Fracture toughness evaluation of interstitial free steel sheet using Essential Work of Fracture (EWF) method

Accepted Manuscript Fracture Toughness Evaluation of Interstitial Free Steel Sheet using Essential Work of Fracture (EWF) Method S.K. Chandra, R. Sarkar, A.D. Bhowmick, P.S. De, P.C. Chakraborti, S.K. Ray PII: DOI: Reference:

S0013-7944(18)30577-0 https://doi.org/10.1016/j.engfracmech.2018.09.026 EFM 6161

To appear in:

Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

4 June 2018 4 September 2018 17 September 2018

Please cite this article as: Chandra, S.K., Sarkar, R., Bhowmick, A.D., De, P.S., Chakraborti, P.C., Ray, S.K., Fracture Toughness Evaluation of Interstitial Free Steel Sheet using Essential Work of Fracture (EWF) Method, Engineering Fracture Mechanics (2018), doi: https://doi.org/10.1016/j.engfracmech.2018.09.026

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Fracture Toughness Evaluation of Interstitial Free Steel Sheet using Essential Work of Fracture (EWF) Method a

a

a

a

S.K. Chandra , R.Sarkar , A.D. Bhowmick , P. S. Deb, P. C. Chakrabortia*and S.K. Ray a

Metallurgical and Material Engineering Department, Jadavpur University, Kolkata 700032 b Department of Engineering Design, Indian Institute of Technology, Madras 600036 *

Corresponding Author: e-mail: [email protected]

Abstract Essential Work of Fracture (EWF) method has been used to evaluate ductile tearing resistance of interstitial-free (IF) steel sheet of 1 mm thickness. Experiments have been done on double edge notched tensile (DENT) specimens with and without fatigue precracking at three different quasistatic ramp rates. The fracture in all cases was flat with maximum specimen necking of 36% of thickness. The estimated crack tip opening angle (CTOA,

) was found to be essentially

independent of , and also of deformation rate. The critical CTOA (

) was optically estimated

using a precracked DENT specimen. Also, the tearing resistance parameters obtained from EWF tests have been compared with that determined using 3-dimentional Finite Element Modeling (FEM). Results of FEM, along with a chosen traction-separation law, were used in a 3dimentional Cohesive Zone Model (CZM) for flat fracture to simulate crack growth behavior in the DENT specimens.

Key words: Interstitial free (IF) steel; Tearing resistance; EWF method; Crack tip opening angle; Cohesive zone model.

1

Nomenclature plastic component of work done machine elastic compliance critical value of the -integral for crack initiation plastic component of ligament length measurement basis for CTOA maximum ligament length minimum ligament length strain hardening exponent load maximum load correlation coefficient of linear fitting specimen thickness maximum cohesive strength actuator displacement final actuator displacement (data at 90% load drop) final gauge/extensometer displacement (data at 90% load drop) specimen free length extension specimen width essential work of fracture specific work of fracture (total work of fracture/ligament area) total work of fracture non-essential work of fracture plastic zone shape factor 2

actual critical CTOD estimate of the critical CTOD determined from EWF testing crack extension notch root radius critical notch root radios critical notch root radios for proper measurement of critical notch root radius for proper measurement of net sectional stress ultimate tensile strength yield strength mean of the net sectional stress Cohesive energy crack tip opening angle, CTOA (optically measured) critical value of CTOA (optically measured) estimate of critical CTOA determined from EWF testing CTOA crack tip opening angle CTOD crack tip opening displacement DENT double-edge notched tension EWF

essential work of fracture

FEM

Finite element modelling

FPZ

fracture process zone

YS

yield stress

CZM

cohesive zone model

3

1. Introduction Standardization of fracture toughness parameter plays an essential role in application of fracture mechanics methods to structural integrity assessment of different engineering components and structures. The stress intensity factor ( ), -integral and crack-tip opening displacement (CTOD) are the commonly used standard fracture toughness parameters to characterize the tearing resistance in case of plain strain or near plain strain regimes[1]. However, presence of extensive plasticity at the crack tip renders the applicability of these parameters uncertain in case of thin sheets, cf. Ref [2]. Crack tip opening angle, CTOA

is the

only standardized parameter to characterize ductile tearing resistance of thin sheet specimen [3].It has been effectively employed, both experimentally and numerically, to characterize fracture resistance of thin structures in pipelines[4],aircrafts[5]and automotive industry[6]. The experimental procedure to obtain CTOA involves simultaneous measurement of the angle made by crack faces for continuous propagation of crack of large size fatigue precracked compact tension (CT) or middle tension (MT) specimen[7, 8].Such measurement requires operator skill and the experimentation is rather sophisticated for routine quality control purposes in industry. Essential work of fracture (EWF) is another method that provides a much simpler route in terms of experimentation and analysis to characterize tearing resistance of thin sheets. Consequently, it is no surprise that the EWF method has been used extensively for polymeric materials[9, 10], and also for a number of ductile metallic materials[11-15].However,for automotive grade steel sheets only a few reports on this topic exist in open literature [16]. Based on idea of Broberg[17], Cotterell and Reddel[18]proposed the EWF method to separate out the essential work of fracture, that characterizes a material property for the process of crack propagation, from the total work of fracture. The necessary criteria for proper implementation of the EWF concept have been given by Cotterell and Reddel[18] as: (a) The ligament of the double-edge notched tensile (DENT) specimen is under plane state of stress. (b) The plastic zone is confined to the notched ligament. (c) The ligament is fully yielded prior to crack initiation and small enough to avoid edge effects.

