-cracked panel (CCP) with o/W = 0.75 and n = 10 (from Fig. 8) normalized by J$. .I$ is given by (29).
800
SUN JUN 0.6
0
2
1
3
LWWl~~,,/O,)l Fig. 16. Crack tip normal stress
for an edge-cracked bend specimen (ECBS) with a/W = 0.75 and n = 10 (from Fig. 8) normalized by Jc.. J& is given by (29).
triaxiality constraints along the crack tip ligaments of the specimens, which is very significant for one to develop a plane strain fracture criterion independent of specimen geometry, particularly in large scale yielding. In addition, when (0,/o,) = (a,@,),,, in (28) or (a&,), = (a,/~,)~~~ in (29), (28) and (29) reduce to J;= J; (30) namely, J-dominance of crack tip fields is valid as a single parameter. In order to verify the validity of J$ characterizing crack tip stresses, it is necessary to examine the relationship between the constraints defined by Jz, and those of “real” specimens. Substituting (10) and (17) into (14) gives (31) By employing (26), one can obtain the triaxiality constraint required by the parameter J$:
where (a,,,/~,) is the triaxiality constraint presented at the crack tip ligaments of specimens. The analysis above again shows that the sufficient and necessary condition for one to ensure the validity 0.9 0.8
r
I-
-
HRR field.
Center-crrckcd
panel
n-3
Fig. 17. Crack tip normal stress for a center-cracked panel with a/W = 0.75 and n = 3 (from Fig. 8) normalized by .I$. $,, is given by (29).
Stress triaxiality constraint and crack tip parameters
0.8
no I) ‘2 s$
0.6
4
0.3
-
HRR field.
Edge-cracked
bend rpsciman
801
n=3
0.5 0.4
0.2
0
1
2
3
LoglXl(j,/o,)l Fig. 18. Crack tip normal stress for an edge-cracked bend specimen with a/W’ = 0.75 and n = 3 (from Fig. 8) normalized by Jc.. J;, is given by (29).
of a single parameter characterizing crack tip stresses is that the crack tip triaxiality constraints defined by the characterizing parameters should be satisfied by those of “real” specimens. 5. CRACK TIP TRIAXIALITY
CONSTRAINT
AND FRACTURE
CRITERIA
Studies performed recently [13,24] showed that, in the plane stress case, the triaxiality constraints in crack tip regions of real thin specimens are able to meet the very low constraint requirements defined by the HRR solution and are independent of the specimen geometry, the crack depth and the load level for a given material state, which ensures the validity of J-dominance in the plane stress case. Consequently, the plane stress Jlc and J-R curves can be considered as suitable fracture criteria to characterize the material resistance against crack initiation and stable growth based on the specimen geometry independence of J,c and J-R curves. However, “plane strain” is an immediate situation. Up to now, no substantial improvement along these lines has yet appeared to develop a plane strain fracture criterion to be a material constant independent of the specimen geometry for a given material state. Equation (29) may be helpful for one to further discuss the problem mentioned above. When crack growth initiation occurs in the crack tip regions of a specimen, from (29), we get:
J;y = (J,y 1,~.
(33)
Because the crack tip stresses on crack tip ligaments of different specimens can be uniquely determined by J$ given by (29), (Jz),, should be a material constant independent of the specimen geometry. At the same time, for some specimen we can obtain the critical values: J = Ji
(34)
and (%I/~,), = (%I/~,),,.
