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Extended investigation of the test specimen thickness (TST) effect on the fracture toughness (Jc) of a material in the ductile-to-brittle transition temperature region as a difference in the crack tip constraint - What is the loss of constraint in the TST effects on Jc?
Toshiyuki Meshiia, Kai Lu b'*, Yuki Fujiwara b
a Faculty of Engineering, University ofFukui, 3-9-1 Bunkyo, Fukui, Fukui, Japan b Graduate School of Engineering, University ofFukui, 3-9-1 Bunkyo, Fukui, Fukui, Japan
ARTICLE INFO ABSTRACT
This paper answers a question that was raised from our recent work [1] for the non-standard single-edge notched bend (SE(B)) specimens, which exhibited the test specimen thickness effect on Jc (TST effect on Jc) together with the bounded nature for increasing TST in the ductile-to-brittle transition temperature region. The question was ''what is the loss of constraint in the TST effect on Jc, because the crack opening stress r22 distribution scaled with CTOD at fracture load was approximately same for different TSTs?''
To answer this question, several well-known constraint parameters were investigated to determine their correlations with the TST effect and the bounded nature of Jc for increasing TST. Based on the elastic-plastic finite element analysis results, it was demonstrated that the well-known stress triaxiality factor 0 = (hydrostatic stress)/(von Mises stress), measured at 45tc (crack-tip opening displacement at fracture load Pc), exhibited a good correlation with the decreasing and subsequently bounded nature of Jc for increasing TST. 0 started to decrease at some load level before fracture for relatively thin specimens, and this was the loss of constraint in the TST effects on Jc.
© 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-SA license (http://creativecommons.org/licenses/by-nc-sa/3XI/).
1. Introduction
The measured fracture toughness Jc of ferritic materials in the ductile-to-brittle transition (DBT) temperature region exhibits test specimen size effects, even tested using the standardized specimens [2]. For example, the Jc differed for the specimens with different planar geometries [3-6], which is usually known as the ligament or in-plane size effect on Jc (hereafter named as in-plane size effect on Jc). Another well-known size effect is the test specimen thickness (TST) effect on Jc, which is empirically formulated as Jc / B~1/2 (B = TST) [7]. Wallin [7] has proposed two physical explanations for explaining these specimen size effects: (i) the loss of crack-tip constraint (or the loss of stress triaxiality) effect and (ii) the statistical weakest link (hereafter designated SWL) size effect.
* Corresponding author. Fax: +81 776 27 9764. E-mail address: kai_lu@u-fukui.ac.jp (K. Lu).
Contents lists available at ScienceDirect
Engineering Fracture Mechanics
journal homepage: www.elsevier.com/locate/engfracmech
CrossMark
Article history:
Received 15 April 2014
Received in revised form 20 July 2014
Accepted 21 July 2014
Available online 10 August 2014
Keywords:
Fracture toughness
Test specimen thickness effect
Transition temperature
Stress triaxiality
Crack-tip constraint
http://dx.doi.org/10.1016/j.engfracmech.2014.07.025 0013-7944/© 2014 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-SA license (http://creativecommons.org/licenses/by-nc-sa/3.0/).
Nomenclature
B specimen thickness
J J-integral
Jo Jc average fracture toughness and its average from the experimental results
Jc FEA
Kc Ki Ko
T„ __ W a di n
r, e X
bii, b33 dt dtc eeq
Om rMises
Oys O22c
J obtained at the fracture load Pc via FEA SIF corresponding to the fracture load Pc local mode-I stress intensity factor nominal SIF for the elastic analysis fracture load Q-parameter
ratio of r33 over (r-n + r22) T-stresses specimen width crack length
distance from the crack-tip to the Hmax location
strain-hardening exponent in the Ramberg-Osgood fitting
in-plane polar coordinates
crack tip local coordinates (j = 1, 2, 3)
Ramberg-Osgood fitting parameters
normalized T-stresses
crack-tip opening displacement (CTOD)
CTOD corresponding to the fracture load Pc
equivalent strain
triaxiality factor
maximum value of H
critical stresses (Fig. 7)
principal stresses (i = 1, 2, 3)
stresses components (i, j = 1, 2, 3)
hydrostatic stress
von Mises stress
true yield stress
critical crack opening stress
Regarding the in-plane size effect, as shown in the left side of Fig. 1, previous studies [8-12] indicated that the effect could be explained as the difference in the in-plane crack-tip constraint or the hydrostatic stress triaxiality, which J fails to describe. The later works by Bilby et al. [13] presented some results of the in-plane constraint loss using modified boundary layer analyses, as shown in the right side of Fig. 1. In concrete, the displacements were specified as the boundary conditions with a constant stress intensity factor (SIF) but different Tn-stress. The results indicated that the crack opening stress level r22 was influenced, especially for the negative T11, which can be understood as the in-plane constraint loss [13].
