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Plastic deformation behavior in SEB specimens with various crack length to width ratios
Engineering Fracture Mechanics xxx (2017) xxx–xxx
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Plastic deformation behavior in SEB specimens with various crack length to width ratios Tomoya Kawabata a,⇑, Tetsuya Tagawa b, Yoichi Kayamori c, Mitsuru Ohata d, Yoichi Yamashita e, Masao Kinefuchi f, Hitoshi Yoshinari g, Shuji Aihara a, Fumiyoshi Minami h, Hiroshi Mimura i,1, Yukito Hagihara j,1 a
Department of Systems Innovation, The University of Tokyo, Tokyo 113-8656, Japan Steel Research Laboratory, JFE Steel Corporation, Chiba 260-0835, Japan Steel Research Laboratories, Nippon Steel & Sumitomo Metal Corporation, Amagasaki 660-0891, Japan d Department of Manufacturing Science, Osaka University, Suita 565-0871, Japan e Research Laboratory, IHI Corporation, Yokohama 235-8501, Japan f Materials Research Laboratory, Kobe Steel Ltd., Kobe 651-2271, Japan g National Maritime Research Institute, Mitaka 181-0004, Japan h Joining & Welding Research Institute, Osaka University, Suita 567-0047, Japan i Department of Mechanical Engineering, Yokohama National University, Yokohama 240-8501, Japan j Department of Mechanical Engineering, Sophia University, Tokyo 102-8554, Japan b c
a r t i c l e
i n f o
Article history: Received 30 July 2016 Received in revised form 11 January 2017 Accepted 20 March 2017 Available online xxxx Keywords: Toughness testing Test standards Crack tip opening displacement Plastic hinge model Rotational factor Strain hardening
a b s t r a c t The authors have already proposed a new crack tip opening displacement (CTOD) calculation formula based on the plastic hinge model considering the unique crack tip blunting behavior due to strain hardening of the material. This formula is limited to the standard length-to-width ratio (a0/W) range of between 0.45 and 0.55. However, CTOD calculations are necessary for various other a0/W conditions when the critical CTOD for a specific microstructure must be evaluated in a welded joint. ISO15653, which prescribes CTOD test methods for welds, covers a wide range of a0/W between 0.10 and 0.7 by applying a J-integral based CTOD derived from ASTM E1290 for a0/W from 0.10 to 0.45 and a plastic hinge based CTOD for a0/W from 0.45 to 0.70. This procedure leads to a discontinuity in the evaluated CTOD at a0/W = 0.45. This problem results from the fact that the J-integral based CTOD is inconsistent with the conventional plastic hinge based CTOD, especially in low yield-to-tensile ratio materials. Thus, a single CTOD calculation method for the wide range of a0/W is required for rational evaluation. In this study, the plastic deformation behavior of SEB specimens was analytically examined in the a0/W range from 0.05 to 0.70, and the capacity of the plastic hinge based CTOD was discussed with the aim of establishing a unified CTOD calculation formula for SEB specimens with the wide range of a0/W. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction On the basis of the CTOD concept proposed by Wells [1] in 1961, the British Standard Institution published in 1972 ‘‘Methods for crack opening displacement testing” [2] as a ‘‘Draft for Development 19,” or the so-called DD19. Since ⇑ Corresponding author. 1
E-mail address: [email protected] (T. Kawabata). Formerly.
http://dx.doi.org/10.1016/j.engfracmech.2017.03.029 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Nomenclature a crack length a0 initial crack length B specimen thickness CTOD, d crack tip opening displacement del elastic component of CTOD dpl plastic component of CTOD CTODBS, dBS CTOD in BS7448-1:1991 CTODJWES, dJWES CTOD proposed by Japan Welding Engineering Society CTODFEM, dFEM CTOD numerically calculated from displacement between two ±45° intersection points from crack tip on crack opening profile CMOD crack mouth opening displacement E Young’s modulus f correction factor for plastic component of CTODJWES F correction factor for thickness J J-integral K stress intensity factor N strain hardening exponent of Swift type stress-strain relation P load r rotational factor rp plastic rotational factor R inverse number of yield-to-tensile ratio, ruts/rys Vg crack mouth opening displacement Vp plastic component of crack mouth opening displacement W specimen width x, y coordinates in same thickness plane YR yield-to-tensile ratio, rys/ruts z height of knife edge a fitting parameter of Swift type stress-strain relation c non-dimensional critical clip gage displacement ep equivalent plastic strain g plastic eta factor for calculation of J m Poisson’s ratio r stress rys yield strength rY effective yield strength, (rys + ruts)/2 ruts ultimate tensile strength r equivalent stress
DD19, CTOD has been calculated by a hinge model of rotational deformation centered at a point in a ligament. Although the formulation in DD19 was supported by experimental measurements with various ratios of crack depth to specimen width (a0/W), the a0/W range from 0.45 to 0.55 had been a standardized measurement in BS5762 [3] established in 1979, in which CTOD is calculated by Eq. (1).
dBS ¼ del þ dpl ¼
K2 2rys E0
þ
r p ðW a0 Þ Vp rp ðW a0 Þ þ a0 þ z
ð1Þ
The same calculation by Eq. (1) was continued in BS7448 [4], which replaced BS5762 in 1991 with further modifications. CTOD calculated with a rotational hinge model has widely prevailed in the field of steel structures through the long history since DD19. The authors have already proposed the new crack tip opening displacement (CTOD) calculation formula shown in Eq. (2) based on the plastic hinge model considering the unique crack tip blunting behavior due to strain hardening of the material [5]. This formula is limited to the standard length-to-width ratio (a0/W) range between 0.45 and 0.55.
dJWES ¼ del þ dpl ¼
K2 r p ðW aÞ þ f ðYR; BÞ Vp r p ðW aÞ þ a þ z mðYRÞrY E0
ð2Þ
mðYRÞ ¼ 4:9 3:5YR f ðYR; BÞ ¼ FðBÞ f ðYRÞ@B¼25 FðBÞ ¼ 0:8 þ 0:2f0:019ðB 25Þg f ðYRÞ@B¼25 ¼ 1:4ðYRÞ2 þ 2:8ðYRÞ 0:35 rp ¼ 0:43 Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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BS7448 also provides a CTOD evaluation procedure for welds in Part 2. When the critical CTOD at a specific microstructure in the heat affected zone (HAZ) is evaluated, the fatigue crack tip shall be introduced so as to hit the target location. However, because the CTOD calculation method in BS7448 Part 2 is only applicable in the a0/W range from 0.45 to 0.55, critical CTOD evaluation at the specific microstructure at a shallow position in a butt joint is impossible. ISO 15653 [6], which replaces BS 7448 Part 2, provides a shallow crack option to counter the above-mentioned problem. ISO15653 covers the wide range of a0/W between 0.10 and 0.70 by applying a J-integral based CTOD derived from ASTM E1290 [7] for a0/W from 0.10 to 0.45 and a plastic hinge based CTOD for a0/W from 0.45 to 0.70. However, it has been pointed out that this procedure leads to a discontinuity in the evaluated CTOD at a0/W = 0.45 [8]. This problem results from an inconsistency between the J-integral based CTOD and the conventional plastic hinge based CTOD, especially in low yield-to-tensile ratio materials [9]. Thus, a unified CTOD calculation method for the wide range of a0/W has been desired. The J-integral based CTOD, which was transferred to ASTM E1820 [10] due to the abolishment of E1290, covers a0/W from 0.10 to 0.70. However, the BS-CTOD method has prevailed so widely that the plastic hinge based CTOD may not be easily replaced by the J-integral based CTOD. Several previous works have investigated the plastic hinge approximation for crack opening deformation and the rotational factor, rp in specimens with various a0/W. The procedures to define the factor, rp were divided into the three methods shown in Fig. 1. Method (a) is a geometrical procedure assuming a linear crack opening profile. Ingham et al. [11] measured the plastic crack opening behavior of various kinds of steels and configurations, including some a0/W conditions, but only observed crack opening deformation, as the aim was to clarify the value of the rotational factor of the formula. The method is shown schematically in Fig. 1(a). It should be noted that the plastic deformation behavior of the whole specimen was not investigated in that study. Lin et al. [12], Tanaka et al. [13] and Tsukamoto [14] also investigated crack opening deformation and rotational factors under various a0/W by the same methodology. While their experiments should be highly evaluated, they were still far from a sufficient resolution from the viewpoints of total plastic deformation and appropriateness as a rotational factor. On the other hand, deformation in the ligament region of SEB specimens was examined by slip line field analysis by Wu [15,16] and Matsouka [17] and by strain gauge measurement by Thiess et al. [18]. This method is shown schematically in Fig. 1(b). However, their observations were limited to the surface and thus provided no information about the mid-thickness position, which is more critical for toughness evaluation. The third method for clarification of the rotational factor is simple back-calculation based on CTOD values obtained by FEM, etc. This method is shown in Fig. 1(c), and was adopted by Kirk et al. [19], Donato et al. [20] and Kayamori et al. [21]. In this method, the rotational factor does not have a physical meaning. The aim of the present study was to obtain a thorough understanding of the plastic deformation behavior in SEB specimens. For this, a sufficiently validated FE analysis methodology was used because the more critical evaluation position is the mid-thickness plane, but it is difficult to observe the rotational deformation of this region experimentally. The obtained information is to be applied to a rational calculation formula for CTOD evaluation covering various a0/W conditions. 2. Numerical procedures 2.1. Models and method for FE analyses In order to obtain universal findings which can be applied to many materials, a constitutive equation for ordinary and normal 490 MPa steel is used for FE analysis. As representative of material property variations, strain hardening exponents are changed for parametric studies. The detailed configuration of the stress-strain curve is based on actual tensile test results
Fig. 1. Schematic illustration showing definitions of rotational center in various methods.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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[22] and is extrapolated to the high strain range by using the Swift equation. Also, when performing the parametric study of various strain hardening coefficients, the material constants which describe the configuration of the Swift equation (Eq. (3)) are changed in conjunction each other. A tensile strength of a constant value, namely, 520 MPa, is used, and the yield ratios (YR) are set to 0.6 and 0.9, as shown in Table 1. Although Lüders strain is sometimes observed in actual steels, here, no Lüders strain is assumed, which means strain hardening occurs immediately after yielding. The relationships between the equivalent stress and plastic equivalent strain of the constitutive equation are shown in Fig. 2.
