Received: 11 December 2018 Revised: 13 February 2019 Accepted: 19 February 2019

OR I G I NAL CONTR I BUT I ON

DOI: 10.1111/ffe.13005

A CTOD equation based on the rigid rotational factor withthe consideration of crack tip blunting due to strainhardening for SEN(B)

WeeLiam Khor

Faculty of Engineering and PhysicalSciences, University of Surrey, Guildford,Surrey, GU2 7XH, UK

CorrespondenceWeeLiam Khor, Faculty of Engineeringand Physical Sciences, University ofSurrey, Guildford, Surrey GU2 7XH, UK.Email: w.khor@surrey.ac.uk

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This is an open access article under the terms of the C

original work is properly cited.

© 2019 The Authors Fatigue & Fracture of Enginee

Nomenclature: a0, original crack length, mmthe crack tip, W − a0, mm; CMOD, crack mostrain energy around the crack, Nmm−1; K, strfactor); mJWES, m factor corrected for the effecuncracked ligament ahead of the crack tip; rpof the clip gauge opening displacement, mm; Vthe crack mouth where displacement is medisplacement, CTOD, mm; δel, elastic compofrom the middle thickness of the FE model bcalculated based on the strain hardening corσuts, tensile ratio, describing material strain h

Fatigue Fract Eng Mater Struct. 2019;1–9.

Abstract

Crack tip opening displacement (CTOD) from national and international

standards was shown to give different values. This paper investigates the feasibil-

ity of CTOD determined based on the concept of rigid rotational factor in single‐

edge notched bend (SENB) specimens. Based on validated modelling methods,

finite element (FE) models were simulated for crack ratios 0.3 ≤ a0/W ≤ 0.7and yield‐to‐tensile ratio 0.44 ≤ σys/σuts ≤ 0.98. This covers cases of shallow todeeply cracked specimens and a wide range of strain hardening properties.

CTOD obtained from the FE models was used as the basis of a newly imple-

mented strain hardening corrected rotational factor, which considers the effects

of crack tip blunting due to strain hardening, rp sh. An improved equation consid-

ering strain hardening was implemented based on the rp sh. The equation gives

accurate estimation of CTOD from the FE models compared with the equation

from BS 7448‐1, ASTM E1820, and WES 1180.

KEYWORDS

CTOD, rotational factor, strain hardening, varied crack length

1 | INTRODUCTION

The crack tip opening displacement (CTOD) is one of thebest known strain‐based elastic‐plastic fracture toughness

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reative Commons Attribution Lice

ring Materials & Structures Publish

; a0/W, crack length–specimenuth opening displacement, mmess intensity factor, Nmm−1.5; mts of strain hardening, used in

sh, semiempirical strain harden

g1, Vg2, lower and upper clip gasured, mm; z1, z2, z corresponent of CTOD, mm; δ45, CTOased on the opening of the orrected rotational factor, rp sh;ardening properties; ν, Poisson

parameter in fracture mechanics. The concept was origi-nally developed in TWI1 in the 1960s, commonly usedin the oil and gas industry. CTOD is advantageous incases where the concept of linear elastic fracture

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nse, which permits use, distribution and reproduction in any medium, provided the

ed by John Wiley & Sons Ltd

width ratio; B, specimen thickness, mm; B0, remaining ligament ahead of; E, modulus of elasticity, MPa; FE, finite element; J, path independent, factor relating CTOD to J or K (sometimes referred to as a “constraint”WES 1108; rp, rotational factor, describing the “hinge” point within theing corrected rp; SENB, single‐edge notched bend; Vp, plastic componentauge respectively, mm; W, specimen width, mm; z, vertical height abovending to the lower and upper clip gauge, mm; δ, crack tip openingD based on the 45° intercept at the crack face; δFE, CTOD measurediginal crack tip, mm; δpl, plastic component of CTOD, mm; δsh, CTODσuts, ultimate tensile stress, MPa; σys, 0.2% yield/proof stress, MPa; σys/ratio

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https://orcid.org/0000-0001-7699-8194mailto:w.khor@surrey.ac.ukhttp://creativecommons.org/licenses/by/4.0/https://doi.org/10.1111/ffe.13005http://wileyonlinelibrary.com/journal/ffe

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mechanics is insufficient to account for ductile deforma-tion, such as in pressure vessels, offshore platforms, andpipelines. Other well‐known fracture toughness parame-ters includes (but not limited to) the J‐integral, stressintensity factor, and crack tip opening angle. CTOD isoften preferred in cases where the component is loadedprimarily in strain, ie, offshore pipe reeling.