(35)
Thus, one has the critical form of (29) for the specimen:
namely, although the J, and (a&,), values are different for various types of specimens, the (J$>,, value is the same for a given material state (n-value), where the reason for applying (a,&,), to characterize the critical situation of the crack tip triaxiality constraints of the specimens is that the fracture is always initiated at the zone with maximum value of the stress triaxiality on the crack tip ligaments [22,23,25,26], because the separation of materials, i.e. cavity nucleation and development, occurs most easily within the zone due to maximum volume dilatation of materials EFM 44,5-K
SUN JUN
802
caused by (a,/a,), [25]. Hence, it becomes clear that this region is the most suitable for formulating a fracture criterion even in the case of small scale yielding [25], and the (~,/a,)~~ value for a specimen can be reasonably utilized to characterize the critical case of crack tip triaxiality constraint on the onset of crack growth of the specimen. If there was a nonlinear elastic cracked solid with a large enough size, on the crack tip ligament of which the triaxiality constraint defined by the HRR solution could be met, one would have (0, I, ),, = (a, /a, LRR 1 i.e. the quantity in the brackets of (36) is unity. Thus, the plane strain fracture toughness J,, could be obtained:
(J,:.),, = Jr, or equivalently
(37)
by taking account of (36) J,,
=
J,
tam/ae)mc + l/d n ’ [ (amm/a,h + l/J51
(38)
holds for some real specimen. This means that the plane strain fracture toughness J,, cannot be measured directly from real laboratory testing specimens with finite sizes because the crack tip triaxiality constraints on the crack tip ligaments of the real elastic-plastic cracked specimens are all lower than those defined by the HRR solution at different degrees [12-141, particularly in large scale yielding, as discussed previously, even though the ASTM E813-87 Standard’s requirements for measuring Jlc were fully met [27]. Similar results can be obtained in measuring the critical CTOD. Substituting (27) into (7) gives the JCTOD relationship: a0 4
J=T
[
tam/a,),+
l/Js
(a,/a,)HRR + l/$
1’
(39)
which is clearly geometry or constraint dependent. When the constraint requirements of the HRR solution are met, namely (am/a,), = (am/ee)nRRT J and CTOD are equally valid crack tip characterizing parameters and a unique relationship between them exists:
at the initiation of crack growth, and J,, = ? 6,,
(41)
n
10 -
n=lO
n-3
10 -
P2 l-7
2 x co
S-
00.4
I 0.6
I 0.8
I 1.0
Fig. 19. Predicted relationship between ratio of J value at initiation of crack growth to plane strain fracture toughness and crack tip triaxiality constraint.
n=3
n-5
II=10
S-
0 0.4
I
I
I
0.6
0.8
1 .o
Fig. 20. Predicted relationship between ratio of CTOD value at initiation of crack growth to plane strain fracture toughness and crack tip triaxiality constraint.
Stress triaxiality constraint and crack tip parameters
Fig. 21. Schematic representation
of specimen geometries and dimensions (in mm).
at the same time; for (39) we have the J-CTOD
relationship:
1Introducing
803
(42)
(41) and (42) into (38) yields:
where Jlc and 6,, are only dependent on the n-value and are material constants independent of specimen geometry, even though (a&,), , Ji and ai are different for various specimens. Figures 19 and 20 display the relationship between Ji(Si) and the corresponding (b,/o,),,,,. The results show that the Ji and ai values are very sensitive to differences of (O&J,), at the crack tip ligaments of various specimens from one geometry to another, in particular for materials with lower strain hardening (larger n-value). Therefore, generally, the plane strain fracture toughness “JIc” and a,,, measured from standard laboratory test specimens, are geometry dependent and not the “real” Jlc due to losses in the crack tip triaxiality constraints. In addition, from (38), for two types of specimens, i.e. A and B, with the same material state and different crack tip triaxiality constraints, the following relationship holds:
+ l/J5 JiA (6nl~e)mB Jis= [ (am/a,),, + l/Jr
1’ n
WI
which may be of great importance for one to predict Ji of some cracked component with a crack tip triaxiality constraint of (cr,/~,),,,, from the results measured experimentally from a laboratory test specimen for a given material state. In order to further discuss the dependence of the Ji value on (cr,,,/a,),,,, an experimental investigation along these lines has been performed, and is described below. The material used in this study was a normalized C-Mn steel for petroleum casing, in the form of a rolled plate. The chemical composition and room temperature tensile properties normal to the rolling direction have been given elsewhere [24,28]. The yield strength a0 = 400 MPa and the strain hardening exponent n = 5. The plates were machined into four types of specimen geometry: center-cracked panel (CCP), single edge-cracked (SEC), double edge-cracked (DEC) and compact cracked tension (CCT) specimens, as shown in Fig. 21, with thicknesses of B = 12.5, 25.4 and 50.8 mm. In order to discuss the problem under the condition of “pure plane strain” crack tip constraint environments and to avoid the effect of the shear lip, all of the specimens were side-grooved with 45” V sharp notches to a depth of 25%. The percentage side-groove depth is given by percentage side-groove depth = (B, - B,)/B, x 100, where Bg = gross thickness and B,, = net thickness. The fatigue pre-cracking was carried out under load control and at room temperature to give a final crack length to width ratio of about 0.5.
SUN JUN
804 0 l
CCP SEC
I
0 DEC . CCT 200
200
180 P 2E
150 I
$,
100
e;-
P
1
1 50
0
I
10
P
ii
20
30
Thickness
40
50
B(mm)
g
140
d
120
z i-
100
0 12.5 . 25.4
O
I-
160
T
i
r
0
0
0
50.8
i3
80
2.6
2.4
60
2.8
3.0
3.2
3.4
B(mm)
Fig. 22. Effect of thickness for side-grooved specimens with four types of geometry on J values at the onset of crack
Fig. 23. Dependence of J values at crack growth initiation of four specimen geometries with different thicknesses on crack tip triaxiality constraints.