However, for the TST effect on Jc, the widely accepted interpretation is that the SWL size effect is dominant, even thoughJc does not decrease indefinitely with thickness [14], which contradicts the prediction from the SWL size effect described as Jc / B-1/2 [7], as shown in Fig. 2. To solve this contradiction predicted from the SWL formulation, the TST effect onJc has been
Fig. 1. In-plane size effect on Jc, including the ligament size effect, from the standpoint of the in-plane crack-tip constraint [8-13].
S* oo o
Jc k B-1/2
SWL size effect
for B -»• 00
Contour of Constant of
Fig. 2. TST effect on Jc explained as the SWL size effect [7,14].
investigated from the standpoint of the out-of-plane constraint by assuming this TST effect is induced by the difference in the out-of-plane crack-tip constraint. Several constraint parameters, such as Tz [15-23] and T33 [1,24-36], have been extensively studied to characterize this out-of-plane constraint effect.
Using T33 as a relevant parameter to express the out-of-plane constraint, one of the authors [1] conducted both fracture toughness tests and elastic-plastic (EP) finite element analysis (FEA) on the non-standard single-edge notched bend (SE(B)) specimens, which were designed with identical planar configurations but various thickness-to-width ratios, B/W = 0.25-1.5 (Fig. 3 left). From the experimental results, the TST effect on Jc and, in particular, the bounded nature of Jc for large TST (which cannot be predicted from the SWL formulation) was observed for JIS S55C (0.55% carbon steel), and the observations were reproduced via EP FEA [1]. Another finding was that the SIF, Kc, which was calculated from the fracture load and the measured crack depth, was approximately independent of the TST, even though Jc exhibited a strong dependence on the TST [1]. All these results validated the contribution of the out-of-plane crack-tip constraint to the TST effect on Jc regarding the point that the bounded nature of Jc cannot be predicted by the SWL approach.
Because it was believed that the TST effect on Jc was explained as the difference in the crack-tip constraint, the famous (4dt, r22c) failure criterion [3], which was used to explain the crack depth dependence on Jc, was motivated to be applied to explain the TST effects on Jc [1]; this explanation was determined to be appropriate, as shown in the right side of Fig. 3. Note that this (4dt, r22c) criterion states that the failure occurs when the crack opening stress at 4dt (dt: crack-tip opening displacement (CTOD)) exceeds a critical value r22c [3]. The fact that the (4dt, r22c) criterion successfully explained the TST effect and the bounded nature of Jc for increasing TST raises a question: ''the crack opening stress r22 level at the fracture load does not change, although the out-of-plane crack-tip constraint changed due to the TST; this result is different from the in-plane constraint loss as introduced in Fig. 1[13], correct?''
Thus, this paper provides answers to the question raised by our recent work [1]: what is the constraint loss in the TST effects on Jc? Several well-known constraint parameters were studied to investigate whether they can be used to describe the TST effect on Jc together with the bounded nature of Jc for increasing TST.
2. The in-plane and out-of-plane stress distributions
Before proceeding to select a constraint parameter that best describes the TST effect on Jc, the in-plane and out-of-plane stress distributions at the mid-plane of the specimens at fracture load were compared using our previous EP FEA results [1].
Lower Bound
log (B/W)
B/W 1.5 (T„ c * 0)
B/W 1.0 (T33 c < 0)
B/W 0.5 (T33 c < 0)
B/W 0.25 (t33 c < 0)
T33c decreases
Fig. 3. TST effect on Jc from the standpoint of the out-of-plane constraint [1].
Because the (4dt, r22c) criterion could be applied to explain the TST effects on Jc, the stress distribution along the x^axis was considered. The mid-plane was considered because fracture initiated at the specimen mid-plane for all B/Ws.