4

To meet the above conditions, a range of valid ligament length ( ) values, e.g. , where

is the specimen width, was empirically proposed by Cottrell[18]. This ligament

length range is generally considered as the “rule of thumb” for selecting ligament lengths for DENT specimens in EWF testing. In this equation, the lower limit

is necessitated because in

sheets, cracking initiates under a complex 3D stress state, and there is a transition to a quasiplane stress state only after a small extent of crack growth. The upper limit

is meant to

ensure full yielding of the ligament before onset of fracture. When these criteria are fulfilled, an autonomous Fracture Process Zone (FPZ) can be identified inside the large plastic zone near the crack (Fig. 1). The critical process of necking and fracture is confined in this zone (FPZ). Subsequently, the total energy can be partitioned into two components: the work of fracture ( associated with FPZ, and the plastic work ( While

)

) dissipated in the outer plastic zone.

which scales with ligament area depends only on ligament length,

depends on

volume of the plastic zone. Accordingly, the total energy can be expressed as follows (1) Here is specimen (original) thickness and

is a shape factor denoting shape of the plastic zone.

After normalizing by the ligament area, , the specific work of fracture,

can be written as (2)

Where

is the Essential Work of Fracture. The following important relationship between the

final displacement,

and ligament length

has also been proposed by Cotterell[19] (see also

Ref [18]): (3) where

was identified as the inferred average critical crack tip opening displacement and

the inferred CTOA from EWF testing. The EWF method thus yields four parameters: (Eq. (2)), and

and

as ,

(Eq. (3)) pertaining to tearing resistance. To obtain these parameters

DENT specimens of different ligament length ( ) are ramp loaded to complete fracture.The focus of research in EWF method has been on the parameter

, and also its relation with

(the critical

value of the -integral for crack initiation).Considerably less attention has been paid to the

5

parameter

and its relation with

(the critical value of CTOD for crack initiation) and

virtually little attention has been paid to

and its relation with critical CTOA (

).

The starting point of EWF testing is the selection of valid ligament length ( ) range for the any material; any significant deviation that violates the basic conditions required for this method can lead to significant inconsistency in EWF parameters [20, 21]. The “thumb rule” proposed by Cotterell and Reddel [18] has been found unsatisfactory for some materials. For example, the lower limit (

) was reported to be much higher than the proposed value of

for low

density polymeric film [22, 23]. For metallic materials (Al and Zn alloys) also Marchal et al. [24] found (

as

–

. On the other hand, the upper bound of the proposed ‘thumb rule’

)has been considered to be conservative by several researchers, cf.[25, 26]. EWF testing on thin sheets of different materials [11, 12] reveals that

adequately

characterizes the fracture toughness of the material for a given sheet thickness at least for operational purpose. The dependence of the measured

on the notch tip radius,

is well

recognized in the EWF literature[27]. As indicated earlier, a few researchers [12, 28] even recognized

as a measure of critical value of -integral for crack initiation (

with determining a -independent independent

By analogy

[29], this leads to the possibility that for determining a -

it will be necessary that

, where

is a critical value that depends on

material and possibly on sheet thickness. Initially,

was identified as critical crack tip opening displacement or CTOD for crack

initiation ( )[18]. However subsequently, it was pointed out that

is the opening across the

FPZ at which separation occurs in a fully propagating crack and this definition differs in normal usage from that of CTOD[19, 30]. It is possible that a critical tip radius criterion applies for the parameter as well. The parameter

is the angle of opening of the FPZ for a growing crack;

from a computational model with an initially precracked specimen, it was proposed that , the critical CTOA, only if the FPZ is infinitesimally small as compared to the outer plastic zone, and

otherwise [19, 30].

6

Figure 1: Double-edge notched tensile (DENT) specimen geometry. A practical method for determining -independent EWF parameters, would be to test fatigue precracked DENT specimens. But this would require some experimental skill, and considerably increase the duration of a test campaign.In this context, researchers from the authors’ laboratory have argued [31] that for industrial applications, -independent EWF parameters are necessary only for integrity assessment. By analogy with the very common Charpy V-notch impact testing for routine material qualification and quality control, it should be adequate to test DENT specimens with a pre-fixed

which is small, but achievable by machining, and precracking of

the specimens should not be required. These authors also noted that among the EWF parameters determined by testing notched DENT specimens, statistically

was the best determined

parameter for a DP590 grade dual-phase steel sheet. It was speculated that since

deals with a

propagating crack, it may not be seriously affected by the constraint for crack initiation, i.e., the value of , and as such should qualify also for integrity assessment. Sarkar[32] confirmed this idea in case of a DP780 grade dual-phase steel sheet (

mm) and showed that the

calculated from EWF matched closely with the optically measured critical CTOA ( DENT specimens.

