the effect of specimen dimensions on mixed mode ductile fracture

$: The effect of specimen dimensions on mixed mode ductile fracture$
Engineering Fracture Mechanics 75 (2008) 4394–4409

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Engineering Fracture Mechanics

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The effect of specimen dimensions on mixed mode ductile fracture

D.J. Smith a,*, T.D. Swankie b, M.J. Pavier a, M.C. Smith c

a University of Bristol, Bristol BS8 1TR, UKb Advantica Ltd., Loughborough LE11 3GR, UKc British Energy Generation Ltd., Gloucester GL4 3RS, UK

a r t i c l e i n f o

Article history:Received 23 May 2007Received in revised form 25 February 2008Accepted 8 April 2008Available online 15 April 2008

0013-7944/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.engfracmech.2008.04.005

* Corresponding author. Tel.: +44 1179288212; faE-mail address: [email protected] (D.J. S

a b s t r a c t

The results from a series of experiments are presented to determine the effect of specimendimensions on the ductile tearing resistance of A508 Class 3 forged steel at ambient tem-perature. Single edge notch tension specimens were subjected to Mode I, Mode II and com-bination of Modes I and II. Mode I tests on various specimen sizes reveal characteristicfeatures found in earlier work, such as decreasing slope of the tearing resistance withincreasing constraint (or specimen size). In contrast, for Mode II the tearing resistance isshown to be independent of specimen size, although dependent on initial crack length.The tests show that there is a competition between void growth and shear localisationas mechanisms for ductile crack extension. The dominance of one mechanism over theother is shown to be related to the local Mode I and Mode II components of the J-integral.

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1. Introduction

There is a wealth of experimental data that demonstrates that the resistance to Mode I ductile tearing in steels is depen-dent on specimen dimensions [1,2] and crack depth [3–5]. In general, initiation toughness JIc does not vary significantly whenspecimen thickness, B, width, W and the ratio of the initial crack length, a0 to specimen width change. However, tearing resis-tance, or the slope of the R-curve, dJ/da, varies considerably, especially when W and a0/W are different. These features areoften quantified in terms of constraint which in itself is a measure of a structure’s resistance to plastic deformation. The de-gree of constraint depends on the geometry and the type of loading.

In the application of structural integrity assessments there are circumstances where combinations of tensile (Mode I) andshear (Mode II) loading need to be considered. Practical examples include drive shafts for pumping systems and support py-lons for wind turbines. For Mode II and combined Mode I and II conditions the experimental evidence for the effects of load-ing mode on ductile fracture of steels is conflicting. Some report a Mode II initiation toughness equal to or higher than theMode I toughness [6–8]. Others report either a lower Mode II initiation toughness or a lower mixed mode or Mode II tearingresistance [9–12]. The difficulties associated with mixed mode and Mode II testing may contribute to the variation in ob-served behaviour. This is partly because there is no universally agreed testing standard for mixed mode loading and no stan-dardised method for estimating the mixed mode crack driving force. Some studies make no attempt to evaluate an elastic–plastic crack driving force [9,10]. Most of the more recent studies use J as the correlating parameter. However, because anumber of different test specimens are used, each study tends to use their own J-calibration.

The calculation of J under mixed mode loading is more complex than pure Mode I or Mode II [13], so there is increasedscope for uncertainty in crack driving force. Nevertheless, more recent studies all show a change in the resistance curve usingdifferent specimen designs with the lowest curve observed for Mode II loading. The decrease in toughness is associated with

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Nomenclature

a0 initial crack length, mmDa crack extension (or crack growth), mmfI (a, a0/W) Mode I elastic geometry functionfII(a, a0/W) Mode II elastic geometry functionklll local load line stiffness, N/mmklcp local stiffness parallel to crack, N/mmklcn local stiffness normal to crack, N/mmA a constant for resistance to ductile tearingA2 amplitude of additional stresses in a three term series expansion of stressesB specimen thickness, mmE Young’s modulus, GPaJT total J-integral, MPa mmJI Mode I J-integral (or toughness), MPa mmJIc initiation Mode I J-integral (or toughness), MPa mmJII Mode II J-integral (or toughness), MPa mmPmax maximum load attained in a testQ normalised value of inelastic hydrostatic stressT first non-singular stress term in elastic series expansion, MPaUe elastic component of area under load vs. displacement curve, N mmUp plastic component of area under load versus displacement curve, N mmUT total area under load vs. displacement curve, N mmUk extraneous system energy, N mmW specimen width, mma angle of loading, a = 0� for Mode I, a = 90� for Mode IIb a constant for resistance to ductile tearingge elastic geometry factor related to energy Ue

gp plastic geometry factor related to energy Up

Dlll local load line displacement, mmDlcn local displacement normal to crack, mmDlcp local displacement parallel to crack, mmSZW stretch zone width

D.J. Smith et al. / Engineering Fracture Mechanics 75 (2008) 4394–4409 4395

a change in mode of fracture, with Mode I crack growth occurring by microvoid coalescence and Mode II crack growth dom-inated by shear localisation occurring at the sharpened end of the crack tip.

The purpose of the work reported in this paper was to explore systematically the effects of specimen dimensions, B and Wand relative crack size, a0/W, on Mode II and mixed Mode I and II ductile fracture, using A508 Class 3 ferritic steel at ambienttemperature. These tests were an extension of those performed by Davenport and Smith [12] and Davenport [14]. Tests werecarried out using single edge notched specimens and a special fixture that allowed combinations of Mode I and Mode II load-ing to be applied.

