Path independent limiting criteria in sheet metal forming

G Model

ARTICLE IN PRESS

JMP-348; No. of Pages 13

Journal of Manufacturing Processes xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro

Technical paper

Path independent limiting criteria in sheet metal forming Surajit Kumar Paul ∗ R&D, Tata Steel Limited, Jamshedpur 831007, India

a r t i c l e

i n f o

Article history: Received 30 September 2014 Received in revised form 26 June 2015 Accepted 30 June 2015 Available online xxx Keywords: Strain path dependency Non-proportional loading Stress based Forming Limit Diagram Polar Effective Stress Diagram Polar Effective Plastic Strain Diagram

a b s t r a c t The Forming Limit Diagram (FLD) is a conventional failure diagram to estimate necking limits of sheet metal for proportional loading conditions. Previous studies reveal that the FLD is not suitable for predicting the influence of nonlinear strain paths. This paper presents methodical comparison among all common available strain path independent strain/stress based limiting criteria. All the strain path independent strain based limiting criteria (Traditional Failure Diagram (TFD), Extended Forming Limit Diagram (XFLD), Extended Stress Ratio Based Forming Limit Diagram (ESRFLD), and Polar Effective Plastic StrainDiagram (PEPSD)) and stress based limiting criteria (Traditional Stress based Failure Diagram (TFSD), Stress Based Forming Limit Diagram (FLSD), Stress Ratio and Stress Based Forming Limit Diagram (SRFLSD), Extended Stress Based Forming Limit Diagram (XFLSD), and Polar Effective Stress Diagram (PESSD)) are approximately path-independent for smaller amount of pre-straining and path dependent for large prestraining conditions. From advance image correlation technique precisely determination of local strains near necked area is possible today. However direct determination of local stresses near necked area is not possible. Therefore, local stresses and equivalent stress are determined by employing both yield criterion and strain-hardening law. Similarly equivalent strain is calculated by the use of yield criterion. Therefore, the choice of yield criterion has an impact on the results for TFD, XFLD, ESRFLD and PEPSD. However, selections of both yield criterion and strain-hardening law have substantial influence on the results for TFSD, FLSD, SRFLSD, XFLSD and PESSD. The inherent calculation error can be minimized by generation of experimental data for each material and then selection of representable yield criterion and strain-hardening law. Improvement of experimental techniques and generation of rigorous material data bank in various strain paths may help researchers to diagnose and resolve the issue. TFD, TFSD and XFLSD have inherent variables to take care the effect of through thickness stress, however rigorous experimental verification is needed before the field application. © 2015 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction Remarkable efforts have been devoted to understand the physical nature of forming limits in sheet metal forming operations due to the wide use of Forming Limit Diagram (FLD) in sheet metal forming industry. The forming limit is generally defined as the ability of metal to deform without necking or fracture into desired shape. The sheet metal can be deformed successfully without failing only up to a certain limit, which is normally known as forming limit curve (FLC). FLC is generally governed by localized necking (an instability phenomenon), which eventually leads to the ductile fracture [1,2]. The FLC can be divided into two branches: “left branch” and “right

∗ Present address: School of Engineering, Deakin University, Pigdons Rd, Waurn Ponds 3217, VIC, Australia. Tel.: +61 431362497; E-mail address: [email protected]

branch”. The “right branch” of FLC is valid for positive major and minor strains (i.e. stretching), while the “left branch” of FLC is applicable for positive major and negative minor strains (i.e. drawing). The FLD is a plot of limit strains in principal strain space under linear (i.e. proportional) strain paths. The FLD is experimentally determined through limiting dome height examinations where a hemispherical punch is used to deform a sheet until localized neck or fracture observed. The state of strain near the necked region is called the forming limit strain. The limiting strains along different strain paths (e.g. uniaxial, plane strain and equi-biaxial) can be achieved by altering the initial blank size. The forming limit strains are predicted by the onset of localized necking. Localized necking is observed during ductile fracture of the materials due to void nucleation, coalescence and growth with strain paths ranging from uniaxial tension to equi-biaxial tension. However in case of materials with low ductility, fracture often occurs without any obvious necking phenomenon. The sheet metal forming is different

http://dx.doi.org/10.1016/j.jmapro.2015.06.025 1526-6125/© 2015 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model JMP-348; No. of Pages 13

