Dependence of ductile crack formation in tensile tests on stress triaxiality, stress and strain ratios

Dependence of ductile crack formation in tensile tests on stress triaxiality, stress and strain ratios

Engineering Fracture Mechanics 72 (2005) 505–522 www.elsevier.com/locate/engfracmech Dependence of ductile crack formation in tensile tests on stress...

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Engineering Fracture Mechanics 72 (2005) 505–522 www.elsevier.com/locate/engfracmech

Dependence of ductile crack formation in tensile tests on stress triaxiality, stress and strain ratios Yingbin Bao

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Impact and Crashworthiness Laboratory, Massachusetts Institute of Technology, Room 5-218, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Received 17 August 2003; received in revised form 16 April 2004; accepted 18 April 2004 Available online 17 July 2004

Abstract An experimental and numerical study on ductile crack formation in tensile tests was conducted. Five different specimens including flat specimens, smooth round bars, notched bars (two types) and flat-grooved plates were investigated. Von Mises equivalent strain to crack formation, stress triaxiality, and stress and strain ratios at critical locations, were obtained. Accuracy of the Bridgman formulas for stresses in necked round bars, and McClintock’s model for flat-grooved plates, were studied. A relationship between the stress triaxiality and equivalent strain to crack formation was determined in a high stress triaxiality range for Al 2024-T351. More importantly, it was found that equivalent strain and stress triaxiality are the two most important factors governing crack formation, while stress and strain ratios cause secondary effects. It appears possible to make a good prediction of crack formation with equivalent strain and stress triaxiality. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Stress triaxiality; Equivalent strain; Stress and strain ratio; Experiment; Numerical simulation; Ductile crack formation

1. Introduction In many practical problems, equivalent strain to crack formation at the critical location in tensile specimens is taken as a measure of ductility [1–5]. In this paper, ductility is defined as the ability of a material to accept large amounts of deformation without crack formation. Different types of tensile test pieces are often used, such as flat and round specimens. However, equivalent strain to crack formation is dependent of the stress state, which is related to the shape of specimens, and clearly it is not the same in tensile specimens with different geometries. Consequently, different specimens do not necessarily have the same equivalent strain to crack formation. Clausing [1] observed from a series of tests on uniaxially loaded wide, flat, double face-grooved plates that tensile ductility of structural steels was substantially reduced when the strain state was changed from axisymmetric (round specimens) to plane strain (flat-grooved

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Tel.: +1-617-253-6055; fax: +1-617-253-8125. E-mail address: [email protected] (Y. Bao).

0013-7944/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2004.04.012

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plates). The degree of difference was found dependent on the hardness of the steels. The interpretation of the difference was qualitative and limited to the degree of physical constraint. However, for engineers the outputs of structural analysis are the components of stress and strain tensors. Clearly, a qualitative explanation is not sufficient. In tensile tests on specimens with different cross-sections, the stress triaxiality, and stress and strain ratios are likely to be different, especially for flat plates and round bars. McClintock [3] and Rice and Tracey [6], by studying growth of long cylindrical voids and spherical voids, respectively, have shown that fracture strain of ductile metals is strongly dependent on hydrostatic stress. Atkins [5,7] also pointed out that ductile crack formation should depend on hydrostatic stress. Quantitative studies of the effect of stress triaxiality on crack formation for metals performed in the past were conducted mainly using pre-notched round tensile bars. For example, Hancock and Mackenzie [2] carried out a series of tensile tests on pre-notched steel specimens. It was found that ductility depended markedly on the triaxiality of stress states. In their study, stress triaxiality was calculated using the Bridgman’s [8] formula. Recently, Mirza et al. [9] performed an experimental and numerical study on three different materials (pure iron, mild steel and aluminum alloy BS1474) over a wide range of strain rates (103 –104 s1 ). Equivalent strain to crack formation for all the three materials was found to be strongly dependent of the level of stress triaxiality. The dependence was different for different materials. In comparison, although some of the empirical fracture criteria proposed are related to stress and strain ratios (e.g. [10,11]), to the best of the author’s knowledge, the dependence of ductile crack formation on stress and strain ratios has not been well understood. Detailed study on ductile crack formation in tensile specimens with different cross-sections for the same material will certainly contribute to a more complete picture. In this study, tensile tests on flat specimens, flat-grooved plates, and smooth and pre-notched round bars, were conducted. All the specimens were cut from a same block of Al 2024-T351. Those tests give not only different stress triaxialities, but also different stress and strain ratios. Certainly, they provide a good way to study the effects of stress triaxiality, and stress and strain ratios, on crack formation.