2. Experiments

2.1. Materials and specimens

The material for the experiments was A508 Class 3 forged steel. The chemical composition, in wt% is; 0.16 C, 1.34 Mn,0.007 S, 0.004 P, 0.22 Si, 0.67 Ni, 0.17 Cr, 0.51 Mo, 0.06 Cu, 0.004 Sb, 0.01 Al, 0.004 Sn, 0.019 As, <0.01V, <0.01 Ti and<0.01 Nb. The S–L orientation was chosen for the fracture tests since earlier fracture tests using C(T) specimens [12,14]identified this as the least tough orientation. The basic tensile properties at ambient temperature are 430 MPa and561 MPa for yield and tensile strengths, and 201 GPa for Young’s modulus, E. An equi-axed pearlitic grain structure wasfound throughout the A508 steel forging. The average grain size was approximately 16.0 lm and the lamellae thicknessof ferrite within the pearlite grains was approximately 1.5 lm, while the cementite lamellae were slightly thinner, approx-imately 0.75 lm.

Plain sided single edge notched (SEN) specimens were tested. Sidegrooves were not introduced in the specimens andconsequently the direction of crack growth was not predefined. A total of 149 tests with variations in specimenthickness (B = 10 mm, 20 mm and 40 mm), width (W = 20 mm, 40 mm and 80 mm) and crack length (a0/W = 0.1, 0.5 and0.7) were used. The matrix of tests is shown in Table 1. Many specimens were needed and to minimise time andexpense of specimen manufacture, each specimen was notched using electro-discharge machining (EDM) and not fatiguepre-cracked.

Table 1Summary of mixed mode tests

W a/W = 0.1 a/W = 0.5 a/W = 0.7

B = 10 B = 20 B = 40 B = 10 B = 20 B = 40 B = 10 B = 20 B = 40

a = 0.0� 20 2 2 3 7 4 4 340 X 4 5 180 X X 5 3 X 3 X

a = 22.5� 20 540 380 3 X X

a = 45.0� 20 2 8 4 340 X 480 X X 3 X 3

a = 67.5� 20 540 X 480 X X 3 X

a = 90.0� 20 4 3 8 4 4 4 440 4 4 380 X 4 4 X 3

Numbers with italics indicate number of successful tests. X boxes indicate that tests could not be done using a 500 kN test rig.

4396 D.J. Smith et al. / Engineering Fracture Mechanics 75 (2008) 4394–4409

2.2. Test fixture and procedure

Mixed Mode I/II loading was applied using a loading fixture, shown in Fig. 1. An overall arrangement is shown in Fig. 1aand a photograph of the fixture, together with displacement extensometers is shown in Fig. 1b. This fixture was similar tothat designed by Davenport [14]. The specimens were designed so that the loading holes where located directly in line withthe crack tip and the load was applied directly through the crack tip. The loading fixture, manufactured from EN24T steel,comprised four sections which bolted together as two halves around an SEN specimen which was located within the fixtureusing two EN24T hardened pins.

All fracture tests were carried out in air at ambient temperature in a 500 kN servohydraulic test machine. The tests weredone under displacement control at a constant rate of approximately 0.6 mm/min. In each case, applied load and global axialdisplacement were measured and recorded. During the mixed mode experimental tests additional extensometers were usedto measure local load line displacement Dlll and local displacements normal to the crack (Mode I, Dlcn) and parallel to thecrack (Mode II, Dlcp). The arrangement of these extensometers is shown in Fig. 1.

Prior to the fracture studies, a stiffness test was performed for each test geometry to enable the elastic system energyassociated with flexure of the loading assembly to be calculated. The stiffness functions were calculated from load versuslocal specimen displacement records of solid, uncracked A508 calibration specimens of each specimen size.

In general, either three or four specimens were used to characterise the crack growth resistance of a particular specimensize and crack length for each combination of tension and shear loading. This method of testing is commonly referred to as amultiple-specimen technique [15].

A digital XY table with magnifying optics (at X200) was used to measure the stretch zone width (SZW) and crack growth(Da) attributed to each tested specimen, to an accuracy of ±1 lm. For the thin specimen tests (B = 10 mm), SZW and Da mea-surements were recorded at 0.5 mm intervals along the crack front on both crack faces. For the thicker specimens (B = 20 mmand 40 mm), measurements were recorded at 1.0 mm intervals on both crack faces. The average values of crack growth werethen determined.

3. Analysis

To interpret the experimental results two sets of analyses were conducted: an elastic analysis to determine elastic stressintensity factors and elastic geometry factors and an elastic–plastic analysis to determine plastic geometry factors. Theseanalyses are summarised in this section. The J-integral for each test was determined from the load versus displacement re-cord. The J-integral also required evaluation of geometry functions for the various specimen dimensions.

3.1. Calculation of J-integral

The elastic–plastic fracture toughness, J of each specimen was calculated from the area beneath the loading portion of theapplied load versus displacement curves, using a method developed by Sumpter and Turner [16]. Local load line displace-ment (Dlll) and local displacements normal to the crack (Dlcn) and parallel to the crack (Dlcp) were measured. Consequently,the J-integral was determined using these displacements. The area, U, beneath each loading curve was calculated using thetrapezoidal rule

Local load line lll,

extensometer

Local parallel to crack

lcpextensometer

Local normal to crack

lcnextensometerdisplacement, Δ

displacement, Δ

displacement, Δ

Fig. 1. Mixed mode fracture test fixture: (a) general arrangement of mixed mode fracture test fixture and (b) arrangement of extensometers on test fixturefor a mixed mode load test at a = 45�.


Using the load line displacements the total J-integral, JT, was calculated from the sum of the elastic and plastic compo-nents where,

JT ¼geUe

BðW � a0Þþ

gpUp

BðW � a0Þ: ð1Þ

The subscripts ‘e’ and ‘p’ refer to the elastic and plastic components of the total area UT under the load–displacement curves.The geometry dependent functions, ge and gp, which are sensitive to mode of loading and crack length, were determinedfrom finite element analyses described later.