ARTICLE IN PRESS

2

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx 0.40

0.40

(a)

0.35

0.25 0.20 As received Biaxial pre-straining 0.03 0.085 0.13 0.17

0.15 0.10

0.25

0.15

0.05

Al alloy A1 6111 T4

0.00 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Pre-straining perpendicular to RD Tested perpendicular to RD 0.00 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Minor strain (ε2)

Minor strain (ε2) 0.14

0.40

Major strain (ε1)

0.095 0.14 Plane strain pre-straining 0.05 0.12

0.20 0.15

(c)

(d) 0.12 0.10

Major strain (ε1)

As received Pre-straining parallel to RD Uniaxial pre-straining 0.35 Tested perpendicular to RD 0.05

0.25

As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

0.20

0.10

Tested perpendicular to RD

0.30

(b)

0.30

Major strain (ε1)

Major strain (ε1)

0.30

0.05

Al alloy A1 6111 T4

0.35

0.10

0.08 0.06

Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

0.04

0.05

0.02

Al alloy A1 6111 T4 0.00 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Al alloy AA5083 0.00 -0.06 -0.04 -0.02

Minor strain (ε2)

Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

0.00

0.02

0.04

0.06

0.08

Minor strain (ε2)

Fig. 1. Forming Limit Diagram (FLD): major strain (ε1 ) vs. minor strain (ε2 ) plot: (a) biaxial pre-strain paths of Al alloy A1611T4 [4] (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 [4] (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 [4] (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083 [7].

Forming Limit Diagram (FLD): Major strain (ε1) vs. Minor strain (ε2)

Yield function

Traditional Failure Diagram (TFD): Equivalent strain (εeq) vs. stress triaxiality (η) Extended Forming Limit Diagram (XFLD): Equivalent strain (εeq) vs. strain rate ratio (β)

Determination of equivalent strain (εeq)

Extended Stress Ratio Based Forming Limit Diagram (ESRFLD): Equivalent strain (εeq) vs. stress ratio (α) Polar Effective Plastic Strain Diagram (PEPSD)

Hardening law Stress Based Forming Limit Diagram (FLSD): Major stress (σ1) vs. Minor stress (σ2)

Determination of equivalent stress (σeq)

Traditional Stress Based Failure Diagram (TFSD): Equivalent stress (σeq) vs. stress triaxiality (η) Extended Stress Based Forming Limit Diagram (XFLSD): Equivalent stress (σeq) vs. hydrostatic/mean stress (σm) Extended Stress Ratio Based Forming Limit Diagram (ESRFLD): Equivalent strain (εeq) vs. stress ratio (α) Polar Effective Stress Diagram (PESSD)

Fig. 2. Schematic diagram of procedure to determine path independent criteria.

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model

ARTICLE IN PRESS

JMP-348; No. of Pages 13

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx

0.40

420

(a)

410

0.35

Equivalent stress (σeq)

400

0.30

Major strain (ε1)

3

0.25 0.20 0.15 0.10

390

von Mises- El Magd von Mises- Power law Hill- El Magd Hill- Power law

(a)

380 370 360 350 340 330 320 310

0.05 Al alloy AA6014

300

0.00 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

290 -0.4

Minor strain (ε2)

Al alloy AA6014 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

Strain rate ratio (β)

0.7

350

(b)

(b)

340

0.6

Major stress, MPa (σ1)

Equivalent strain (εeq)

330 0.5 0.4 0.3 0.2 0.1 0.0 -0.4

von Mises Hill -0.2

0.0

Al alloy AA6014 0.2

0.4

0.6

0.8

1.0

Strain rate ratio (β)

320 310 300 290 280 von Mises- El Magd von Mises- Power law Hill- El Magd Hill- Power law

270 260 250 -50

0

50

100

150

Al alloy AA6014 200

250

300

350

Minor stress, MPa (σ2)

Fig. 3. For aluminum alloy AA6014 (a) as received FLD [25] (b) equivalent strain (εeq ) vs. strain rate ratio (ˇ) plot with two yield criteria, von Mises and Hill normal anisotropy.

400

Fig. 5. After transformation of as received FLD for aluminum alloy AA6014 by two different yield criteria (von Mises and Hill normal anisotropy) and two different strain-hardening law (El-Magd and power law); their combination are (a) equivalent stress (␴(eq ) vs. strain rate ratio (ˇ) plot (b) FLSD, major stress ( 1 ) vs. minor stress (␴2 ).