r ¼ rY 1 þ
ep a
N ð3Þ
In this study, the mesh design around the crack tip was completely the same as that in the previous study and could give a smooth deformation configuration of the crack face even under a condition of large deformation, for example, CTOD of 0.2 mm. In order to estimate accurately the coordination of the rotational center, the element size along the crack path in the ligament is set to be sufficiently small, 0.1 mm. The overall appearance of the FE analysis for a0/ W = 0.5 is shown in Fig. 3. In this study, thickness, B is a constant value of 25 mm, which is the most widely used in steel structure fields where the CTOD concept is required. a0/W is changed from 0.05 to 0.70 in increments of 0.05, as exemplified in Fig. 4. Nodal reaction force, nodal displacement and strain are outputted from the FEM calculation results, and CTODFEM, which is defined by 45° method [5] using the opening profile of nodes backward from the crack tip, is calculated in each loading step. The calculation process is ended at the CTOD of 0.2 mm, which is an approximate indication of the ductile crack initiation that was observed on the unloaded section in a previous work [22] by the authors. This limitation is based on the philosophy that CTOD is no longer directly corresponding to the initial crack tip blunting after ductile crack initiation. 2.2. Mesh design In the previous report [5], where the CTOD calculation formula for B 2B and a0/W = 0.5 specimens was used, it was noted that the mesh configuration around the crack tip is quite important for the FE analysis results of the deformation around the crack tip. In this study, completely the same mesh configuration is given to the crack tip regions of all the specimens with various a0/W ratios as shown in Fig. 5. The justification for application of this mesh division to other a0/W conditions, especially shallower a0/W conditions has been already proven by experimental method, silicone rubber technique [9].
Table 1 Constitutive equation used in FEM study. Mark
rys [MPa]
ruts [MPa]
YR ( = rys/ruts)
YR60 YR90
312 468
520
0.6 0.9
Swift parameters
a
N
9.09E03 1.82E02
2.27E01 1.10E01
Fig. 2. Constitutive equation used for FEM study.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 3. General dimensions of FE model for SEB specimen.
(a) a0/W=0.05
(b) a0/W=0.70
Fig. 4. Example of FE models for SEB specimen with various a0/W.
3. Investigation of plastic deformation behavior in forward area from crack tip in SEB specimens 3.1. Plastic strain distribution in ligament of SEB specimens in mid-thickness plane In order to investigate the manner of plastic deformation, the first approach is investigation of the contour maps of the equivalent plastic strain of 14 kinds of specimens with elevated conditions of a0/W in the mid-thickness plane. This can give the general tendency of the plastic deformation behavior in the ligament region. Fig. 6 shows examples of material of YR = 0.6 at the CTODFEM of 0.1 mm (Fig. 6(a)) and 0.2 mm (Fig. 6(b)). The figures clearly indicate that the plastic deformation in all the specimens shows rotational behavior with a central focus on a point, as this contour is constructed in logarithmic classification so there will be scarcely any plastic strain in the blue or black colored areas. Also under the condition of a0/W 5 0.15, it can be seen that the highly plastically strained area is growing to the crack side edge of the specimen. This means that, under the shallow condition of a0/W 5 0.15, the plastic deformation behavior is essentially similar to bending of a simple beam without a crack. For accurate determination of the coordination of the rotational center position, the strain distribution in the ligament region was investigated. Fig. 7 is a schematic explanation of the plastic strain distribution in the ligament region in a SEB specimen and the definition of the rotational center position by the strain distribution in the crack propagation plane. In order to establish a CTOD formula with the plastic hinge model, a quantitative knowledge of the rotational center
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 5. Mesh designs around crack tip.
coordination and its change under increasing loading is important. In the context of Section 1, this determination procedure can be categorized as method (b) in Fig. 1. In Fig. 8, the open circle marks show the distribution of the plastic strain increment in the opening direction distribution of YR = 0.9, a0/W = 0.5 on the mid-thickness plane at the CTODFEM of 0.05 mm, 0.1 mm, 0.15 mm and 0.2 mm. Considering that the formula of the CTOD calculation equation is divided into two terms, i.e., an elastic (small scale yielding) and a plastic term, and the plastic hinge model is related only to the plastic term, the rotational deformation should be specified only in plastic deformation. Therefore, the plastic strain distribution is important for establishing the formula that is the objective of this study. An incremental value is used for more accurate evaluation of the coordination of the neutral point by bending. Fig. 8 also indicates there is a more or less zero-plastic-strain region around the center of the ligament. Here, the horizontal axis shows the non-dimensional distance of the ligament, which is the value obtained by dividing the distance from the crack tip by the initial ligament length. By this arrangement, it can be shown that the rotational center point is located inside the zero-plastic-strain region; however, the exact position remains unclear. Even if a large amount of deformation is applied, the elastic region that exists around the center of the ligament still remains. This means the rotational center cannot be computed only by the plastic strain distribution as explained above. Therefore, the incremental value of the whole strain component, including the elastic term, is introduced instead of the increment of the plastic strain, and the strain distribution is also summarized as shown by the solid triangle marks in Fig. 8. By using the whole strain increment, it is possible to determine a unique value of the coordination of the rotational center, as the whole strain distribution diminishes monotonically from the tension side to the compression side. In contrast, in the case of CTODFEM = 0.05 mm, there is a large difference between the distribution of the plastic strain increment and that of the whole strain increment, as almost all of the specimen remains elastic, and in case of CTODFEM > 0.05 mm, the two strain increments are almost the same and a pure elastic region is limited. This indicates that there is no critical problem in definition of the rotational center by the whole strain distribution in the opening direction. Fig. 9 shows the comparison of whole strain distribution for several a0/W conditions from 0.1 to 0.7. This Fig. simply shows that higher a0/W conditions indicate nearer non-dimensional positions of the bending neutral point from the crack tip, xneutral. Thus, xneutral is calculated from the whole strain distribution in the opening direction under all conditions, that is, all the combinations of 14 a0/W conditions and 2 kinds of materials shown in Table 1. Fig. 10 shows
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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(a) CTODFEM 0.1mm
(b) CTODFEM 0.2mm Fig. 6. Distribution of equivalent plastic strain in mid-thickness plane of SEB specimens with various a0/W.
the change of the non-dimensional coordination of the bending neutral position, xneutral/(W a0) for all the conditions until the deformation level of CTODFEM = 0.20 mm. There seems to be a relatively larger amount of change of xneutral/(W a0) in the extreme cases of a0/W = 0.05 or 0.7, but even in those cases, the amount of the change is limited to less than 5%. This mild fluctuation of the coordination of the rotational center during loading is more pronounced in comparison with that of method (a) described in Fig. 1. Here, in order to make a quantitative comparison with the fluctuation of method (a), the index of fluctuation, |Fluct.| was introduced, as shown in Eq. (4). Fig. 11 shows that this method (method (b)) is far superior to method (a) from the viewpoint of fluctuation during loading, in spite of the fact that the coordination of the rotational center defined in the BS5762 specification was experimentally investigated by method (a). With the BS equation, |Fluct.| changes with various a0/W.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 7. Schematic illustration showing definition of rotational center in this study.
Fig. 8. Comparison of plastic strain distribution and total strain distribution along centerline of ligament (YR = 0.9, a0/W = 0.50).
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 9. Comparison of non-dimensional total strain distribution along centerline of ligament (YR = 0.6, CTODFEM = 0.2 mm).
Fig. 10. Change of neutral point position determined by total strain increment under various a0/W for increasing deformation.
Fig. 11. Comparison of fluctuation in range of CTOD from 0.05 to 0.2 with methods (a) and (b).