2. Theoretical consideration 2.1. Flat specimen For ductile metals, necking in a thin sheet under uniaxial tension usually occurs before fracture. Obviously, necking is an important mechanism in thin sheets under tension for fracture prediction. The analysis for necking under plane stress was conducted by many authors such as Hill [12], and McClintock and Zheng [13]. For a thin sheet in uniaxial stress subjected to an axial load, P , necking occurs when P reaches a maximum. The necking is spread over a length of the order of the width w shown in Fig. 1 while the rest of the specimen remains prismatic. This is called diffuse necking. The maximum load, at which necking begins, occurs when the slope of the equivalent stress–strain curve satisfies the condition d r  ¼r de

ð1Þ

For a power-law material, which has the relation,  ¼ r0en r

ð2Þ

 and e are equivalent stress and strain, respectively; and r0 and n are two material constants for where, r power law materials. The corresponding equivalent strain at the onset of necking is

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Fig. 1. Tensile necking in a flat specimen (after [13]).

e ¼ n

ð3Þ

If diffuse necking continues, localized necking, also shown in Fig. 1 may occur over a length of the order of the sheet thickness. The localized necking occurs when the slope of the equivalent stress–strain curve reaches [13]  d r r ¼ de 2

ð4Þ

For a power-law material, the corresponding equivalent strain is e ¼ 2n

ð5Þ

The surroundings of the localized neck, the so-called shoulders, remain rigid and the strain parallel to the localized neck varnishes. From Mohr’s circle of strain the localized neck turns out to be inclined to the maximum principal stress at an angle / that depends on a function sr of the stress ratio [13] 1 þ r2 =r1 sr ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1  r2 =r1 þ ðr2 =r1 Þ

ð6Þ

from which " # 1 1 sr 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / ¼ p þ sin 4 2 3ð1  sr2 Þ For a uniaxial tensile test (r1 6¼ 0 and r2 ¼ r3 ¼ 0)   p 1 1 1 / ¼ þ sin  54:7° 4 2 3

ð7Þ

ð8Þ

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However, the above analysis for necking is not sufficient for fracture prediction since, in most cases for ductile metals, a crack forms at a certain stage after localized necking occurs. 2.2. Round bar Round bars are most often used in tensile tests to get material properties, including fracture constants. For ductile materials, necking also occurs in round bars before fracture initiates. The necking observed in round bars, shown in Fig. 2 is similar to the diffuse necking in thin sheets. Consequently, the same condition (Eq. (1)) holds when necking occurs. Obviously, the region of the neck is the most highly stressed and strained region and therefore it is also the critical location for this type of specimen. In order to obtain the local stress and strain in a necked round bar, Bridgman [14] developed a semi-empirical analysis in terms of the radius of curvature of the neck and the radius of the minimum cross-section. The main feature of Bridgman’s analysis is that the equivalent strain is constant across the minimum cross-section, but the radial, hoop and axial stresses (rrr , rhh and rzz , respectively) vary. The stress and strain components are determined as follows, referring to the geometry given in Fig. 2. a 0 e ¼ 2 ln ð9Þ a

 2  a þ 2aR  r2  1 þ ln rzz ¼ r ð10Þ 2aR   ln rrr ¼ rhh ¼ r rm 1 ¼ þ ln  3 r



a2 þ 2aR  r2 2aR

a2 þ 2aR  r2 2aR

 ð11Þ

 ð12Þ

 and rm are axial, radial, hoop, equivalent, and mean normal stress, respectively, e is the where, rzz , rrr , rhh , r equivalent strain, a and R are the radius of the minimum cross-section and the radius of the circumferential notch, a0 is the initial value of a.

