The effect of specimen dimensions on mixed mode ductile fracture

Engineering Fracture Mechanics 75 (2008) 4394–4409

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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 b c

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

a r t i c l e

i n f o

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

a b s t r a c t The results from a series of experiments are presented to determine the effect of specimen dimensions on the ductile tearing resistance of A508 Class 3 forged steel at ambient temperature. Single edge notch tension specimens were subjected to Mode I, Mode II and combination of Modes I and II. Mode I tests on various specimen sizes reveal characteristic features found in earlier work, such as decreasing slope of the tearing resistance with increasing constraint (or specimen size). In contrast, for Mode II the tearing resistance is shown 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 localisation as mechanisms for ductile crack extension. The dominance of one mechanism over the other is shown to be related to the local Mode I and Mode II components of the J-integral. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction There is a wealth of experimental data that demonstrates that the resistance to Mode I ductile tearing in steels is dependent on specimen dimensions [1,2] and crack depth [3–5]. In general, initiation toughness JIc does not vary significantly when specimen thickness, B, width, W and the ratio of the initial crack length, a0 to specimen width change. However, tearing resistance, or the slope of the R-curve, dJ/da, varies considerably, especially when W and a0/W are different. These features are often quantified in terms of constraint which in itself is a measure of a structure’s resistance to plastic deformation. The degree 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) and shear (Mode II) loading need to be considered. Practical examples include drive shafts for pumping systems and support pylons for wind turbines. For Mode II and combined Mode I and II conditions the experimental evidence for the effects of loading mode on ductile fracture of steels is conflicting. Some report a Mode II initiation toughness equal to or higher than the Mode I toughness [6–8]. Others report either a lower Mode II initiation toughness or a lower mixed mode or Mode II tearing resistance [9–12]. The difficulties associated with mixed mode and Mode II testing may contribute to the variation in observed behaviour. This is partly because there is no universally agreed testing standard for mixed mode loading and no standardised 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 a number 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 increased scope for uncertainty in crack driving force. Nevertheless, more recent studies all show a change in the resistance curve using different specimen designs with the lowest curve observed for Mode II loading. The decrease in toughness is associated with

* Corresponding author. Tel.: +44 1179288212; fax: +44 1179294423. E-mail address: [email protected] (D.J. Smith). 0013-7944/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2008.04.005

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

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Nomenclature a0 initial crack length, mm Da crack extension (or crack growth), mm fI (a, a0/W) Mode I elastic geometry function fII(a, a0/W) Mode II elastic geometry function local load line stiffness, N/mm klll klcp local stiffness parallel to crack, N/mm klcn local stiffness normal to crack, N/mm A a constant for resistance to ductile tearing A2 amplitude of additional stresses in a three term series expansion of stresses B specimen thickness, mm E Young’s modulus, GPa JT total J-integral, MPa mm Mode I J-integral (or toughness), MPa mm JI JIc initiation Mode I J-integral (or toughness), MPa mm JII Mode II J-integral (or toughness), MPa mm Pmax maximum load attained in a test Q normalised value of inelastic hydrostatic stress T first non-singular stress term in elastic series expansion, MPa Ue elastic component of area under load vs. displacement curve, N mm plastic component of area under load versus displacement curve, N mm Up UT total area under load vs. displacement curve, N mm Uk extraneous system energy, N mm W specimen width, mm a angle of loading, a = 0° for Mode I, a = 90° for Mode II b a constant for resistance to ductile tearing ge elastic geometry factor related to energy Ue gp plastic geometry factor related to energy Up Dlll local load line displacement, mm Dlcn local displacement normal to crack, mm Dlcp local displacement parallel to crack, mm SZW stretch zone width