To begin with, the in-plane normal stresses r11 and r22, and the out-of-plane normal stress r33 distributions from our previous FEA results for each B/W were plotted in Fig. 4. As expected, it was observed from Fig. 4(c) that r33 exhibited a remarkable decreasing tendency for the relatively thin specimen with B/W = 0.25, although the variation of r33 was negligible for B/W p 1. The decrease in r33 at the constant Kc for our non-standard SE(B) specimens is similar to the results that we obtained involving the decreasing r22 for the in-plane constraint loss [13] (as shown in Fig. 1, right). This decrease in r33 for increasing B/W seems to be explained as the loss in the out-of-plane crack-tip constraint.
Unexpectedly, the in-plane stress components r11 (Fig. 4 (a)) exhibited a change with B/W for a relatively thin specimen of B/W = 0.25; however, overall, the in-plane stress distribution was only slightly affected by B/W. These results seemed to validate our approach of designing the non-standard test specimens considering the elastic T-stresses, that is, the in-plane constraint does not change for the non-standard SE(B) specimens, but the out-of-plane constraint significantly changes for B/ W [1].
3. Constraint parameters
Although the constraint effect on the fracture toughness Jc has been studied for years, to the best of our knowledge, a definite measure of the crack-tip constraint magnitude, especially for the EP issues, does not exist. Because the traditional approaches based on the in-plane T11-stress or Q-parameter, which successfully describe the in-plane crack-tip constraint, are not accurate in describing the out-of-plane crack-tip constraint, some well-known parameters were used in this work to investigate whether they have correlations with the observed decreasing and bounded behavior of Jc for increasing TST. Another difficulty encountered is determining the location at which to measure the stress triaxiality. Thus, the distributions of the different constraint parameters at the specimen mid-plane under Pc were considered as follows.
3.1. Tz parameter
Based on the results of the stress distributions as described in Section 2, the constraint parameter Tz = r33/(r11 + r22) [15,16], defined as the ratio of the out-of-plane stress r33 to the sum of the in-plane stresses r11 and r22, was considered next. For this purpose, the distributions of Tz at the specimen mid-plane under Pc were compared for four B/Ws, as shown in Fig. 5. Tz exhibited a strong dependence on B/W, as expected. The region of Tz > 0.45, i.e., the red zone, gradually expanded with increasing B/W, which means the out-of-plane constraint level is highly strengthened as the TST increases. However, the highest Tz zone did not coincide with the x1-axis and could not be correlated with the fracture location predicted by the (4dt, r22c) criterion if fracture is to initiate at the highest stress triaxiality location. Nevertheless, Tz at the location 4dtc (CTOD at Pc) seemed to be able to correlate with the TST effect as well as with the bounded nature of Jc for increasing TST, as shown in Fig. 6.
In summary, it appears that Tz cannot be directly correlated with the TST effect on Jc; however, once the measured location is specified, Tz could explain the difference in the out-of-plane stress triaxiality.
3.2. Triaxiality factor &
In the following, the well-known stress triaxiality factor & [37], defined as the ratio between the hydrostatic stress rm and the von-Mises stress ffMises, was considered.
& = rm =_(r1 + r2 + r3)/3__(1)
rMises " r2)2 + (r2 - r3)2 + (r3 - r )2/P2
02468 10 02468 10 02468 10
r/St r/St n'4
(a) <xn distributions (b) <J22 distributions [1] (c) a}3 distributions
Fig. 4. The in-plane normal stresses r11 and r22, and the out-of-plane normal stress r33 distributions taken at the specimen mid-plane for the non-standard SE(B) specimens under fracture load Pc (W = 25 mm and a/W = 0.5).
B/W=0.25 B/W=0.5 B/W=1.0 B/W=1.5 0.5 r
Fig. 5. Tz around the crack-tip taken at the specimen mid-plane for the non-standard SE(B) specimens under fracture load Pc (W = 25 mm and a/W = 0.5).
Fig. 6. Tz distributions at the location 4dtc for the non-standard SE(B) specimens (W = 25 mm and a/W = 0.5).
Here, r (i = 1, 2, 3) are the principal stresses.