7

value

) values on

However, in the context of comparing the relations between ( -independent)

,

data in metallic sheets with their fracture mechanics counterparts ( ,

, the fracture

,

,

and

morphology of sheet metals [33] should also be considered.It is well reported that in sheet DENT specimens ductile crack typically starts with an initially flat crack, but may twist into a slant crack after some amount of crack extension. In such a situation, the initiation values pertain to flat mode in which crack initiates; but their comparison with

and

and

respectively

determined by back extrapolating of data pertaining to dominantly slant mode of fracture appears to be of doubtful validity. It must also be emphasized that this limitation does not apply for comparison of

and

, because the relevant test data for both pertain to the same fracture

mode. In fracture mechanics analysis of thick specimens, finite element modeling (FEM) based numerical simulations, and cohesive zone modeling (CZM) for simulating crack growth have played an important supportive role [34, 35]. In contrast, reports on application of 3 dimensional (3-D) (to replicate specimen necking) FEM[12, 30]and CZM based simulations[36-38]for sheet specimens are relatively few. In the reported 3-D CZM modeling [12, 19, 39, 40], experimentally identified crack initiation point had been used for tuning the cohesive parameters for crack growth simulations in sheet DENT specimens. It appears that there is considerable scope of research in this area. The brief survey presented above underscores the potential of the EWF method for both routine quality control and material qualification, as well as integrity assessment of automotive and similar grade steels. Previous research from the authors’ laboratory in this area dealt with two dual-phase automotive grade steels [31, 32]. These studies pointed to the need of extending the studies to relatively high ductility steels with higher work hardening rates. Interstitial-Free steel offers a good choice as a candidate material for such verification. The main aim of the present investigation was to assess the usefulness of EWF approach in case of an interstitial-free (IF) steel of thickness ( ) 1 mm and to verify the consistency in using the EWF parameters for characterization of tearing resistance. The influence of test parameters e.g. deformation rate and notch root radius on the EWF parameters were also examined. In addition, for notched specimens (

mm), a finite element simulation using 3-D cohesive zone model (CZM)

was adopted for simulating the crack growth. For this model, the tuning parameters were 8

determined by identifying the crack initiation points by comparing test data with results from a 3-D FEM formulation for non-growing crack. The EWF analyses of test data and data generated by the CZM based simulation for growing cracks were also compared.

2. Experimental details 2.1. Material The investigated material is an interstitial-free (IF) steel sheet of 1 mm thickness ( ). The tensile properties obtained at three different nominal strain rates are reported in Table 1. The tensile test records did not show any evidence of strain ageing. With increase in nominal strain rate, the yield strength and ultimate tensile strength increased slightly, whereas the percentage of elongation to failure decreased. Table 1: Tensile properties at different strain rate Strain Rate Yield Strength

Tensile Strength

Total Elongation

Strain hardening

(s-1)

(MPa)

(MPa)

(%)

exponent ( )

10-4

162

324

44

0.241

10-3

168

338

43

0.247

10-2

184

348

42

0.248

2.2. Testing of DENT specimens DENT specimens (Figure 1) of 10 different ligament lengths ( ) ranging from 2-10 mm were machined in longitudinal (L-T) orientation having a dimension of 90mm x 30mm x 1mm. All specimens had same notch tip radius (

of 0.1 mm. The EWF testing of DENT specimens were

carried out in INSTRON8501R servo-hydraulic universal testing machine. Three different ramp rates of 0.6,0.06 and 0.006 mms-1, corresponding to nominal strain rates of 10-2,10-3and 10-4s-1, were chosen for tensile loading of the specimens. An extensometer of 25 mm gage length and 50% range was used for the EWF tests. All the tests were stopped at 90% load drop of the peak load. This was done to prevent any potential damage of the extensometer and the fracture surface that was later subjected to fractographic observation. Six specimens were fatigue precracked to varying ligament lengths ranging from = 4.4 mm to 8.3 mm and were tested at a ramp rate of 0.006 mm.s-1. Fatigue precracking of DENT specimen was carried out under load-control mode 9

with a sinusoidal waveform following ASTM E1820 standard [41]. The specimens were flipped (horizontally and vertically) several times to maintain the symmetry of the precrack. To determine the critical CTOA (

