An approach to predicting the forming limit stress components from mechanical properties

Journal of Materials Processing Technology 229 (2016) 758–768

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An approach to predicting the forming limit stress components from mechanical properties B.S. Levy a , C.J. Van Tyne b,∗ a b

B.S. Levy Consultants, 1700 E. 56th St., Suite 3705, Chicago, IL 60637, USA Department of Metallurgical and Materials Engineering Colorado School of Mines Golden, CO 80401, USA

a r t i c l e

i n f o

Article history: Received 31 January 2015 Received in revised form 19 October 2015 Accepted 22 October 2015 Available online 2 November 2015 Keywords: Forming limit curve Stress-based forming limit curve Instability point in a tensile test Plane-strain forming limit

a b s t r a c t Forming limit curves (FLCs) are used to determine the amount of deformation that can be applied to a sheet metal before the onset of a localized neck. Most FLCs are shown in strain space, and stress-based FLCs have advantages because they are often strain-path independent. The current study develops a method to calculate a stress-based forming limit curve. The necessary data for this calculation can be obtained from a uniaxial tensile test. The calculations depend on the Z parameter, which can be considered to be the point of instability during a tensile test. With the use of the Keeler–Brazier equation, the effective stress in plane strain at the forming limit is shown to be a function of the Z parameter and thickness. Data from 4 experimental studies are shown to be consistent with this function. With the generally accepted observation that the left side of the strain-based FLC is a line with slope of –1 and an appropriate constitutive model for the stress-strain behavior of the material, the stress-based FLC corresponding to the left side of the strain-based FLC can be calculated. Comparison of the calculated stress-based FLC for three steels with the stress-based FLC determined directly from the strain-based FLC shows good agreement. The calculated stress-based FLC is 15–20 MPa below the data generated directly from the strain-based FLC. © 2015 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Description of a strain-based forming limit diagram A strain-based forming limit describes the locus of in-plane principal strains at which a critical local neck forms. A critical local neck is the failure criterion for a forming limit diagram (FLD). A typical forming limit diagram shows the major in-plane strain on the vertical axis and the minor in-plane strain on the horizontal axis. The forming limit diagrams typically employed in the press shop use engineering strains, although in most research studies, true strains are used. The forming limit curve (FLC), which is the locus of critical points on the FLD, is the point where necking occurs, which leads to ductile fracture in sheet metal deformation with minimal additional strain. Limit strains in forming limit curves can be higher than uniform elongation, UE, in a tensile test because during biaxial sheet deformation, geometrical constraints prevent diffuse necking. For

∗ Corresponding author. Fax: +1 3032733795. E-mail addresses: [email protected] (B.S. Levy), [email protected] (C.J. Van Tyne). http://dx.doi.org/10.1016/j.jmatprotec.2015.10.027 0924-0136/© 2015 Elsevier B.V. All rights reserved.

an FLC, failure is a critical local neck. The damage process initiates at the point where local necking starts and continues until ductile failure occurs. Keeler and Backofen (1963) described a local neck as a narrow band of deformation where the incremental principal strain component, d2 , equals zero along the axis of the local neck, and an angle between the orientation of the local neck and the largest principal stress component,  1 , can be computed. Levy and Green (2002) showed qualitative agreement between calculated and experimental results. Marciniak and Kuczynski (1978) described a local neck as a groove where the strain in the groove accelerates along a non-linear strain path where the normal strain component perpendicular to the axis of the groove continues to increase, and the normal strain component along the axis of the groove goes to zero. In order to have a failure criterion, the authors introduced the concept of an incipient notch that initiates failure. Many authors consider the incipient notch to be the result of metallurgical damage. While there has been considerable additional work on the mechanics associated with the formation and subsequent failure of local necks, further discussion is excluded from the current study, because the emphasis is on failure limits due to a critical local neck.

B.S. Levy, C.J. Van Tyne / Journal of Materials Processing Technology 229 (2016) 758–768

1.2. Predicting strain-based FLCs Keeler and Brazier (1977) developed a relationship between the plane-strain forming limit, FLC0 , and the strain hardening exponent, n, and thickness, t using regression analysis. Keeler (1989) showed that the initial regression analysis is only valid for thickness values of up to 3.1 mm, that between 3.1 and 3.5 mm there is a gradual decrease in the effect of thickness, and above 3.5 mm there is no effect of thickness. The Keeler–Brazier equation is FLC0 = (23.3 + 14.13t)

 n  0.21

(1)

where n ≤ 0.21, t ≤ 3.1 and t is in mm. With the advent of highly formable interstitial free steels, a study by the North American Deep Drawing Research Group, in which one of the current authors (Levy) participated, showed that higher values of n could be used in Eq. (1). Hiam and Lee (1978) found a thickness effect on FLCs between 0.86 to 4.32 mm for cold rolled low carbon rimmed and capped steels, hot rolled low carbon steels, and higher carbon HSLA steels. In contrast to Hiam and Lee (1978), Kleemola and Kumpulainen (1980b) studied hot and cold rolled AKDQ steels that were 0.97, 1.95, 3.00, and 4.65 mm in thickness and concluded that the thickness effect on FLCs is due to erroneous measuring techniques and definitions of limit strains. Cayssails (1998) indicated that the Keeler and Brazier (1977) approach is inadequate for steels that are more than 1.5 mm in thickness. However, two different methods were used to measure FLC limit strains. Cayssails (1998) uses the Bragard (1989) method to measure FLC limit strains, which measures strains on either side of a fracture and interpolates to determine the actual FLC limit strain. More recently, a subgroup of the International Deep Drawing Research Group (Monford, 1999) improved the interpolation method initially used by Bragard. In contrast, the data used to formulate the Keeler and Brazier (1977) equation (i.e., Eq. (1)) is based on a method in which strain is measured over an incipient local neck (i.e., the North American method). An example of this approach is shown in the work of Levy and Green (2002). The difference between the Keeler and Brazier (1977) results and the Cayssails (1998) results may be due to the different methods of measuring FLC strain. Such differences may be more apparent for thicker steels where the FLC strains are larger. Several research studies have presented evidence supporting Eq. (1), even when FLC0 was determined by different experimental methods. The work by Shi (1995) used a hemispherical punch; the work by Konieczny (2001) used the Nakazima et al. (1968) method with a spherical punch; and the work by Levy and Green (2002) used the double blank method of Marciniak and Kuczynski (1967), which requires a flat punch. The results from these studies are consistent with Eq. (1). Cayssails (1998) developed a method for predicting the FLC, based on plastic instability and a damage model which has critical variables of strain hardening, strain rate hardening, and thickness. Cayssails (1998) used a damage model from Schmitt and Jalinier (1982), which is based on the growth of cavities formed during rolling. The volume fraction of cavities and the ratio of sheet thickness to cavity diameter are critical parameters in the Cayssails (1998) model. Cayssails and LeMoine (2005) extended the initial Cayssails (1998) model to ultra-high strength steels. They indicate that for ultra-high strength steels, the original Cayssails (1998) method must be upgraded to include consideration of a transition from a ductile to a brittle failure mode, a more complex understanding of void growth, and an improved understanding of the effect of thickness.