Each of the load vs. displacement curves generated from experiment comprised not only of the elastic and plastic energyassociated with growing a crack but also the extraneous system energy, Uk corresponding to flexure of the loading fixtureand test machine. Using stiffness calibration functions measured for each size of specimen and mode of loading [17] itwas possible to remove the extraneous system energy and hence determine the energy associated only with crack growthusing,

JT ¼geðUeÞ

BðW � a0Þþ

gpðUT � Ue � UkÞBðW � a0Þ

; ð2Þ

where Up in Eq. (1) has been replaced by the elastic energy components subtracted from the total energy. UT corresponded tothe total area beneath the loading portion of the load versus displacement curve and Ue was calculated from the elastic stressintensity factor using finite element analysis described later.

It can be shown [17] that for plain sided specimens Eq. (2) can be rewritten as

JT ¼P2

maxð1� m2Þ fI a; a0W

� �2 þ fII a; a0W

� �2n o

B2WE1�

gp

ge

� �þ

gp

BðW � a0ÞUT �

P2max

2klll

( ); ð3Þ

where Pmax is the maximum load attained in the test. Values of ge, gp and the elastic geometry functions, fI(a,a0/W) andfII(a,a0/W), are dependent on the angle of loading, a and a0/W. Also the load line stiffness, klll of the specimen and load fixtureis dependent on angle of loading. The subscripts I, and II refer to Mode I and Mode II.

The J-integrals, Jlcn and Jlcp, using the components of crack opening (Dlcn) and crack sliding (Dlcp) displacement were alsodetermined. To enable the individual contributions to be calculated, the load P was resolved into normal, Pcosa and parallelPsina components with respect to the mixed mode loading angle, a and analysed with respect to the corresponding, Dlcn andDlcp displacements.

For mixed mode loading the individual components of the J-integral are

Jlcn ¼ðPmax cos aÞ2ð1� m2Þ fI

a0W

� �2n o

B2WE1�

gpI

geI

� �þ

gpI

BðW � a0ÞUlcn �

ðPmax cos aÞ2

2klcn

( ); ð4Þ

Jlcp ¼ðPmax sin aÞ2ð1� m2Þ fII

a0W

� �2n o

B2WE1�

gpII

geII

� �þ

gpII

BðW � a0ÞUlcp �

ðPmax sin aÞ2

2klcp

( ): ð5Þ

Here, ge and gp are dependent on either Mode I or Mode II (indicated by the subscripts I and II), i.e., when the loading angle acorresponds to 0� and 90� respectively. klcn and klcp are the corresponding stiffness of the specimen and test fixture for dis-placements normal and parallel to the crack plane.

3.2. Finite element studies

The principle purpose of the study was to obtain stress intensity factors and elastic and plastic values of ge and gp. Theelastic and plastic geometry functions used in Eqs. (3)–(5) were obtained from finite element analyses of the test specimenswithin the test fixture. In order to avoid a complex 3D analysis a full scale 2D plane strain FE model of a single edge notch(SEN) specimen located within the mixed mode loading fixture was developed using the ABAQUS finite element code. As willbe seen later in the experimental results the levels of plasticity (and hence J) were high and plane strain conditions not nec-essarily appropriate.

For the FE analyses the fixture and specimen were assumed to be as a single unit with a rigid connection between theinternal edges of the loading fixture and the outer surface of the specimen. Eight-noded plane strain quadrilateral elements(type CPE8R) were used to construct the mesh. Stress and plastic strain data for A508 [13] were converted to true stress andlog plastic strain, and used in the ABAQUS input file to define the plastic behaviour of the specimen material.

For convenience the crack tip was modelled as a key-hole notch with a radius of 0.05 mm. The crack tip region was mod-elled using 24 rows of 32 circumferential elements, biased so that the size of each element increased with increasing dis-tance from the crack tip. For each ratio of crack depth to specimen width used in the study (a0/W = 0.1, 0.5 and 0.7) 24rows of elements were always used to define the core region.

For each mode mixity, stress intensity factors, KI and KII were calculated [17]. Then elastic geometry functions, fI(a,a0/W),and fII(a,a0/W) were calculated from the Mode I and II stress intensity factors by inverting

K I ¼Pmax

BffiffiffiffiffiffiWp fI a;

a0

W

� �; ð6Þ

K II ¼Pmax

BffiffiffiffiffiffiWp fII a;

a0

W

� �: ð7Þ

The elastic and plastic values of ge and gp were also determined from FE analysis by evaluating the J-integral. ge isconstant regardless of applied load and was calculated from the initial load increment, where Up was zero and Eq. (1) rear-

Mixed Mode Loading Angle, α - degrees

0.0˚ 22.5˚ 45.0˚ 67.5˚ 90.0˚

η *Fa

ctor

0.0

1.0

2.0

3.0ηe, ao/W=0.1ηe, ao/W=0.5ηe, ao/W=0.7

ηp, ao/W=0.1ηp, ao/W=0.5ηp, ao/W=0.7

Fig. 2. Comparison of mixed mode g factors for different a0/W.


ranged to find ge. Having calculated ge, Eq. (1) was rearranged further to determine gp for increasing plastic deformationwhere gp is

gp ¼JBðW � a0Þ � geUe

Up: ð8Þ

As the load increased, Up increased and the terms from the elastic part became relatively small in comparison. Values of gp

were calculated from Eq. (8) as the load was incremented. The initial value of gp was large but decreased with increasingplasticity until a steady state value was achieved. It is the steady state value of gp, and the corresponding constant ge, thatwere used to calculate J from the area beneath a load vs. displacement curve.