350

True stress, MPa

300 250 200 150 100 50 0 0.0

Tensie test Bulge test El Magd Power law

Al alloy AA6014 0.1

0.2

0.3

0.4

0.5

0.6

0.7

True strain Fig. 4. Tensile true stress–strain curve extrapolation by El-Magd & power law, and verification from the flow curve recorded during biaxial bulge test. (experimental data are collected from Ref. 24 and 25)

from bulk forming. The sheet metal forming is mainly driven by stretching and generally plane stress condition exists, whereas bulk forming is driven by compression and 3D stress state is evident. The concept of the FLD has been extensively applied to predict formability to the sheet metal forming industry because of its simplicity, straightforwardness and comfort of use. The FLC is dynamic in nature, whose shape and position significantly alter with strain paths [3–9]. Therefore, the use of conventional FLD might be restrictive because the sheet metals experience nonlinear strain paths during the real industrial forming operations. This path dependence of conventional FLC confines its application in single step forming operations and same time it argues the necessity of further path independent forming limit criteria. The path independent forming limit criterion is mandatory in multi-step forming and hydroforming processes. Over the last two decades number of path independent forming limit criteria are presented. In the present investigation a comparative assessment is conducted among them with the help of experimental data collected from literatures for better understanding their applicability, benefits and drawbacks. Apart from FLC,

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model JMP-348; No. of Pages 13

ARTICLE IN PRESS

4

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx

[14,15]. In the present work, our investigation is restricted on forming limit criteria only.

400

B

350 6.3% increase

True stress, MPa

300

2. Forming Limit Diagram (FLD): A path dependent limiting criterion

A

250

Keeler and Backofen [16] and Goodwin [17] have developed the concept of FLD to investigate the formability of sheet metals. The locus of the necking limit in the in-plane principal strain space of sheet metals is generally known as FLC and the diagram is known as FLD. With advancement of digital image correlation (DIC) analysis methods, accurate measurement of strain near necking zone is possible today. Thus experimentally construction of FLD with DIC method also becomes accurate. Wang et al. [18] discussed in details about several DIC procedures to generate FLD. The FLD is affected by many factors, such as strain rate (the forming speed), strain hardening property, normal and planar anisotropy of the sheet metals [19], thickness of sheets, through thickness stress [20,21] and lubrication condition. Some investigators used fracture criteria in place of FLD to define the formability of materials. Chung et al. [22,23] explained that the sheets are so ductile that sheet forming more often fails after abruptly severe strain localization, especially in the thinning mode. In such a case, measuring the fracture property is impractical and forming limit criteria cannot be replaced by fracture criteria. This is possible for limited number of cases where material fails suddenly without any observable necking. The strain paths in most metal forming applications particularly in the first draw die of axisymmetric and uniform sectioned parts are satisfactorily linear and the FLD for the as-received condition can be used comfortably in making formability assessments.

200 150 100 50 0 0.0

50% increase

Al alloy AA6014 0.1

0.2

0.3

0.4

0.5

0.6

0.7

True strain Fig. 6. Tensile true stress–strain curve extrapolation by El-Magd law; to move from point A to point B around 50% increase in true strain is noticed whereas increase of stress is approximately 6.3% only.

other failure criteria are also used to define the limit in sheet metal forming, they are: wrinkling limit diagram—presence of compressive stress [10,11], fracture limit diagram—useful for incremental sheet forming [12] and high strength material like TWIP steel which is fractured without noticeable necking [13], hole expansion ratio—is also defined as fracture limit of the edge of hub hole 0.55

0.55

(a)

0.45

0.45

0.40

0.40

0.35 0.30 0.25 0.20 As received Biaxial pre-straining 0.03 0.085 0.13 0.17

0.15 0.10 0.05 0.00 0.25

0.30

0.35

Al alloy A1 6111 T4

0.45

0.50

0.55

0.60

0.35 0.30 0.25 0.20 0.15 0.10 0.05

Tested perpendicular to RD 0.40

(b)

0.50

Equivalent strain (εeq)

Equivalent strain (εeq)

0.50

0.65

As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

0.00 0.25

0.30

0.35

(c)

0.50

Equivalent strain (εeq)

Equivalent strain (εeq)

0.55

0.60

0.65

(d)