) value, precracked DENT specimens of ~ 21 mm were

tested adopting the essence of ASTM E2472 standard [3]. CTOA value was determined at three different ramp rates of 0.6, 0.06 and 0.006 mm.s-1. For each ramp rate, the DENT specimen was loaded up to a predetermined displacement level, the cross-head motion was arrested to allow load relaxation, and then the specimen was unloaded to a load value of 1 kN. This load-relaxunload sequence was repeated several times and the whole load-displacement curve was generated. During the each holding (relaxation) step, the crack tip visible at the specimen’s surface was photographed by using a light microscope integrated with a digital camera. In order to achieve high quality micrographs, the side surfaces of the specimen were polished before the test. By this technique, a series of micrographs was obtained for each test. It may be noted that tunneling effect is a major concern for testing metallic sheets [7, 8]. Accordingly, the unloading step was included to have an alternative means to calculate the crack length using elastic compliance method, if optical measurement of crack tip were found unsatisfactory.

3. Results and discussion 3.1 Determining the valid ligament length range For determination of valid range, the linearity of the present study. The

and

and

with was considered in the

plotsin Fig. 2 show good linearity even at the ligament

length range of 2-10 mm, though the minimum ligament, proposed by Cottrell [18].For the determination of

clearly violates the “thumb rule”

, a stress criterion based on Hill’s theory

[42] derived from an ideal elastic-plastic material model was considered. According to this theory, when a DENT specimen is fractured under plane stress condition then: (1) the net sectional stress,

(load divided by cross section area along ligament) at maximum load (

becomes independent of ; and (2) 44] show that can deviate from

attains a value of ~

)

. Existing literatures [21, 43,

is nearly independent of in pure plane stress domain, though the value of . Considering that Hill’s criterion may not be suitable for a strain

10

hardening material, based on experimental results Clutton [45] suggested the use of the mean of the net sectional stress(

) at

Figure 2: (a)

for a stress based criterion.

and (b)

Figure 3 depicts the variation of

plots to determine valid range.

with ligament length ( ) of the specimens used in the

present study. It is evident that the value of

becomes independent of , specifically in the range

mm.Further the value is much higher than

11

for all the ramp rates and also for

precracked specimens. Hill’s criterion (

~

) thereby becomes unimportant for such high

strain hardening material. An empirical limit

appears to be a better choice for the

present test material, especially for precracked specimens (Figure 3b).However, this is open to verification for other materials covering a wide range of strain hardening materials.

12

Figure 3: plots to determine valid range for (a) notched specimen at deformation -1 rate0.006 mm.s , (b) precracked specimen at deformation rate 0.006 mm.s-1,(c) notched specimen at deformation rate 0.06 mm.s-1and (d) notched specimen at deformation rate 0.6 mm.s1 . According to the criterion proposed by Clutton, a mixed mode stress state develops in the specimen with 1

mm for all deformation rates, especially for the higher rates, viz. 0.06 mms-

and 0.6 mms-1, as shown in Fig. 3(c) and 3(d), respectively. Therefore, this ligament length was 13

excluded for plane stress analysis. Moreover, the number of specimens to be tested and their distribution over the valid ligament range has also been of interest to researchers. Some results on polymeric materials indicate that more than 20 specimens are required for satisfactory standard deviation over mean value of

< 0.1 [10, 45]. However, for metallic material much

smaller number of specimen (6-8) was found adequate for sufficiently good statistics[11, 19]. 3.2 Effect of notch radius on the tearing resistance parameters evaluated from EWF testing The -dependence of

and

has been shown in Figs. 4(a) and 4(b) respectively, for both

notched and precracked specimens. It is observed that both the parameters follow satisfactory linear relationships with ligament length. Results from these linear fits are shown in Table 2.From the results it can be inferred that the value of the parameters

and

reduced on reducing notch tip radius ( ) by precracking: the value of

are significantly

is reduced by ~48% and

by ~15%. Similar observation has been reported by several other researchers [46, 47].In the present instance, the value of

is also increased by ~17% with precracking. In contrast,

is

found to be independent of . Therefore, for integrity assessments (in terms of -independence), is a better parameter than and

and

, at least for the present test material. Now,

areassociated with the crack initiation phenomenon, whereas

characterizes a growing

crack. Therefore, the effect of crack tip blunting will be more pronounced for the parameters and

than for

because after certain amount of crack growth the effect of blunting becomes

insignificant. Interestingly, the findings indicate that critical notch root radius for

,

is much less sensitive to

and the critical notch root radius for

the same. Therefore, the percentage of variation in that

,

and

than

should not be with

suggests

and

) when

.