There are several different definition for CTOD, but alldescribes CTOD as a material parameter relative to thephysical opening of the crack at the crack tip region. Ingeneral, CTOD is described as the displacement‐basedfracture toughness of a material at the point of maximumload, initiation of stable crack extension, and unstablecrack extension. In lab conditions, specimens withnotches representing cracks are loaded, in which theload‐displacement data and specimen set‐ups details arerecorded for the calculation of CTOD. In addition to thetypical specimen set‐up, ie, C(T) and SEN(B), practicalfull‐sized specimen set‐up, ie, PRN(B), had been devel-oped in the recent years, which reduces machining andthe effects because of subsized specimens.2

Fracture toughness testing (often described simply as“CTOD testing”) became standardised in the 1970s andis currently represented in a number of standards includ-ing BS 7448, ISO 12135, ASTM E1820, and WES1108.3-6

However, different assumptions about the determinationof CTOD are used in each, which can give differentvalues of CTOD.

It is important to define the value of CTOD withaccuracy, particularly when CTOD is being used todetermine tolerable flaw sizes for structural integrityassessments. Underestimates of CTOD would lead toeither rejection of the material or unnecessary repairs,increasing the cost of operation and/or fabrication. Onthe other hand, overestimates of CTOD might lead topotentially unsafe structures being assessed as fit forservice.7

2 | THE CONCEPT OF RIGIDROTATIONAL FACTOR FOR THEDETERMINATION OF CTOD

When a SEN(B) specimen is loaded under 3‐point bend-ing, the crack tip would experience tensile stress,whereas there would be a region in the uncracked liga-ment that experiences compression. In the calculation ofthe plastic component of CTOD, BS 7448‐1, ISO 12135,WES1108, early versions of ISO 12135, ASTM E1290‐93,8 and E1820‐019 assumed that the specimen flangesrotate about a stationary point within the uncracked lig-ament ahead of the crack tip, a distance equal to rp × B0,where rp is the rotational factor and B0 is the remainingligament ahead of the crack tip. This concept was intro-duced by Dawes10 in 1979, using a 2‐D plastic hingemodel with assumptions of plane strain condition atthe crack tip.11 Based on the argument that there is atensile and compressive region ahead of the crack tip,the idea of a rotational hinge point had also beenapplied in the calculation of the crack tip opening angle(CTOA).12

According to the assumption of slip line theory, theflanges of the specimen would deform as rigid arms rotat-ing about a rotational point.13 Green's observation onphoto‐elastic images of notched bend specimens showedthat the yielding pattern at the crack tip region is similarto that predicted in a slip‐line field.14,15 Digital image cor-relation observation on a SEN(B) specimen showed thatthe strain distribution near the crack tip is similar to thatdescribed in the slip line field theory by Green.16

The determination of the rotational factor is based onthe geometrical analysis of a specimen under 3‐pointbend loading. Consider a deformed SEN(B) specimen(Figure 1), the distance of the rotational point from thecrack mouth opening displacement (CMOD), H is definedin terms of rotational11 factor, rp(W − a) + a.

FIGURE 1 Diagram for the evaluationof the geometrical based crack tip opening

displacement

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From Figure 1, it is shown that

rp ¼ H − aW − aVp ¼ 2 rp W − að Þ þ aþ z

� �sin θp

δp ¼ 2rp W − að Þ sin θp:

Relating Vp into δp gives11

δpl ¼ rp W − að ÞVprp W − að Þ þ aþ z� �;

where δpl is the plastic component of CTOD, whereδ = δel + δpl. Anderson et al found that rp is insensitiveto the specimen geometry for the same material.17 Wellsshown that in the case of the SEN(B) specimen geometry,initially, the rotational point would be close to the cracktip (rp < 0.1), which extends and converge to a pointwithin the unbroken ligament ahead of the crack tip(rp ≈ 0.45) with increasing load and after general yield-ing.18 BS 7448 assumed a constant value of rp = 0.4 inthe calculation of CTOD. The basis for the constant rpfor SEN(B) is not in the public domain, but it wasunderstood that this value is determined throughextensive experiments and should underestimate theactual CTOD.19

3 | METHODOLOGY

3.1 | Finite element modelling techniques

A total of 50 quarter Bx2B single‐edge notched bend(SENB) specimen models were generated to investigateCTOD for a range of a0/W and σys/σuts using ABAQUSv6.14. A blunted crack tip of 0.03‐mm radius was used inthe models, which allows better plastic deformation at

FIGURE 2 Idealised true stress‐straincurve based on the modified Ramberg‐

Osgood power law [Colour figure can be

viewed at wileyonlinelibrary.com]

the crack tip region.7,16,20,21 The modelling technique usedin this paper was shown to give good prediction of CTOD,further described in previous published works.7,16,20 Thecrack tip was designed to blunt continuously as the crackopen without stable ductile tearing and thus leads to lowervalues of CTOD.7,20

Ten different strain hardening properties were gener-ated using the modified Ramberg‐Osgood power law for0.44 ≤ σys/σuts ≤ 0.98, given below22,23:

ε ¼ σEþ α σ

σys

� �n−1; (1)

where α = 0.002, E = 207GPa, and σys = 400 MPa. Thedecrease of n in Equation 1 gives increasing strain hard-ening properties (decreasing σys/σuts). The true stressstrain curves obtained using Equation 1 were used todescribe the material tensile properties of the models(Figure 2).

In addition to the different strain hardening proper-ties, five different crack length models were generatedto cover 0.3 ≤ a0/W ≤ 0.7. These cover the cases of shal-low crack up to deeply cracked specimen set‐up.Figure 3 shows the symmetry boundary conditions speci-fied on a typical model and technique used to extract theCTOD values.

3.2 | The determination of rp based on thesimilar triangles method

The concept of rigid rotational point has been success-fully implemented for the determination of CTOD withvarious degree of accuracy.7,16,20,21 To investigate therotational factor, rp for SEN(B) specimens, an openedcrack was investigated geometrically based on the similar

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FIGURE 3 Quarter single edge notched bend model for a0/W = 0.5, showing boundary conditions and measurement technique for cracktip opening displacement, adjusted for the 0.03‐mm crack radius used in the models [Colour figure can be viewed at wileyonlinelibrary.com]

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triangles concept (Figure 4). This method extrapolates thecrack face angles into the unbroken ligament ahead ofthe crack tip, where the intersection of the angles isdescribed as the rotational point. Similar concept withminor modifications had been employed by BS 8571,ExxonMobil, and DNV in their calculation of CTOD.24-26

Figure 4 shows the analytical diagram for the determi-nation of the rp. A similar method had been used

FIGURE 4 The determination of rp based on double clip gaugeset‐up

successfully by Robinson and Tetelman to determine therp.

27 The following terms were used to simplify the deri-vation, rpB0 = Y, z1 + a0 = C, and a0 + z2 = D. To relatethe lower and upper clip gauge opening, Vg1 and Vg2respectively to the point of rotation,

sinθ ¼ Vg1C þ Y ¼

Vg2Dþ Y

leading to

Vg2Vg1

¼ Dþ YC þ Y :

Expanding D and factoring C + Y gives

Vg2Vg1

¼ C þ Y þ z2 − z1ð ÞC þ Y ¼ 1þ

z2 − z1ð ÞC þ Y :

Rearranging the equation leads to

Y ¼ rpB0 ¼ z2 − z1Vg2

�Vg1

� �− 1

0@

1A − z1 þ a0ð Þ;

where the rotation factor, rp, based on Vg1 and Vg2 isgiven as

rp ¼ z2 − z1Vg2

�Vg1

� �− 1

0@

1A − z1 þ a0ð Þ

24

35 × 1

B0: (2)

Equation 2 allows the rotational factor to be calculatedbased on two clip gauges positioned at different heightsabove the crack mouth in standard laboratory tests.Similarly, the rotational factor based on the plastic

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displacement can be obtained by simply replacing thelower and upper clip gauge displacement, Vg1 and Vg2with the plastic lower and upper clip gauge displacement,Vp1 and Vp2.