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Raghavan et al. (1992) developed a predictive equation for FLC0 based on hemispherical punch tests using thickness, t, and total elongation, TE, as independent variables. The equation is FLC0 = 2.78 + 3.24t + 0.892TE

(2)

where the coefficient of determination, R2 , equals 0.93, t is in mm, and TE is transverse total elongation in a 50.8 mm gauge length tensile test using ASTM A646. Their complete FLC curves have a different shape than the standard North American FLC curve. Total elongation depends on both strain hardening and strain rate hardening. Strain rate hardening has a pronounced effect on post-uniform elongation. Using the North American approach to measuring FLC0 , the use of total elongation with a fixed specimen size can be seen to be a reasonable tensile property for predicting FLC0 . Abspoel et al. (2011a) have shown that necking strain in plane-strain tension has a linear correlation with total elongation. Abspoel et al. (2012, 2013) used 4 strain paths to describe an FLC using the uniaxial tension necking point, the plane-strain point, the intermediate biaxial stretching point, and the equi-biaxial stretch point. They show that the left side of the FLC is a line of pure shear. Their work also shows a thickness effect, which is consistent with the left side of the North American FLC. The studies described above show the importance of tensile properties in predicting strain-based FLCs. The tensile properties include strain hardening, strain hardening exponent, n, strain rate, and total elongation. Total elongation is a function of strain hardening, strain rate hardening, and fracture behavior. It can be seen that all the predictive methods use tensile properties that are related to stress-strain behavior, and as a result, stress is implicit in all the predictive methods. With the exception of Kleemola and Kumpulainen (1980b), all the predictive methods include thickness. In general, thickness and tensile property terms are independent of each other. The Cayssails’ method (1998) is not stated in a simple equation form, but the sense from the paper is that thickness is an independent variable. These prior studies show that thickness is an important variable for predicting forming limit curves. 1.3. Explaining the effect of thickness Abspoel et al. (2012) have shown that once a local neck forms, it grows until failure occurs. Timothy (1989) studied a 1.6 mm aluminum alloy and found that it took 25 s from the initiation of a local neck to failure, though test speed is not given in the paper. Timothy also indicated that the local strain rate increases by two orders of magnitude from just before necking to final failure. It was also found in one alloy that fracture was preceded by initiation and propagation of shear bands. Korbell and Martin (1988) studied a 0.06% C, 0.75 mm aluminum killed steel with rolling strains from 0.1 to 0.9. They indicated that the macroscopic localization of strain originated from micro-shear banding. They state that micro shear banding is caused by crystallographic cross slip that penetrates several grains and propagates across a sample in the form of thin plates. Cayssails (1998) indicates the importance of the ratio of thickness to cavity size in his damage model. If it is assumed that cavity size for a given material is independent of final thickness, then the ratio of thickness to cavity size can explain some or all of the thickness effect. Keeler (1989) suggested that FLC0 increases with thickness because as thickness increases, a local neck becomes more diffuse and more time would be required for a local neck to reach the critical depth that is defined as failure. Furthermore, in the Keeler (1989) method, which is the North American approach, grids over a critical local neck are used to determine failure. Thus, strain rate hardening is important because strain rate hardening allows