0.14

0.40 0.35 0.30 0.25

0.05

0.50

0.16

0.45

0.10

0.45

0.18

0.55

0.15

0.40

Triaxiality (η)

Triaxiality (η)

0.20

Al alloy A1 6111 T4 Pre-straining perpendicular to RD Tested perpendicular to RD

As received Uniaxial pre-straining 0.05 0.095 0.14 Plane strain pre-straining 0.05 0.12

0.00 0.25

0.30

0.35

Al alloy A1 6111 T4 Pre-straining parallel to RD Tested perpendicular to RD

0.40

0.45

0.50

Triaxiality (η)

0.55

0.60

0.65

0.12 0.10 0.08 Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

0.06 0.04

Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

0.02 Al alloy AA5083 0.00 0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

Triaxality (η)

Fig. 7. Traditional Failure Diagram (TFD): equivalent strain (εeq ) vs. stress triaxiality (␩) plot: (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model

ARTICLE IN PRESS

JMP-348; No. of Pages 13

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx 500

500

(a)

400

350

350

300 250 200 150

As received Biaxial pre-straining 0.03 0.085 0.13 0.17

100 50 0 0.25

0.30

0.35

Al alloy A1 6111 T4

300 250 200 150 100 50

Tested perpendicular to RD 0.40

0.45

0.50

0.55

0.60

(b)

450

400

Equivalent stress (σeq)

Equivalent stress (σeq)

450

0.65

As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

0 0.25

0.30

0.35

(c)

450

0.50

0.55

0.60

0.65

(d)

350

Equivalent stress (σeq)

Equivalent stress (σeq)

0.45

400

400 350 300 250 200

50

0.40

450

500

100

Al alloy A1 6111 T4 Pre-straining perpendicular to RD Tested perpendicular to RD

Triaxiality (η)

Triaxiality (η)

150

5

As received Uniaxial pre-straining 0.05 0.095 0.14 Plane strain pre-straining 0.05 0.12

0 0.25

0.30

0.35

Al alloy A1 6111 T4 Pre-straining parallel to RD Tested perpendicular to RD

0.40

0.45

0.50

0.55

0.60

0.65

Triaxiality (η)

300 250 200 Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

150 100

Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

50 Al alloy AA5083 0 0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

Triaxality (η)

Fig. 8. Traditional Stress based Failure Diagram (TFSD): equivalent stress ( eq ) vs. stress triaxiality () plot: (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

However, nonlinear strain paths are common for the complex 3D geometries, multi-stage forming and hydro-forming. The major limitation of the FLD is shown in Fig. 1 which is illustrated by Graf and Hosford [4] and Zhalehfar et al. [7] experimentally. The experimental results reveal that the FLD for bi-linear strain path is not static, rather it is an inherently dynamic limit for sheet metal forming. This result indicates that the FLD for the as-received condition provides no clue about the margin of safety of a specified forming process for bi-linear/non-linear strain paths. Therefore, the strain path independent criteria are necessary to predict the failure or the safe forming in sheet metal forming industry. 3. Path independent limiting criteria Several path independent limiting criteria are developed to overcome the strain path dependency of the FLD. Fig. 2 represents the schematic diagram of available types of path independent limiting criteria in sheet metal forming. Fortunately, the major (ε1 ) and minor (ε2 ) strains and from there the strain rate ratio (ˇ) can be determined directly from the experiment. To determine equivalent strain (εeq ), stress ratio (˛) and triaxiality () a suitable yield criterion is required. Furthermore, equivalent stress ( eq ), major ( 1 ) and minor ( 2 ) stresses and hydrostatic/mean stress ( m ) can be determined by utilization of yield criterion and strain-hardening law of the material simultaneously. Before going to in details predictive capability analysis of individual path independent limiting criteria, the effect of yield criterion and material strain-hardening law on the mapping process (like major and minor strain space to major and minor stress space) are discussed in this section. For this purpose, the material data are collected form Werber et al. [24,25].