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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jFluct:j ¼
Xd¼0:050:20mm ½r p i ½rp av : ½dFEM i ½dFEM i1 i ð0:20 0:05Þ
100 ½%
ð4Þ
Fig. 12 shows the summarized relationship between a0/W and the representative value of each condition, which is computed by averaging the xneutral/(W a0) histories over the range of CTODFEM = 0.05–0.2 mm. This figure also shows the results of the slip line field analysis of the surface of the three point bending specimen conducted by Wu et al. [15] and the strain distribution measurement by Theiss [18]. This figure clearly indicates that decreasing a0/W (in other words, a shallower specimen) increases xneutral/(W a0) and its dependency is almost the same between two materials. This tendency is similar to the tendency reported by Wu et al. [15]. As a summary of this method, the numerical function of xneutral/(W a0) can be drawn by Eq. (5), which is a regression curve by least square fitting without the effect of YR.
a 6 a 5 a 4 a 3 a 2 a xneutral 0 0 0 0 0 0 þ 113 97:5 þ 38:5 6:45 þ 0:13 ¼ 49:6 þ 0:50 ðW a0 Þ W W W W W W
ð5Þ
3.2. Plastic strain distribution in ligament and coordination of rotational center of SEB specimens in whole thickness For the establishment of a calculation formula covering various thickness conditions, it is also important to investigate the plastic deformation behavior in the whole thickness. Fig. 13 shows contour figures of plastic equivalent strain at the deformation level of CTODFEM = 0.1 mm. This figure is illustrated from the crack face side toward half of the SEB model (whole figure in thickness direction). These contour figures are constructed in logarithmic classification, as in Fig. 6, so
Fig. 12. Summary of change of neutral point position calculated from strain increment distribution along ligament by 3D-FEM.
Fig. 13. Comparison of plastic equivalent strain distribution in whole model (YR = 0.6, CTOD = 0.1 mm).
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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the black and blue regions show a substantially zero-plastic-strain region. The zero-plastic-strain region is located near the midpoint of the ligament and is almost uniform in the thickness direction. Fig. 14 summarizes the change of xneutral/(W a0) under increasing deformation for several a0/W conditions, and shows that the coordination is almost constant regardless of the thickness positions and the loading level. Fig. 15 shows the average value of xneutral/(W a0), [xneutral/(W a0)]av. at loading levels from CTOD = 0.05 mm to 0.2 mm. In case of a0/W = 0.3, 0.5 and 0.7, [xneutral/(W a0)]av. is slightly smaller at the surface than that at deeper positions in the thickness direction, whereas at a0/W = 0.1, [xneutral/(W a0)]av is almost constant in the thickness direction. This characteristic behavior is based on the fact that a specimen with a shallower condition than a0/W = 0.15 is plastically deformed as a simple bending beam without a crack, as described in Section 3.1. The reason for the slight drop of [xneutral/(W a0)]av. at the surface position under the deep crack conditions should be examined in the future. However, the authors feel that this tendency is related to some extent to plastic dent deformation at the surface. 3.3. CTOD distribution in thickness direction The authors are taking the stand that the CTOD formula should calculate the actual CTOD at mid-thickness, as critical events have mostly occurred from the vicinity of the mid-thickness position. In a previous report [15], various coefficients were set so as to accurately predict a large number of CTODFEM at the mid-thickness position. However, in order to obtain a thorough grasp of the deformation behavior of SEB specimens, it is important to investigate the CTOD distribution in the thickness direction. Fig. 16 shows the CTOD distribution in the thickness direction in case of YR = 0.6 material. The four kinds
Fig. 14. Change of neutral point position determined by total strain of surface and mid-thickness of SEB specimens for various a0/W conditions.
Fig. 15. Distribution of neutral point position determined by total strain in thickness direction for various a0/W conditions.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 16. CTOD distribution in thickness direction under several deformation steps for various a0/W conditions (YR = 0.6).
of plots show the loading levels where CTOD at mid-thickness corresponds to 0.05, 0.1, 0.15 and 0.2 mm. Under all the conditions, as the loading level increases, the difference of CTOD between the mid-thickness and surface also increases; in other words, the configuration of the distribution changes from a flat shape to a round shape. The difference depending on a0/W conditions was examined while keeping this general tendency. In case of a0/W = 0.1, the change of CTOD in the thickness direction is milder than that under the other conditions with deeper a0/W. In order to clarify the reason for this difference, the following discussion will focus on the ratio of CTOD at the surface position to CTOD at the mid-thickness position. Fig. 17 summarize this ratio under increasing loading conditions. Interestingly, the conditions of a0/W = 0.3, 0.5 and 0.7 show almost the same constant value of 0.35–0.4, and only the condition of a0/W = 0.1 shows a much higher value. This is also thought to be strongly related to the fact that plastic deformation as a simple bending beam without a crack occurs under shallower conditions than a0/W = 0.15, as described in Section 3.1. 4. Investigation of plastic deformation behavior in backward area from crack tip in SEB specimens 4.1. Importance of crack face displacement in SEB specimen for CTOD evaluation It seems rational that the conventional plastic hinge type CTOD calculation formula should be extended to various a0/W conditions from the formula for the standard a0/W range of 0.45–0.55 proposed in the authors’ previous report [5]. This philosophy is based on the observation in Section 3 that there is a fixed position which is the center of the rotational deformation determined only by a0/W, regardless of the loading level and strain hardening exponent. Here, formulation of the plastic hinge relationship is important from the viewpoint of establishing the calculation formula. That is, the crack edge point should be located on an extrapolated line passing through the rotational center position and the 45° line crossing point for determination, because mouth opening displacement, Vg, which is the only parameter for deformation that an experimenter can measure, may be used for the determination of the plastic term of the CTOD calculation formula. Referring to Fig. 18(a), the following will explain the CTOD calculation procedure in which the plastic component of opening displacement at the edge of the specimen is applied as an experimental variation. It is noted that only the plastic component is expressed in Fig. 18. The plastic component dpl (called dp here) is obtained by the relationship of similar triangles, where the rotational center is allocated as the vertex of the isosceles triangle, as shown in Eq. (6): Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 17. Change of ratio of CTODsurface to CTODcenter with increasing deformation for various a0/W conditions.
Fig. 18. Schematic diagram of determination of plastic component of CTOD by plastic hinge model.
dp
BS
¼
r p ðW a0 Þ Vp a0 þ rp ðW a0 Þ
ð6Þ
This relationship was first proposed by Ingham et al. [11] and is based on the assumption that the corner points of two similar triangles are linearly connected. It is important to consider the crack tip blunting shape due to the strain hardening exponent when these similar triangles are clarified. The authors have pointed out that the simple plastic hinge assumption leads to overestimation of dp, and the amount of overestimation depends on the strain hardening exponent [22]. Therefore, the multiplier exponent f (basically less than 1) is introduced in the plastic term in the calculation formula (Eq. (2)). This is to say, true dp is smaller than the value calculated by the conventional calculation formula (Eq. (1)). Fig. 18(b) is the corrected illustration considering this situation. Based on this background, it is quite important to determine if dp_BS can be estimated by the measured Vp and the ratio of a0 and rp(W a0). If this is in fact possible, then the calculation formula for dp will be established by introducing the exponent f, as shown in Eq. (7).
dp ¼ f dp
BS
ð7Þ
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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4.2. Effect of a0/W on opening displacement at edge of specimen and formation of plastic hinge relationship with rotational center In order to investigate this issue, the FEM results are analyzed in detail. The problem is that the deformation given by FEM contains both elastic and plastic components, and these two components cannot be intrinsically distinguished. The candidate alternative simple methods for extraction of the plastic component are listed below. (a) Unloading on FEM analysis (b-1) Subtract elastic deformation component by elastic FEM analysis from whole deformation profiles [23] (b-2) Subtract elastic deformation component by solution of SEB displacement from the Handbook of Stress Intensity Factors from whole deformation profiles [24] However, these methods are not appropriate or do not have sufficient accuracy for the following reasons: (a) The unloading process of a cracked specimen leads to reversal yielding around the crack tip, and this is different from the true plastic component in the monotonically loading aspect. (b) Elastic analysis never contains yielding behavior and maintains a profile which is linear even at the vicinity of the crack tip. These points are contrary to the fact that small scale yielding must be assumed in the elastic term. In this study, considering the match for the constitution of the calculation formula, the plastic component is computed by the following procedure, which is shown schematically in Fig. 19.
Fig. 19. Method for derivation of plastic component of nodal deformation on crack face by assuming elastic component line.
Fig. 20. Crack opening profiles in specimens with various a0/W at plastic term of CTOD = 0.1 mm (YR60).
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(x) Letting the opening displacements at each of the nodes in the backward position from the FEM analysis results and the opening displacement at the crossing point by the 45° line to the deformation profile be [Vg]i and [Vg]0, respectively. (y) Defining the elastic component at the edge of the opening mouth, [Ve]edge as the construction method on the P-Vg curve, which is obtained by subtracting the plastic component given by the initial elastic gradient line from the whole displacement, and letting the elastic term in Eq. (2), de be [Ve]0 . The plastic component at an arbitrary position from the tip to the edge, [Ve]i is defined as a point on the line which is drawn by connecting two points, [Ve]0 and [Ve]edge. (z) The plastic component of the arbitrary node position is defined as the result of subtraction of [Ve]i from the opening displacement, [Vg]i at each node. The plastic components of the opening displacement at each node for various a0/W, as computed by the method described above, are summarized in Figs. 20–22. The horizontal axis of these figures is the non-dimensional distance from the intercept point by the 45° line and is designed so that the point of the rotational center clarified in Section 3 corresponds to (1, 0). Also, the deformation status of these figures is at the timing of dp/f = 0.1 mm, where f is given as the same value as in Eq. (2). In sum, if point R, the intercept point to the vertical axis of line OQ, where Q is the plastic component of the edge displace-
Fig. 21. Expansion of Fig. 20 (a0/W = 0.1).
Fig. 22. Crack opening profiles in specimens with various a0/W at plastic term of CTOD = 0.1 mm (YR90).