The J, values, at the onset of crack growth initiation, were determined from J-R-Au curves, using the procedure described in ASTM E8 13-87 [27] (which assumes a 2a, Au blunting line) and the methods presented in Gibson et aZ.‘s research [29] for four types of specimen geometry. The (am/ae)mc values, the triaxiality constraints at the location of X = J/a, on the crack tip ligaments of the four specimens, were calculated by the FEM [12-14,201, related to the corresponding Ji value of each specimen. Figure 22 shows the experimental results of J, values for the four types of specimen geometries with different thicknesses. In all cases, Ji shows a small decrease with increasing specimen thickness due to side-grooving. However, Ji evidently depends on the specimen geometry even for different specimen thickness, which is easy to understand from the viewpuint of the crack tip triaxiality constraints. As shown in Fig. 23, the J, of the specimen with lower crack tip constraint is much larger than that with higher triaxiality, the mechanical mechanisms of which have been revealed in previous work [ 13,141. As a proper fracture criterion, the parameter should be a material constant independent from the specimen geometry for a given material state. Obviously, the present J, (or “JIc”) is unable to meet this requirement. As analyzed earlier, (JG),, = Jlc should become geometry independent, where J,, is “real” and not that measured directly from the laboratory test specimens. Figure 24 illustrates the case of (JTy),c = J,c as the fracture criterion at crack growth initiation, which was calculated by (37) and (38) by taking the data from Fig. 23. It can be seen that (J$),c and Jlc are indeed material constants independent of specimen geometry with different crack tip triaxiality constraints. On the other hand, the “real” plane strain fracture toughness JIc value is only about one-third of the Ji value,
B(mm)
26
$j 2 7 .I! -x *a.
0 12.5 l 25.4 0 50.8
0
22’
0
0
l
-VA.
18
- ‘2.4
0
0
•I
2.6
2.8
3.0
3.2
3.4
Rl’=,L
Fig. 24. Specimen geometry independence of plane strain fracture toughness for a given material state.
Stress triaxiality constraint and crack tip parameters
805
or “JIc” as is usually considered, measured from compact tensile specimens. Commonly, the compact tensile specimens were suggested to measure the Jlc value of materials only because they have higher triaxiality constraints in crack tip regions than those of other types of specimens and closer to that required by HRR singularity, namely such “JIc” measured from CCT specimens are just lower than that from other specimens but are still not the “real” J,c of the materials. In other words, the plane strain fracture toughness has been overestimated by “J,c” measured from the standard CCT specimens [l 11, or at least it can be thought that the ASTM E813-87 size requirements are not restrictive enough [30]. 6. CONCLUSIONS (1) A quantitative relationship between the crack tip triaxiality constraints on the ligaments of specimens and those defined by the HRR solution in the plane strain case was derived from the simple correction of the crack tip stress-strain field presented by Anderson. It was found that the crack tip triaxiality given by the relationship corresponds well with that of the location X = J/cq,( - 26,) on the crack tip ligaments of a center cracked panel. (2) An analytical expression was established and verified, which describes a sufficient and necessary condition for the validity of a single parameter characterizing crack tip stresses. The crack tip triaxial constraints required by the single parameter should be fully satisfied by those presented at the crack tip ligaments of specimens. (3) A characterizing parameter of crack tip constraints was introduced into the crack tip parameter proposed by Anderson, and then the validity of the single parameter characterizing crack tip stresses was extended to situations of different specimen geometry. A corrected relationship between J and CTOD was given which clearly shows its dependence (4) on the specimen geometry and load level by taking account of different losses in triaxiality constraint for various specimens with increased load level. relationship between Ji(Si) and the corresponding (rr,/e,),,,, was (5) The interdependent presented, which showed that Ji(Si) is very sensitive to differences of crack tip triaxiality constraints for specimens from those defined by the HRR solution, particularly in the case of low strain hardening (large n-value). Based on the validity of Jz, uniquely characterizing the crack tip stresses of different specimen (6) geometries, a new elastic-plastic fracture criterion was developed and proven experimentally to be a material constant independent of specimen geometry for a given material in the plane strain case, which is the “real” plane strain fracture toughness Jlc. (7) From the viewpoint of crack tip triaxiality constraints, it is impossible to obtain the plane strain Jlc value directly from the laboratory test specimens because the crack tip triaxiality constraints of various specimens are unable to meet those required by the HRR solution, even in small scale yielding. Acknowledgement-The acknowledged.
financial support of Fok Ying Tung Education
Foundation
for this research is gratefully
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