The calculated g factors changed with loading angle, a, but the results were relatively insensitive to changes in a0/W asillustrated in Fig. 2 for W = 40 mm. Results are plotted as a function of a for a0/W = 0.1, 0.5 and 0.7. The results (not shown)for W = 20 and 80 mm were very similar to those for W = 40 mm. For a0/W = 0.1, ge is higher throughout the range of mixedmode loading compared with a0/W = 0.5 and 0.7.

4. Experimental results

The experimental load, displacement and crack length test results were used to calculate the J-integral using Eqs. (3)–(5).Experimental results are shown in Figs. 3–10. In all figures the total J-integral (determined using Eq. (3)) is shown as a func-tion of the average crack growth, Da, measured from the two fracture surfaces of each specimen. The SZW was used as ameasure of blunting up to and including crack initiation. Detailed results in terms of the individual components of the J-inte-gral are not shown here but are provided in [17]. The discussion later uses some of the results from the individual compo-nents to explore the outcome of the results in terms of failure mechanisms.

Fitted curves to experimental results are also illustrated in Figs. 3–10. The curves were obtained by fitting a least squarespower law expression to the experimental data, so that

JT ¼ AðDaÞb; ð9Þ

where A and b are fitted parameters and are given in Tables 2–4. Also shown in Tables 2–4 are results derived from Eq. (9) forthe initiation toughness Jinit corresponding to Da = 0.2 mm and the tearing resistance, dJ/da, of the R-curves at Da = 1 mm.

Results for pure Modes I and II at a0/W = 0.5 are first examined followed by results where the variation in a0/W are stud-ied. Then results for mixed mode loading at a0/W = 0.5 are shown. Finally, the effect of changing a0/W on the R-curves whenspecimens were subjected to mixed mode loading are reported.

Some of the specimens were sectioned through the mid-plane and then polished to reveal the details of the mechanismsof ductile tearing. The observations from these sections are also described below.

4.1. Pure Mode I and pure Mode II tearing at a0/W = 0.5

A summary of pure Mode I fracture tests for a0/W = 0.5 is shown in Fig. 3. Two sets of data are illustrated. Smaller symbolsillustrate results for measured stretch zone widths (SZW) and larger symbols show results from the average of the measuredaverage crack growth, Da. Also shown are curves corresponding to Eq. (9) fitted to crack growth data. In general, regardless ofW and B, the SZW at crack initiation was constant at approximately 0.2 mm.

Average Crack Growth, Δa - mm

0 1 2 3 4 5

Mod

eII

Tou

ghne

ss,J

II-

MPa

mm

0

200

400

600

800

1000

Average SZW measurements

SZW Δa

B=10.0mm, W=20.0mmB=10.0mm, W=40.0mmB=10.0mm, W=80.0mm

B=20.0mm, W=20.0mmB=20.0mm, W=40.0mm

B=40.0mm, W=20.0mmB=40.0mm, W=20.0mm

Mode II tests

Mode I C(T) B=25.0mm Bn=20.0mm

Fig. 4. Influence of specimen dimensions on pure Mode II ductile tearing resistance curves for a0/W = 0.5.


0 1 2 3 4 5

Mod

eI

Tou

ghne

ss,J

I-

MP

am

m

0

1000

2000

3000

4000


SZW Δa

B=10.0mm, W=20.0mmB=10.0mm, W=40.0mmB=10.0mm, W=80.0mm

B=20.0mm, W=20.0mmB=20.0mm, W=40.0mm

B=40.0mm, W=20.0mm

C(T) B=25mm Bn=20mm

Fig. 3. Influence of specimen dimensions on pure Mode I ductile tearing resistance curves for a0/W = 0.5.


Results in Fig. 3 show that for W = 20 mm specimens, tested in Mode I, the effect of thickness (10 mm 6 B 6 40 mm) onthe R-curves was relatively small compared to the increase in toughness during further ductile tearing. For example, the ef-fect of B on crack initiation toughness, Jinit, was such that it varied from 550 to about 900 MPa mm. Jinit was determined atDa = 0.2 mm by extrapolating fitted curves back to Da corresponding to the average SZW = 0.2 mm. Similarly tearing resis-tance, dJ/da (corresponding to Da = 1 mm) ranged from about 1900 to 2600 MPa for 10 mm 6 B 6 40 mm with W = 20 mm.In contrast, for a given B (i.e., B = 10 mm), dJ/da decreased significantly from about 2500 MPa to 540 MPa as W increasedfrom 20 to 80 mm.

Also shown in Fig. 3 are results from earlier tests by Davenport for the same material using sidegrooved C(T) specimens(with net section thickness Bn = 20 mm). In common with findings from others [3,5] the test results for B and W = 10 and20 mm, B and W = 20 and 20 mm and B and W = 40 and 20 mm show that substantially higher R-curves were obtained fromthe plain sided SEN specimens compared to the C(T) specimens. However, in the case of SEN tests with B and W = 10 and


0 1 2 3 4

Mod

eI

Tou

ghne

ss,J

I-

MPa

mm

0

500

1000

1500

2000

2500

3000

3500


ao/W=0.5 ao/W=0.7

B=10.0mm, W=20.0mmB=10.0mm, W=80.0mm

B=20.0mm, W=20.0mm

Fig. 5. Influence on specimen dimensions and a0/W on pure Mode I ductile tearing resistance curves.


0 1 2 3 4

Mod

eII

Tou

ghne

ss,J

II-

MP

am

m

0

100

200

300

400

500

600

700

Mode IIao/W=0.1 ao/W=0.5 ao/W=0.7

B=10.0mm, W=80.0mm B=10.0mm, W=20.0mm


B=40.0mm, W=20.0mm

B=20.0mm, W=40.0mm B=20.0mm, W=20.0mm

Fitted curves

Fig. 6. Influence on specimen dimensions and a0/W on pure Mode II ductile tearing resistance curves.