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deformation in the grid that covers the local neck, which results in increased measured strain at failure. Hence, there are mechanistic reasons for the thickness effect on the forming limit curve. A critical local neck is a forming limit criterion that applies prior to separation. A critical local neck is used to evaluate stampings, because separation usually occurs shortly after the formation of a local neck, especially for the high strength sheet materials. Given the rapidity of separation after formation of a critical local neck, the additional stress required for separation is likely to be nominal. Thus, in practical terms, there is very little difference between a critical local neck and separation. 1.4. Stress-based FLCs Many studies have shown that strain path and changes in strain path affect FLC limit strains. Kleemola and Pelkkikangas (1977) showed strain path effects in steels, copper and brass sheets. Kleemola and Kumpulainen (1980a,b) studied the various parameters that affect strain-based FLCs. They indicate that strain path or prior deformation is one of the factors that needs to be considered. Wilson (1989) showed the effected of strain path on the stretch side of the strain-based FLD. Shi and Gerdeen (1991) indicated that through thickness strain gradients in the sheet will affect the strainbased FLC. Zhao et al. (1996) provide analytical methods to calculate the strain-based FLC for linear, bilinear and trilinear strain paths. Rojek et al. (2013) showed that strain-based FLCs will change if the strain path is complex. Specifically, these studies show that the applicability of normal strain-based FLCs is restricted to proportional loading. To overcome this limitation stress-based FLCs or path independent strain-based FLCs have been promoted. Arrieux et al. (1982) experimentally developed FLC limit stresses for several strain paths. They calculated the FLC limit stresses using a stress-strain relationship and the Pradtl and Resuss equations describing material flow. Their analysis showed that that a stressbased FLC is independent of strain path. Brunet and Chehade (1991) experimentally determined FLC limit strains with a flat punch for a titanium based ULC steel and for XD340. They used a stress-based FEA analysis and found very good agreement between those experimental results and FEA simulation. Stoughton (2000) used data for FLC limit strains for cold rolled aluminum killed steel and 2008-T4 aluminum that had been deformed with changes in strain path. Stoughton found that the stress-based FLC is path independent. He found that when an appropriate constitutive model was used, the variation in the strain-based FLCs that were due to changes in strain path mapped into a narrow band when plotted in stress space. The width of the band was comparable to the uncertainty in the data used to generate the strain-based FLC. The result is a stress-based FLC with axes of the first principal stress component,  1 , and the second principal stress component,  2 . For a power law material, the key values in determining  1 are the strength coefficient, K, the strain hardening exponent, n, and forming limit strains. Stoughton and Zhu (2004) extended the original work by reviewing several theoretical models of sheet metal forming instability; specifically, bifurcation analysis of diffuse and through thickness neck formation, the Marciniak and Kuczynski (MK) (1978) model and microscopic void damage models. The equations governing all these models are shown to be independent of strain path. These results provide a theoretical basis for the stress-based approach. They also showed that “deformation paths are naturally embedded in the final state of stresses because of the incremental nature of plastic flow.” Following the work of Stoughton and Zhu (2004), Zeng et al. (2008) developed a strain-based FLC that is independent of strain path. The Zeng et al. (2008) FLC is equivalent to a stress-based FLC

in all theoretical aspects but differs in presentation. The reason for a strain-based approach is that stamping plant personnel are more comfortable with strain, because only strain can be determined from stamped parts. Stoughton and Yoon (2011, 2012) developed a polar effective plastic strain diagram (PEPS) as another way to present a forming limit diagram that shows the failure criteria in strains in a way that is not path dependent. The limit failure conditions are derived from a conventional strain-based FLC, the strength coefficient, K, and the strain hardening exponent, n. Other work shows that stress path independence may be more ambiguous. Kuroda and Tvergaard (2000) used 4 different anisotropic plasticity models to evaluate non-proportional loading. They found that the presence of unloading between deformation steps significantly affects predictions of stress-based FLCs. Their results show significant differences between proportional and non-proportional loading for deformation paths with or without unloading between deformation steps. Yoshida et al. (2007) analyzed two loading conditions using the Marciniak and Kuczynski (MK) (1978) method for two linear strain paths, case A-unloading between deformation steps and case B-no unloading between deformation steps. For case A, they found that the resulting stress-based FLC described both proportional and non-proportional loading. In contrast, for case B, non-proportional loading does not apply to a stress-based FLC developed for proportional loading. Yoshida et al. (2007) also found that initial imperfections, the exponent of the yield function, plastic anisotropy and the flow rule do not affect results. These results show the importance of stress relaxation and reloading on subsequent stress-strain behavior. The Kuroda and Tvergaard (2000) and Yoshida et al. (2007) results show the importance of stress relaxation and reloading on non-proportional loading. The effect of stress relaxation and unloading depend on crystal structure and the metallurgical condition of the material being deformed. Yoshida and Suzuki (2008) found that the applicability of stress-based FLCs for non-proportional loading can depend on Bauschinger effects or cross-hardening behavior. The effect of cross hardening depends on the angular relation between the first and second deformation step. They conclude that the applicability of stress-based FLCs for non-proportional loading depends on whether changes in strain path affect stress-strain behavior. Olander and Miller (1988) used an improved imperfection and growth model with a set of constitutive equations that include both isotropic and kinematic hardening processes for pure aluminum and 2024-T7 aluminum. They found that with a descriptive mathematical model that adequately describes dislocation behavior, strain-based FLCs can be calculated for non-proportional loading. Such a strain-based FLC can be used to calculate a stress-based FLC. The Olander and Miller (1988) results also show the importance of understanding the fundamental basis of stress-strain behavior. Other researchers have used combined axial loads and internal pressure to experimentally examine the applicability of nonproportional loading to stress-based forming limits for steel and aluminum tubes. Yoshida et al. (2005) examined 5000 series aluminum tubes. They concluded that for the deformation paths evaluated in their study, the stress-based FLCs apply to nonproportional loading. For steel tubes, Yoshida and Kuwabara (2007) concluded that the applicability of non-proportional loading to stress-based FLCs strongly depends on strain hardening behavior for a given loading path. Korkolis and Kyriakides (2009) studied 6260-T4 aluminum tubes where the results seem to indicate that stress-based FLCs are not path independent. However, their results are based on failure strains. For stress-based forming limits to apply to non-proportional loading, stress-strain behavior must remain consistent between deformation steps. Consistent stress-strain behavior depends on

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the metallurgical and crystallographic characteristics of specific sheet material and the specific strain histories. For example, the importance of cross hardening depends on the magnitude of the angle between the two deformation paths. Thus, a difference of 20◦ between two deformation steps probably has a minimal effect. In contrast, a 120◦ difference between two deformation paths could have a significant effect. Similar issues exist with Bauschinger effects, which normally imply a complete reversal of a strain path, because as strain in the second deformation step increases, the Bauschinger effect has progressively less impact on stress-strain behavior as the forming limit is approached. Stress relaxation determines the importance of unloading between deformation steps on the continuity of stress-strain behavior. The importance of stress relaxation depends on the individual material. Consequently, when examining the applicability of nonproportional loading for a given deformation process, the details of the strain path must be carefully examined. When such analysis is done, there will be applications where applying stress-based forming limits for non-proportional loading is appropriate. When deformation continues past the forming limit, fracture occurs quickly, especially for high strength sheet material. So, if the stressbased FLCs developed in the current study are considered a fracture criterion, they still have significant utility in many situations. 1.5. Using the Z parameter for predicting stress-based FLCs The Z parameter has been shown to be a measure of damage accumulation in sheet metal that is subject to deformation. Levy and Van Tyne (2012) and Levy et al. (2013) have shown that the Z parameter is an important factor in determining limit strains in sheared edge stretching. The Z parameter is a measure of instability that can be related to a formability limit. It should also be noted that Z as a measure of instability is related to Z as a damage parameter, because metallurgical damage and instability are interrelated. Z is defined as the strain hardening rate, d/d, at uniform elongation, UE, in a tensile test. Considere’s criterion states that at maximum load, d/d is equal to the true stress at uniform elongation in a tensile test. Hence, Z can be calculated by