Table 2 (set 1) demonstrates negligible variation in the parameters (

,

,

calculated using either extensometer or actuator data. Therefore, all these parameters ( and

, so the

,

,

) could be determined by EWF tests using the actuator data, at least for the material

under investigation. Rink et al [44] made similar observation and suggested to measure the displacement using the actuator cross-head travel. A practical implication of this observation is: use of extensometer may altogether be avoided in EWF tests. 14

Since the tests were stopped before final fracture (90% load drop from

), the

displacements at fracture has been approximated using linear extrapolation from the last segment of the load displacement plots. But the choice of a linear relation for extrapolation introduces a degree of subjectivity, though Table 2 (Set 1 and Set 2) depicts that extrapolated data (up to 100% load drop) and last data point from experiment (90% load drop from

) give almost

same result. Thus, the final data point has been considered as the data point of final separation in this study.

Figure 4: (a) and (b) deformation rate 0.006 mm.s-1.

plots of notched and pre cracked DENT geometries at

15

Table 2: Comparison of , , and for notched and precracked DENT specimen calculated from extensometer and actuator data, with and without extrapolation ( Results from extensometer data) (MJ.m-3)

DENT specimen

(kJ/m2)

(mm)

Notched

0.997

44.26

213.62

0.999

Pre cracked

0.997

51.83

132.7

0.967

(

0.1562 = 17.9o)

0.572

(

0.157 = 17.99o)

0.487

(Results from actuator data) Fitting Equation (MJ.m-3)

DENT specimen

DATA SET 2 Extrapolated data (100% load drop from )

DATA SET 1 Test Data (90% load drop from

)

Fitting Equation

(kJ/m2)

(mm)

Notched

0.997

44.26

213.7

0.999

Pre cracked

0.996

51.83

133.2

0.967

(

0.1558 = 17.85o)

0.578

(

0.1573 = 18.03o)

0.492

(Results from extensometer data)

Fitting Equation (MJ.m-3)

DENT specimen

(kJ/m2)

(mm)

Notched

0.996

44.26

214.12

0.999

Pre cracked

0.997

51.68

133.8

0.967

(

0.156 = 17.87o)

0.580

(

0.1575 = 18.04o)

0.490

(Results from actuator data)

Fitting Equation DENT specimen

(MJ.m-3)

16

(kJ/m2)

(mm)

Notched

0.996

44.26

214.2

0.999

Pre cracked

0.996

51.48

134.8

0.967

(

0.1552 = 17.79o)

0.590

(

0.158 = 18.1o)

0.496

Additionally, (shape factor) is a material dependent factor, that in combination with yields the total energy consumed in screening plastic zone[18]. For determination of knowledge of

is required. A circular

, the

factor is usually associated with metallic material

[12].However, it is observed from Fig. 5 that the plastic zone is more of an elliptical shape for longer ligaments and circular for shorter ligaments. Similar observation has also been reported for zinc sheets [11].Thus factor determination is difficult as a proper measurement of plastic zone height is required. As locating the plastic zone boundaries from the photographs (Fig.5a and b) is difficult,

is used instead of

throughout the present study.

Figure5: Shape of plastic zone for (a)

mm and (b)

mm.

3.3Effect of ramp rate on the tearing resistance parameters evaluated from EWF testing It has been observed that the ramp rate has a significant effect on the shape of the loaddisplacement traces of notched DENT specimens: the maximum load ( increase of ramp rate. The plots of

and

) rose up with

for three different ramp rates of 0.6, 0.06

and 0.006 mm.s-1 are shown in Fig.6a and b, respectively. To ensure that the selected data points are in plane stress, ligament length ( )

has been chosen as discussed earlier in Section 3.1.

17

Summaries

of

,

,

and

for

different

ramp

Table3.FromTable3, no clear trend in the variation of although

rates

have

been

presented

in

with deformation rate is discernible,

is observed to increase with the increase in ramp rate. This observation is

consistent with the findings of other researchers [46, 48] for polymeric materials. The values of depict clear increase with increase in ramp rate. The independent (

parameter is found to be virtually

variation) of deformation rate in the specific range of 0.6 to 0.006 mm.s-1.

However, it should be noted that the same conclusions may not hold good for dynamic deformation rates (where inertial effects would be important).

18

Figure 6: (a) rates.

and (b)

plots for notched DENT geometries at three different ramp

Table 3: Effect of deformation rate on tearing resistance parameters Fitted equation: Specimen

DENT (Notched)

Ramp rate (mm) (mm s-1)

3-10

(MJ.m-3) (kJ/m2)

(mm)

0.006

0.997

44.26

213.62

0.999

0.1562 ( = 17.9o)

0.572

0.06

0.997

55.14

210.1

0.999

(

0.158 = 18.10o)

0.64

0.6

0.997

58.10

238.10

0.999

(

0.159 = 18.22o)

0.70

3.4 SEM of the fractured specimens The scanning electron microscopy (SEM) reveals four different zones associated with the steps of fracture process, Fig.7. The machined notch is followed by the fatigue precrack and then appears the stretched zone, where crack tip blunting takes place, prior to the final step of ductile 19

fracture. It is observed from Fig.7 that the magnitude of necking increases quickly over a short distance from the starting notch tip. Beyond this distance, necking reaches a constant value (maximum ~ 36% of t) over the rest of the ligament. Moreover, it may be noted that the extent of tunneling was not severe in the present study (Fig.7), justifying the measurements of crack extension optically on the specimen surface. Figure 8 shows a predominant flat fracture of the ligament. Clearly, there is no transition from flat to slant fracture in the present test material. The same observation has been made for all the ligament lengths. Apparently, the extent of necking was adequate to restrain the initiated crack to grow in the flat mode.