a change in mode of fracture, with Mode I crack growth occurring by microvoid coalescence and Mode II crack growth dominated 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 W and relative crack size, a0/W, on Mode II and mixed Mode I and II ductile fracture, using A508 Class 3 ferritic steel at ambient temperature. These tests were an extension of those performed by Davenport and Smith [12] and Davenport [14]. Tests were carried out using single edge notched specimens and a special fixture that allowed combinations of Mode I and Mode II loading 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 and 561 MPa for yield and tensile strengths, and 201 GPa for Young’s modulus, E. An equi-axed pearlitic grain structure was found throughout the A508 steel forging. The average grain size was approximately 16.0 lm and the lamellae thickness of ferrite within the pearlite grains was approximately 1.5 lm, while the cementite lamellae were slightly thinner, approximately 0.75 lm. Plain sided single edge notched (SEN) specimens were tested. Sidegrooves were not introduced in the specimens and consequently the direction of crack growth was not predefined. A total of 149 tests with variations in specimen thickness (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 and 0.7) were used. The matrix of tests is shown in Table 1. Many specimens were needed and to minimise time and expense of specimen manufacture, each specimen was notched using electro-discharge machining (EDM) and not fatigue pre-cracked.

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Table 1 Summary of mixed mode tests W

a/W = 0.1 B = 10

a = 0.0°

20 40 80

a = 22.5°

20 40 80

a = 45.0°

20 40 80

a = 67.5°

a = 90.0°

a/W = 0.7

B = 40

B = 10

B = 20

B = 40

B = 10

B = 20

3 4 5

7 5 3

4 1 X

4

3

X

2 X X

2

5 3 3 2 X

20 40 80 20 40 80

a/W = 0.5 B = 20

X 4

X X

8 4 3

X X

5 4 3

3 X

8 4 4

3

B = 40

X

X

X

4

3

X

3

X 4 4 4

4 3 X

4

4

3

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

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. 1a and a photograph of the fixture, together with displacement extensometers is shown in Fig. 1b. This fixture was similar to that designed by Davenport [14]. The specimens were designed so that the loading holes where located directly in line with the 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 fixture using two EN24T hardened pins. All fracture tests were carried out in air at ambient temperature in a 500 kN servohydraulic test machine. The tests were done under displacement control at a constant rate of approximately 0.6 mm/min. In each case, applied load and global axial displacement were measured and recorded. During the mixed mode experimental tests additional extensometers were used to measure local load line displacement Dlll and local displacements normal to the crack (Mode I, Dlcn) and parallel to the crack (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 energy associated with flexure of the loading assembly to be calculated. The stiffness functions were calculated from load versus local 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 specimen size and crack length for each combination of tension and shear loading. This method of testing is commonly referred to as a multiple-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 measurements were recorded at 0.5 mm intervals along the crack front on both crack faces. For the thicker specimens (B = 20 mm and 40 mm), measurements were recorded at 1.0 mm intervals on both crack faces. The average values of crack growth were then determined. 3. Analysis To interpret the experimental results two sets of analyses were conducted: an elastic analysis to determine elastic stress intensity factors and elastic geometry factors and an elastic–plastic analysis to determine plastic geometry factors. These analyses are summarised in this section. The J-integral for each test was determined from the load versus displacement record. 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 the applied load versus displacement curves, using a method developed by Sumpter and Turner [16]. Local load line displacement (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 the trapezoidal rule

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Local load line displacement, Δ lll, extensometer

Local normal to crack displacement, Δ lcn extensometer

Local parallel to crack displacement, Δ lcp extensometer

Fig. 1. Mixed mode fracture test fixture: (a) general arrangement of mixed mode fracture test fixture and (b) arrangement of extensometers on test fixture for 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 components where,

JT ¼

gp U p ge U e þ : BðW  a0 Þ BðW  a0 Þ

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 determined from finite element analyses described later.