Z = TS 1 +

UE 100



(3)

where TS is the ultimate tensile strength and UE is uniform elongation in percent. Z can be used as a proportionality factor to relate d to d at tensile instability by d = Zd

(4)

All terms in Eq. (4) are in macro stress or macro strain. As Z increases, for an equal increment in d, d decreases. If d represents a failure condition, the larger the Z value, the lower the limit strain and the larger the limit stress.Since Z is measured in effective stress, it is particularly useful for predicting stress-based FLCs. As previously shown, thickness is also needed to predict the effective stress at FLC0 . The response surface for the effective stress at FLC0 as a function of Z and t can be written as: ¯ atFLC0 = a0 + a1 Z + a2 t

(5)

2. Experimental approach 2.1. Phase 1—parameters values for Eq. (5) The first phase of the current study relates Z and thickness to FLC0 using Eq. (1). Z was calculated from Eq. (3) using tensile property data for a wide range of steel grades that are taken from a study by Sriram et al. (2003). The steel grades were IF, DDQ+, AKDQ,

761

a range of bake hardening steels, IF RePhos, HSLA 350, HS440W, DP 500, DP600, DP800, DP980, and TRIP600 steels. Values of 0.0, 0.7, 1.4, and 2.1 mm were used for the calculation of FLC0 in Eq. (1).Table 1 shows the tensile properties for the steels used in this analysis. Table 2 gives the calculations of FLC0 for each of the steels based on Eq. (1). The values of atFLC ¯ 0 were obtained by



1 + R¯



FLC0 ln 1 + atFLC ¯ 0 =K  100 ¯ 1 + 2R



n

(6)

where K is the strength coefficient of the material and R¯ is the normal anisotropy. The basis of Eq. (6) is normal anisotropy and planar isotropy (i.e. Hill, 1948) and power law hardening. Power law hardening is ¯ = K ¯

n

(7)

where the values of K and n are calculated using regression analysis for the individual points on a logarithmic stress versus logarithmic strain plot. The use of more complex constitutive models is possible but is outside the scope of the current study. The term in brackets in Eq. (6) converts the engineering FLC stress for plane strain to true strain and then uses Hill (1948) to convert the principal strain component to effective. Power law hardening approximates the stress-strain behavior for most of the steels in the current study. The approximation can cause greater errors in the derivative of the stress–strain curves, but exhibits smaller errors in calculating stress from strain so that it is a reasonable model for the purpose of the current study. Table 3 shows the values for atFLC ¯ 0 and the Z parameter, which was calculated from Eq. (3), for each of the steels. 2.2. Phase 2—validation of Eq. (5) In the second phase of the work, experimental FLC0 , tensile properties, and thickness values were taken from 4 studies: (1) Levy and Green (2002) who studied aluminum killed drawing quality (AKDQ), a bake hardened (BH210), a high strength low alloy (HSLA) and DP600 steels, (2) Shi (1995) who examined interstitial free (IF), AKDQ, BH and a high strength steel (HSS), (3) Raghavan et al. (1992) who tested various cold rolled (CR) and drawing quality (DQ) steels with a variety of coatings including galvannealed (GA), hot dipped galvanized (HDG) and electrogalvanized (EG) and (4) Konieczny (2001) who examined two dual phase (DP) steels each with two different thicknesses. From the data in these studies, values of atFLC ¯ 0 were calculated using Eq. (6) and were compared to the values calculated from Eq. (5). Table 4 shows the tensile properties from these 4 studies. 3. Results 3.1. Parameter values for Eq. (5) A regression analysis was used to determine the constant and coefficients for the response surface of Eq. (5) from the phase 1 data. The values are a0 = 20.30 ± 3.5, a1 = 1.002 ± 0.005, and a2 = 34.25 ± 1.26 with t in mm, where R2 = 0.997, the standard error is 11.5 MPa, and the number of data points is 136. It should also be noted that the constant is statistically significant, but is not physically important. These results essentially validate the Keeler–Brazier equation, but are only a partial validation of Eq. (5). Fig. 1 shows the results of atFLC ¯ 0 as a function of Z for thickness values of 0.0 and 2.1 mm. The lines are from Eq. (5) and the data points are from Table 3. Fig. 1 shows that Eq. (5) well represents

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Table 1 Tensile properties of phase 1 steels. Thickness (mm)

Tensile strength (MPa)

Uniform elongation (%)

Strength coefficient, K (MPa)

Strain hardening exponent, n

Normal anisotropy, R¯

DQSK

0.77 1.19 0.70 1.19 0.70 0.93 0.71 1.00 1.04 0.74 1.02 0.63 0.89 0.66 0.81 1.24 1.29 1.16 1.21 1.62 1.24 1.58 0.96 1.19 1.39 1.23 1.64 1.49 1.40 1.60 1.20 1.59 1.15 1.52

314 316 295 296 359 353 424 402 401 366 356 359 355 528 555 483 414 468 501 445 483 468 624 636 671 676 583 675 673 680 837 785 1037 1024

21.50 21.00 23.80 23.50 19.30 19.00 17.90 16.00 19.50 18.00 21.20 22.00 22.20 18.90 17.40 16.50 18.50 19.10 16.10 14.60 16.70 16.70 16.01 15.72 16.20 13.90 18.50 13.80 19.90 19.30 10.70 10.50 5.80 6.00

