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Procedia Materials Science 3 (2014) 57 - 62

20th European Conference on Fracture (ECF20)

Extended investigation of test specimen thickness (TST) effect on the fracture toughness (Jc) of a material in the transition temperature region as a difference in the crack tip constraint

-what is a loss in constraint in the TST effect on Jc?

Toshiyuki Meshiia*, Kai Lub and Yuki Fujiwarab

aFaculty of Engineering, University of Fukui, 3-9-1 Fukui, 910-8507, Japan bGraduate Student, University of Fukui, 3-9-1 Fukui, 910-8507, Japan

Abstract

This paper is an extension of our recent work (Meshii et al., 2013), which demonstrated through experiments and elastic-plastic (EP) finite element analysis (FEA) that the test specimen thickness (TST) effect on the fracture toughness of a material Jc in the ductile-to-brittle transition temperature region, together with the bounded nature of Jc for large TST, had a correlation with the out-of-plane constraint parameter T33-stress. Because a definite measure of the crack tip constraint magnitude, especially for EP issues, does not exist, several well-known constraint parameters were tested from the standpoint of whether they have correlations to the decreasing and then the bounded nature of Jc for increasing TST. The results clearly indicated that Tz = a33/(ah+ <r22) at the specimen mid-plane could not be directly correlated with the TST effect on Jc, based on the observation that the highest Tz did not coincide with the fracture location predicted by the (4^t, cr22c) criterion, in which cr22c denotes the critical crack opening stress value at a distance ahead of the crack tip that is equal to four times the crack tip opening displacement 8t. On the other hand, the results indicated that the well-known constraint parameter 0 = (hydrostatic stress)/(von Mises stress), measured at 4£t, has an ability to monitor the loss in constraint in the TST effect on Jc. 0 at 4^t had a linear relationship with the cr22 up to the fracture load Pc for thick specimens of thickness-to-width ratio of B/W = 1.0 and 1.5, while 0 at 4St for thin specimens of B/W = 0.25 and 0.5 began to decrease before reaching Pc. In addition, 0 at 4^t for Pc exhibited a good correlation with the TST effect and bounded nature of Jc for increasing TST.

Keywords: fracture toughness; transition temperature; test specimen thickness effect; crack tip constraint; stress triaxiality; loss in constraint; out-of-plane constraint; triaxiality parameter; Tz

* Corresponding author. Tel.: +81-776-27-9764; fax: +81-776-27-9764. E-mail address: meshii@u-fukui.ac.jp

2211-8128 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.013

Nomenclature

B, W, a specimen thickness, width and crack length

Jc fracture toughness from the experimental results

Jc FEA J obtained for the fracture load Pc via FEA

K SIF corresponding to the fracture load Pc

Ku Ko local and nominal mode-I stress intensity factor

Pc fracture load

Tii T33 T-stresses

Ai, A3 normalized T-stresses

4 4 crack-tip opening displacement (CTOD) and CTOD corresponding to the fracture load Pc

^eq equivalent strain

triaxiality parameter and its maximum

oc a-^ a, yield stress, critical stress, stress components (i, j =1, 2, 3) and principal stresses

1. Introduction

It is well-known that the cleavage fracture toughness Jc of a material in the ductile-to-brittle transition (DBT) temperature region exhibits size effects, even during tests of standardized test specimens (Petti and Dodds, 2004). For example, Jc obtained from a shallow cracked specimen exhibits a higher value than that obtained from a deep cracked specimen (Dodds et al., 1991; etc.), as shown in Fig. 1 (a). Another known size effect is the test specimen thickness (TST) effect on Jc, which is described as Jc <x BTm (B = TST) by Wallin (1985), as shown in Fig. 1 (b).

(a) Planar size effect (b) TST size effect

Fig. 1 Size effects on Jc

The two most physically logical explanations in general are the loss in crack tip constraint (or the loss of stress triaxiality effect) and the statistical weakest link (SWL) size effect. Previous studies (Al-Ani and Hancock, 1991; etc.) indicated that the difference in Jc obtained with a different planar specimen configuration, including the crack depth and specimen type, was explained as the difference in the crack tip constraint or the hydrostatic stress triaxiality, which J fails to describe. However, the TST effect on Jc has been explained in terms of the SWL size effect being dominant, even though Jc does not decrease indefinitely with thickness (Anderson et al., 1994), which contradicts the prediction from the SWL size effect described as Jc <x B-1/2 (Wallin, 1985).