FLD for as received condition of aluminium alloy AA6014 is collected from Werber et al. [24,25] and depicted in Fig. 3(a). The equivalent strain (εeq ) and strain rate ratio (ˇ) are computed from von Mises and Hill normal anisotropy criteria [26]. The details of the calculations are illustrated in Appendices A–C. Fig. 3(b) shows that the equivalent strains computed from two yield criteria are dissimilar. The variations are more prominent in the stretching region. Therefore, it can be concluded that the selection of yield criterion has prominent influence on the equivalent strain calculation. Accordingly, the selection of yield criterion has noticeable effect on all path independent equivalent strain based criteria. The error can be minimized by selecting a proper yield criterion which describes the experimental yielding behaviour (uniaxial, plane strain to equibiaxial stress state) of the material accurately. For all path independent stress based criteria, first equivalent strain is determined with appropriate yield criterion and then equivalent stress is determined from suitable strain-hardening law (Fig. 2). Therefore, all path independent stress based criteria should be influenced by both yield criterion and strain-hardening law. Paul [27] showed that only few empirical equations are able to successfully extrapolate the true stress–strain curve beyond the uniform elongation of the material. Normally, diffuse necking initiates at the point of uniform elongation (i.e. maximum load bearing capacity of the material) during tensile test, while the FLC is based on localize necking. Localize necking starts in sheet material after diffuse necking. Therefore, material models which will be used for sheet metal forming operations should describe the stress–strain behaviour of the material beyond the uniform elongation. Paul [27] showed that El-Magd, Swift-Voice and Hockett/Sherby equations are able to effectively extrapolate the true stress–strain curve beyond the uniform elongation and he validates it by bulge test data. Similarly,

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model JMP-348; No. of Pages 13

ARTICLE IN PRESS

6

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx 0.55

0.45

0.45

0.40

0.40

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.75

As received Biaxial pre-straining 0.03 0.085 0.13 0.17

-0.50

-0.25

0.35 0.30 0.25

0.15

0.50

0.75

1.00

Al alloy A1 6111 T4

0.05 Pre-straining perpendicular to RD Tested perpendicular to RD 0.00 -0.75 -0.50 -0.25 0.00 0.25 0.50

Tested perpendicular to RD 0.25

1.25

0.75

1.00

1.25

Strain rate ratio (β)

Strain rate ratio ( β) 0.55

0.18

0.50

(d)

0.16

(c)

0.40 0.35 0.30 0.25 0.20 0.15 0.10 Al alloy A1 6111 T4 0.05 Pre-straining parallel to RD Tested perpendicular to RD 0.00 -0.75 -0.50 -0.25 0.00 0.25

As received Uniaxial pre-straining 0.05 0.095 0.14 Plane strain pre-straining 0.05 0.12

0.50

0.75

1.00

1.25

Strain rate ratio (β)

0.14

Equivalent strain ( εeq)

0.45

Equivalent strain ( εeq)

As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

0.20

0.10 Al alloy A1 6111 T4

0.00

(b)

0.50

Equivalent strain ( εeq)

Equivalent strain ( εeq)

0.55

(a)

0.50

0.12 0.10 0.08 Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

0.06 0.04

Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

0.02 Al alloy AA5083 0.00 -0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Strain rate ratio ( β)

Fig. 9. Extended Forming Limit Diagram (XFLD): equivalent strain (εeq ) vs. strain rate ratio (ˇ) plot: (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

tensile and bulge test data are collected form Werber et al. [24,25]. Up to uniform elongation both El-Magd and power law equations successfully predict the true stress–strain curve as shown in Fig. 4, however large deviation is observed after uniform elongation of the material. Fig. 5(a) shows mapping of equivalent stress ( eq ) vs. strain rate ratio (ˇ) plot from FLD for as received condition. It is clear from Fig. 5(a) that for two different strain-hardening law (El-Magd and power law), the equivalent stresses are also different. Four distinguished curves are observed in Fig. 5(a) for four different combinations of yield criterion and strain-hardening law. Therefore, it can be concluded that all path independent stress based criteria are influenced by choice of yield criterion and strain-hardening law. Inappropriate selection of yield criterion and strain-hardening law can produce significant error. Accepting large inherent calculation error in the computation of path independent strain/stress based criteria to avoid path dependency of FLD is not desirable. Single source of error (due to yield criterion) in the calculation of path independent strain based criteria is noticed, whereas double sources of error (due to both yield criterion and strain-hardening law) in the calculation of path independent stress based criteria are observed. Similarly, the effect of selection of yield criterion and strain-hardening law on Stress Based Forming Limit Diagram (FLSD) is also depicted in Fig. 5(b). Both yield criterion and strainhardening law have considerable influence on FLSD. It is discussed in the following sections in this article that the path independent stress based criteria always look better (falling in narrow band) in comparison with path independent strain based criteria. The reason behind this observation is discussed in Fig. 6. The blue line represents true stress–strain curve of the material. Two red points (A and B) are selected at true strain of 0.3 and 0.45, respectively. To move from point A to point B, around 50% increase