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

JT ¼

gp ðU T  U e  U k Þ ge ðU e Þ þ ; BðW  a0 Þ BðW  a0 Þ

ð2Þ

where Up in Eq. (1) has been replaced by the elastic energy components subtracted from the total energy. UT corresponded to the total area beneath the loading portion of the load versus displacement curve and Ue was calculated from the elastic stress intensity factor using finite element analysis described later. It can be shown [17] that for plain sided specimens Eq. (2) can be rewritten as

n  2  2 o  ( )  P 2max ð1  m2 Þ fI a; aW0 þ fII a; aW0 gp gp P 2max ; 1 UT  JT ¼ þ ge BðW  a0 Þ 2klll B2 WE

ð3Þ

where Pmax is the maximum load attained in the test. Values of ge, gp and the elastic geometry functions, fI(a,a0/W) and fII(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 fixture is 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 also determined. To enable the individual contributions to be calculated, the load P was resolved into normal, Pcos a and parallel Psin a components with respect to the mixed mode loading angle, a and analysed with respect to the corresponding, Dlcn and Dlcp displacements. For mixed mode loading the individual components of the J-integral are

Jlcn

Jlcp

n   o ( ) 2   ðPmax cos aÞ2 ð1  m2 Þ fI aW0 gpI gpI ðP max cos aÞ2 ; þ U lcn  1 ¼ geI BðW  a0 Þ 2klcn B2 WE n   o ( ) 2   ðPmax sin aÞ2 ð1  m2 Þ fII aW0 gpII gpII ðP max sin aÞ2 1 U lcp  ¼ þ : geII BðW  a0 Þ 2klcp B2 WE

ð4Þ

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 a corresponds to 0° and 90° respectively. klcn and klcp are the corresponding stiffness of the specimen and test fixture for displacements 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. The elastic and plastic geometry functions used in Eqs. (3)–(5) were obtained from finite element analyses of the test specimens within 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 will be seen later in the experimental results the levels of plasticity (and hence J) were high and plane strain conditions not necessarily appropriate. For the FE analyses the fixture and specimen were assumed to be as a single unit with a rigid connection between the internal 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 and log 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 modelled using 24 rows of 32 circumferential elements, biased so that the size of each element increased with increasing distance 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) 24 rows 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

Pmax  a0  K I ¼ pffiffiffiffiffiffi fI a; ; W B W   Pmax a0 K II ¼ pffiffiffiffiffiffi fII a; : W B W

ð6Þ ð7Þ

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

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

η* Factor

3.0

2.0

1.0

0.0 0.0˚

22.5˚

45.0˚

67.5˚

90.0˚

Mixed Mode Loading Angle, α - degrees 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 deformation where gp is

gp ¼

JBðW  a0 Þ  ge U e : 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 increasing plasticity until a steady state value was achieved. It is the steady state value of gp, and the corresponding constant ge, that were 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 as illustrated 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 mixed mode 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 function of the average crack growth, Da, measured from the two fracture surfaces of each specimen. The SZW was used as a measure of blunting up to and including crack initiation. Detailed results in terms of the individual components of the J-integral are not shown here but are provided in [17]. The discussion later uses some of the results from the individual components 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 squares power law expression to the experimental data, so that b

JT ¼ AðDaÞ ;

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) for the 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 studied. Then results for mixed mode loading at a0/W = 0.5 are shown. Finally, the effect of changing a0/W on the R-curves when specimens 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 mechanisms of 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 symbols illustrate results for measured stretch zone widths (SZW) and larger symbols show results from the average of the measured average crack growth, Da. Also shown are curves corresponding to Eq. (9) fitted to crack growth data. In general, regardless of W and B, the SZW at crack initiation was constant at approximately 0.2 mm.

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Δa

SZW

4000

Mode I Toughness, JI - MPa mm

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

3000

B=40.0mm, W=20.0mm Average SZW measurements C(T) B=25mm Bn=20mm

2000

1000

0 0

1

2

3

4

5

Average Crack Growth, Δa - mm Fig. 3. Influence of specimen dimensions on pure Mode I ductile tearing resistance curves for a0/W = 0.5.