Recently, the crack tip constraint approach to the TST effect on Jc has been studied under the assumption that the effect is a result of the difference in the out-of-plane crack tip constraint. Guo (1993) introduced a parameter Tz = 033/(0"11+a'22) to characterize the out-of-plane crack tip constraint and has extensively worked with co-workers to express the crack tip stress field using the stress intensity factor (SIF) K or J-Tn-T2. Niemitz and Galkiewicz (2006) utilized a three parameter J-Q-Tz approach to explain the constraint effect on Jc. Whether explicitly expressed or not, Gao (1992), Wang et al. (2003), Gonzalez-Albrnxech et al. (2011) and Meshii et al. (2010-2014) focused their attentions on the out-of-plane T-stress, T33, as a measure of the out-of-plane crack tip constraint.

The authors believe, as illustrated in Fig. 2 left-above, that the contribution of the out-of-plane crack tip constraint to the TST effect on Jc could be demonstrated if the TST effect (especially the bounded nature of Jc with

increasing TST) was demonstrated using a series of non-standard test specimens, whose planar configurations are identical, but whose thickness-to-width ratios, B/W, are changed to realize different thickness specimens, and if the test results were confirmed using finite element analysis (FEA). This use of non-standard test specimens was prompted because the bounded nature of Jc cannot be predicted by using the SWL formulation. This prediction was thought to be enabled by these non-standard specimens because the out-of-plane crack tip constraint (represented by P33 = T33 (xa)m/K0) will increase and saturate with increasing B/W, but the in-plane crack tip constraint (represented by Pw = T„ (aa)1/2/ K0) will not change (Meshii et al., 2013).

Based on the information presented above, fracture toughness tests and elastic-plastic (EP) FEA were conducted for the non-standard 3 point bending (3PB) specimens with B/W = 0.25 - 1.5 (the planar configuration was designed to be identical) (Meshii et al., 2013). The TST effect on Jc together with the bounded nature of Jc for large TST was observed with in the tests for 0.55% carbon steel S55C in the DBT temperature region, and the observations could be reproduced via EP-FEA. All these results validated the contribution of the out-of-plane crack tip constraint to the TST effect on Jc because the bounded nature of Jc cannot be predicted by the SWL approach.

If the TST effect on Jc can be explained as a difference in the out-of-plane crack tip constraint, then it was thought that some failure criterion which can be applied to explain the TST effect exists. For the candidate of the criterion, the famous (44, c22c) failure criterion (Dodds et al., 1991), which was used to explain the crack depth dependence on Jc, was considered to be a candidate, because the TST effect on Jc effect is also a crack tip constraint issue. This (44, o"22c) criterion judges the occurrence of failure when the crack opening stress at 44 (<5t: crack tip opening displacement (CTOD)) exceeds a critical value c22c. As expected, (44, o"22c) failure criterion successfully explained the TST effect on Jc, as shown in Fig. 2 left-below (Meshii et al., 2013). The fracture load level described in the terms of SIF: Kc was approximately independent of the TST.

Fig. 2 Comparison of the test specimen size effects on Jc on the standpoint of (a) out-of-plane (Meshii et al., 2013) and (b) in-plane (Bilby et al., 1986) constraint difference; out-of-plane constraint decreased under the condition Kc = constant, but cr22 showed negligible dependence

However, the fact that the (44, o"22c) criterion applied for explaining the TST effect and bounded nature of Jc for increasing TST indicated that the crack opening stress level at fracture load did not decrease, although the out-of-plane crack tip constraint changed due to the TST, as shown in Fig. 2 left-below. This result is different from what we have experienced for the in-plane constraint loss (Bilby et al., 1986), as shown in Fig. 2 right. So what is the constraint loss in the TST effect on Jc?

This paper is an extension of our recent work (Meshii et al., 2013) on the point that the goal set in this work is to investigate whether some well-known constraint parameters can be used to describe the TST effect on Jc together with the bounded nature of Jc for increasing TST, and become able to explain what the loss in constraint for the TST

effect on Jc is

2. The loss in constraint in the TST effect on Jc

Because a definite measure of the crack tip constraint magnitude, especially for EP issues, does not exist, and because the traditional approaches based on the in-plane Tn-stress or g-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 to the observed decreasing and bounded behavior of Jc for increasing TST. An additional 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 the fracture load Pc were considered from the FEA results in our previous work (Meshii et al., 2013).