40 mm, B and W = 10 and 80 mm and B and W = 20 and 40 mm, the R-curves at small levels of crack extension (less than1 mm) were very similar to the results from the earlier C(T) tests.

As expected, all Mode I tests revealed significant crack tip blunting prior to crack initiation, with subsequent crack growthin the plane of the initial crack through a combination of void growth and their coalescence. Fig. 3 also shows that some ofthe data points for the B = 10 mm specimens lie outside the general trends. Examination of the fracture surfaces revealed thatthe main cause for this was the presence of inclusions in the path of the propagating crack leading to higher amounts of tear-ing compared with other test results.


0.0 0.5 1.0 1.5 2.0

Tou

ghne

ss,J

T-

MPa

mm

0

500

1000

1500

2000

2500

3000

3500

Mode I, α=0.0˚α=22.5˚α=45.0˚α=67.5˚

Mode II, α=90.0˚

Fig. 7. Influence of mixed mode loading on ductile tearing resistance for B = 10 mm, W = 20 mm and a0/W = 0.5.

Average Crack Growth, Δa - mm0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Tou

ghne

ss,J

T-

MP

am

m

0

500

1000

1500

2000

2500

3000

3500

Mode I, α=0.0˚α=22.5˚α=45.0˚α=67.5˚

Mode II, α=90.0˚

Fig. 8. Influence of mixed mode loading on ductile tearing resistance for B = 10 mm, W = 40 mm and a0/W = 0.5.


A summary of pure Mode II test results is shown in Fig. 4 and illustrates that irrespective of specimen size a single mastercurve describes the ductile tearing behaviour. The average SZW from all tests was 0.1 mm. It is also notable that overall theresistance to tearing for Mode II was less than for any of the Mode I test data shown in Fig. 3, including the R-curve fromhighly constrained Mode I C(T) tests. For example the lowest initiation toughness for Mode I was about 200 MPa mm, whilefor Mode II Jinit was 100 MPa mm. There was also a significant difference in the slope of the curve, dJ/da, with Mode II testsexhibiting values at about 170 MPa in contrast to about 540 MPa for B = 10 mm, W = 80 mm Mode I test results.

Polished sections extracted from Mode II tests revealed anti-symmetric blunting of the crack tip prior to crack initiation. Ashear crack initiated at the tip of the sharpened region of the blunted tip. Plastic deformation at the tip of the initiated shearcrack caused the grains to deform and re-orientate themselves and the crack grew in the plane of the initial crack through amechanism of shear localisation and decohesion [17].


0 1 2 3 4 5 6

Tou

ghne

ss,J

T-

MPa

mm

0

500

1000

1500

2000

2500

3000

3500

Mode I, α=0.0˚α=22.5˚α=45.0˚α=67.5˚

Mode II, α=90.0˚

Fig. 9. Influence of mixed-mode loading on ductile tearing resistance for B = 10 mm, W = 80 mm and a0/W = 0.5.

Average Crack Growth, Δa - mm0 1 2 3 4

Tou

ghne

ss,J

T-

MP

am

m

0

500

1000

1500

2000

B=10.0mm, W=20.0mmB=10.0mm, W=80.0mm

ao/W=0.5 ao/W=0.7

Fig. 10. Influence on specimen dimensions and a0/W on mixed-mode loading (a = 45�) ductile tearing resistance curves.


4.2. Influence of a0/W on pure Mode I and pure Mode II tearing

As well as the tests at a0/W = 0.5, tests were conducted at a0/W = 0.1 and 0.7. However, pure Mode I experiments at initialcrack lengths of a0/W = 0.1 did not exhibit ductile crack growth. This was because specimens exhibited gross yieldingthroughout the specimen section. Experimental results for the a/W = 0.7 tests are shown in Fig. 5 along with data for testsusing a/W = 0.5. It is evident that the difference between results with a0/W = 0.5 and 0.7 is small for relatively small speci-mens (B = 10 and 20 and W = 20 mm). For example the average initiation toughness and slope dJ/da for these specimens was817 MPa mm and 2075 MPa.

For larger specimen dimensions (B = 10, W = 80 mm) experiments for a0/W = 0.7, produced a resistance curve with a sig-nificantly lower slope (dJ/da = 542 MPa) compared to the W = 20 mm specimens (dJ/da = 2300 MPa). Nevertheless, the R-curve for a/W = 0.7 was not as low as that for a/W = 0.5.

Table 2Fitted parameters for R-curve data for a/W = 0.5

Loading angle B (mm) W (mm) A b J (MPa mm) at Da = 0.2 mm dJ/da (MPa) at Da = 1 mm

Mode I 10.0 20.0 2741.3 0.9327 611.0 2556.8a = 0.0� 10.0 40.0 1081.0 1.2404 146.8 1340.9

10.0 80.0 792.8 0.6846 263.4 542.820.0 20.0 2772.7 0.6777 931.6 1879.120.0 40.0 1223.3 1.0784 215.7 1319.240.0 20.0 2313.1 0.8932 549.4 2066.1

a = 22.5� 10.0 20.0 1581.6 0.4011 829.4 634.410.0 40.0 1735.7 0.4441 849.3 770.810.0 80.0 1607.1 0.4288 806.0 689.1

a = 45.0� 10.0 20.0 1329.4 0.4151 681.6 551.810.0 40.0 1141.3 0.4151 585.1 473.810.0 80.0 1269.0 0.4780 588.0 606.640.0 20.0 1229.1 0.4327 612.6 531.8

a = 67.5� 10.0 20.0 460.1 0.4864 210.3 223.810.0 40.0 593.3 0.4092 307.1 242.810.0 80.0 568.0 0.4327 283.1 245.8