Although H is usually used in considering ductile fracture [37], it was thought that the (4dt, r22c) criterion resembles the classical material strength theory, which can predict elastic fracture under high H levels, and thus is effective for the present work. An example is shown in Fig. 7 for the case that the principal stress r2 is equal to r3, and proportional loading is assumed; the maximum principal stress r1 might reach the critical stress (rc) before yielding occurs for high H, thereby making this case effective for considering our problem.
Fig. 8(a) shows the relationship between r22 and H at the fixed location 4dtc for increasing load up to Pc. Note that, r22 is approximately equal to the maximum principal stress r1.
As observed, H was not high enough to cause elastic fracture at a low load level, and then yielding occurred at a certain load for all B/Ws (P/Pc > 0.45).
After yielding, for the cases of relatively thick specimens of B/W =1.0 and 1.5, H monotonously increased in proportion to the crack opening stress r22 until r22 reached the critical value r22c. In contrast, for the cases of the relatively thin specimens
°Mises °YS
Fig. 7. An example of the relationship between the principal stresses and the triaxiality factor H for the case of CTj > r2 = r3.
(a) | 2
Jeq "YS
(°Í1=°33) 0
□ B/W = 1.0 - O B/W = 1.5
elastic fracture
°22 c'°YS
0eq=0YS
(o11=a33) 0
"@4.Stc
□ B/W = 1.0 O B/W = 1.5
ŒE5 sssssb®® D
@@4Stc
B/W = 0.25 B/W = 0.5
P/Pc=0.98
- P/Pc=0.55 J /
®@4Stcc
Fig. 8. Results at the fixed location 4dtc for increasing load (W = 25 mm and a/W = 0.5) (a) normalized a22 vs. 0 and (b) eeq vs. 0.
°22 c' °YS
of B/W = 0.25 and 0.5, 0 started to decrease at some load level (P/Pc > 0.55 for B/W = 0.25 and P/Pc > 0.98 for B/W =0.5) before r22 reached r22c. This decreasing tendency of the triaxiality factor 0 at 4dtc for the specimens with B/W = 0.25 and 0.5 at some load level could be interpreted as the constraint loss in the TST effect on Jc.
The loss in constraint leads to a sudden increase in the equivalent strain eeq before fracture, as seen in Fig. 8(b), but eeq at fracture load was small (up to 3.1%) to ensure the cleavage fracture. Moreover, from Fig. 9, eeq at 4dtc exhibited the decreasing and bounded behavior with increasing TST, which seemed to be consistent with the tendency obtained for the fracture toughness Jc.
The idea to measure the triaxiality factor 0 at 4dtc was reasonable because the maximum 0 values located on the x1-axis, and the distance d1 of 0max ahead of the original crack-tip was in the range of approximately (2-4)dtc, as shown in Fig. 10, which appeared to be in agreement with the critical distance 4dt included in the famous (4dt, r22c) failure criterion [3].
Then, 0 at 4dtc for all B/Ws were plotted together with Jc FEA, as summarized in Fig. 11. Clearly, the increasing tendency and bounded behavior for increasing TST agreed with the relationship between Jc FEA and B/W, regarding the point that both Jc FEA and 0 at 4dtc exhibited a bounded nature for large TST.
In summary, 0 measured at the location 4dtc was a good measure to understand both the loss in the crack-tip constraint in the TST effect and the bounded nature of Jc for increasing TST.
0 0.25 0.5 0.75 1 1.25 1.5
Fig. 9. Equivalent strain £eq measured at 4dtc for the non-standard SE(B) specimens (W = 25 mm, a/W = 0.5, specimen mid-plane and 0 = 0).
B/W=0.25
B/W=0.5
B/W=1.0
0.125 mm
V2t d 0,
0.125 mm *
¿1=1-74
di=3.4^tc
¿1=3.04
B/W=1.5
0.125 mm
¿1=3.0^tc
3.25 3 2 1
Fig. 10. The triaxiality factor H around the crack-tip observed at the specimen mid-plane for the non-standard SE(B) specimens under fracture load Pc (W = 25 mm and a/W = 0.5).
—.....
■ H Jc FEA 0@4S tc
Fig. 11. The TST effect on Jc FEA and H at 4dtc observed at the specimen mid-plane for the non-standard SE(B) specimens under fracture load Pc (W = 25 mm and a/W = 0.5).