Fig. 2. Tensile necking in a round bar.

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Bridgman’s method clearly depends on the measurements of the internal horizontal radius of the minimum cross-section and the external vertical radius of curvature of the assumed circumferential notch, a and R. In fact, this objective is difficult to achieve during the tests because (1) the position of neck is not known before the test; (2) it needs a series of tests with different loadings, or pausing the test several times to obtain the history of R and a; and (3) measuring R is not straightforward. In addition, the stress state in the neck given by the Bridgman solution essentially has not been verified. As reviewed by McClintock and Argon [15], ‘‘Marshall and Shaw [16] ran tensile tests on specimens which were machined to arbitrary values of longitudinal curvature at different stages. They found that the results obtained could be correlated into a smooth curve by applying the Bridgman correction. On the other hand, Parker et al. [17] determined the stress distribution in the neck of a tensile specimen by unloading a necked specimen, calculating the stress change on unloading from elasticity theory and then boring out the specimen to determine the residual stress. They disagreed with Bridgman’s results’’. Recently, Alves and Jones [18] performed a finite element analysis for notched round bars under tensile loading. Stress triaxiality and equivalent strain at the neck were compared between numerical simulations and the Bridgman solution and large differences were found. However, it should be noted that in their study, the change of the radius of curvature of the neck due to the deformation was not considered in calculating stress and strain components using the Bridgman solution. A more precise comparison was conducted in this study and is given in Section 4. 2.3. Flat-grooved plate Compared to thin sheets and round bars, flat-grooved plates are not widely used. Causing [1] investigated the fracture ductility of steel under uniaxial loading in plane strain experimentally using the type of specimen shown in Fig. 3. During his tests, the crosshead was stopped at frequent intervals after the specimen yielded. The load and the crosshead position just prior to the stopping of the crosshead were recorded. With the crosshead stopped, the thickness of the minimum section was measured. Then the strain was calculated each time from e ¼ ln

t 0

ð13Þ

t

where t0 is the original thickness in the gauge section, and t is the current thickness of the minimum section. The strain was plotted against the crosshead displacement and the resulting curve was extrapolated to the known value of the crosshead displacement when final fracture occurred. Finally, the corresponding strain at final fracture was read from the curve. This method gives an estimation of the strain. However, it clearly

Fig. 3. Flat-grooved plate.

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transition zone

TPS zone

Fig. 3 Flat-grooved plate Uniaxial 54.70 stress

l0

Necking

σps w ps

σpn

w pn w0

Fig. 4. McClintock’s plane strain zone boundaries (a) and stress distribution (b) in gauge section of the flat-grooved specimen.

depends on the accuracy of Eq. (13) and on the extrapolation. In fact, Eq. (13) is only valid when the strain is uniform within the minimum section, but this condition is difficult to achieve because of necking. McClintock and Zheng [13] studied a similar specimen and found that the stresses and strains were uniform across most of the width, which defines the so-called transverse plane strain (TPS) zone shown in Fig. 4. The specimen was approximately modeled with two distinct zones: plane stress and TPS zones (Fig. 4). To represent the uniaxial and transition effects, it was proposed that a uniaxial stress, rps , be assumed to exist in the transition zones near the edges, with pffiffiffi rps ¼ rpn 3=2; and ð14Þ L0 tan 54:7° ð15Þ 2 where rpn is the stress in the TPS zone, wps is the width of each edge-transition zone, and l0 is the original length of the gauge section (Fig. 4).  and strain e in the TPS region were expressed by the With those assumptions, the equivalent stress r following two equations. pffiffiffi 3 P =wt 2 ¼ r ð16Þ pffiffiffi !  L0 3 1 1 tan 54:7° 2 w0 wps ¼

2 L  t0 e ¼ pffiffiffi ln 3 L 0  t0

ð17Þ

where, P is the load, w, t and L are the current width, thickness and length, respectively, and w0 , t0 and L0 are the initial values of w, t and L, respectively. As described above, the analytical models by Hill, Bridgman, Clausing and McClintock provide a clear picture of the plastic deformation of flat specimens, round bars and flat-grooved plates under uniaxial

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tension, respectively. However, those models involve assumptions, such as the uniform strain distribution along the minimum cross-section, which may lead to errors in strain and stress calculations. For some of the models, no individual components of stress and strain are given. An alternative way to get accurate detailed stresses and strain is to perform numerical simulations. In the present paper, both experiments and parallel numerical simulations were carried out on flat plates, round bars and flat-grooved plates under uniaxial tensile loading.