Mode II 10.0 20.0 248.2 0.5303 105.7 131.6a = 90.0� 10.0 40.0 263.7 0.6363 94.7 167.8

10.0 80.0 249.7 0.6838 83.1 170.720.0 20.0 288.8 0.6658 98.9 192.320.0 40.0 268.5 0.6035 101.7 162.040.0 20.0 266.8 0.6294 96.9 167.940.0 40.0 260.3 0.7920 72.8 206.2Mode II master curve 269.1 0.6278 98.0 168.9



Mode I 10.0 20.0 2993.6 0.7669 871.3 2295.8a = 0.0� 10.0 80.0 1470.6 0.4861 672.6 714.9

20.0 20.0 2426.0 0.6454 858.6 1565.7

a = 45.0� 10.0 20.0 1177.8 0.4569 564.6 538.110.0 80.0 1456.2 0.2747 935.9 400.0

Mode II 10.0 20.0 186.9 0.4833 85.9 90.3a = 90.0� 10.0 80.0 277.0 0.5992 105.6 166.0

20.0 20.0 251.3 0.6252 91.9 157.1

Mode II master curve 254.4 0.6803 85.1 173.1



Mode II 10.0 20.0 357.9 0.3889 191.4 139.2a = 90.0� 40.0 20.0 442.2 0.4744 206.1 209.8

Mode II master curve 420.8 0.4836 193.2 203.5


Similar to the microstructural features observed for Mode I a/W = 0.5 tests the polished sections of a/W = 0.7 also revealedthat crack growth was by a process of microvoid growth and coalescence.

Unlike the Mode I a/W = 0.1 tests pure Mode II tests with a0/W = 0.1, did exhibit ductile tearing. The test results are shownin Fig. 6 and compared with data for a/W = 0.5 and 0.7. Data at longer crack lengths (a0/W = 0.7) were very similar to themaster curve described earlier for pure Mode II tests at a0/W = 0.5 with an initiation toughness at about 90 MPa mm and val-ues of dJ/da around 170 MPa. However, experiments with a0/W = 0.1 produced at higher pure Mode II tearing resistance withthe initiation toughness and dJ/da increased to about 200 MPa mm and 200 MPa, respectively.

In all the Mode II tests at different initial crack lengths the cracks grew by shear localisation and decohesion.

4.3. Mixed Mode I and II tearing at a0/W = 0.5

The effects of mixed mode loading on ductile tearing are shown in Figs. 7–9. The results correspond to given values of Band W at constant a0/W = 0.5. The range of thickness tested in pure Mode I and pure Mode II demonstrated that thickness


effects were negligible. For each combination of tensile and shear loading (a = 22.5�, 45.0� and 67.5�), specimens withW = 20 mm, 40 mm and 80 mm were tested and results are shown in Figs. 7–9, respectively.

An additional four tests were conducted for a = 45.0� on B = 40 mm specimens (W = 20 mm). Although the results are notillustrated they confirmed the small effect of thickness seen in the pure mode studies.

For B = 10 mm, W = 20 mm, Fig. 7 shows that the R-curves decreased with increasing shear loading (Mode II), with a sub-stantial drop in toughness when the loading angle, a, was greater than 67.5�. For example, the initiation toughness for Mode Iand a = 22.5� and 45� was greater than about 600 MPa mm, while for Mode II and a = 67.5� the initiation toughness was low-er than about 200 MPa mm. Also, typically the average SZW width decreased with increasing Mode II from about 0.2 mm forMode I to 0.1 mm for Mode II. Most notably the slope of the R-curve reduced dramatically from 2560 MPa to 634 MPa whenthe loading angle was moved from Mode I to 22.5�.

With a change in W from 20 to 40 mm (at constant B = 10 mm) results illustrated in Fig. 8 show that increasing the shearloading initially increased the tearing initiation resistance from 147 MPa mm for Mode I to 850 MPa mm for a = 22.5, butreduced the slope of the R-curve from 1340 MPa to 770 MPa. Then interestingly, the initiation toughness and slope reduced

Fig. 11. Schematic of the blunting and crack growth characteristics for mixed mode loading.


with increasing loading angle, so that for example, the initiation toughness reduced by from 850 MPa mm for a = 22.5–94 MPa mm for pure Mode II.

By increasing W to 80 mm and for the same thickness (B = 10 mm), results shown in Fig. 9, reveal that the R-curve for ModeI loading lies between the R-curves for mixed mode loading at a = 45.0� and 67.5�. As with the results shown in Fig. 8, initiallyenlarging the shear component increased the mixed mode initiation toughness Jinit starting at 263 MPa mm for Mode I andincreasing to 806 MPa mm at a = 22.5�. This resulted in the highest R-curve for W = 80 mm, B = 10 mm and a0/W = 0.5. Forincreasing load angles (and increasing amounts of shear loading) the R-curves decreased below the Mode I curve.

4.4. Influence of a0/W on mixed mode tearing

Experiments also explored the influence of initial crack length on mixed mode ductile fracture but only when a = 45�. Aswith the earlier tests for pure Mode I at short crack lengths (a0/W = 0.1) ductile tearing was not observed and gross yieldingbeing the main failure mechanism. Additional tests at a = 45� and at longer crack lengths a0/W = 0.7 produced R-curves sim-ilar to those a0/W = 0.5 as shown in Fig. 10. As results in Tables 2–4 illustrate the slopes of the various R-curves shown in Fig.10 were similar with an average value of 520 MPa.

For mixed Mode I/II (with a = 45�) anti-symmetric blunting of the initial crack tip was observed to be similar to the behav-iour of Mode II tests. However, it was evident that there was a significant contribution from crack opening displacement(Mode I) in addition to crack sliding displacement (Mode II). Once again crack initiation and propagation occurred in theplane of the initial crack, from the tip of the sharpened corner of the blunted crack tip. Examinations of specimens with sig-nificantly more crack growth demonstrated that the preferred direction for crack growth was in the plane of the initial crackwith crack growth through shear localisation.