3.3 | Strain hardening corrected rp basedon finite element modelling, rp sh

The SEN(B) models were modelled under 3‐point bendloading and the apex of the crack face opening was iden-tified as the hinge location. Findings from experimentalwork and finite element (FE) modelling shows that asthe crack open due to loading, the effects of strain hard-ening due to plasticity cause the crack tip to deform awayfrom the crack face plane.7,28-30 To account for the effectsof crack tip blunting due to strain hardening, rp wasextracted based on the intersection line extrapolated fromthe CMOD and CTOD to the symmetry line (Figure 5).This rp accounts for crack tip blunting and should givelower rp than that obtained from crack face angles, there-after described as rp sh.

7,21

To extract rp sh from the finite element (FE) models,Equation 2 was modified based on the CTOD and CMOD,described as

FIGURE 5 The effect of crack tip blunting due to strain hardening

rp sh ¼ a0CMODδ

� �− 1

0BB@

1CCA

2664

3775 × 1B0: (3)

4 | RESULTS

4.1 | Strain hardening rp, rp sh extractedfrom the models

To obtain a better resolution of the rp sh distribution dueto the effects of strain hardening, data were extractedfrom the FE models and processed using Equation 3.CTOD was extracted for 0.2 mm < δFE < 1.0 mm to min-imise the influence of the elastic CTOD and large defor-mation.7,21 Figure 6 shows rp sh for the increasing FECTOD, δFE.

Figure 6 shows the dependency of rp sh on tensile ratio,where in general, rp sh decreases with the increase of strainhardening. With the increase of CTOD, rp sh wouldincrease gradually with decreasing gradient. To exhibitthe distribution of rp sh due to strain hardening and cracklength‐specimen width ratio (a0/W), rp sh was plotted toσys/σuts (Figure 7). In general, the decrease of a0/W from0.7 to 0.3 shows an overall decrease of rp sh. As σys/σutsmovetowards 1, rp sh for all a0/W tends to converge and give asimilar value. Models from a0/W = 0.5 were validatedexperimentally7,21 and thus were used as the baseline forthe fitting. A linear line was fitted to the rp sh data fromtheir respective a0/W, while passing through point σys/σuts = 0.98, rp sh = 0.51. This is based in the assumptionwhere rp sh converges at a fixed constant value formaterialsbehaving in an elastic–perfect plastic behaviour. Data fromσys/σuts = 0.98 would be exhibiting elastic dominant prop-erties where a blunted crack tip could give a significantlarger CTOD and therefore was not used in the fitting.31,32

Based on the fitting of the data, the rp sh equation canbe described using the following equation:

rp sh ¼ R1 σys=σuts� �

þ R2 (4)

R1 ¼ −0:62 a0=Wð Þ þ 0:76R2 ¼ 0:61 a0=Wð Þ − 0:22:

5 | DISCUSSION

Equation 2 shown that based on the principles of similartriangles and assuming that the crack tip location would

FIGURE 6 Strain hardening rotational factor vs finite element (FE) crack tip opening displacement (CTOD) for a0/W = 0.3, 0.4, 0.5, 0.6,and 0.7 (A, B, C, D, and E, respectively) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 7 Relationship between thestrain hardening rotational factor, rp sh,

and tensile ratio based on 0.2 mm < FE

CTOD < 1.0 mm [Colour figure can be

viewed at wileyonlinelibrary.com]

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be in line with the crack face plane, the rotational factorcan be determined directly from the test specimen.Conceptually, this allows the rp to be extracted at anyinstance of the test for the calculation of CTOD.

This method requires two clip gauge measurements,positioned at different heights above the crack mouth.