3. Experiment Tensile tests on flat specimens, smooth round bars, notched round bars with two external notch radii, and flat-grooved plates, were conducted using a universal testing machine (Model 45G, MTS System Corporation, Eden Prairie, MN) with a 200 kN load cell at a loading rate of 0.2 mm/s, as shown in Fig. 5. All the specimens were cut from a same block of 2024-T351 aluminum alloy. The dimensions of the specimens are listed in Table 1. An extensometer with 25.4 mm gauge length was used to measure the displacement change in length since the measurement of crosshead travel does not accurately measure the deformation of specimens. Load–displacement responses were recorded using TestWorks software (Sintech Division, MTS). Fracture occurred inside the gauge section for all the five cases. Two or three samples were tested for each case. Quite repeatable results were obtained, as shown in Fig. 6 for the notched bar with a 12 mm external radius of notch as an example. The crack formation was associated with a sudden drop of the load.

Fig. 5. Initial set up of tensile tests on different specimens.

Table 1 Dimensions of tensile specimens (units: mm) Specimen

Gauge length

Width

Thickness

Diameter

External radius of notch

Flat specimen Smooth round Round bar with a large radius notch Round bar with a small radius notch Flat-grooved plate

25.4 25.4 N/A N/A 8

12.5 N/A N/A N/A 50

3 N/A N/A N/A 1.6

N/A 9 8 8 N/A

N/A 1.8 12 4 N/A

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Fig. 6. Force–displacement responses of notched specimens (R ¼ 12 mm).

4. Numerical simulation Numerical simulations of the five tensile specimens under tensile loading were performed using ABAQUS STANDARD 6.2. In all simulations, a finite displacement rate was applied to one end of the model while the other end was fixed. One coarse and one fine mesh for each case were developed to study mesh size sensitivity. The number of elements of the fine meshes was twice that of the coarse meshes. The difference of forces, stresses and strains between the coarse meshes and fine meshes used in this study was found small (within 4%) for all five cases. The shape of the true stress–strain curve up to necking was directly obtained from the tensile test on the round bar as follows: P pa2   l e ¼ ln l0 ¼ r

ð18Þ ð19Þ

where P is the load, and l0 and l are the initial and current gauge lengths, respectively. Measuring the changing diameter of test pieces during tests is a customary way to determine the true stress–strain curve beyond necking. However, this method clearly depends on the uniformity of the stress and strain distribution across the neck. In fact, the error caused by the assumption of uniform strain distribution and the Bridgman correction could be as much as 10% and 40%, respectively (see the section of round bars). Also, it is difficult to measure the changing diameter and neck profile as indicated in Section 2. In this study, the shape of the curve after necking was determined by a trial-and-error method until the numerical calculation of the load and necking deformation corresponded well with the test data on the round bar. A large number of runs were needed to get the correct points after necking. The post-necking iteration procedure is summarized as follows: (1) Calculate the pre-necking, stress–strain curve from the force–displacement response of the round bar from Eqs. (18) and (19). Extrapolate the curve using a power law fit. (2) Perform numerical simulations with the stress–strain curve obtained in Step (1) and compare the calculated load–displacement response with experiments. Calculate the relative error of the force.

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Fig. 7. Stress–strain curve for Al 2024-T351.

(3) Adjust the stress–strain curve based on the relative error obtained in Step (2) and replace the extrapolation of the stress–strain curve with the updated one in Step (2). (4) Repeat Step (2) and (3) until the relative error becomes satisfactory. The stress–strain curve determined for Al 2024-T351 is shown in Fig. 7. 4.1. Flat specimen The specimen was modeled by 4-node elements. As shown in Fig. 8, the numerical simulation successfully captured diffuse necking and localized necking, which are the main features of interest for tensile tests on flat specimens, and are clearly displayed in the tested specimen. Final width of the minimum crosssection obtained from the numerical simulation was 10.6 mm, which was close to 10.4 mm measured from the test. Amazingly, even the angle of the shear band from the test and the numerical simulation was found

Fig. 8. Deformed flat specimen.