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(a)
p/f=0.1mm
(b)
p/f=0.2mm
Fig. 23. Deviation from geometric similarity of complete formation of plastic hinge relationship.
ment, is located exactly on point P, (0, 0.1), a perfect plastic hinge relationship would be established. However, the actual intercept point, R is different from point P. Fig. 21 shows an enlargement of the area around dp. It is clearly shown that this graph is drawn at the loading step at the crossing point of line OP, which corresponds to a side of the similar triangle in the plastic hinge model shown in Fig. 18(b), and the crack profile. In this case, due to the low YR condition, the actual dp is much lower than the crossing point. This is expressed by coefficient f in Eq. (2). Viewing the results of the four kinds of a0/W in both case of (a)YR60 and (b)YR90, a certain amount of gap, shown as distance RP, is observed under a wide range of a0/W and YR conditions, except under the condition around a0/W = 0.5. This is also evidence that the plastic hinge relationship is completely satisfied in case of a0/W = 0.5, and Eq. (2) maintains good accuracy under the standard a0/W conditions. These gaps are summarized in Fig. 23, which is shown as the correlation with a0/W. These figures mean that a smaller a0/W results in a larger gap. Fortunately, no effect of YR is observed in the relationship between the gap and a0/W. Also, under the condition of a0/W = 0.1, a larger gap is observed than the extended line from the other a0/W conditions. This may be because the deformation behavior is different from that in the other cases, where plastic deformation is expanded to the edge of the specimen like simple beam bending. Although these gaps do not seem particularly large, these quantitative features of the gaps from the perfect plastic hinge solution will be very helpful for establishing a calculation formula which covers various a0/W conditions.
5. Concluding remarks In this study, plastic bending behavior not only in the ligament region but also in crack opening profiles behind the crack tip, which is an important fundamental part of the CTOD calculation formula based on the plastic hinge assumption, is investigated with the final goal of establishing a unified formula which will be applicable to a wide range of a0/W. A 3D dynamic elasto-plastic FEM analysis for various a0/W conditions from 0.05 to 0.7, which was already validated by actual CTOD measurements by the silicone casting method, is used in order to gain a thorough grasp of the deformation behavior of SEB specimens. The conclusions obtained are listed below. (1) Under every a0/W condition, the fact that the ligament area is rotationally deformed at a certain point is clearly observed. This can be a strong motivation for continuing to use the plastic hinge type calculation formula in determinations of CTOD. (2) It is confirmed that for a/W < 0.15, the deformation mode transitions from rotation only in the ligament to whole specimen rotation as a simple bending beam without a crack. (3) Coordination of the bending neutral position can be described as a simple regression polynomial equation regardless of YR and loading level as expressed in Eq. (5). This coordination function can be applied as rp in the plastic hinge type CTOD calculation formula, however, in our final recommended CTOD calculation formula, rp is set to be constant value in order to avoid complication [25].
a 6 a 5 a 4 a 3 a 2 a xneutral 0 0 0 0 0 0 þ 113 97:5 þ 38:5 6:45 þ 0:13 ¼ 49:6 þ 0:50 ðW a0 Þ W W W W W W
ð5Þ
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(4) Deformation in the thickness direction can be classified into two groups, i.e., shallower than a0/W of 0.15 and others. The bending neutral position and CTOD at the surface are quite constant under shallower conditions than a0/W = 0.15. This is also due to the transition of the deformation mode to whole bending behavior under shallow conditions. (5) The plastic component of opening displacement is clarified in detail from the viewpoint of satisfaction of the plastic hinge relationship by using the coordination of the actual rotational center. As a result, the perfect plastic hinge relationship is satisfied only in case of the standard a0/W condition. However, it is suggested that the plastic hinge calculation method can be applied after a small correction of dp. The correction coefficient may be a function of a0/W and independent of the strain hardening exponent. In our final recommended CTOD calculation formula this correction factor is employed [25].
Acknowledgement The authors wish to acknowledge CTE committee of the Japanese Welding Engineering Society for their dedicated discussions of this study. References [1] Wells Alan A. Unstable crack propagation in metals: cleavage and fast fracture. In: Proceedings of the crack propagation symposium, Cranfield, UK, vol. 2; 1961. p. 2010. [2] British Standards Institution. Methods for crack opening displacement testing – COD-DD19; 1972. [3] British Standards Institution. Methods for crack opening displacement (COD) testing. BS5762; 1979. [4] British Standard BS7448 – Part 1. Fracture mechanics toughness tests – Part 1. Method for determination of KIC, critical CTOD and critical J values of metallic materials; 1991. [5] Tomoya Kawabata, Tetsuya Tagawa, Takahiro Sakimoto, Yoichi Kayamori, Mitsuru Ohata, Yoichi Yamashita, et al. Proposal of a new CTOD calculation formula. Eng Fract Mech 2016;159:16–34. [6] International Organization for Standardization. Method of test for the determination of quasistatic fracture toughness of welds. ISO15653; 2010. [7] American Society for Testing and Materials. Standard test method for crack-tip opening displacement CTOD fracture toughness measurement. ASTM E1290-1989; 1989. [8] Kayamori Yoichi, Inoue Takehiro, Tagawa Tetsuya. Changes in ISO 15653-based CTOD for specimens of a0/W = 0.45. J Solid Mech Mater Eng 2012;vol. 6 (6):645–53 [special issue on Advanced Technology in Experimental Mechanics]. [9] Tagawa Tetsuya, Kawabata Tomoya, Sakimoto Takahiro, Kayamori Yoichi, Ohata Mitsuru, Yamashita Yoichi, et al. Experimental measurements of deformed crack tips in different yield-to-tensile ratio steels. Eng Fract Mech 2014;128:157–70. [10] American Society for Testing and Materials. Standard test method for measurement of fracture toughness. ASTM E1820-2013; 2013. [11] Ingham T, Egan GR, Elliott D, Harrison TC. The effect of geometry on the interpretation of COD test data. Inst Mech Eng 1971;C54/71:200. [12] Lin IH, Anderson TL, De Witt R, Dawes MG. Displacements and rotational factors in single edge notched bend specimens. Int J Fract 1982;20:R3–7. [13] Tanaka Hidehiko, Ando Kotoji, Sakai Yuzuru, Ogura Nobukazu. Brittle crack initiation from shallow notch and hot strain embrittlement. In: Proceedings of Fracture Mechanics Symposium. The Society of Materials Science, Japan, Committee on Fracture Mechanics; 1981. p. 147–51. [14] Tsukamoto Mitsuo. Effect of initial crack length on critical COD of unstable fracture with stable fibrous crack growth, memoirs of the faculty of education, Kumamoto University. Science 1994;43:153–62. [15] Wu Shang-Xian, Brian Cotterell, Mai Yiu-Wing. Slip line solutions for three-point notch-bend specimen. Int J Fract 1988;37:13–29. [16] Wu Shang-Xian. Evaluation of CTOD and J-integral for three point bend specimen with shallow cracks. In: Proceedings of the 7th international conference on fracture, ICF 7, Houston, Texas, vol. 1; March 1989. p. 517–24. [17] Matsoukas et al. On the plastic rotation constant used in standard COD tests. Int J Fract 1984;26(2):R49–53. [18] Theiss TJ, Rolfe SJ, Shum DKM. Shallow-crack toughness results for reactor pressure vessel steel. Shallow Crack Fracture Mechanics, Cambridge, UK; 23–24 September 1992 [paper 37]. [19] Kirk Mark T, Dodds Jr Robert H. J and CTOD estimation equations for shallow cracks in single edge notch bend specimen. J Test Eval 1993;21:223–38. [20] Donato Gustavo HB, Ruggieri Claudio. Estimation procedures for J and CTOD fracture parameters using three-point bend specimens. In: 2006 International pipeline conference. Volume 3: materials and joining; pipeline automation and measurement; risk and reliability, Parts A and B, Calgary, Alberta, Canada; September 25–29, 2006. [21] Kayamori Yoichi, Inoue Takehiro, Hagihara Yukito. Plastic rotational factor calculation for shallow notched SEB specimen. Mater Sci Forum 2014;783– 786:2322–6. [22] Tagawa Tetsuya, Kayamori Yoichi, Ohata Mitsuru, Handa Tsunehisa, Kawabata Tomoya, Yamashita Yoichi, et al. Comparison of CTOD standards: BS 7448-Part 1 and revised ASTM E1290. Eng Fract Mech 2010;77(2):327–36. [23] Tagawa Tetsuya, Takashima Yasuhito, Ohata Mitsuru, Minami Fumiyoshi. Numerical investigation on BS-CTOD and ASTM-CTOD. IIW doc, X-1676-10. [24] Nishitani Hironobu, Noguchi Hiroshi, Mori Kazuya, Handa Tsunehisa, Kawabata Tomoya, Yamashita Yoichi, et al. Analysis of single-edge-cracked specimen under three- or four-point bending by body force doublet method. Trans Jpn Soc Mech Eng 1986;52(474):539–43. [25] Tomoya Kawabata, Tetsuya Tagawa, Yoichi Kayamori, Mitsuru Ohata, Yoichi Yamashita, Masao Kinefuchi, Hitoshi Yoshinari, Shuji Aihara, Fumiyoshi Minami, Hiroshi Mimura, and Yukito Hagihara, Applicability of new CTOD calculation formula to various a0/W conditions and B x B configuration, Eng Fract Mech, accepted for publication.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Plastic deformation behavior in SEB specimens with various crack length to width ratios Tomoya Kawabata a,⇑, Tetsuya Tagawa b, Yoichi Kayamori c, Mitsuru Ohata d, Yoichi Yamashita e, Masao Kinefuchi f, Hitoshi Yoshinari g, Shuji Aihara a, Fumiyoshi Minami h, Hiroshi Mimura i,1, Yukito Hagihara j,1 a