20

Figure 7: Representative micrograph of different zones on the precracked fractured ligament (

7.96 mm).

Figure 8: SEM image showing predominant flat fracture(

21

7.96 mm).

3.5 CTOA

determination by optical measurements

The ASTM standard (ASTM E2472[3]) recommends the use of large sized C(T) or M(T) geometry with anti-buckling guide for CTOA determination. In the present study specimens with DENT geometry have been used as it does not exhibit any buckling problem and thus the use of anti-buckling fixture was avoided. Precracked DENT geometries (

21 mm) were tested under

displacement-control mode at three different ramp rate (0.6, 0.06 and 0.006 mm.s-1). Following the test method described in Section 2, a series of micrographs was obtained for each test. From each micrograph, CTOA was measured along with the crack extension,

,on the specimen

surface. In the present study ,the two-point method described in ASTM E2472 has been followed [4], where the crack tip is always included in the calculation of the CTOA. Thus firstly the crack tip in the micrographs has been located, and then pair of points along the crack profile (in the range 0.5–1.5 mm behind the crack tip) from the tip has been selected for CTOA determination (Fig.9) using the following equation: (4) where

is the distance between the two points located at the position i, and

is the distance

between the crack tip and the location (Fig.9).

Figure 9: A representative micrograph showing measurement of CTOA. It should be noted that an accurate measurement of CTOA requires proper selection of length

[49].A high value of

may introduce size and geometric effects on CTOA 22

measurement. In this analysis,

values within the range from 0.5 mm to 1.5 mm were chosen

according to ASTM E2472 [3] and within this range, three measurements were done for each micrograph. The average was taken as the measure of CTOA. From Fig.10 it can be observed that the relation for critical CTOA vs.

remains unaffected by the choice of

Figure 10: Choice of CTOA measurement basis

.

for DENT geometries.

For convenience, subsequent CTOA values (presented in this paper) were determined 1.5 mm behind the crack tip. The CTOA vs

curves for different deformation rates have been plotted

in Fig.11. The plot follows the usual trend that from high initial values, CTOA decreases, and as the crack growth stabilizes it reaches a fairly constant value. It may be noted that the transition distance for all the cases is ~5.5 mm with negligible variation of less than 2%. This is the distance over which the crack is traversed to attain a steady CTOA value,

The comparison between optically determined critical CTOA (

.

) and the

estimated from

linear extrapolation of the EWF test data of DENT specimens having ligament length, mm has been shown in Table 4 (Data Set 1) for different ramp rates. The comparative results indicate that

values are identical to the

values within a maximum variation of 4%. This 23

leads to the possibility of using

as a measure of critical CTOA

at least for the present test

material. Evidently, EWF testing of notched DENT specimen can be used as an alternate approach to determine the critical CTOA (

).

Figure 11: CTOAopt-∆a plots for DENT geometries at different deformation rate Table 4: Comparison of Critical CTOA Ramp rate (mm s ) ( )

with

Data set 1

-1

(mm)

0.006

from EWF

Data set 2 (mm)

17.54o

17.90

0.06

3-10

0.6

18.10 18.22

from EWF

6-10

17.54o 17.91o

Similar comparison has also been made for the DENT specimens having ligament length, mm (Data Set 2, Table 4). This ligament length range was purposefully chosen such that the ligament lengths exceeded the minimum transition distance (~5.5 mm) evaluated from the CTOAvariation of

curve (Fig.7 and 8). It is found that for this ligament length range the maximum with critical CTOA (

) is further reduced to 2%. It would be interesting to

24

examine this aspect with higher number of specimens with ligament lengths higher than the maximum value used in the present study. 3.6 Finite element modeling The finite element modeling in support of the research reported above had two steps. In the first step, 3-dimensional (3D) FE simulations for non-growing crack had been carried out for all the notched DENT specimens. In the second step, 3-D cohesive zone model (CZM) simulations were done by using the parameter determined from the first simulation. The objective was to examine if such CZM can successfully simulate the deformation and crack growth, and thus reproduce the EWF test results. However, an important conclusion from the results already presented is: strictly, an extensometer is not necessary for the EWF tests, and actuator displacement can very well be used in place of gauge extension. Thus it makes sense to carry out the FE simulations without requiring gage extension values. Now, replacing gauge extension by actuator displacement is justified for EWF tests because “machine” (i.e., the segment of the load train excluding the free length of the specimen) deformation is elastic, and for a test conducted from zero load to zero load (i.e., complete specimen fracture), it will have no effect on the computed measured

or

. But this does not apply for intermediate points in the load-displacement traces well

before fracture. Therefore, it was decided to carry out the FE simulations using specimen free length extension. The machine free length extension values

for the experimental test data were

determined from the actuator displacement values using the equation: is the “machine” elastic compliance. In the present study