525 524 508 508 582 569 673 619 652 582 592 604 599 850 874 750 663 756 773 669 752 729 961 975 1037 1004 934 1001 1100 1102 1169 1092 1290 1281

0.195 0.191 0.213 0.211 0.176 0.174 0.165 0.148 0.178 0.166 0.192 0.199 0.200 0.173 0.160 0.153 0.170 0.175 0.149 0.136 0.154 0.154 0.149 0.146 0.150 0.130 0.170 0.129 0.181 0.176 0.102 0.100 0.056 0.058

1.93 1.98 2.25 2.15 1.64 1.75 1.13 1.47 1.00 2.08 1.54 1.90 1.83 0.83 0.92 1.04 1.60 1.07 1.15 1.09 0.90 0.94 0.86 0.85 0.87 0.93 1.01 1.00 0.93 0.89 0.85 0.85 0.80 0.80

DDQ+ BH 210 BH 280

UlcBH340 IF Rephos DP 500 BH 300 HSLA 350

HS 440 W DP 600

TRIP 600 DP800

Effective Stress at FLC0 (MPa)

DP980

1200 t = 2.1 mm 1000 t = 0.0 mm 800 600 400 400

600

800

1000

1200

Z parameter (MPa) Fig. 1. Effective stress at FLC0 (atFLC ¯ 0 ) as a function of the Z parameter for thicknesses, t, of 0.0 and 2.1 mm for phase 1 steels.

the relative effects of Z and thickness on atFLC ¯ 0 . It is also observed that Z has a much greater effect on atFLC ¯ 0 than thickness.

Experimentral Effective Stress at FLC 0 (MPa)

Material

1200 1000 800 600 400 400

600

800

1000

1200

Calculate Effective Stress at FLC0 (MPa)

Fig. 2. Effective stress at FLC0 (atFLC ¯ 0 ) from experimental data versus effective ¯ stress at FLC0 (atFLC 0 ) from Eq. (5) for phase 2 steels.

3.2. Validation of Eq. (5) The validation of Eq. (5) is based on the 45 experimental FLC0 data points obtained from 4 different investigations as described in phase 2 of the experimental procedure section. Table 5 shows the values of atFLC ¯ 0 based on the experimental data from these 4 studies. Table 5 also gives the calculated value of the Z parameter based on Eq. (3). The experimental (or measured) values are calculated from Eq. (6) and the calculated values are determined from

Eq. (5). The differences between experimental and calculated values of atFLC ¯ 0 are also shown in Table 5. The average value of these differences is 0.94 ± 16 MPa Fig. 2 shows a plot of the experimental values versus the calculated values foratFLC ¯ 0. The close agreement between experimental values and calculated values from Eq. (5) in Fig. 2 provides a validation of Eq. (5) for a wide range of forming quality steels, bake hardening steels,

B.S. Levy, C.J. Van Tyne / Journal of Materials Processing Technology 229 (2016) 758–768 Table 2 Plane strain forming limit, FLC0 (%) for various thicknesses, t (mm) (from Eq. (1)).

DQSK DDQ+ BH 210 BH 280

UlcBH340 IF Rephos DP 500 BH 300 HSLA 350

HS 440 W DP 600

TRIP 600 DP800 DP980

t = 0.0

t = 0.7

t = 1.4

t = 2.1

21.61 21.15 23.69 23.42 19.58 19.30 18.27 16.47 19.77 18.36 21.33 22.06 22.24 19.21 17.80 16.94 18.83 19.39 16.56 15.12 17.14 17.14 16.48 16.20 16.66 14.44 18.83 14.34 20.14 19.58 11.28 11.08 6.26 6.47

30.79 30.13 33.75 33.37 27.90 27.50 26.03 23.46 28.16 26.17 30.40 31.44 31.69 27.37 25.36 24.14 26.83 27.63 23.60 21.54 24.41 24.41 23.48 23.08 23.74 20.57 26.83 20.44 28.69 27.90 16.07 15.78 8.91 9.21

39.97 39.12 43.81 43.32 36.22 35.70 33.79 30.46 36.56 33.97 39.46 40.81 41.14 35.53 32.92 31.34 34.83 35.87 30.64 27.97 31.69 31.69 30.48 29.96 30.81 26.71 34.83 26.53 37.24 36.22 20.86 20.49 11.57 11.96

49.14 48.10 53.88 53.26 44.53 43.90 41.55 37.45 44.96 41.77 48.52 50.18 50.59 43.69 40.48 38.54 42.83 44.11 37.67 34.39 38.97 38.97 37.48 36.84 37.89 32.84 42.83 32.62 45.80 44.53 25.65 25.20 14.23 14.70

rephosphorized steels, micro-alloyed steels, carbon-manganese high strength steels, and dual phase steels. 4. Discussion of results 4.1. Physical basis of Eq. (5) The constant in Eq. (5) is 20.3 ± 3.5. While the constant is statistically different from zero, physically the constant is not significant and the physical basis for it is not known. The coefficient of the Z term in Eq. (5) is 1.002 ± 0.005, which is essentially 1.0. Given that the constant is not physically significant, it can be seen that if thickness equals zero (i.e., t = 0), atFLC ¯ 0 approximately equals Z. This result suggests that the Z parameter is directly related to the effective stress necessary to initiate a critical local neck, which is the point of instability or the forming limit in plane strain. Shi and Gerdeen (1991) used the size of a damage parameter based on the MK analysis to predict FLC limits. It is suggested that the size of the Shi and Gerdeen (1991) damage parameter can be a surrogate for voids at hard phase/soft phase boundaries. Wilson and Acselrad (1984) showed that for inclusions less than 10 ␮m in size, the role of cavitation is less clear. It is possible that in some cases, cavitation associated with small particles produce voids, but that these voids are too small for void growth and subsequent link up. It would seem more likely that the source of the voids that lead to instability and eventually to ductile fracture is at the hard phase/soft phase interfaces. Wilson and Acselrad (1979) showed the importance of cavity formation at poorly bonded oxide and sulfide inclusions to the failure process. Hiam and Lee (1978) showed the importance of elongated sulfide inclusions to the failure process. While these results are examples of hard phase/soft phase interfaces, recent advances in steelmaking have improved internal cleanliness to such

763

Table 3 ¯ Plane strain forming limit stress, atFLC 0 (MPa) for various thicknesses, t (mm) (from Eq. (6)) and Z values (MPa) (from Eq. (3)).