2.1. Tz parameter

Because our non-standard 3PB specimens were designed to increase the elastic out-of-plane constraint with increasing TST, Tz = cr33/(on+ cr22), defined as the ratio of the EP out-of-plane stress cr33 to the sum of in-plane stresses ct11 and cr22 (Guo, 1993) was considered first. For this purpose, Tz taken at the specimen mid-plane under fracture load Pc was compared for four B/Ws in Fig. 3. The mid-plane was considered because fracture initiated at the specimen mid-plane for all B/Ws. It is seen from Fig. 3 that Tz exhibited 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 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 (44, c22c) criterion if fracture is to initiate at the highest stress triaxiality location.

Based on these results, it appears that Tz cannot be directly correlated with the TST effect on Jc.

Fig. 3 Tz around the crack tip taken at the specimen mid-plane for the non-standard 3PB specimens under fracture load Pc (W=25 mm, a/W=0.5)

2.2. Triaxiality parameter ©

The well-known constraint parameter © (Henry and Luxmoore, 1997), defined as the ratio between the hydrostatic stress cm and the von-Mises stress crMises, was considered next.

__fa +^2 +^3 )/3_

rMises -^faj -C2)2 +fa2 -C3)2 +fa3 -CTj)2 /V2"

Here, cr1, a2 and c3 are the principal stresses.

Although © is usually used in correlation with ductile fracture (Henry and Luxmoore, 1997), it was thought that the (44, ct22c) criterion resembles the classical material strength theory, which can predict elastic fracture under high © level, and thus is effective for the present work. For example in the case that the principal stress c2 = crn is equal to c3 = c33, and proportional loading is assumed, it can read from Fig. 4 that the maximum principal stress cr1 = c22 might reach the critical stress (cc) before yielding for a case of high © equal to 4 and elastic fracture occurs. Thus it was thought that this case can be effective for considering our problem.

The relationship between c22 and © at the fixed location 44c (4c = CTOD at fracture) located in the specimen mid-plane were plotted for increasing load up to Pc, as shown in Fig. 4 left. It was observed that © was not high enough to cause elastic fracture at a low load level and yielding occurred when P/Pc > 0.45 for all B/Ws. After yielding, for the cases of B/W = 1.0 and 1.5, © monotonously increased in proportion to the crack opening stress c22 until c22 reached the critical value c22c. In contrast, for the cases of B/W = 0.25 and 0.5, © started to decrease at a certain load before c22 reached c22c, and this loss in stress triaxiality leads to a sudden increase in the equivalent plastic strain seq before fracture, as shown in Fig. 4 right. This loss in stress triaxiality is the loss in constraint in the TST effect on Jc and could be monitored by © at 44c. Although the loss in constraint resulted in an increasing equivalent strain, seq at fracture load was a small value of up to 3.1 % to ensure cleavage fracture.

Fig. 4 Results at the fixed location 4iSc for increasing load (W=25 mm, a/W=0.5) (a) normalized ct22 vs © (b) equivalent strain £"eq vs ©

© at 44 for all B/Ws were plotted together with Jc FEA, as shown in Fig. 5. 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 © at 44 exhibited bounded nature for large TST.

■ JcFEA ■ ®@4Jtc

Fig. 5 The TST effect on Jc FEA and © at 4$ observed at the specimen mid-plane for the non-standard 3PB specimens under fracture load Pc

(W = 25 mm, a/W = 0.5)

In summary, © at 44 was a good measure to understand the loss in crack tip constraint in the TST effect and the bounded nature of Jc for increasing TST.

3. Conclusions

In this work, based on the results of non-standard 3PB test specimens obtained from our previous work (Meshii et al., 2013), several well-known constraint parameters were selected to investigate their correlations to the decreasing and subsequently bounded nature of Jc for increasing TST. The conclusions of this work are summarized as follows.

1) Tz cannot be directly correlated with the TST effect on Jc unless the location is specified.

2) The constraint parameter © measured at 44c had an ability to monitor the loss in constraint in the TST effect on Jc, i.e., ©at 4^c had a linear relationship with the 022 up to the fracture load Pc for thick specimens of thickness-to-width ratio of B/W = 1.0 and 1.5, while © at 43c for thin specimens of B /W = 0.25 and 0.5 started to decrease before reaching Pc. This decrease in © with load increase was the loss in constraint in the TST effect on Jc.

3) © measured at 43c exhibited good correlation with the TST effect and the bounded nature of Jc for increasing TST.

Acknowledgment

Part of this work was supported by JSPS KAKENHI Grant Number 2456103. Their support is greatly appreciated.

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