For mixed mode loading the mechanism of crack initiation for the mixed mode tests was dependent on specimen widthand the degree of shear loading. For example, it was observed that for B = 10 mm, W = 20 mm specimens there was a changefrom the common microvoid coalescence mechanism typical for Mode I loading to shear localisation when the loading anglewas increased from zero (Mode I) to 22.5�. However, the switch in mechanism occurred at 45� for the larger width(W = 80 mm) specimens. These observations are summarised using schematic diagrams in Fig. 11.

5. Discussion

5.1. Mode I

The Mode I experiments, Fig. 3, demonstrated that an increase in specimen thickness (B = 10, 20 and 40 mm) had a neg-ligible effect on Jinit and dJ/da in Mode I. This is in agreement with Joyce and Link [18] who tested HSLA HY100 steel usingSEN specimens up to 50 mm thick. Also shown in Fig. 3 is the effect of specimen width. The results illustrate that changes inspecimen width had a significant effect on both Jinit and dJ/da.

Similar effects of specimen size on Jinit and dJ/da in Mode I loading were observed in the deeper cracked specimens, cor-responding to an a0/W of 0.7, Fig. 5. Test data obtained from SEN specimens of HSLA HY80 steel over a range of a0/W ratiosfrom 0.13 to 0.83 were obtained by Joyce and Link [18]. They reported nearly constant values of Jinit but with widely varyingdJ/da which decreased as a0/W increased, until 0.7 when dJ/da began to increase. The results in Fig. 5 for W = 20 mm agreewith their observations.

The Mode I results shown in Figs. 3 and 5 illustrate the well known phenomena of loss of constraint [3,4] wherein theresistance to ductile tearing is increased through dissipation of the additional plastic deformation. Consequently, all ofthe Mode I tests for a/W = 0.5, with the exception of the B = 10 mm, W-80 mm tests, exhibited significantly higher R-curvescompared to data obtained by Davenport [14] using C(T) specimens. It is beyond the scope of the paper to undertake a con-straint based correction to the experimental data shown here. Nonetheless a number of schemes [19–22] for constraint cor-rection are available, all of which introduce a second parameter in addition to J to characterise initiation and growth ofductile fracture. For example, Chao and Zhu [22] suggest that a second parameter, called A2, is used in conjunction with Jto characterise ductile tearing resistance curves. Here, A2 represents the amplitude of the additional stress terms in a threeterm series expansion of the near crack tip stresses. Unlike the application of Q or T [20,21] to correct for constraint Chao andZhu [22] claim that A2 is load independent under fully plastic deformation. In contrast to the use of T, Q or A2, Brocks andSchmitt suggest that a measure of local triaxiality, such as the ratio of the hydrostatic stress to the yield (or effective stress)can be used. The advantage of this approach is that in-plane (relative to the crack front) and out-of-plane constraint are bothincluded.

Irrespective of which method is used, all require detailed computational analyses to determine the necessary parameters.While catalogues for T solutions see Ref. [21] are available for a range of specimens this is not the case for the other param-eters. For the SEN specimen a normalised T-stress at a0/W = 0.5 is equal to �0.6. The T-stress is normalised with respect tothe nominal stress across the cracked section. For a C(T) specimen at the same crack length the normalised T-stress is 6.0.Consequently, based on a two parameter analysis using J and T, high values of T together with corresponding low values of Jfor C(T) specimens lead to lower tearing resistance compared to high tearing resistance of SEN specimens where low (andnegative values of T) corresponded to high values of J.


When the width of the uncracked ligament in an SEN specimen and the depth of the crack are sufficient to constrain plas-tic flow the ductile tearing resistance curves were shallower as evidenced by the test results for B = 10 mm, W = 80 mm, asshown in Figs. 3 and 5. Normalised T-stress values for deeply cracked SEN specimens are greater than about 2 which is sig-nificantly greater than for a/W = 0.5 but less than for C(T) specimens at a/W = 0.5. Notably for all Mode I tests the same mech-anism of crack initiation by microvoid growth and coalescence prevailed.

5.2. Mode II and mixed Mode I/II

The Mode II experimental results (Figs. 4 and 6) demonstrated that specimen dimensions had a negligible effect on Jinit

and dJ/da, with changes in the tearing resistance occurring only when a0/W = 0.1. Changing the specimen dimensions wouldbe expected to change the degree of stress triaxiality (or constraint). It is apparent that the lack of influence of constraint ontearing resistance in Mode II also placed the tearing resistance at lower values than any of the Mode I or mixed mode resis-tance curves.

A range of theoretical models using finite element (FE) analysis tend to support the experimental findings for Mode IItearing. For example, Ayatollahi et al. [23] studied the effect of constraint on ductile fracture and, provided that the mech-anism is shear localisation, they showed that there was no effect of constraint for Mode II conditions. This is entirely in con-trast to the influence of constraint in Mode I ductile fracture. Other FE studies examining the effects of ductile damageformation under small scale yielding conditions [24–26] illustrate that the lower part of the crack sharpens to the extentthat strain controlled crack growth occurs before ductile crack growth at the blunted upper part of the crack. This is in com-plete correspondence to the observations obtained from sectioned samples. However, the FE results from small scale yieldingshould be treated with caution when applied to practical tests.