Department of Systems Innovation, The University of Tokyo, Tokyo 113-8656, Japan Steel Research Laboratory, JFE Steel Corporation, Chiba 260-0835, Japan Steel Research Laboratories, Nippon Steel & Sumitomo Metal Corporation, Amagasaki 660-0891, Japan d Department of Manufacturing Science, Osaka University, Suita 565-0871, Japan e Research Laboratory, IHI Corporation, Yokohama 235-8501, Japan f Materials Research Laboratory, Kobe Steel Ltd., Kobe 651-2271, Japan g National Maritime Research Institute, Mitaka 181-0004, Japan h Joining & Welding Research Institute, Osaka University, Suita 567-0047, Japan i Department of Mechanical Engineering, Yokohama National University, Yokohama 240-8501, Japan j Department of Mechanical Engineering, Sophia University, Tokyo 102-8554, Japan b c
a r t i c l e
i n f o
Article history: Received 30 July 2016 Received in revised form 11 January 2017 Accepted 20 March 2017 Available online xxxx Keywords: Toughness testing Test standards Crack tip opening displacement Plastic hinge model Rotational factor Strain hardening
a b s t r a c t The authors have already proposed a new crack tip opening displacement (CTOD) calculation formula based on the plastic hinge model considering the unique crack tip blunting behavior due to strain hardening of the material. This formula is limited to the standard length-to-width ratio (a0/W) range of between 0.45 and 0.55. However, CTOD calculations are necessary for various other a0/W conditions when the critical CTOD for a specific microstructure must be evaluated in a welded joint. ISO15653, which prescribes CTOD test methods for welds, covers a wide range of a0/W between 0.10 and 0.7 by applying a J-integral based CTOD derived from ASTM E1290 for a0/W from 0.10 to 0.45 and a plastic hinge based CTOD for a0/W from 0.45 to 0.70. This procedure leads to a discontinuity in the evaluated CTOD at a0/W = 0.45. This problem results from the fact that the J-integral based CTOD is inconsistent with the conventional plastic hinge based CTOD, especially in low yield-to-tensile ratio materials. Thus, a single CTOD calculation method for the wide range of a0/W is required for rational evaluation. In this study, the plastic deformation behavior of SEB specimens was analytically examined in the a0/W range from 0.05 to 0.70, and the capacity of the plastic hinge based CTOD was discussed with the aim of establishing a unified CTOD calculation formula for SEB specimens with the wide range of a0/W. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction On the basis of the CTOD concept proposed by Wells [1] in 1961, the British Standard Institution published in 1972 ‘‘Methods for crack opening displacement testing” [2] as a ‘‘Draft for Development 19,” or the so-called DD19. Since ⇑ Corresponding author. 1
E-mail address: [email protected] (T. Kawabata). Formerly.
http://dx.doi.org/10.1016/j.engfracmech.2017.03.029 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Nomenclature a crack length a0 initial crack length B specimen thickness CTOD, d crack tip opening displacement del elastic component of CTOD dpl plastic component of CTOD CTODBS, dBS CTOD in BS7448-1:1991 CTODJWES, dJWES CTOD proposed by Japan Welding Engineering Society CTODFEM, dFEM CTOD numerically calculated from displacement between two ±45° intersection points from crack tip on crack opening profile CMOD crack mouth opening displacement E Young’s modulus f correction factor for plastic component of CTODJWES F correction factor for thickness J J-integral K stress intensity factor N strain hardening exponent of Swift type stress-strain relation P load r rotational factor rp plastic rotational factor R inverse number of yield-to-tensile ratio, ruts/rys Vg crack mouth opening displacement Vp plastic component of crack mouth opening displacement W specimen width x, y coordinates in same thickness plane YR yield-to-tensile ratio, rys/ruts z height of knife edge a fitting parameter of Swift type stress-strain relation c non-dimensional critical clip gage displacement ep equivalent plastic strain g plastic eta factor for calculation of J m Poisson’s ratio r stress rys yield strength rY effective yield strength, (rys + ruts)/2 ruts ultimate tensile strength r equivalent stress
DD19, CTOD has been calculated by a hinge model of rotational deformation centered at a point in a ligament. Although the formulation in DD19 was supported by experimental measurements with various ratios of crack depth to specimen width (a0/W), the a0/W range from 0.45 to 0.55 had been a standardized measurement in BS5762 [3] established in 1979, in which CTOD is calculated by Eq. (1).
dBS ¼ del þ dpl ¼
K2 2rys E0
þ
r p ðW a0 Þ Vp rp ðW a0 Þ þ a0 þ z
ð1Þ
The same calculation by Eq. (1) was continued in BS7448 [4], which replaced BS5762 in 1991 with further modifications. CTOD calculated with a rotational hinge model has widely prevailed in the field of steel structures through the long history since DD19. The authors have already proposed the new crack tip opening displacement (CTOD) calculation formula shown in Eq. (2) based on the plastic hinge model considering the unique crack tip blunting behavior due to strain hardening of the material [5]. This formula is limited to the standard length-to-width ratio (a0/W) range between 0.45 and 0.55.
dJWES ¼ del þ dpl ¼
K2 r p ðW aÞ þ f ðYR; BÞ Vp r p ðW aÞ þ a þ z mðYRÞrY E0
ð2Þ
mðYRÞ ¼ 4:9 3:5YR f ðYR; BÞ ¼ FðBÞ f ðYRÞ@B¼25 FðBÞ ¼ 0:8 þ 0:2f0:019ðB 25Þg f ðYRÞ@B¼25 ¼ 1:4ðYRÞ2 þ 2:8ðYRÞ 0:35 rp ¼ 0:43 Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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BS7448 also provides a CTOD evaluation procedure for welds in Part 2. When the critical CTOD at a specific microstructure in the heat affected zone (HAZ) is evaluated, the fatigue crack tip shall be introduced so as to hit the target location. However, because the CTOD calculation method in BS7448 Part 2 is only applicable in the a0/W range from 0.45 to 0.55, critical CTOD evaluation at the specific microstructure at a shallow position in a butt joint is impossible. ISO 15653 [6], which replaces BS 7448 Part 2, provides a shallow crack option to counter the above-mentioned problem. ISO15653 covers the wide range of a0/W between 0.10 and 0.70 by applying a J-integral based CTOD derived from ASTM E1290 [7] for a0/W from 0.10 to 0.45 and a plastic hinge based CTOD for a0/W from 0.45 to 0.70. However, it has been pointed out that this procedure leads to a discontinuity in the evaluated CTOD at a0/W = 0.45 [8]. This problem results from an inconsistency between the J-integral based CTOD and the conventional plastic hinge based CTOD, especially in low yield-to-tensile ratio materials [9]. Thus, a unified CTOD calculation method for the wide range of a0/W has been desired. The J-integral based CTOD, which was transferred to ASTM E1820 [10] due to the abolishment of E1290, covers a0/W from 0.10 to 0.70. However, the BS-CTOD method has prevailed so widely that the plastic hinge based CTOD may not be easily replaced by the J-integral based CTOD. Several previous works have investigated the plastic hinge approximation for crack opening deformation and the rotational factor, rp in specimens with various a0/W. The procedures to define the factor, rp were divided into the three methods shown in Fig. 1. Method (a) is a geometrical procedure assuming a linear crack opening profile. Ingham et al. [11] measured the plastic crack opening behavior of various kinds of steels and configurations, including some a0/W conditions, but only observed crack opening deformation, as the aim was to clarify the value of the rotational factor of the formula. The method is shown schematically in Fig. 1(a). It should be noted that the plastic deformation behavior of the whole specimen was not investigated in that study. Lin et al. [12], Tanaka et al. [13] and Tsukamoto [14] also investigated crack opening deformation and rotational factors under various a0/W by the same methodology. While their experiments should be highly evaluated, they were still far from a sufficient resolution from the viewpoints of total plastic deformation and appropriateness as a rotational factor. On the other hand, deformation in the ligament region of SEB specimens was examined by slip line field analysis by Wu [15,16] and Matsouka [17] and by strain gauge measurement by Thiess et al. [18]. This method is shown schematically in Fig. 1(b). However, their observations were limited to the surface and thus provided no information about the mid-thickness position, which is more critical for toughness evaluation. The third method for clarification of the rotational factor is simple back-calculation based on CTOD values obtained by FEM, etc. This method is shown in Fig. 1(c), and was adopted by Kirk et al. [19], Donato et al. [20] and Kayamori et al. [21]. In this method, the rotational factor does not have a physical meaning. The aim of the present study was to obtain a thorough understanding of the plastic deformation behavior in SEB specimens. For this, a sufficiently validated FE analysis methodology was used because the more critical evaluation position is the mid-thickness plane, but it is difficult to observe the rotational deformation of this region experimentally. The obtained information is to be applied to a rational calculation formula for CTOD evaluation covering various a0/W conditions. 2. Numerical procedures 2.1. Models and method for FE analyses In order to obtain universal findings which can be applied to many materials, a constitutive equation for ordinary and normal 490 MPa steel is used for FE analysis. As representative of material property variations, strain hardening exponents are changed for parametric studies. The detailed configuration of the stress-strain curve is based on actual tensile test results
Fig. 1. Schematic illustration showing definitions of rotational center in various methods.