in strain is observed whereas only 6.3% increase in stress is noticed. This is why the point in path independent stress based criteria falls in narrow band and looks better than path independent strain based criteria. 4. Traditional Failure Diagram (TFD) Stress triaxiality (), the ratio of hydrostatic/mean stress ( m ) to equivalent stress ( eq ) is widely used as a conventional damage parameter in many classical damage models like Gurson–Tvergaard–Needleman (GTN) model for ductile damage [28], Pirondi and Bonora (PB) continuum damage mechanics (CDM) model [29], Johnson–Cook fracture model (JC) [30] etc. Xue–Wierzbicki model (XW) [31,32] has related the locus of fracture equivalent strain to stress triaxiality and load angle parameter. Johnson–Cook fracture model [30] has described equivalent strain (εeq ) vs. stress triaxiality () plot to characterize fracture locus at different stress states. Paul [33] also has used same equivalent strain (εeq ) vs. stress triaxiality () diagram to describe necking limit of materials. To access the predictive capability of the TFD, the experimental data are collected from literatures. Uniaxial and biaxial pre-straining forming limits data are collected for Al alloy A1611T4 [4] and Al alloy AA5083 [7] for as received condition. The same sets of data are utilized to access the usefulness of various strain path independent stress/strain based forming limiting criteria in this article. The equivalent strain (εeq ) vs. stress triaxiality () plots for Al alloy A1611T4 and Al alloy AA5083 are depicted in Fig. 7 for various pre-straining conditions. The stress state in sheet metal forming operation can be considered as plane stress as sheet thickness which is very less in comparison with other dimensions. Therefore,

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model

ARTICLE IN PRESS

JMP-348; No. of Pages 13

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx 0.55

0.45

0.45

0.40

0.40

0.35 0.30 0.25 0.20 0.15

As received Biaxial pre-straining 0.03 0.085 0.13 0.17

0.10 0.05 Al alloy A1 6111 T4 Tested perpendicular to RD 0.00 -0.25 0.00 0.25 0.50

0.75

1.00

(b)

0.50

Equivalent strain (εeq)

Equivalent strain (εeq)

0.55

(a)

0.50

7

1.25

0.35 0.30 0.25 As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

0.20 0.15 0.10

Al alloy A1 6111 T4

0.05 Pre-straining perpendicular to RD Tested perpendicular to RD 0.00 -0.25 0.00 0.25 0.50 0.75

1.00

1.25

Stress ratio (α)

Stress ratio (α) 0.55

0.18

(c)

0.50

(d)

0.16

0.45

Equivalent strain (εeq)

0.35 0.30 0.25 0.20

As received Uniaxial pre-straining 0.05 0.095 0.14 Plane strain pre-straining 0.05 0.12

0.15 0.10 Al alloy A1 6111 T4 0.05 Pre-straining parallel to RD Tested perpendicular to RD 0.00 0.00 0.25 0.50 -0.25

0.75

1.00

Equivalent strain (εeq)

0.14

0.40

1.25

Stress ratio (α)

0.12 0.10 0.08 Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

0.06 0.04

Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

0.02 Al alloy AA5083 0.00 -0.2

0.0

0.2

0.4

Stress ratio (α)

0.6

0.8

1.0

Fig. 10. Extended Stress Ratio Based Forming Limit Diagram (ESRFLD): equivalent strain (εeq ) vs. stress ratio (˛) plot: (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