DQSK DDQ+ BH 210 BH 280

UlcBH340 IF Rephos DP 500 BH 300 HSLA 350

HS 440 W DP 600

TRIP 600 DP800 DP980

t = 0.0

t = 0.7

t = 1.4

t = 2.1

Z

404 405 391 390 448 441 515 484 493 456 451 463 458 642 668 578 512 574 599 523 577 560 740 752 797 787 710 786 828 831 943 882 1109 1097

429 430 418 417 474 466 543 507 522 480 480 493 488 679 703 607 541 607 629 547 606 588 776 788 837 821 750 820 877 879 975 912 1130 1119

449 449 439 437 494 485 564 525 543 499 501 516 511 706 729 629 562 632 651 565 629 610 803 815 867 847 780 846 914 916 999 934 1146 1135

464 464 455 453 509 500 581 539 560 514 518 534 528 728 750 647 579 651 668 579 646 627 825 837 890 867 803 866 944 944 1018 952 1159 1148

382 382 365 366 428 420 500 466 479 432 431 438 434 628 652 563 491 557 582 510 564 546 724 736 780 770 691 768 807 811 927 867 1097 1085

a degree that these inclusions should no longer control the value ofatFLC ¯ 0. It can be seen that the Z parameter represents the effective stress necessary to initiate void growth and crack initiation at hard phase/soft phase interfaces that begin the process that ultimately leads to an instability that causes a critical local neck. The final step in the process is sufficient propagation of the initial micro crack through the thickness for a critical local neck to form. The coefficient of the thickness term, a2 , in Eq. (5) describes the quantitative effect of the final stage of the failure process in increasing atFLC ¯ 0. A variety of qualitative explanations have been presented on why a thickness effect should occur, but from the available evidence the value of a2 is an empirical result. One normally expects that when a parameter such as Z increases, a measure for the failure should decrease. Various equations such as the Keeler and Brazier (1977) and Raghavan et al. (1992) equations have this form. Thus, for the Keeler–Brazier equation, as thickness and the n value decrease, FLC0 decreases. Similarly for the Raghavan et al. (1992) equation, as thickness and total elongation decrease, FLC0 decreases. In contrast, for Eq. (5), as thickness and Z value increase, the limit effective stress increases. While it is unusual for a failure criterion to increase as the parameters (such as Z) increase, this is a consequence of Z being both an instability parameter and a measure of strength; that is, Z is the tensile strength in true stress in a tensile test. Z is also the effective stress at which significant void nucleation and micro cracking occurs leading to instability, especially in the higher strength steels. The steels in the current study are ductile, so it is reasonable to expect that as their strength increases, their failure stress should increase. While thickness is a dimensional parameter, its coefficient converts it to effective stress. The effect of increasing thickness is to

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Table 4 Tensile properties of phase 2 steels. Material

Thickness (mm)

Tensile strength (MPa)

Uniform elongation UEb (%)

Levy and Green (2002) AKDQ 0.72 0.7CR 0.7EG 0.73 1.2CR 1.20 1.2EG 1.22

325 306 317 314

21.90 23.49 21.77 22.75

BH 210 0.7EG 0.8EG

0.73 0.78

366 351

HSLA 0.8HDGa 1.2CR

0.79 1.17

Strain hardening exponent, n

Normal anisotropy, R¯

550 525 535 533

0.212 0.211 0.209 0.205

1.70 1.78 1.65 1.56

19.48 16.88

595 548

0.178 0.156

1.83 1.55

460 493

16.53 14.80

723 752

0.168 0.149

1.05 1.06

618

14.00

924

0.136

1.12

303 312 314 364 385

26.87 26.11 25.11 23.24 2.02

541 552 549 622 425

0.238 0.232 0.224 0.209 0.020

1.89 1.84 1.57 1.62 1.49

313 308 316 321 409 402 397 407 432 517 486 441 630 607 610 528 708

27.25 25.99 21.90 22.63 22.75 20.56 21.90 17.94 23.00 13.54 14.11 16.77 17.94 18.77 25.86 5.23 2.43

561 544 531 544 695 663 667 646 736 763 725 687 1000 976 1077 647 793

0.241 0.231 0.198 0.204 0.205 0.187 0.198 0.165 0.207 0.127 0.132 0.155 0.165 0.172 0.230 0.051 0.024

1.11 1.21 1.28 1.47 1.42 1.30 1.31 0.96 0.99 1.04 1.03 1.28 1.02 0.96 0.85 0.76 0.78

0.71 0.76 0.94 0.81 0.76 0.86 0.69 0.81

300 308 310 286 304 311 276 296

22.02 21.17 18.29 17.94 24.11 21.53 21.65 18.77

505 512 495 454 525 520 462 476

0.199 0.192 0.168 0.165 0.216 0.195 0.196 0.172

1.51 1.80 1.40 1.67 1.52 1.34 1.63 1.64

0.81 0.71 0.94

315 290 293

26.24 25.73 24.23

558 511 507

0.233 0.229 0.217

1.42 1.77 1.59

1.20 1.80 1.20 1.60

590 592 837 777

16.65 17.12 12.19 13.20

918 928 1204 1139

0.154 0.158 0.115 0.124

1.02 0.99 0.88 0.87

DP600 1.18 1.2HDG Shi (1995) IF-I 0.82 0.81 IF -II 0.80 AKDQ 0.78 BH 0.82 BIW HSS Rhaghavan et al. (1992) CR DQ 0.81 DQ 0.81 0.91 DQ 0.94 DQSK 0.74 B37PK 1.37 40PK 40R 0.81 0.74 CA40 B40PO 1.32 0.94 HH Temper 1.27 B50XK B50XK 1.17 0.76 CA60 CA60M 0.76 0.69 60D 1.88 B60RO 1.73 B60RK HDG DQSK G90 G60 A40 G60 G60 G60 G60 G60 EG DQSK 61/0 61/20 61/20 Konieczny (2001) DP590 DP590 DP780 DP780 a

Strength coefficient, Kc (MPa)

Yield point elongation of 5.9%.

increase the time for an initial micro crack at a hard phase/soft phase interface to propagate to where instability allows a critical local neck to form. The increase in time allows further strain hardening to occur, which increases atFLC ¯ 0 . As shown in Fig. 1, thickness has only a modest effect on increasing atFLC ¯ 0.