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[22] and is extrapolated to the high strain range by using the Swift equation. Also, when performing the parametric study of various strain hardening coefficients, the material constants which describe the configuration of the Swift equation (Eq. (3)) are changed in conjunction each other. A tensile strength of a constant value, namely, 520 MPa, is used, and the yield ratios (YR) are set to 0.6 and 0.9, as shown in Table 1. Although Lüders strain is sometimes observed in actual steels, here, no Lüders strain is assumed, which means strain hardening occurs immediately after yielding. The relationships between the equivalent stress and plastic equivalent strain of the constitutive equation are shown in Fig. 2.
r ¼ rY 1 þ
ep a
N ð3Þ
In this study, the mesh design around the crack tip was completely the same as that in the previous study and could give a smooth deformation configuration of the crack face even under a condition of large deformation, for example, CTOD of 0.2 mm. In order to estimate accurately the coordination of the rotational center, the element size along the crack path in the ligament is set to be sufficiently small, 0.1 mm. The overall appearance of the FE analysis for a0/ W = 0.5 is shown in Fig. 3. In this study, thickness, B is a constant value of 25 mm, which is the most widely used in steel structure fields where the CTOD concept is required. a0/W is changed from 0.05 to 0.70 in increments of 0.05, as exemplified in Fig. 4. Nodal reaction force, nodal displacement and strain are outputted from the FEM calculation results, and CTODFEM, which is defined by 45° method [5] using the opening profile of nodes backward from the crack tip, is calculated in each loading step. The calculation process is ended at the CTOD of 0.2 mm, which is an approximate indication of the ductile crack initiation that was observed on the unloaded section in a previous work [22] by the authors. This limitation is based on the philosophy that CTOD is no longer directly corresponding to the initial crack tip blunting after ductile crack initiation. 2.2. Mesh design In the previous report [5], where the CTOD calculation formula for B 2B and a0/W = 0.5 specimens was used, it was noted that the mesh configuration around the crack tip is quite important for the FE analysis results of the deformation around the crack tip. In this study, completely the same mesh configuration is given to the crack tip regions of all the specimens with various a0/W ratios as shown in Fig. 5. The justification for application of this mesh division to other a0/W conditions, especially shallower a0/W conditions has been already proven by experimental method, silicone rubber technique [9].
Table 1 Constitutive equation used in FEM study. Mark
rys [MPa]
ruts [MPa]
YR ( = rys/ruts)
YR60 YR90
312 468
520
0.6 0.9
Swift parameters
a
N
9.09E03 1.82E02
2.27E01 1.10E01
Fig. 2. Constitutive equation used for FEM study.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 3. General dimensions of FE model for SEB specimen.
(a) a0/W=0.05
(b) a0/W=0.70
Fig. 4. Example of FE models for SEB specimen with various a0/W.
3. Investigation of plastic deformation behavior in forward area from crack tip in SEB specimens 3.1. Plastic strain distribution in ligament of SEB specimens in mid-thickness plane In order to investigate the manner of plastic deformation, the first approach is investigation of the contour maps of the equivalent plastic strain of 14 kinds of specimens with elevated conditions of a0/W in the mid-thickness plane. This can give the general tendency of the plastic deformation behavior in the ligament region. Fig. 6 shows examples of material of YR = 0.6 at the CTODFEM of 0.1 mm (Fig. 6(a)) and 0.2 mm (Fig. 6(b)). The figures clearly indicate that the plastic deformation in all the specimens shows rotational behavior with a central focus on a point, as this contour is constructed in logarithmic classification so there will be scarcely any plastic strain in the blue or black colored areas. Also under the condition of a0/W 5 0.15, it can be seen that the highly plastically strained area is growing to the crack side edge of the specimen. This means that, under the shallow condition of a0/W 5 0.15, the plastic deformation behavior is essentially similar to bending of a simple beam without a crack. For accurate determination of the coordination of the rotational center position, the strain distribution in the ligament region was investigated. Fig. 7 is a schematic explanation of the plastic strain distribution in the ligament region in a SEB specimen and the definition of the rotational center position by the strain distribution in the crack propagation plane. In order to establish a CTOD formula with the plastic hinge model, a quantitative knowledge of the rotational center
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 5. Mesh designs around crack tip.
coordination and its change under increasing loading is important. In the context of Section 1, this determination procedure can be categorized as method (b) in Fig. 1. In Fig. 8, the open circle marks show the distribution of the plastic strain increment in the opening direction distribution of YR = 0.9, a0/W = 0.5 on the mid-thickness plane at the CTODFEM of 0.05 mm, 0.1 mm, 0.15 mm and 0.2 mm. Considering that the formula of the CTOD calculation equation is divided into two terms, i.e., an elastic (small scale yielding) and a plastic term, and the plastic hinge model is related only to the plastic term, the rotational deformation should be specified only in plastic deformation. Therefore, the plastic strain distribution is important for establishing the formula that is the objective of this study. An incremental value is used for more accurate evaluation of the coordination of the neutral point by bending. Fig. 8 also indicates there is a more or less zero-plastic-strain region around the center of the ligament. Here, the horizontal axis shows the non-dimensional distance of the ligament, which is the value obtained by dividing the distance from the crack tip by the initial ligament length. By this arrangement, it can be shown that the rotational center point is located inside the zero-plastic-strain region; however, the exact position remains unclear. Even if a large amount of deformation is applied, the elastic region that exists around the center of the ligament still remains. This means the rotational center cannot be computed only by the plastic strain distribution as explained above. Therefore, the incremental value of the whole strain component, including the elastic term, is introduced instead of the increment of the plastic strain, and the strain distribution is also summarized as shown by the solid triangle marks in Fig. 8. By using the whole strain increment, it is possible to determine a unique value of the coordination of the rotational center, as the whole strain distribution diminishes monotonically from the tension side to the compression side. In contrast, in the case of CTODFEM = 0.05 mm, there is a large difference between the distribution of the plastic strain increment and that of the whole strain increment, as almost all of the specimen remains elastic, and in case of CTODFEM > 0.05 mm, the two strain increments are almost the same and a pure elastic region is limited. This indicates that there is no critical problem in definition of the rotational center by the whole strain distribution in the opening direction. Fig. 9 shows the comparison of whole strain distribution for several a0/W conditions from 0.1 to 0.7. This Fig. simply shows that higher a0/W conditions indicate nearer non-dimensional positions of the bending neutral point from the crack tip, xneutral. Thus, xneutral is calculated from the whole strain distribution in the opening direction under all conditions, that is, all the combinations of 14 a0/W conditions and 2 kinds of materials shown in Table 1. Fig. 10 shows
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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(a) CTODFEM 0.1mm
(b) CTODFEM 0.2mm Fig. 6. Distribution of equivalent plastic strain in mid-thickness plane of SEB specimens with various a0/W.
the change of the non-dimensional coordination of the bending neutral position, xneutral/(W a0) for all the conditions until the deformation level of CTODFEM = 0.20 mm. There seems to be a relatively larger amount of change of xneutral/(W a0) in the extreme cases of a0/W = 0.05 or 0.7, but even in those cases, the amount of the change is limited to less than 5%. This mild fluctuation of the coordination of the rotational center during loading is more pronounced in comparison with that of method (a) described in Fig. 1. Here, in order to make a quantitative comparison with the fluctuation of method (a), the index of fluctuation, |Fluct.| was introduced, as shown in Eq. (4). Fig. 11 shows that this method (method (b)) is far superior to method (a) from the viewpoint of fluctuation during loading, in spite of the fact that the coordination of the rotational center defined in the BS5762 specification was experimentally investigated by method (a). With the BS equation, |Fluct.| changes with various a0/W.
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Fig. 7. Schematic illustration showing definition of rotational center in this study.
Fig. 8. Comparison of plastic strain distribution and total strain distribution along centerline of ligament (YR = 0.9, a0/W = 0.50).
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Fig. 9. Comparison of non-dimensional total strain distribution along centerline of ligament (YR = 0.6, CTODFEM = 0.2 mm).
Fig. 10. Change of neutral point position determined by total strain increment under various a0/W for increasing deformation.
Fig. 11. Comparison of fluctuation in range of CTOD from 0.05 to 0.2 with methods (a) and (b).
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jFluct:j ¼
Xd¼0:050:20mm ½r p i ½rp av : ½dFEM i ½dFEM i1 i ð0:20 0:05Þ
100 ½%
ð4Þ
Fig. 12 shows the summarized relationship between a0/W and the representative value of each condition, which is computed by averaging the xneutral/(W a0) histories over the range of CTODFEM = 0.05–0.2 mm. This figure also shows the results of the slip line field analysis of the surface of the three point bending specimen conducted by Wu et al. [15] and the strain distribution measurement by Theiss [18]. This figure clearly indicates that decreasing a0/W (in other words, a shallower specimen) increases xneutral/(W a0) and its dependency is almost the same between two materials. This tendency is similar to the tendency reported by Wu et al. [15]. As a summary of this method, the numerical function of xneutral/(W a0) can be drawn by Eq. (5), which is a regression curve by least square fitting without the effect of YR.
a 6 a 5 a 4 a 3 a 2 a xneutral 0 0 0 0 0 0 þ 113 97:5 þ 38:5 6:45 þ 0:13 ¼ 49:6 þ 0:50 ðW a0 Þ W W W W W W
ð5Þ
3.2. Plastic strain distribution in ligament and coordination of rotational center of SEB specimens in whole thickness For the establishment of a calculation formula covering various thickness conditions, it is also important to investigate the plastic deformation behavior in the whole thickness. Fig. 13 shows contour figures of plastic equivalent strain at the deformation level of CTODFEM = 0.1 mm. This figure is illustrated from the crack face side toward half of the SEB model (whole figure in thickness direction). These contour figures are constructed in logarithmic classification, as in Fig. 6, so
Fig. 12. Summary of change of neutral point position calculated from strain increment distribution along ligament by 3D-FEM.