Mode II Toughness, JII - MPa mm

1000

800

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

600 Δa

SZW

Mode II tests B=10.0mm, W=20.0mm B=10.0mm, W=40.0mm B=10.0mm, W=80.0mm

400

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

200

B=40.0mm, W=20.0mm B=40.0mm, W=20.0mm Average SZW measurements

0 0

1

2

3

4

5

Average Crack Growth, Δa - mm Fig. 4. Influence of specimen dimensions on pure Mode II 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) on the R-curves was relatively small compared to the increase in toughness during further ductile tearing. For example, the effect of B on crack initiation toughness, Jinit, was such that it varied from 550 to about 900 MPa mm. Jinit was determined at Da = 0.2 mm by extrapolating fitted curves back to Da corresponding to the average SZW = 0.2 mm. Similarly tearing resistance, 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 increased from 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 and 20 mm, B and W = 20 and 20 mm and B and W = 40 and 20 mm show that substantially higher R-curves were obtained from the plain sided SEN specimens compared to the C(T) specimens. However, in the case of SEN tests with B and W = 10 and

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

4401

3500

Mode I Toughness, JI - MPa mm

3000

2500

2000

1500

1000

ao/W=0.5

ao/W=0.7

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

500

B=20.0mm, W=20.0mm Average SZW measurements

0 0

1

2

3

4

Average Crack Growth, Δa - mm Fig. 5. Influence on specimen dimensions and a0/W on pure Mode I ductile tearing resistance curves.

700

Mode II Toughness, JII - MPa mm

600

500

400

300

Mode II

ao/W=0.1 ao/W=0.5 ao/W=0.7

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

200

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

100

Fitted curves Average SZW measurements

0 0

1

2

3

4

Average Crack Growth, Δa - mm 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 than 1 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 growth in the plane of the initial crack through a combination of void growth and their coalescence. Fig. 3 also shows that some of the data points for the B = 10 mm specimens lie outside the general trends. Examination of the fracture surfaces revealed that the main cause for this was the presence of inclusions in the path of the propagating crack leading to higher amounts of tearing compared with other test results.

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3500 Mode I, α=0.0˚ α=22.5˚ α=45.0˚ α=67.5˚ Mode II, α=90.0˚

Toughness, JT - MPa mm

3000

2500

2000

1500

1000

500

0 0.0

0.5

1.0

1.5

2.0

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

3500 Mode I, α=0.0˚ α=22.5˚ α=45.0˚ α=67.5˚ Mode II, α=90.0˚

Toughness, JT - MPa mm

3000

2500

2000

1500

1000

500

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Average Crack Growth, Δa - mm 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 master curve describes the ductile tearing behaviour. The average SZW from all tests was 0.1 mm. It is also notable that overall the resistance to tearing for Mode II was less than for any of the Mode I test data shown in Fig. 3, including the R-curve from highly constrained Mode I C(T) tests. For example the lowest initiation toughness for Mode I was about 200 MPa mm, while for Mode II Jinit was 100 MPa mm. There was also a significant difference in the slope of the curve, dJ/da, with Mode II tests exhibiting 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. A shear crack initiated at the tip of the sharpened region of the blunted tip. Plastic deformation at the tip of the initiated shear crack caused the grains to deform and re-orientate themselves and the crack grew in the plane of the initial crack through a mechanism of shear localisation and decohesion [17].

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3500 Mode I, α=0.0˚ α=22.5˚ α=45.0˚ α=67.5˚ Mode II, α=90.0˚

Toughness, JT - MPa mm

3000

2500

2000

1500

1000

500

0 0

1

2

3

4

5

6

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

Toughness, JT - MPa mm

2000

1500

1000

ao/W=0.5

ao/W=0.7

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

0 0

1

2

3

4

Average Crack Growth, Δa - mm 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 initial crack lengths of a0/W = 0.1 did not exhibit ductile crack growth. This was because specimens exhibited gross yielding throughout the specimen section. Experimental results for the a/W = 0.7 tests are shown in Fig. 5 along with data for tests using 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 specimens (B = 10 and 20 and W = 20 mm). For example the average initiation toughness and slope dJ/da for these specimens was 817 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 significantly lower slope (dJ/da = 542 MPa) compared to the W = 20 mm specimens (dJ/da = 2300 MPa). Nevertheless, the Rcurve for a/W = 0.7 was not as low as that for a/W = 0.5.