, where has been

determined using the calibration procedure described by [50] and the estimated accuracy of the computed free length extension values was about 3 m. 3.6.1. FEM modeling for non-growing crack For the first step, 3-dimensional (3D) time independent large strain FE simulations for nongrowing crack had been carried out for all the notched DENT specimens ( range: 3 to 10 mm) for ramp rate of 0.006 mm.s-1. For this purpose, hexahedral elements with mesh size 0.2 mm around the notches, and 1 mm in the rest of the specimens were used, and element thickness was kept as 0.125 mm (i.e., the sheet specimen was divided in 8 slices). The commercial finite 25

element software, ABAQUS 6.10, was used for these simulations. The simulations were continued well beyond the maximum load in each case. By comparing the results for 0.2 mm and 0.1 mesh sizes around the notch for the specific case of

mm, it was found that for the

intended application, 0.2 mm mesh size is adequate. The details are not presented here for the sake of brevity. As mentioned above, these simulations were done to identify the crack initiation points. Figure 12 shows an example; it compares the experimental, and the simulated load-specimen free length extension plots of DENT specimen having ligament length,

mm. As the figure

shows, the simulation results match well with the early portion of the experimental curve, but deviates from the experimental plot at Point ‘b1’.As this simulation does not incorporate damage, this deviation can be ascribed to crack initiation. This method of identifying the initiation point ‘b1’ involves some subjectivity, which, for the present resolution of test data and analyses, should be small. In this figure it is also noted that crack initiation point appears before reaching the maximum load (Point ‘a’). Similar results were obtained for all the ligament lengths. During the course of the test with

mm, the crack tip was also photographed at

several intermediate stages. The point of first appearance of a surface crack was at a slightly higher displacement than the corresponding b1 point. This is consistent with crack initiation in the mid-section, which then spreads to the surface.

26

Figure 12:FEM simulated load-displacement plot superimposed on the experimental curve for mm, point ‘b1’ is the crack initiation point. Considering isotropic hardening and using the initial notch tip radius,

= 0.1 mm, the CTOD

obtained from FE simulation at point ‘b1’in Fig.12 was ~ (0.86 – 0.2) = 0.66 mm (Fig.13b). The optically measured CTOD value just before the cracking was ~ 0.62 mm (Fig.13a). CTOD is defined here as the distance between two points situated on the intersection of the crack faces with two perpendicular planes at 45º from the crack plane minus the initial opening [11]. The CTOD values for crack initiation (

for different ligament lengths have been shown in Fig.14.

Figure 13: (a) Optically measured CTOD at the crack tip of the precracked specimen ( =15.2 mm) prior to crack initiation; and (b) CTOD determined at point ‘b1’ of the Figure 14 (a). value was calculated at the crack initiation point‘b1’ using the following equation (5) In the above equation of work done

, and

for DENT specimens [1]. The plastic component

was determined by subtracting the elastic component of

initial elastic compliance). The

(obtained using the

values at crack initiation thus determined for the different

ligament lengths have also been shown in Fig. 14.

27

Figure 14 shows slight variation of the computed

at initiation and

lengths. Further the variation is more for lower ligament lengths (

values with ligament

3 and 4 mm). Part of the

variation could arise from the subjectivity in identifying the initiation point as mentioned earlier. It is also possible that higher constraint for crack initiation for the two smallest ligaments leads to lower initiation values, compared to those for the higher ligament lengths. But, this requires further scrutiny. Considering the entire data set, the average reasonable

agreement

with

the

for

notched

value of 199.8 kJ/m2 shows

specimens

(~213.6

kJ/m2

with

extensometer/actuator travel data, Table 2). This suggests that necking energy contribution to is relatively small in the present instance. It will be interesting to carry out such comparison for precracked specimens. However, the mean CTOD, considerably higher as compared to the

value from Fig.14 (0.64 mm) is

values determined for notched specimens (0.57 - 0.58

mm depending upon extensometer/actuator travel data, Table 2). It is possible that the difference possibly reflects the difference in the definitions of these two parameters; here too, such comparison appears desirable for precracked specimens.