Fig. 13. Comparison of plastic equivalent strain distribution in whole model (YR = 0.6, CTOD = 0.1 mm).
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the black and blue regions show a substantially zero-plastic-strain region. The zero-plastic-strain region is located near the midpoint of the ligament and is almost uniform in the thickness direction. Fig. 14 summarizes the change of xneutral/(W a0) under increasing deformation for several a0/W conditions, and shows that the coordination is almost constant regardless of the thickness positions and the loading level. Fig. 15 shows the average value of xneutral/(W a0), [xneutral/(W a0)]av. at loading levels from CTOD = 0.05 mm to 0.2 mm. In case of a0/W = 0.3, 0.5 and 0.7, [xneutral/(W a0)]av. is slightly smaller at the surface than that at deeper positions in the thickness direction, whereas at a0/W = 0.1, [xneutral/(W a0)]av is almost constant in the thickness direction. This characteristic behavior is based on the fact that a specimen with a shallower condition than a0/W = 0.15 is plastically deformed as a simple bending beam without a crack, as described in Section 3.1. The reason for the slight drop of [xneutral/(W a0)]av. at the surface position under the deep crack conditions should be examined in the future. However, the authors feel that this tendency is related to some extent to plastic dent deformation at the surface. 3.3. CTOD distribution in thickness direction The authors are taking the stand that the CTOD formula should calculate the actual CTOD at mid-thickness, as critical events have mostly occurred from the vicinity of the mid-thickness position. In a previous report [15], various coefficients were set so as to accurately predict a large number of CTODFEM at the mid-thickness position. However, in order to obtain a thorough grasp of the deformation behavior of SEB specimens, it is important to investigate the CTOD distribution in the thickness direction. Fig. 16 shows the CTOD distribution in the thickness direction in case of YR = 0.6 material. The four kinds
Fig. 14. Change of neutral point position determined by total strain of surface and mid-thickness of SEB specimens for various a0/W conditions.
Fig. 15. Distribution of neutral point position determined by total strain in thickness direction for various a0/W conditions.
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Fig. 16. CTOD distribution in thickness direction under several deformation steps for various a0/W conditions (YR = 0.6).
of plots show the loading levels where CTOD at mid-thickness corresponds to 0.05, 0.1, 0.15 and 0.2 mm. Under all the conditions, as the loading level increases, the difference of CTOD between the mid-thickness and surface also increases; in other words, the configuration of the distribution changes from a flat shape to a round shape. The difference depending on a0/W conditions was examined while keeping this general tendency. In case of a0/W = 0.1, the change of CTOD in the thickness direction is milder than that under the other conditions with deeper a0/W. In order to clarify the reason for this difference, the following discussion will focus on the ratio of CTOD at the surface position to CTOD at the mid-thickness position. Fig. 17 summarize this ratio under increasing loading conditions. Interestingly, the conditions of a0/W = 0.3, 0.5 and 0.7 show almost the same constant value of 0.35–0.4, and only the condition of a0/W = 0.1 shows a much higher value. This is also thought to be strongly related to the fact that plastic deformation as a simple bending beam without a crack occurs under shallower conditions than a0/W = 0.15, as described in Section 3.1. 4. Investigation of plastic deformation behavior in backward area from crack tip in SEB specimens 4.1. Importance of crack face displacement in SEB specimen for CTOD evaluation It seems rational that the conventional plastic hinge type CTOD calculation formula should be extended to various a0/W conditions from the formula for the standard a0/W range of 0.45–0.55 proposed in the authors’ previous report [5]. This philosophy is based on the observation in Section 3 that there is a fixed position which is the center of the rotational deformation determined only by a0/W, regardless of the loading level and strain hardening exponent. Here, formulation of the plastic hinge relationship is important from the viewpoint of establishing the calculation formula. That is, the crack edge point should be located on an extrapolated line passing through the rotational center position and the 45° line crossing point for determination, because mouth opening displacement, Vg, which is the only parameter for deformation that an experimenter can measure, may be used for the determination of the plastic term of the CTOD calculation formula. Referring to Fig. 18(a), the following will explain the CTOD calculation procedure in which the plastic component of opening displacement at the edge of the specimen is applied as an experimental variation. It is noted that only the plastic component is expressed in Fig. 18. The plastic component dpl (called dp here) is obtained by the relationship of similar triangles, where the rotational center is allocated as the vertex of the isosceles triangle, as shown in Eq. (6): Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029
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Fig. 17. Change of ratio of CTODsurface to CTODcenter with increasing deformation for various a0/W conditions.
Fig. 18. Schematic diagram of determination of plastic component of CTOD by plastic hinge model.
dp
BS
¼
r p ðW a0 Þ Vp a0 þ rp ðW a0 Þ
ð6Þ
This relationship was first proposed by Ingham et al. [11] and is based on the assumption that the corner points of two similar triangles are linearly connected. It is important to consider the crack tip blunting shape due to the strain hardening exponent when these similar triangles are clarified. The authors have pointed out that the simple plastic hinge assumption leads to overestimation of dp, and the amount of overestimation depends on the strain hardening exponent [22]. Therefore, the multiplier exponent f (basically less than 1) is introduced in the plastic term in the calculation formula (Eq. (2)). This is to say, true dp is smaller than the value calculated by the conventional calculation formula (Eq. (1)). Fig. 18(b) is the corrected illustration considering this situation. Based on this background, it is quite important to determine if dp_BS can be estimated by the measured Vp and the ratio of a0 and rp(W a0). If this is in fact possible, then the calculation formula for dp will be established by introducing the exponent f, as shown in Eq. (7).
dp ¼ f dp
BS
ð7Þ
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4.2. Effect of a0/W on opening displacement at edge of specimen and formation of plastic hinge relationship with rotational center In order to investigate this issue, the FEM results are analyzed in detail. The problem is that the deformation given by FEM contains both elastic and plastic components, and these two components cannot be intrinsically distinguished. The candidate alternative simple methods for extraction of the plastic component are listed below. (a) Unloading on FEM analysis (b-1) Subtract elastic deformation component by elastic FEM analysis from whole deformation profiles [23] (b-2) Subtract elastic deformation component by solution of SEB displacement from the Handbook of Stress Intensity Factors from whole deformation profiles [24] However, these methods are not appropriate or do not have sufficient accuracy for the following reasons: (a) The unloading process of a cracked specimen leads to reversal yielding around the crack tip, and this is different from the true plastic component in the monotonically loading aspect. (b) Elastic analysis never contains yielding behavior and maintains a profile which is linear even at the vicinity of the crack tip. These points are contrary to the fact that small scale yielding must be assumed in the elastic term. In this study, considering the match for the constitution of the calculation formula, the plastic component is computed by the following procedure, which is shown schematically in Fig. 19.
Fig. 19. Method for derivation of plastic component of nodal deformation on crack face by assuming elastic component line.
Fig. 20. Crack opening profiles in specimens with various a0/W at plastic term of CTOD = 0.1 mm (YR60).
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(x) Letting the opening displacements at each of the nodes in the backward position from the FEM analysis results and the opening displacement at the crossing point by the 45° line to the deformation profile be [Vg]i and [Vg]0, respectively. (y) Defining the elastic component at the edge of the opening mouth, [Ve]edge as the construction method on the P-Vg curve, which is obtained by subtracting the plastic component given by the initial elastic gradient line from the whole displacement, and letting the elastic term in Eq. (2), de be [Ve]0 . The plastic component at an arbitrary position from the tip to the edge, [Ve]i is defined as a point on the line which is drawn by connecting two points, [Ve]0 and [Ve]edge. (z) The plastic component of the arbitrary node position is defined as the result of subtraction of [Ve]i from the opening displacement, [Vg]i at each node. The plastic components of the opening displacement at each node for various a0/W, as computed by the method described above, are summarized in Figs. 20–22. The horizontal axis of these figures is the non-dimensional distance from the intercept point by the 45° line and is designed so that the point of the rotational center clarified in Section 3 corresponds to (1, 0). Also, the deformation status of these figures is at the timing of dp/f = 0.1 mm, where f is given as the same value as in Eq. (2). In sum, if point R, the intercept point to the vertical axis of line OQ, where Q is the plastic component of the edge displace-
Fig. 21. Expansion of Fig. 20 (a0/W = 0.1).
Fig. 22. Crack opening profiles in specimens with various a0/W at plastic term of CTOD = 0.1 mm (YR90).