Under conditions of mixed mode loading Figs. 7–9 illustrate that, depending on specimen width, increasing the shearloading could either decrease or increase the tearing resistance. As shown by Ayatollahi et al. [23] there is only a limitedinfluence of constraint (or triaxiality) on shear localisation and in contrast the mechanism of void growth and coalescencefor Mode I tearing is strongly influenced by constraint (stress triaxiality), [19]. It is therefore apparent from the experimentalresults shown in Figs. 7–9 that the introduction of external shear loading modifies the local state of triaxiality. It wouldtherefore be necessary to determine for each mixed mode loading condition the level of constraint through an appropriateparameter such as those discussed earlier for Mode I conditions. However, it is not currently possible to quantify the degreeof constraint for the Mode II and mixed mode tests since further numerical work is required. Nevertheless, for pure Mode IIloading Ayatollahi et al. [27] determined values of QII for the SEN specimen within the loading fixture and demonstrated thatQII depended on the details of the boundary conditions between the specimen and the fixture. In the case of combined pinand contact loading between the SEN specimen and the test fixture values of QII were found to be about �0.4 at the max-imum load. Values of QII in Mode II also depend on load similar to the case for Mode I [19]. However, the test results for pureMode II show that there is no influence on specimen size (and constraint).

Although the level of constraint can not be directly quantified it is evident from the experimental results shown in Figs. 7–9 that constraint played an important role. For example, when W = 20 mm results in Fig. 7 show there was a decrease in bothJinit and dJ/da with increasing Mode II loading until the limiting condition of pure Mode II was reached. The microstructuralobservations indicated that additional shear loading generated crack initiation and growth by shear localisation.

The relative positions of the R-curves for SEN specimens with B = 10 mm, W = 80 mm and a0/W = 0.5 are significantly dif-ferent in Fig. 9 compared to those in Fig. 7. As noted earlier (Fig. 3) the Mode I R-curve for W = 80 mm was the lowest R-curvefor all Mode I tests. Results in Fig. 9 show that by increasing the external shear loading (to a = 22.5�) the constraint wasapparently decreased and the tearing resistance increased, with initiation values increased from 260 MPa mm to 800 MPamm. Further increases in shear loading led to a change in failure mechanism which was retained until pure Mode IIconditions.

As Fig. 10 shows, the R-curves under mixed mode conditions with a = 45� were essentially independent of specimenwidth. This is not surprising since all specimens exhibited crack growth through shear localisation. Nonetheless, the low con-straint of these specimens allowed additional plastic energy to be dissipated so that the resulting slope of R-curve (at about520 MPa) was significantly greater than that for pure Mode II (as at 170 MPa).

5.3. J and changes in mechanism

The separation of the local components of displacement into perpendicular and parallel components permitted the indi-vidual components of J to be divided into Mode I (local component normal to the crack) and II (local component parallel tothe crack), Jlcn and Jlcp using Eqs. (4) and (5). The resulting R-curves using these components were used to estimate initiationvalues of Jlcn and Jlcp for different mixed mode loading conditions (or different loading angles). The tearing resistance dJ/da,determined from the local values of J, are shown in Figs. 12 and 13. Also shown in Figs. 12 and 13 are the loading angleswhere the two observed failure mechanisms of ductile tearing through void growth and coalescence and shear localisationand decohesion dominate. The normal component dJlcn/da was the greatest when the dominant failure mechanism was voidgrowth and coalescence. In contrast failure through shear localisation and decohesion was pronounced when dJlcp/da wasgreater than dJlcn/da. The cross over of the two sets of curves in Figs. 11 and 12 coincided approximately with the observedchange in failure mechanism.

Mode IIMixed Mode Loading Angle, α - degrees

0.0˚ 22.5˚ 45.0˚ 67.5˚ 90.0˚

Tea

ring

Res

ista

nce,

dJ/d

a-

MPa

0

250

500

750

1000Sheardecohesion Jlcn

Jlcp

Mode I

Microvoid coalescence

Fig. 13. Relationship between change in failure mechanism and slope dJ/da (at Da = 1 mm) for components parallel, Jlcp and normal, Jlcn to the crack plane.B = 10 mm, W = 80 and a0/W = 0.5.

Mixed Mode Loading Angle, α - degees

0.0˚ 22.5˚ 45.0˚ 67.5˚ 90.0˚

Tea

ring

Res

ista

nce,

dJ/d

a-

MP

a

0

200

400

600

8002200

2400

2600

2800

Jlcn

Jlcp

Mode I Mode II

Microvoidcoalescence

Sheardecohesion

Fig. 12. Relationship between change in failure mechanism and slope dJ/da (at Da = 1 mm) for components parallel, Jlcp and normal, Jlcn to the crack plane.B = 10 mm, W = 20 and a0/W = 0.5.


6. Conclusions

A comprehensive matrix of tests using a special loading fixture enabled combinations of Mode I and Mode II loading to beapplied directly through the crack tip of SEN specimens. Specimens of different dimensions and crack depths ratio revealedchanges in the ductile crack initiation and tearing resistance of a C–Mn steel at ambient temperature.

As expected for Mode I loading, specimens with small specimen dimensions (i.e., exhibiting low constraint) resulted inhigh resistance to tearing. In contrast, the Mode II tearing resistance was not influenced by specimen dimensions (i.e., con-straint). When SEN specimens were subjected to Mode II loading the tearing resistance was lower than for Mode I. For mixedmode loading, the transition from ductile fracture (Mode I) to shear localisation (Mode II) was a strong function of specimendimensions.

The mixed mode ductile fracture characteristics of a C–Mn steel revealed that there were two failure mechanisms: voidgrowth and coalescence, and shear localisation and decohesion. The dominance of one mechanism over another has beenshown to strongly related to the local Mode I, Jlcn and Mode II, Jlcp components of the J-integral.

Acknowledgements

This work was supported by British Energy and the UK Engineering and Physical Science Research Council. David Smithalso acknowledges the support of a Royal Society Wolfson Merit Award which prompted the development of this work.


References

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the effect of specimen dimensions on mixed mode ductile fracture

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