4.2. Comparing predictive approaches Raghavan et al. (1992) used regression analysis to relate FLC0 limit strain to thickness and total elongation. Both Eq. (5) and the Raghavan et al. (1992) equation have a thickness term. Raghavan et al. (1992) use total elongation as the second term, while Z is the

B.S. Levy, C.J. Van Tyne / Journal of Materials Processing Technology 229 (2016) 758–768

765

Table 5 ¯ Values of atFLC 0 from experimental data and from Eq. (5). Material

Expt. FLC0 (%)

Expt. atFLC ¯ 0 (MPa)

Z parameter (MPa)

Eq. (5) atFLC ¯ 0 (MPa)

Difference Expt.−calculated (MPa)

Levy and Green (2002) AKDQ 0.7CR 0.7EG 1.2CR 1.2EG

30.2 33.7 35.5 36.6

438 427 439 441

396 378 386 385

442 424 448 448

−4 3 −9 −8

BH 210 0.7EG 0.8EG

24.4 26.6

476 454

437 410

484 458

−8 −4

HSLA 0.8HDG# 1.2CR

22.2 27.0

567 621

536 566

584 627

−18 −6

25.0

771

705

767

4

39.6 40.0 NA 33.0 33.6

445 457 NA 503 416

384 393 393 449 393

434 442 NA 497 442

12 15 NA 7 −26

44.0 44.0 39.0 37.0 30.5 35.0 27.5 32.0 37.0 27.5 28.5 28.5 23.0 25.5 27.0 19.0 11.0

458 449 442 450 553 549 524 535 597 650 616 571 790 774 796 595 753

398 388 385 394 502 485 484 480 531 587 555 515 743 721 768 556 725

447 437 437 447 549 553 533 527 598 641 619 576 791 769 813 641 806

10 12 5 3 4 −4 −9 8 −1 9 −4 −5 −1 6 −17 −47 −53

37.0 41.0 41.0 43.0 44.0 41.5 42.5 43.0

419 439 428 399 444 440 395 416

366 373 367 337 377 378 336 352

411 420 420 386 424 428 380 400

8 19 8 13 19 11 15 16

42.5 43.0 41.5

460 429 424

398 365 364

446 410 417

14 19 7

30.0 26.0 22.0 28.0

764 753 1014 973

688 693 939 880

751 777 1002 956

13 −24 12 16

DP600 1.2HDG Shi (1995) IF-I IF -II AKDQ BH BIW HSS Rhaghavan et al. (1992) CR DQ DQ DQ DQSK B37PK 40PK 40R CA40 B40PO HH temper B50XK B50XK CA60 CA60M 60D B60RO B60RK HDG DQSK G90 G60 A40 G60 G60 G60 G60 G60 EG DQSK 61/0 61/20 61/20 Konieczny (2001) DP590 DP590 DP780 DP780

key parameter in the current study. While total elongation can be a valuable parameter for an empirical relationship, the Z term in Eq. (5) provides a fundamental relationship to the ductile failure process that leads to a critical local neck. The coefficient of the Raghavan et al. (1992) thickness term is 3.24 compared to 34.25 in this study. The coefficient of the thickness term in both regression equations is empirical. However, quantitative comparisons are not possible, because FLC0 in the Raghavan ¯ et al. (1992) equation is in engineering strain while atFLC 0 in Eq. (5) is in effective stress.

Cayssails (1998) has another method of predicting forming limit curves that is based on fundamental relations. While the independent variables are described, the actual predictive relationships have not been published, so direct comparisons are not possible. What is clear is that the Cayssails model requires a number of metallurgical properties for its use. The advantage of the Z parameter is that it requires knowledge of only the tensile strength and the uniform elongation from a tensile test. The Z parameter is also easy to determine, and could be used to characterize steel grades.

B.S. Levy, C.J. Van Tyne / Journal of Materials Processing Technology 229 (2016) 758–768

First Principal Stress Component (MPa)

766

900

HSLA Steel 0.8 mm thick From Strain-Based FLC

800

700 From Z parameter and Mechancial Properties

600

500 0

100

200

300

400

500

Second Principal Stress Component (MPa)

Fig. 3. Strain-based forming limit curve (FLC) in true strains with the plane-strain strain path and an arbitrary strain path indicated.

First Principal Stress Component (MPa)

Fig. 5. Stress-based forming limit curve for HSLA steel based on the strain-based FLC and calculated from the Z parameter and mechanical properties.

shows the left side of an FLC in true strain. Fig. 3 showsthat the left side of the FLC originates at plane strain, FLC0 . FLC0 is designated 1−0 in true strain,and the major strain increases as the minor strain becomes more negative. So along the left side of the FLC

800

AKDQ Steel 0.7mm thick From Strain-Based FLC

700

1 = −2 + 1−0 600

Fig. 3 shows a strain path from the origin to the value of 1 on the FLC. The strain path is designate as ˇ, which is the ratio of 2 to 1 . It should be noted that ˇ is unrelated to R¯ because it is an imposed strain path and does not involve the flow response of the material. Along this strain path

From Z parameter and Mechancial Properties

500

(8)

2 = ˇ 1

400 0

100

200

300

400

500

Second Principal Stress Component (MPa)

Fig. 4. Stress-based forming limit curve for AKDQ steel based on the strain-based FLC and calculated from the Z parameter and mechanical properties.