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Table 2 Fitted 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 a = 0.0°

10.0 10.0 10.0 20.0 20.0 40.0

20.0 40.0 80.0 20.0 40.0 20.0

2741.3 1081.0 792.8 2772.7 1223.3 2313.1

0.9327 1.2404 0.6846 0.6777 1.0784 0.8932

611.0 146.8 263.4 931.6 215.7 549.4

2556.8 1340.9 542.8 1879.1 1319.2 2066.1

a = 22.5°

10.0 10.0 10.0

20.0 40.0 80.0

1581.6 1735.7 1607.1

0.4011 0.4441 0.4288

829.4 849.3 806.0

634.4 770.8 689.1

a = 45.0°

10.0 10.0 10.0 40.0

20.0 40.0 80.0 20.0

1329.4 1141.3 1269.0 1229.1

0.4151 0.4151 0.4780 0.4327

681.6 585.1 588.0 612.6

551.8 473.8 606.6 531.8

a = 67.5°

10.0 10.0 10.0

20.0 40.0 80.0

460.1 593.3 568.0

0.4864 0.4092 0.4327

210.3 307.1 283.1

223.8 242.8 245.8

Mode II a = 90.0°

10.0 20.0 10.0 40.0 10.0 80.0 20.0 20.0 20.0 40.0 40.0 20.0 40.0 40.0 Mode II master curve

248.2 263.7 249.7 288.8 268.5 266.8 260.3 269.1

0.5303 0.6363 0.6838 0.6658 0.6035 0.6294 0.7920 0.6278

105.7 94.7 83.1 98.9 101.7 96.9 72.8 98.0

131.6 167.8 170.7 192.3 162.0 167.9 206.2 168.9

Table 3 Fitted parameters for R-curve data for a/W = 0.7 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 a = 0.0°

10.0 10.0 20.0

20.0 80.0 20.0

2993.6 1470.6 2426.0

0.7669 0.4861 0.6454

871.3 672.6 858.6

2295.8 714.9 1565.7

a = 45.0°

10.0 10.0

20.0 80.0

1177.8 1456.2

0.4569 0.2747

564.6 935.9

538.1 400.0

Mode II

10.0 10.0 20.0

20.0 80.0 20.0

186.9 277.0 251.3

0.4833 0.5992 0.6252

85.9 105.6 91.9

90.3 166.0 157.1

Mode II master curve

254.4

0.6803

85.1

173.1

A

b

J (MPa mm) at Da = 0.2 mm

dJ/da (MPa) at Da = 1 mm

357.9 442.2 420.8

0.3889 0.4744 0.4836

191.4 206.1 193.2

139.2 209.8 203.5

a = 90.0°

Table 4 Fitted parameters for R-curve data for a/W = 0.1 Loading angle

B (mm)

Mode II a = 90.0°

10.0 20.0 40.0 20.0 Mode II master curve

W (mm)

Similar to the microstructural features observed for Mode I a/W = 0.5 tests the polished sections of a/W = 0.7 also revealed that 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 shown in 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 the master curve described earlier for pure Mode II tests at a0/W = 0.5 with an initiation toughness at about 90 MPa mm and values of dJ/da around 170 MPa. However, experiments with a0/W = 0.1 produced at higher pure Mode II tearing resistance with the 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 B and W at constant a0/W = 0.5. The range of thickness tested in pure Mode I and pure Mode II demonstrated that thickness

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effects were negligible. For each combination of tensile and shear loading (a = 22.5°, 45.0° and 67.5°), specimens with W = 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 not illustrated 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 substantial drop in toughness when the loading angle, a, was greater than 67.5°. For example, the initiation toughness for Mode I and a = 22.5° and 45° was greater than about 600 MPa mm, while for Mode II and a = 67.5° the initiation toughness was lower than about 200 MPa mm. Also, typically the average SZW width decreased with increasing Mode II from about 0.2 mm for Mode I to 0.1 mm for Mode II. Most notably the slope of the R-curve reduced dramatically from 2560 MPa to 634 MPa when the 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 shear loading initially increased the tearing initiation resistance from 147 MPa mm for Mode I to 850 MPa mm for a = 22.5, but reduced 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.