An idealised case was investigated based on the typi-cal double clip gauge set‐up used in experiments, where

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the lower and upper clip gauges, z1 and z2, were posi-tioned 2 and 12 mm above the crack mouth, respec-tively. Equation 2 shows that the clip gauge ratio, Vg2/Vg1, is the main determinant of the resultant rp. Forthe rotational point to fall ahead of the crack tip, Vg2must be larger than Vg1 at all stages of loading.Figure 8 shows the effect of various Vg2/Vg1 on theresulting rp. As Vg2/Vg1 moves towards 1, rp tends tomove towards infinity (Figure 8, top). In real specimens,rp lies in the unbroken ligament ahead of the crack tip,which falls between 0 < rp < 1.

14,15,18 Figure 8B showsthat within the unbroken ligament, rp decreases withthe increase of the clip gauge ratio. Therefore, in labora-tory tests, it should be noted that negative rp and rp ≥ 1are indications of false reading and do not reflect theactual rp of the specimen.

7

Generally, the increasing σys/σuts (decreasing strainhardening) results in increasing rp sh in a linear relation-ship for each respective a0/W set‐up (Figure 7). As strainhardening decreases, the material tensile responseexhibits increasing similarity to an elastic–perfect plastictensile response (Figure 2). This implies that as strainhardening decreases (decreasing plastic behaviour), therewill be decreased blunting at the crack tip because of theopening of the crack faces.7 In the ultimate case ofelastic–perfect plastic situation (no strain hardening),the crack tip would fall almost in the plane of the crackface.30

The models suggest that as a0/W increases, rp shwould move gradually towards the middle of the mate-rial ahead of the crack tip (rp sh ≈ 0.5). Because of thedifferent a0/W, the maximum difference of rp sh is seenin σys/σuts = 0.44, the lowest difference in σys/σuts = 0.89.The rp sh shows decreasing sensitivity to a0/W as strainhardening decreases (increasing σys/σuts), and the rp shtrend from the different a0/W seemingly intersects at apoint between 0.89 ≤ σys/σuts ≤ 1.00. This is portrayed

FIGURE 8 The relative rotational factor for increasing clip gauge rativiewed at wileyonlinelibrary.com]

by the rp sh at σys/σuts = 0.98, where a0/W = 0.7 gavethe lowest rp sh, followed by a0/W = 0.6, 0.5, 0.4,and 0.3, which is the opposite of that observed in0.44 ≤ σys/σuts ≤ 0.89. It was suspected that thisphenomenon is contributed by the way rp sh wasextracted from the model, where the total CTODwas used instead of separating it into the elastic andplastic.

Given that Equation 4 is able to account for the effectscrack tip blunting due to strain hardening in terms ofrp sh, the equation used in BS 7448‐1 based on the similartriangles approach was modified to account for theeffects of crack tip blunting due to strain hardening,described as

δsh ¼ K2 1 − v2ð Þ

mJWESσysEþ rpshBoVprpshBo þ a0; (5)

where mJWES = 4.9–3.5(σys/σuts).The elastic CTOD from WES 1108 was used as it was

calibrated theoretically to account for the effects ofstrain hardening in the elastic loading region.6,7,16,20,21,29

The error in prediction of CTOD using Equation 5 wasinvestigated by normalising it to the CTOD from themodels (δFE), described as CTOD ERR. For the rangeof 0.44 ≤ σys/σuts ≤ 0.98 and 0.3 ≤ a0/W ≤ 0.7,CTOD between 0.2 and 1.0 mm was considered forthe investigation. CTOD below 0.2 mm is considered“critical” in standards, where the elastic‐based stressintensity factor, K, is more relevant3-5; CTOD beyond1.0 mm is generally large, where some significantstable crack propagation would be expected in realspecimens.7,20,33

The mean and standard deviation of CTOD ERRwere calculated and plotted to a0/W for all CTODequations: CTOD from Equation 5 (δsh), BS 7448‐1,

o A, and the clip gauge ratio for 0 < rp < 1, B [Colour figure can be

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FIGURE 9 Standard deviation and mean of the crack tip opening displacement (CTOD) ERR for A, 0.3 ≤ a0/W ≤ 0.7 and B, 0.44 ≤ σys/σuts ≤ 0.98 [Colour figure can be viewed at wileyonlinelibrary.com]

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WES 1108, and ASTM E1820 (Figure 9). The equationswere compared relative with the respective a0/W andσys/σuts (Figure 9A,B).