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Fig. 9. Comparison of force–displacement response (flat specimen).

to be the same as the analytical result predicted by Eq. (8), which was 54.7° (Fig. 8). In addition, the correlation of the force–displacement response between the numerical simulation and test up to crack formation was almost perfect (Fig. 9). 4.2. Round bars Round bars were modeled by 4-node axisymmetrical elements. The main feature of interest for tensile tests on smooth round bars is the development of the circumferential neck, which is clearly shown in the tested specimen. The numerical simulations captured this phenomenon nicely, as displayed in Fig. 10(a). The correlations of the deformation shapes between the experiment and numerical simulations for the two notched bars were also quite good (Fig. 10(b) and (c)). Besides, the final diameters of the minimum crosssections obtained from the numerical simulations were 7.17, 6.9, and 7.2 mm, for the smooth and the two notched round bars, respectively, all close to the 7.2, 6.8, and 7.15 mm, measured from the experiments. As shown in Figs. 11–13, the comparisons between the force–displacement responses for the numerical simulations and the experiments, up to crack formation were also almost perfect for the three cases. One might argue that it is not surprising that the correlation is perfect for the smooth round bar since the input of the stress–strain curve shown in Fig. 7 was determined from this test. However, it should be noted that the correlation is also perfect for the two notched bars, the flat specimen described before, and the flat-grooved plate, which will be discussed later. This clearly shows that the stress–strain curve determined from the trialand-error method is good, and the simulations performed in this study are satisfactory. To compare the Bridgman solution (Eqs. (9)–(12)) with the numerical simulation, we considered the notched round bar with internal initial radius of the minimum cross-section a0 ¼ 4 mm and initial external radius of the notch R0 ¼ 4 mm, and focused on one deformation stage at which a ¼ 3:46 mm and R ¼ 4:54 mm. Clearly, the dimension of the neck changed during deformation process. The deformed mesh is shown in Fig. 14. Comparisons of the Bridgman solutions and numerical calculations are displayed in Figs. 15 and 16 for equivalent strain and stresses, respectively. The difference of equivalent strain was relatively small (15%), while the difference of stresses was large (30–50%) at the center of the neck, which is the location of crack formation. The error caused by Bridgman formula probably came from the assumptions of the uniform strain distribution and the shape of the neck. It is interesting to see the importance of the change of curvature in the stress triaxiality calculation. At the deformation stage at which a ¼ 3.46 and R ¼ 4.54 mm, the stress triaxiality using the current profile is

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Fig. 10. Deformation shape of round bars. The cracks initiated at the center. The photographs were taken later because crack grew very rapidly and we were not able to stop the test at the time crack just initiated.

Fig. 11. Comparison of the force elongation between experiments and numerical simulations (tensile tests on round bars).

0.66, while the stress triaxiality calculated using the initial geometry is 0.74. Clearly, it is important to take into account the change of curvature for large deformation.

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Fig. 12. Response comparison of experiment and numerical simulations (R ¼ 12 mm).

Fig. 13. Response comparison of experiment and numerical simulations (R ¼ 4 mm).

4.3. Flat-grooved plate We considered 1/8 of the specimen, which was modeled by 8-node solid elements instead of plane strain elements. Diffuse necking, which occurred in the experiment, was also observed in the numerical simulation (Fig. 17). The final width of the minimum cross-section, obtained from both the numerical simulation and the test was 48.6 mm. Correlation of the force–displacement response between the numerical simulation and the test, up to crack formation, was good (Fig. 18). It was found that a plane strain zone (Fig. 17) developed at the center region of the gauge section. This region was highly stressed and strained (Fig. 17). The distribution of the equivalent strain from the edge to the center is shown in Fig. 19. It is seen that the length of the plane strain zone obtained from the simulation was around 70% of the total width, while the value estimated using McClintock and Zheng [13]’s model (Fig. 4 and Eq. (15)) was 80%. McClintock and Zheng’s model is a good approximation for this type of flat-grooved specimens.