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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 Mode I 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, initially enlarging the shear component increased the mixed mode initiation toughness Jinit starting at 263 MPa mm for Mode I and increasing 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. For increasing 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°. As with the earlier tests for pure Mode I at short crack lengths (a0/W = 0.1) ductile tearing was not observed and gross yielding being the main failure mechanism. Additional tests at a = 45° and at longer crack lengths a0/W = 0.7 produced R-curves similar 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 behaviour 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 the plane of the initial crack, from the tip of the sharpened corner of the blunted crack tip. Examinations of specimens with significantly more crack growth demonstrated that the preferred direction for crack growth was in the plane of the initial crack with crack growth through shear localisation. For mixed mode loading the mechanism of crack initiation for the mixed mode tests was dependent on specimen width and the degree of shear loading. For example, it was observed that for B = 10 mm, W = 20 mm specimens there was a change from the common microvoid coalescence mechanism typical for Mode I loading to shear localisation when the loading angle was 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 negligible effect on Jinit and dJ/da in Mode I. This is in agreement with Joyce and Link [18] who tested HSLA HY100 steel using SEN specimens up to 50 mm thick. Also shown in Fig. 3 is the effect of specimen width. The results illustrate that changes in specimen 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, corresponding 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 ratios from 0.13 to 0.83 were obtained by Joyce and Link [18]. They reported nearly constant values of Jinit but with widely varying dJ/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 agree with their observations. The Mode I results shown in Figs. 3 and 5 illustrate the well known phenomena of loss of constraint [3,4] wherein the resistance to ductile tearing is increased through dissipation of the additional plastic deformation. Consequently, all of the Mode I tests for a/W = 0.5, with the exception of the B = 10 mm, W-80 mm tests, exhibited significantly higher R-curves compared to data obtained by Davenport [14] using C(T) specimens. It is beyond the scope of the paper to undertake a constraint based correction to the experimental data shown here. Nonetheless a number of schemes [19–22] for constraint correction are available, all of which introduce a second parameter in addition to J to characterise initiation and growth of ductile fracture. For example, Chao and Zhu [22] suggest that a second parameter, called A2, is used in conjunction with J to characterise ductile tearing resistance curves. Here, A2 represents the amplitude of the additional stress terms in a three term series expansion of the near crack tip stresses. Unlike the application of Q or T [20,21] to correct for constraint Chao and Zhu [22] claim that A2 is load independent under fully plastic deformation. In contrast to the use of T, Q or A2, Brocks and Schmitt 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 both included. 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 parameters. 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 to the 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 J for C(T) specimens lead to lower tearing resistance compared to high tearing resistance of SEN specimens where low (and negative values of T) corresponded to high values of J.