The Keeler and Brazier (1977) relationship is the basis for Eq. (5). In phase 1, the engineering FLC0 strain was calculated using experimental n values and representative thicknesses. The value of FLC0 was converted to atFLC ¯ 0 . Z was calculated using the same tensile data for the steels. Regression analysis was used to determine Eq. (5). Eq. (5) was then verified using other data from the literature. 4.3. Predicting a stress-based FLC using Z for the left side of the strain-based FLC Predicting the stress-based FLC from the left side of a strainbased FLC is based on the assumption that the predicted stress for plane strain is correct. Stress-based FLCs are represented in plots of the principal stress components,  1 and  2 . Eq. (5) predicts the ¯ effective stress for strain path in plane strain (i.e.,atFLC 0 ). If the value of atFLC ¯ 0 is known and the assumption of the slope for the left side of the strain-based FLC in true strain is –1, then the stressbased FLC corresponding to the left side of the strain-based FLC can ¯ be determined. The value of atFLC 0 can be determined from the thickness, the tensile strength, and uniform elongation from Eq. (5). It is generally accepted that the left side of the strain-based FLC is parallel to the line of pure shear; pure shear is when the major true strain, 1 , equals the negative of the minor true strain, 2 . Fig. 3

(9)

So when the strain path reaches a point on the left side of the FLC,

1 = −ˇ1 + 1−0

(10)

and upon rearrangement

1 =

1−0

(11)

1+ˇ

Eq. (11) is valid for any strain path that reaches a point on the left side of the FLC in true strain. With the assumption of Hill (1948) plasticity (i.e., normal anisotropy but planar isotropy), the effective strain for any point on the left side of the strain-based FLC can be determined as 1 + R¯

¯ = 

1 + 2R¯



21 + 22 +

2R¯ 1 + R¯

1 2

(12)

Effective stress can be calculated using the power law hardening constitutive material model as

 n

¯ = K ¯

(13)

The 3 principal strain components on the left side FLC can be determined. For a fixed value of ˇ, 1 is calculated from Eq. (11). The value of 1−0 is determined from Eq. (5) by converting the atFLC ¯ 0 into effective strain via Eq. (13) and inverting Eq. (12) to solve for 1−0 , which is 1 , since 2 = 0 for the plane-strain conditions. The value of 2 is determined from Eq. (9). The value of 3 is determined from constant volume.

First Principal Stress Component (MPa)

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space to stress space. Standard mechanics are then used to extend the effective FLC0 stress to the stress-based FLC.

1100 5. Conclusions

DP600 Steel 1.2 mm thick 1000

From Strain-Based FLC

900

800

From Z parameter and Mechancial Properties

700 0

100

200

300

400

500

600

Second Principal Stress Component (MPa)

Fig. 6. Stress-based forming limit curve for DP600 steel based on the strain-based FLC and calculated from the Z parameter and mechanical properties.

The stress ratio, ˛, between the minor principal stress component,  2 , and the major principal stress component,  1 , and assuming Hill (1948) plasticity can be calculated as



˛=



1 + R¯ ˇ + R¯

(14)

¯ 1 + R¯ + Rˇ

¯



1 + ˛2





(15)

¯ − 2R/1 + R¯ ˛

and 2 = ˛1

The Z parameter, which can be calculated from a uniaxial tensile test, is the key variable in determining the effective stress at the forming limit for the plane-strain condition. It has been shown by knowing the value of the Z parameter and the thickness of the sheet, the plane-strain effective forming limit stress can be calculated. The Z value can be regarded as an instability parameter, but it can also be regarded as a damage parameter because microstructural damage contributes to the instability condition. A method has been developed to predict a stress-based FLC corresponding to the left side of a strain-based FLC using thickness and properties obtained from a tensile test. This method is applicable to conventional steels, but further work is required to determine its applicability to advanced high strength steels. Since stress-based FLCs are often strain path independent, the use of this method could provide a practical approach to analyzing sheet metal stampings with complex strain paths. Acknowledgement Partial support for this work from the Advanced Steel Processing and Products Research Center at Colorado School of Mines is gratefully acknowledged. References

The major principal stress components are 1 =

767

(16)

The values of  1 and ␴2 in Eqs. (15) and (16) are a point on the stress-based FLC associated with the strain path ˇ for the left side of a strain-based FLC. Using a variety of strain paths allows the calculation of the stress-based FLC for the full left side of the strain-based FLC. Strain-based FLCs for a 0.7 mm aluminum killed steel, a 0.8 mm HSLA steel, and a 1.2 mm DP600 from Levy and Green (2002) are used to compare the actual and predicted values for stress-based FLCs. The Levy and Green FLC0 values are used in the verification of Eq. (5), but the data for the left side of the strain-based FLC has not been used previously. These strain-based forming limit curves were converted to stress-based FLCs using the procedure of Stoughton (2000). Figs. 4–6 compare the stress-based FLC calculated from the left side of the strain-based FLC to the stress-based FLC based upon Eqs. (8)–(16) using the value of Z from Eq. (5). Figs. 4–6 show good agreement between experimental and predicted results with the predicted values being only 15–20 MPa less than the experimental values. Arrieux et al. (1982), Brunet and Chelhade (1991), Stoughton (2000), Stoughton and Zhu (2004), Zeng et al. (2008), and Stoughton and Yoon (2011, 2012) have indicated that some stress-based FLCs are independent of the stress path. However, it has been difficult to develop stress-based FLCs. The prediction method described in the current study provides a practical approach for determining the stress-based FLC corresponding to the left side of the strain-based FLC, which should lead to increased use of stress-based FLCs. The use of Z in Eq. (5) with the appropriate coefficients is the critical step in extending the Keeler–Brazier equation from strain

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