the stress triaxiality () is 0.33 for uniaxial tension, 0.57 for plane strain tension and 0.67 for equi-biaxial tension for von Mises yield criterion. For, Hill normal anisotropic yield criterion the stress triaxiality value alters accordingly with coefficient of normal anisotropy (r). In the current investigation, Hill normal anisotropic yield criterion is used for all various strain path independent stress/strain based forming limiting criteria. Except at high equi-biaxial tension, the equivalent strain remains almost unaltered with pre-straining paths. From Fig. 7, the current investigation illustrates that the TFD is not fully strain path independent especially at high prestraining levels. For low pre-strain levels it can be stated that the TFD is roughly strain path independent. The key advantages of the TFD are (a) less strain path dependent in comparison to FLD, (b) only yield criterion is required for computation, and (c) stress triaxiality is a function of through thickness stress also so it can be used for hydroforming, however, rigours experimental verification is required before use it in hydroforming. The main disadvantage of the TFD is the predictive capability which alters with yield criterion. 5. Traditional Stress based Failure Diagram (TFSD) After replacing equivalent strain (εeq ) with equivalent stress ( eq ), the TFD becomes TFSD. The yield criterion is required to compute the equivalent strain and after that an appropriate strainhardening law is necessary to calculate equivalent stress. In this work, Hill normal anisotropic yield criterion and El-Magd strainhardening law are used. The TFSDs for Al alloy A1611T4 and Al alloy AA5083 with various pre-straining conditions are illustrated in Fig. 8. At a glance, the predictive capability of TFSD seems better than TFD, because the all pre-straining limiting lines fall in a narrow band. But, the reason behind this is already discussed in Fig. 6. Practically, there is no significant difference between TFD and TFSD

exist, TFD is in equivalent strain space while TFSD is in equivalent stress space. Like TFD, TFSD also have many leading benefits, they are: (a) less loading path dependent in comparison to FLD, (b) only yield criterion and strain-hardening law are required for calculation, and (c) it can be used in forming applications where through thickness stress is not zero like hydroforming as the stress triaxiality is a function of through thickness stress. The main shortcoming of the TFSD are: (a) dependent on yield criterion, (b) dependent on strain-hardening law, and (c) total error is cumulative of two computation steps, one in determination of equivalent strain from yield criterion and another one is determination of equivalent stress from the strain-hardening law (e.g. suppose 10% error occur in each of the two steps then the total error can be up to 19%). 6. Extended Forming Limit Diagram (XFLD) Zeng et al. [34] have claimed that the equivalent strain (εeq ) against strain rate ratio (ˇ) plot which is also known as XFLD, is independent of strain paths. The effectiveness of XFLD is shown in Fig. 9 with same sets of data of Al alloy A1611T4 and Al alloy AA5083. Similar to TFD, the XFLD is also almost strain path independent except high biaxial pre-straining levels. XFLD is approximately strain path independent for low pre-straining levels. XFLD is roughly strain path independent, only yield criterion is necessary for calculation, not applicable for presence of through thickness stress and error may occur for choosing improper yield criterion. 7. Extended Stress Ratio Based Forming Limit Diagram (ESRFLD) Yoshida et al. [35] have demonstrated in their investigation that equivalent strain (εeq ) vs. stress ratio (˛) plot (ESRFLD) is not strain

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model JMP-348; No. of Pages 13

ARTICLE IN PRESS

8

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx

500

500

(a)

450

400

350 300 250 As received Biaxial pre-straining 0.03 0.085 0.13 0.17

200 150 100

Major stress, MPa (σ1)

Major stress, MPa (σ1)

400

Al alloy A1 6111 T4

50

350 300

0

50

As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

250 200 150 100 50 Al alloy A1 6111 T4

Tested perpendicular to RD 0 -50

(b)

450

0 -50

100 150 200 250 300 350 400 450 500

0

50

100 150 200 250 300 350 400 450 500

Minor stress, MPa (σ2)

Minor stress, MPa (σ2) 500

500

(c)

400

400

350

350

300 250

As received Uniaxial pre-straining 0.05 0.095 0.14 Plane strain pre-straining 0.05 0.12

200 150 100

0 -50

Al alloy A1 6111 T4 0

50

Pre-straining parallel to RD Tested perpendicular to RD

100 150 200 250 300 350 400 450 500

Minor stress, MPa (σ2)

(d)

450

Major stress (σ1)

Major stress, MPa (σ1)

450

50

Pre-straining perpendicular to RD Tested perpendicular to RD

300 250 Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

200 150 100 50 Al alloy AA5083 0 -50

0

50

Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

100 150 200 250 300 350 400 450

Minor stress (σ2)