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(a)
p/f=0.1mm
(b)
p/f=0.2mm
Fig. 23. Deviation from geometric similarity of complete formation of plastic hinge relationship.
ment, is located exactly on point P, (0, 0.1), a perfect plastic hinge relationship would be established. However, the actual intercept point, R is different from point P. Fig. 21 shows an enlargement of the area around dp. It is clearly shown that this graph is drawn at the loading step at the crossing point of line OP, which corresponds to a side of the similar triangle in the plastic hinge model shown in Fig. 18(b), and the crack profile. In this case, due to the low YR condition, the actual dp is much lower than the crossing point. This is expressed by coefficient f in Eq. (2). Viewing the results of the four kinds of a0/W in both case of (a)YR60 and (b)YR90, a certain amount of gap, shown as distance RP, is observed under a wide range of a0/W and YR conditions, except under the condition around a0/W = 0.5. This is also evidence that the plastic hinge relationship is completely satisfied in case of a0/W = 0.5, and Eq. (2) maintains good accuracy under the standard a0/W conditions. These gaps are summarized in Fig. 23, which is shown as the correlation with a0/W. These figures mean that a smaller a0/W results in a larger gap. Fortunately, no effect of YR is observed in the relationship between the gap and a0/W. Also, under the condition of a0/W = 0.1, a larger gap is observed than the extended line from the other a0/W conditions. This may be because the deformation behavior is different from that in the other cases, where plastic deformation is expanded to the edge of the specimen like simple beam bending. Although these gaps do not seem particularly large, these quantitative features of the gaps from the perfect plastic hinge solution will be very helpful for establishing a calculation formula which covers various a0/W conditions.
5. Concluding remarks In this study, plastic bending behavior not only in the ligament region but also in crack opening profiles behind the crack tip, which is an important fundamental part of the CTOD calculation formula based on the plastic hinge assumption, is investigated with the final goal of establishing a unified formula which will be applicable to a wide range of a0/W. A 3D dynamic elasto-plastic FEM analysis for various a0/W conditions from 0.05 to 0.7, which was already validated by actual CTOD measurements by the silicone casting method, is used in order to gain a thorough grasp of the deformation behavior of SEB specimens. The conclusions obtained are listed below. (1) Under every a0/W condition, the fact that the ligament area is rotationally deformed at a certain point is clearly observed. This can be a strong motivation for continuing to use the plastic hinge type calculation formula in determinations of CTOD. (2) It is confirmed that for a/W < 0.15, the deformation mode transitions from rotation only in the ligament to whole specimen rotation as a simple bending beam without a crack. (3) Coordination of the bending neutral position can be described as a simple regression polynomial equation regardless of YR and loading level as expressed in Eq. (5). This coordination function can be applied as rp in the plastic hinge type CTOD calculation formula, however, in our final recommended CTOD calculation formula, rp is set to be constant value in order to avoid complication [25].
a 6 a 5 a 4 a 3 a 2 a xneutral 0 0 0 0 0 0 þ 113 97:5 þ 38:5 6:45 þ 0:13 ¼ 49:6 þ 0:50 ðW a0 Þ W W W W W W
ð5Þ
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(4) Deformation in the thickness direction can be classified into two groups, i.e., shallower than a0/W of 0.15 and others. The bending neutral position and CTOD at the surface are quite constant under shallower conditions than a0/W = 0.15. This is also due to the transition of the deformation mode to whole bending behavior under shallow conditions. (5) The plastic component of opening displacement is clarified in detail from the viewpoint of satisfaction of the plastic hinge relationship by using the coordination of the actual rotational center. As a result, the perfect plastic hinge relationship is satisfied only in case of the standard a0/W condition. However, it is suggested that the plastic hinge calculation method can be applied after a small correction of dp. The correction coefficient may be a function of a0/W and independent of the strain hardening exponent. In our final recommended CTOD calculation formula this correction factor is employed [25].
Acknowledgement The authors wish to acknowledge CTE committee of the Japanese Welding Engineering Society for their dedicated discussions of this study. References [1] Wells Alan A. Unstable crack propagation in metals: cleavage and fast fracture. In: Proceedings of the crack propagation symposium, Cranfield, UK, vol. 2; 1961. p. 2010. [2] British Standards Institution. Methods for crack opening displacement testing – COD-DD19; 1972. [3] British Standards Institution. Methods for crack opening displacement (COD) testing. BS5762; 1979. [4] British Standard BS7448 – Part 1. Fracture mechanics toughness tests – Part 1. Method for determination of KIC, critical CTOD and critical J values of metallic materials; 1991. [5] Tomoya Kawabata, Tetsuya Tagawa, Takahiro Sakimoto, Yoichi Kayamori, Mitsuru Ohata, Yoichi Yamashita, et al. Proposal of a new CTOD calculation formula. Eng Fract Mech 2016;159:16–34. [6] International Organization for Standardization. Method of test for the determination of quasistatic fracture toughness of welds. ISO15653; 2010. [7] American Society for Testing and Materials. Standard test method for crack-tip opening displacement CTOD fracture toughness measurement. ASTM E1290-1989; 1989. [8] Kayamori Yoichi, Inoue Takehiro, Tagawa Tetsuya. Changes in ISO 15653-based CTOD for specimens of a0/W = 0.45. J Solid Mech Mater Eng 2012;vol. 6 (6):645–53 [special issue on Advanced Technology in Experimental Mechanics]. [9] Tagawa Tetsuya, Kawabata Tomoya, Sakimoto Takahiro, Kayamori Yoichi, Ohata Mitsuru, Yamashita Yoichi, et al. Experimental measurements of deformed crack tips in different yield-to-tensile ratio steels. Eng Fract Mech 2014;128:157–70. [10] American Society for Testing and Materials. Standard test method for measurement of fracture toughness. ASTM E1820-2013; 2013. [11] Ingham T, Egan GR, Elliott D, Harrison TC. The effect of geometry on the interpretation of COD test data. Inst Mech Eng 1971;C54/71:200. [12] Lin IH, Anderson TL, De Witt R, Dawes MG. Displacements and rotational factors in single edge notched bend specimens. Int J Fract 1982;20:R3–7. [13] Tanaka Hidehiko, Ando Kotoji, Sakai Yuzuru, Ogura Nobukazu. Brittle crack initiation from shallow notch and hot strain embrittlement. In: Proceedings of Fracture Mechanics Symposium. The Society of Materials Science, Japan, Committee on Fracture Mechanics; 1981. p. 147–51. [14] Tsukamoto Mitsuo. Effect of initial crack length on critical COD of unstable fracture with stable fibrous crack growth, memoirs of the faculty of education, Kumamoto University. Science 1994;43:153–62. [15] Wu Shang-Xian, Brian Cotterell, Mai Yiu-Wing. Slip line solutions for three-point notch-bend specimen. Int J Fract 1988;37:13–29. [16] Wu Shang-Xian. Evaluation of CTOD and J-integral for three point bend specimen with shallow cracks. In: Proceedings of the 7th international conference on fracture, ICF 7, Houston, Texas, vol. 1; March 1989. p. 517–24. [17] Matsoukas et al. On the plastic rotation constant used in standard COD tests. Int J Fract 1984;26(2):R49–53. [18] Theiss TJ, Rolfe SJ, Shum DKM. Shallow-crack toughness results for reactor pressure vessel steel. Shallow Crack Fracture Mechanics, Cambridge, UK; 23–24 September 1992 [paper 37]. [19] Kirk Mark T, Dodds Jr Robert H. J and CTOD estimation equations for shallow cracks in single edge notch bend specimen. J Test Eval 1993;21:223–38. [20] Donato Gustavo HB, Ruggieri Claudio. Estimation procedures for J and CTOD fracture parameters using three-point bend specimens. In: 2006 International pipeline conference. Volume 3: materials and joining; pipeline automation and measurement; risk and reliability, Parts A and B, Calgary, Alberta, Canada; September 25–29, 2006. [21] Kayamori Yoichi, Inoue Takehiro, Hagihara Yukito. Plastic rotational factor calculation for shallow notched SEB specimen. Mater Sci Forum 2014;783– 786:2322–6. [22] Tagawa Tetsuya, Kayamori Yoichi, Ohata Mitsuru, Handa Tsunehisa, Kawabata Tomoya, Yamashita Yoichi, et al. Comparison of CTOD standards: BS 7448-Part 1 and revised ASTM E1290. Eng Fract Mech 2010;77(2):327–36. [23] Tagawa Tetsuya, Takashima Yasuhito, Ohata Mitsuru, Minami Fumiyoshi. Numerical investigation on BS-CTOD and ASTM-CTOD. IIW doc, X-1676-10. [24] Nishitani Hironobu, Noguchi Hiroshi, Mori Kazuya, Handa Tsunehisa, Kawabata Tomoya, Yamashita Yoichi, et al. Analysis of single-edge-cracked specimen under three- or four-point bending by body force doublet method. Trans Jpn Soc Mech Eng 1986;52(474):539–43. [25] Tomoya Kawabata, Tetsuya Tagawa, Yoichi Kayamori, Mitsuru Ohata, Yoichi Yamashita, Masao Kinefuchi, Hitoshi Yoshinari, Shuji Aihara, Fumiyoshi Minami, Hiroshi Mimura, and Yukito Hagihara, Applicability of new CTOD calculation formula to various a0/W conditions and B x B configuration, Eng Fract Mech, accepted for publication.
Please cite this article in press as: Kawabata T et al. Plastic deformation behavior in SEB specimens with various crack length to width ratios. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.03.029