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Fig. 14. Finite element mesh of the notch region.

Fig. 15. Comparison of equivalent strain between Bridgman and numerical simulation.

5. Result and discussion The displacement to crack formation df was determined by both observations during experiments and the force–displacement responses (see for example Fig. 9). There was a significant load drop in the force– displacement responses. This drop was taken as the point of crack formation in this study. In classical tests on thin sheets and round bars, cracks formed at the centers of necks. This observation was reported by a number of studies [2,19–21], etc., with additional sophisticated procedures. In this study, the critical location was determined as the one where both equivalent strain and the stress triaxiality parameter rm = r were the highest along the path of the final crack obtained from the test, since in the high stress triaxiality range, the equivalent strain to crack formation decreases with increase of stress triaxiality. The physical reasons on this behavior have been extensively studied e.g. [2,3,6] etc. It should be noted that for the round

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Fig. 16. Comparison of stresses between Bridgman and numerical simulation.

Fig. 17. Deformed flat-grooved specimen.

Fig. 18. Comparison of force–displacement response (flat-grooved plate).

bar with R ¼ 4 mm, the equivalent strain on the mid-plane of the neck is not a maximum. However, the equivalent strain to crack formation decreases dramatically with increase of the stress triaxiality. The stress

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Fig. 19. Strain distribution along the minimum cross-section of the flat-grooved plate.

triaxiality is highest on mid-plane of the neck and the difference of equivalent strain across the neck is not large. It is reasonable to assume the critical location for this case is also at the mid-plane of the neck. It was found from the numerical simulations that locations of crack formation were at the center of the necks for the flat specimen and the round bars. This result agreed with the classical observations. The critical location was also found to be at the center of the neck for the flat-grooved plates. In addition, crack formation at the center of the neck in the flat specimens and the flat-grooved plates was also observed in the tests. The numerical simulations provided accurate values of stress and strain components at every stage of deformation. The histories of stresses and strains at the locations of crack formation can be expressed as functions of the axial displacement for each case. In this study, we focused on four parameters, namely, equivalent strain, stress triaxiality, and stress and strain ratios. The equivalent strain to crack formation ef was determined as the equivalent strain corresponding to the displacement at crack formation df obtained from tests. Generally, the stress triaxiality at the critical location is not constant. An average stress triaxiality was introduced and calculated, defined by   Z rm 1 ef rm de ð20Þ ¼  av ef 0 r  r  are the hydrostatic and equivalent stress, respectively. where rm and r It should be noted that it was not clear whether an initial, average or final value of stress triaxiality was ðr2 r3 Þ 2 e3 Þ used in the literature. The stress and strain ratios in this study were calculated as ðr and ðe , where, r1 , ðe1 e3 Þ 1 r3 Þ r2 , and r3 are maximum, intermediate, and minimum principal stresses, respectively; e1 , e2 , and e3 are maximum, intermediate, and minimum principal strains, respectively. The ratios were strictly zero for the three round bars. A similar average value was introduced for the flat specimens and flat-grooved plates. The calculated parameters are summarized in Table 2. Although all the specimens were under tensile loading, the stress and strain states were quite different. This was mainly due to the different geometries of the specimens and also the different necking formations. However, it is seen from the table that the flat specimens and smooth round bars had a similar equivalent strain to crack formation and stress triaxiality, though they had different stress and strain ratios. The same phenomenon was also observed in the flatgrooved plates and the notched round bars with R ¼ 12 mm, a ¼ 4 mm. In comparison, the smooth round bars and the two notched round bars had different equivalent strains to crack formation and different stress triaxialities, though they had same stress and strain ratios.

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Table 2 List of the parameters calculated form FEA Specimen

ef

r m  av r

ðr2  r3 Þ ðr1  r3 Þ

ðe2  e3 Þ ðe1  e3 Þ

Smooth flat specimen Smooth round bar Round notched bar (R ¼ 12 mm, a ¼ 4 mm) Round notched bar (R ¼ 4 mm, a ¼ 4 mm) Flat-grooved plate

0.44 0.45 0.28 0.16 0.26

0.38 0.4 0.65 0.95 0.62

0.1 0 0 0 0.45

0.33 0 0 0 0.5

Fig. 20. Relation of stress triaxiality and equivalent strain to crack formation of Al 2024-T351.