4. Discussion
There is an argument that the SWL size effect cannot be ignored. Certainly, we do not deny that the SWL size effect exists to some extent. Nevertheless, the fact that the (4dt, r22c) criterion could be applied to explain the TST effect together with the bounded nature of Jc for increasing TST indicated that the stress level at fracture load did not decrease with TST. One possibility is that the effective volume whose stress level exceeds a threshold did not increase in proportion to the TST, but was approximately constant. For example, if the J-dominant zone, which is known to be (2-6)dt, was considered as the effective zone, it was found to decrease due to the increase in the TST because of the increase in the crack-tip constraint; as a result, the effective area, as shown in Fig. 12, did not increase in proportion to the TST. By comparing the cases of B/W = 0.25 and 1.5, the thickness increased by six times, but the increase in the effective area was only 1.6, and the magnification was not constant. This issue still warrants further study.
The frequently received comment to our previous work [1] is that the TST effect on Jc observed for our non-standard SE(B) specimens [1] could be simply explained as that plane-stress state for a thin specimen approached the plane-strain state for a thick specimen. However, note that there is always some level of crack-tip stress triaxiality, as Hom and McMeeking [38] indicated. In a recently published textbook by Anderson, this finding was reflected in the following statement: ''there is no such thing as ''plane-stress fracture'' except perhaps in very thin foil [39]''. In fact, the r22 distributions at the specimen mid-plane under fracture load Pc for the case of B/W = 0.25 deviated from the plane stress HRR stress distribution but seemed to be close to the plane strain HRR stress distribution [40,41], as shown in Fig. 13. Because fracture initiated at the mid-plane for our non-standard SE(B) specimens, the TST effect observed from our non-standard SE(B) specimens seems to be understood as the difference in crack-tip stress triaxiality, or crack-tip constraint.
Another argument arouse from our results is that whether the coefficient 4 in the (4dt, r22c) failure criterion is definite or not; as seen in Fig. 11, H@45tc seemed to show bounded nature for B/W p 0.5, while the Jc FEA did not. Thus, assuming that fracture initiates at the maximum stress triaxiality location, the relationships between Hmax and Jc FEA vs. B/W were plotted as shown in Fig. 14 for reference. From this figure, Hmax showed bounded nature for B/W p 1.0, which seemed to be more consistent with Jc FEA vs. B/W relationship. The argument on the coefficient 4 is under further study.
The current and our previous paper [1] focused attention on the mechanical view mainly onto the specimen thickness effect (out-of-plane constraint loss), based on the fact that the critical crack opening stress r22c measured at the specimen
Fig. 12. Comparison of the r22 distributions over the range of (2-6)dtc in the thickness direction for the non-standard SE(B) specimens (W = 25 mm and a/W =0.5).
Fig. 13. Comparison of the r22 distributions at the specimen mid-plane with HRR solutions [40,41] for the thin specimen with B/W =0.25 (the Ramberg-Osgood parameters for S55C were approximated as r0 = 394 MPa, n = 4.6 and a = 1.95).
_ Jc FEA
1 II ■
Fig. 14. The TST effect on Jc FEA and 0max for the non-standard SE(B) specimens under fracture load Pc (W = 25 mm and a/W = 0.5).
mid-plane showed only 2.0% variation as thickness changed in the range of B/W = 0.25 to 1.5. In contrast, Dlouhy et al. [42] reported that the critical stress could be affected for a case of cast ferritic steel and in-plane constraint loss due to the crack depth difference. Their result is worth being considered in our future study.
5. Conclusions
In this work, detailed investigations were performed based on our previous EP FEA results [1] for the non-standard SE(B) specimens. Several well-known constraint parameters were studied to investigate their correlations with the TST effect and the bounded nature of Jc for increasing TST. It was concluded that the triaxiality factor 0 at the location 4dtc had an ability to monitor the loss in constraint in the TST onJc. 0 measured at 4dtc exhibited a good correlation with both the TST effect and the bounded nature of Jc for increasing TST and successfully clarified the nature of the constraint loss in the TST effects on Jc.
Acknowledgment
Part of this work was supported by JSPS KAKENHI Grant Number 24561038, and their support is greatly appreciated. References
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