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When the width of the uncracked ligament in an SEN specimen and the depth of the crack are sufficient to constrain plastic flow the ductile tearing resistance curves were shallower as evidenced by the test results for B = 10 mm, W = 80 mm, as shown in Figs. 3 and 5. Normalised T-stress values for deeply cracked SEN specimens are greater than about 2 which is significantly 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 mechanism 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 would be expected to change the degree of stress triaxiality (or constraint). It is apparent that the lack of influence of constraint on tearing resistance in Mode II also placed the tearing resistance at lower values than any of the Mode I or mixed mode resistance curves. A range of theoretical models using finite element (FE) analysis tend to support the experimental findings for Mode II tearing. For example, Ayatollahi et al. [23] studied the effect of constraint on ductile fracture and, provided that the mechanism is shear localisation, they showed that there was no effect of constraint for Mode II conditions. This is entirely in contrast to the influence of constraint in Mode I ductile fracture. Other FE studies examining the effects of ductile damage formation under small scale yielding conditions [24–26] illustrate that the lower part of the crack sharpens to the extent that strain controlled crack growth occurs before ductile crack growth at the blunted upper part of the crack. This is in complete correspondence to the observations obtained from sectioned samples. However, the FE results from small scale yielding should 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 shear loading could either decrease or increase the tearing resistance. As shown by Ayatollahi et al. [23] there is only a limited influence of constraint (or triaxiality) on shear localisation and in contrast the mechanism of void growth and coalescence for Mode I tearing is strongly influenced by constraint (stress triaxiality), [19]. It is therefore apparent from the experimental results shown in Figs. 7–9 that the introduction of external shear loading modifies the local state of triaxiality. It would therefore be necessary to determine for each mixed mode loading condition the level of constraint through an appropriate parameter such as those discussed earlier for Mode I conditions. However, it is not currently possible to quantify the degree of constraint for the Mode II and mixed mode tests since further numerical work is required. Nevertheless, for pure Mode II loading Ayatollahi et al. [27] determined values of QII for the SEN specimen within the loading fixture and demonstrated that QII depended on the details of the boundary conditions between the specimen and the fixture. In the case of combined pin and contact loading between the SEN specimen and the test fixture values of QII were found to be about 0.4 at the maximum load. Values of QII in Mode II also depend on load similar to the case for Mode I [19]. However, the test results for pure Mode 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 both Jinit and dJ/da with increasing Mode II loading until the limiting condition of pure Mode II was reached. The microstructural observations 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 different 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-curve for all Mode I tests. Results in Fig. 9 show that by increasing the external shear loading (to a = 22.5°) the constraint was apparently decreased and the tearing resistance increased, with initiation values increased from 260 MPa mm to 800 MPa mm. Further increases in shear loading led to a change in failure mechanism which was retained until pure Mode II conditions. As Fig. 10 shows, the R-curves under mixed mode conditions with a = 45° were essentially independent of specimen width. This is not surprising since all specimens exhibited crack growth through shear localisation. Nonetheless, the low constraint of these specimens allowed additional plastic energy to be dissipated so that the resulting slope of R-curve (at about 520 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 individual components of J to be divided into Mode I (local component normal to the crack) and II (local component parallel to the crack), Jlcn and Jlcp using Eqs. (4) and (5). The resulting R-curves using these components were used to estimate initiation values 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 angles where the two observed failure mechanisms of ductile tearing through void growth and coalescence and shear localisation and decohesion dominate. The normal component dJlcn/da was the greatest when the dominant failure mechanism was void growth and coalescence. In contrast failure through shear localisation and decohesion was pronounced when dJlcp/da was greater than dJlcn/da. The cross over of the two sets of curves in Figs. 11 and 12 coincided approximately with the observed change in failure mechanism.

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Tearing Resistance, dJ/da - MPa

2800

Microvoid coalescence

Shear decohesion

Jlcn Jlcp

2600 2400 2200 800 600 400 200 0 0.0˚ Mode I

22.5˚

45.0˚

67.5˚

90.0˚ Mode II

Mixed Mode Loading Angle, α - degees

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.

Tearing Resistance, dJ/da - MPa

1000 Microvoid coalescence

Shear decohesion

Jlcn Jlcp

750

500

250

0 0.0˚ Mode I

22.5˚

45.0˚

67.5˚

Mixed Mode Loading Angle, α - degrees

90.0˚ Mode II

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.

6. Conclusions A comprehensive matrix of tests using a special loading fixture enabled combinations of Mode I and Mode II loading to be applied directly through the crack tip of SEN specimens. Specimens of different dimensions and crack depths ratio revealed changes 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 in high resistance to tearing. In contrast, the Mode II tearing resistance was not influenced by specimen dimensions (i.e., constraint). When SEN specimens were subjected to Mode II loading the tearing resistance was lower than for Mode I. For mixed mode loading, the transition from ductile fracture (Mode I) to shear localisation (Mode II) was a strong function of specimen dimensions. The mixed mode ductile fracture characteristics of a C–Mn steel revealed that there were two failure mechanisms: void growth and coalescence, and shear localisation and decohesion. The dominance of one mechanism over another has been shown 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 Smith also acknowledges the support of a Royal Society Wolfson Merit Award which prompted the development of this work.

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