Figure 14:Variation of Jp and

with ligament length

3.6.2 FEM modeling for growing crack: CZM modeling For the mode 1 cohesive traction-separation (

-

) law, the PPR model by Park et al.[51,

52] has been used. This relationship takes the form ( see Alfano et al [53]): 28

In this equation,

= normal cohesive traction I;

opening displacement,

= final value of

= mode I cohesive energy;

at which

= normal

; and α is a shape parameter

defining the softening part of the traction-separation law. m, the non-dimensional exponent, may be written as (7) where

is the critical crack opening displacement at crack initiation, at which

, the

maximum traction. There are thus four independent unknown parameters:

and α, which need to be

determined in order to fully define this cohesive interaction. However, in this report the mode I cohesive energy,

was calculated using the relation given by Cornec et al [54] (8)

This reduces the number of independent parameters to 3. In order to determine the parameters, the damage initiation point has been determined by identifying the point of deviation of the experimental and 3D-FE simulated (for a non-growing crack) curve (point ‘b1’, Figure 12, section 3.6.1). The axial stress at the point of damage initiation (i.e., at point ‘b1’), and the corresponding

were extracted from the simulation results, and taken as

and

respectively’. CZM analyses almost to complete fracture were carried out by implementing the above cohesive law and considering a zero thickness layer of cohesive elements (COH3D8 element) along the crack as a user-defined element (UEL) in the FE code ABAQUS 6.10. Due to the symmetry of the DENT geometry, half specimen has been simulated. The meshing used for these models were the same as that used for the non-growing crack simulations. The shape parameter α was estimated by choosing the α values that led to matching between the experimental and simulated load-displacement plots to large displacement levels. Following this procedure, the parameters that have been determined for DENT geometry having ligament length, = 10 mm are: the cohesive energy

= 300 kJ/m²,

α = 1.15. These values were used in all the 3-D CZM analyses.

29

mm,

MPa, and

The numerically simulated load-displacement curves show reasonably good agreement with the experimental plots, Fig.15. A careful examination, however, shows excellent match for the three highest ligament lengths, but perceptible deviations appear at lower ligament lengths. This needs further careful scrutiny covering the best method of tuning the CZM for both operational study, in terms of practical application of the model, and theoretical aspects. It must be noted here, while the CZM modeling is being used to simulate the crack growth under essentially (quasi-) plane stress condition, the input data for tuning this model correspond to higher constraints, which, referring to Fig.14, are perceptible for the two lowest ligament lengths.

Figure 15: CZM simulated load-displacement plots superimposed on the experimental one for different at deformation rate 0.006 mm s-1. The consequence of mismatch with decreased ligament lengths, particularly

and 4 mm,

was examined by applying EWF analysis for both the sets of data is shown in Fig. 15. The results, presented in Table 5 and Figs.16a and b, demonstrate that the energy based parameters and

were affected, whereas the displacement based parameters

and

obtained for

these two sets showed good correspondence. Apparently, tuning of the CZM can be carried out using the procedure described in this report for a single ligament length, chosen somewhere in the middle of the validity range defined in Fig. 3(a) where

becomes effectively constant.

Also, for EWF analysis using simulated data, the usual method of identifying outliers need to be practiced. This also requires further examination. 30

Table 5: Comparison of simulated and experimental data Fitted equation: SPECIMEN

DENT (Notch )

(MJ.m-3)

Experimental 0.997 Simulated (CZM)

0.994

(kJ/m2)

(mm)

44.26

213.62

0.990

41.6

250.2

0.990

31

(

0.1546 = 17.71o)

(

0.155 = 17.76o)

0.58

0.576

Figure16: Comparison of simulated and experimental (a) deformation rate of 0.006 mm s-1.

and (b)

plots at a

4. Conclusions 1. All the parameters that have been evaluated using EWF tests,

,

,

and

can be

simply determined from the actuator displacement ( ) by testing DENT specimens, an extensometer is strictly not required. This will simplify the EWF testing. 2. For routine material qualification and quality control, the parameters,

,

,

and

determined using specimens with machined notches of a fixed (and sufficiently small) notch tip radius

can be used to characterize ductile tearing behavior of the material. In fact,

for application purpose precracking is not required; thus it will be very effective for industrial application.

3. For potential application in integrity assessments, fatigue precracking of the specimens is necessary for determining the parameters

and

, but not for determining

, possibly if

is sufficiently small (e.g. 0.1 mm for present test material). 4. In the quasistatic deformation rate range 0.6 to 0.006 mm s-1, the exhibit any definite trend, whereas, the parameters strain rate. However,

and

remained constant (a variation of ~

range. 32

parameter did not

increased with increase in in this deformation rate

5. The optically determined critical CTOA (

) values by testing DENT specimen of larger

ligament length ( ) at different ramp rates were comparable to the

values estimated from

the EWF tests. 6. 3-D CZM was used for numerically simulating crack growth in the notched specimens. The results were generally satisfactory, and the analyses led to identification of several areas for further detailed study. Acknowledgement The authors thank M/s Tata Steel, India, for providing the test material.

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38

Highlights



Evaluation of ductile tearing resistance of interstitial-free steel sheet by EWF method.



Dependence of all EWF parameters on strain rate and notch root radius.



Verification of EWF method as an alternative for derivation of critical CTOA value.



Comparisons of experimental results with that obtained from numerical simulation.

39