Furthermore, as shown in Fig. 20, the five testing points follow a simple function in the space of equivalent strain to crack formation versus the average stress triaxiality, ef ¼ 

0:18  rm  av r

ð21Þ

It is recognized that this equation is for Al 2024-T351 in a high stress triaxiality range. Hancock and Mackenzie [2] correlated their experimental results on steels with the Rice–Tracey exponential function.   3 rm ef ¼ c1 exp  ð22Þ  2 r where c1 is a material constant. Wierzbicki and Muragishi [22] found that the following equation  5=3 rm ef ¼ c2  r

ð23Þ

gave a better correlation with Hancock’s experimental data than the Rice–Tracey criterion. Also, it was clearly shown in the work by Mirza et al. [9] (Fig. 10 in their paper) that the dependence of the equivalent strain to crack formation on the stress triaxiality is different for mild steels, pure iron and aluminum alloys. It should be pointed out that the result obtained by Hancock and Mackenzie [2] should be revisited due to the possible inaccuracy in the calculation of stress triaxialities caused by the Bridgman solution.

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It is clear now that the difference of equivalent strain at crack formation in round bars and flat-grooved plates obtained by Clausing [1] was due to the different stress triaxialities that the specimens experienced at the critical locations. It is understandable that the magnitude of the difference was not the same for different steels, since the relationship of the equivalent strain to crack formation and stress triaxiality is different for different materials. It is likely that one can make a good prediction of ductile crack formation with the information of equivalent strain and stress triaxiality. In other words, equivalent strain and stress triaxiality are the two most important factors governing crack formation while strain and stress ratios either do not come in to the picture or probably induce secondary effects. It should be noted that this paper deals with ductile crack formation in uncracked bodies by introducing no-dimensional functions. The effect of specimen size may come from the inhomogeneity of materials. Consequently, the importance of size effect depends on materials. For extrusions such as Al 2024-T351, the non-dimensional criteria calibrated by specimens with the order of mms can be applied in large structures. However, for castings, specimen size plays an important role in fracture prediction. Fracture criterion calibrated by small specimens may overestimate ductility or fracture toughness of large structures. For those materials, it is important to introduce a dimensional parameter.

6. Conclusion A detailed discussion on crack formation in tensile specimens was provided in this study. Tensile tests on five different Al 2024-T351 specimens, including flat specimens, smooth and notched round bars, and flatgrooved plates, were carried out. Parallel numerical simulations were performed using the commercial finite element code ABAQUS. The correlation of experimental and numerical results in both load–displacement responses and deformation shapes was satisfactory. A relationship between stress triaxiality and equivalent strain to crack formation was obtained in a high stress triaxiality range for Al 2024-T351. It was found that the Bridgman formula gave a good estimation on the strain in the neck while the differences of stresses calculated by the Bridgman formula and the numerical simulation, were large. The length of the plane strain zone in the flat-grooved plate estimated by McClintock and Zheng’s model correlated well with the numerical simulation. More importantly, the specimens with the same stress triaxiality had the same equivalent strain to crack formation, while the specimens with different stress triaxialities had different values of equivalent strain to crack formation. It can be concluded that equivalent strain and stress triaxiality are the two most important factors governing crack formation, while the stress and strain ratios cause secondary effects, and that one can make a good prediction of ductile crack formation with equivalent strain and stress triaxiality alone. Acknowledgements The present research was supported by the joint MIT/Industry Consortium on the Ultralight Metal Body Structure and the Volpe Center Grant to MIT. Thanks are due to Professor Tomasz Wierzbicki and Professor Frank A. McClintock for many valuable discussions.

References [1] Clausing DP. Effect of plastic strain state on ductility and toughness. Int J Fract Mech 1970;6:71–85. [2] Hancock JW, Mackenzie AC. On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. J Mech Phys Solids 1976;24:147–69.

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