For 0.3 ≤ a0/W ≤ 0.7, the CTODs considering theeffects of strain hardening (CTOD from Equation 5,WES 1108 and ASTM E1820) gave lower overall errorcompared with BS 7448‐1 (Figure 9A). The mean CTODfrom BS 7448‐1 overestimated CTOD for a0/W = 0.3and 0.4, as it is outside the valid range of0.45 ≤ a0/W ≤ 0.55 specified in the standard. Thethree strain hardening CTODs gave comparable lowvalues for the mean CTOD ERR, while CTOD fromEquation 5 gave the most accurate estimation forthe range of 0.3 ≤ a0/W ≤ 0.7. Likewise, the bestconsistency/dispersion of data was portrayed by CTODfrom Equation 5 with the lowest standard deviation,followed by ASTM E1820 then both WES 1108 and BS7448‐1.

In terms of strain hardening, for the range of0.44 ≤ σys/σuts ≤ 0.98, ASTM E1820 gave the best estima-tion for 0.44 ≤ σys/σuts ≤ 0.54, but the accuracy andconsistency reduce as σys/σuts increases to 1 (Figure 9B).Similar to before, δsh gave best compromise of bothaccuracy and consistency/dispersion. BS 7448‐1 didnot consider strain hardening in the equation; itoverestimated high strain hardening and underestimatedlow strain hardening, similar to that observed in previousworks.7,16,20,21 In general, the lower estimation andaccuracy shown by WES 1108 is contributed by theassumptions used in the calibration of the equation17;ASTM E1820 converts J into δ45,

5,30 which gives lowerCTOD compared with CTOD based on the opening ofthe original crack tip,7,20 which is employed in this work.The results shown that the rigid rotational concept is arobust and reliable technique for the estimation ofCTOD, especially when strain hardening considerationis employed.

6 | CONCLUSIONS

The concept of rigid rotational factor is a robust conceptimplemented for the estimation of CTOD. A series ofFE models generated within the range of 0.44 ≤ σys/σuts≤ 0.98 and 0.3 ≤ a0/W ≤ 0.7 showed that SEN(B) testsdo not exhibit a fixed value of rp but instead increase rpsh in a linear manner from 0.2 to 0.53 with the decreaseof strain hardening (increasing σys/σuts).

The models showed that for the same strain hardeningproperty, rp sh would increase with increasing a0/W from0.3 to 0.7. However, as the material strain hardeningdecreases, rp sh showed reduced sensitivity to a0/W. Datasuggest that that rp sh converges at a point independentof a0/W between 0.89 ≤ σys/σuts ≤ 1.0, when the materialexhibits significant elastic dominant properties. In thecalculation of CTOD, δsh (Equation 5) showed the bestcompromise of both accuracy and dispersion comparedwith BS 7448‐1, ASTM E1820, and WES 1108 for therange of 0.44 ≤ σys/σuts ≤ 0.98 and 0.3 ≤ a0/W ≤ 0.7.

In laboratory tests or modelling works, Equation 2enables the extraction of rp for the calculation of CTODbased on the similar triangles approach. However, cautionshould be exercised to account for human and test equip-ment error, as the estimation is heavily dependent on thedisplacement input at the crack mouth region. It shouldbe noted that this technique does not account for theeffects of crack tip blunting due to strain hardening.

ACKNOWLEDGEMENTS

The author is appreciative of the facilities organised byChris Brown at Brunel University and informative discus-sions with Dr. Philippa Moore and Dr. Henryk Pisarski atTWI. Additionally, the author is grateful for the fundinggranted by the University of Surrey for publication ofthe work.

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ORCID

WeeLiam Khor https://orcid.org/0000-0001-7699-8194

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How to cite this article: Khor WL. A CTODequation based on the rigid rotational factor withthe consideration of crack tip blunting due to strainhardening for SEN(B). Fatigue Fract Eng MaterStruct. 2019;1–9. https://doi.org/10.1111/ffe.13005

https://orcid.org/0000-0001-7699-8194https://doi.org/10.1111/ffe.13005