Fig. 11. Stress Based Forming Limit Diagram (FLSD): major stress ( 1 ) vs. minor stress ( 2 ) plot: (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

path dependent. ESRFLD is very similar to XFLD, XFLD is the diagram of equivalent strain (εeq ) against strain rate ratio (ˇ) while ESRFLD is the diagram of equivalent strain (εeq ) vs. stress ratio (˛). The effectiveness of ESRFLD is shown in Fig. 10 with same sets of data of Al alloy A1611T4 and Al alloy AA5083. Like XFLD, the ESRFLD is also almost strain path independent except high biaxial pre-straining levels. The same advantages and disadvantages of XFLD are also applied for ESRFLD. 8. Stress Based Forming Limit Diagram (FLSD) FLSD is the diagram of major stress ( 1 ) against minor stress ( 2 ). During the analysis of flanging operations involving copper, brass and steel alloys following a draw forming operation Kleemola and Pelkkikangas [36] have reported the limitations of the FLD and they have proposed FLSD. Later Arrieux et al. [37] have elaborately discussed the FLSD and successfully used FLSD for all secondary forming operations. Stoughton [38] has popularized the FLSD in industrial sheet metal forming applications. Afterward many literatures have published in support that the FLSD is strain path independent [7,33,35]. With the same data set, the FLSD is plotted for various pre-straining conditions in Fig. 11. In the present

investigation, Hill normal anisotropic yield criterion is used to compute the equivalent strain and El-Magd strain-hardening law is used to compute equivalent stress from equivalent strain. Afterward, the constitutive relations derived from yield criterion are utilized to determine major stress ( 1 ) and minor stress ( 2 ) from equivalent stress. The details of the calculations are illustrated in Appendices A and B for von Mises yield criterion and Hill normal anisotropic yield criterion, respectively. From Fig. 11, the decision can be made that the FLSD is approximately strain path independent. However, if we zoom Fig. 11(a) for biaxial pre-straining path the judgment may seem biased. The point already has discussed in Fig. 6 that the scatter band will be less in stress space than strain space. Like TFD, XFLD and ESRFLD the FLSD is also witnessed strain path dependency for high biaxial pre-straining condition. Yoshida and Kuwabara [39] have also concluded from experimental investigation that the forming limit stress of the steel tube is not completely path-independent, and that the path dependence of forming limit stress is strongly affected by the strain hardening behaviour of the material for given loading paths. For steel sheets, this type of sophisticated experimental study is not reported in literature up to date. Sufficient amount of database of experimental results will help us to understand the loading path dependency

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model

ARTICLE IN PRESS

JMP-348; No. of Pages 13

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx 500

500

(a)

400 350 300 250 200 150

As received Biaxial pre-straining 0.03 0.085 0.13 0.17

100 50 Al alloy A1 6111 T4 Tested perpendicular to RD 0 -0.25 0.00 0.25 0.50

0.75

1.00

(b)

450

Equivalent stress, MPa (σeq)

Equivalent stress, MPa (σeq)

450

1.25

400 350 300 250 As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

200 150 100

Al alloy A1 6111 T4 50 Pre-straining perpendicular to RD Tested perpendicular to RD 0 -0.25 0.00 0.25 0.50 0.75

1.00

1.25

Stress ratio (α)

Stress ratio (α) 500

450

(c)

400 350 300 250 200 As received Uniaxial pre-straining 0.05 0.095 0.14 Plane strain pre-straining 0.05 0.12

150 100

Al alloy A1 6111 T4 50 Pre-straining parallel to RD Tested perpendicular to RD 0 -0.25 0.00 0.25 0.50

0.75

1.00

1.25

Stress ratio (α)

(d)

400

Equivalent stress, MPa (σeq)

450

Equivalent stress, MPa (σeq)

9

350 300 250 200 Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

150 100

Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

50 Al alloy AA5083 0 -0.2

0.0

0.2

0.4

Stress ratio (α)

0.6

0.8

1.0

Fig. 12. Stress Ratio and Stress Based Forming Limit Diagram (SRFLSD): equivalent stress ( eq ) vs. stress ratio (˛) plot: (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

issue of FLSD and researchers will also receive some clue to overcome such problem. The advantages of FLSD are: (a) less loading path dependent in comparison to FLD, and (b) require both yield criterion and strain-hardening law. The limitations of FLSD are: (a) dependent on yield criterion, (b) dependent on strain-hardening law, (c) total error is cumulative of errors due to yield criterion and strain-hardening law, and (d) restricted use where through thickness stress presents like hydroforming. 9. Stress Ratio and Stress Based Forming Limit Diagram (SRFLSD) SRFLSD is the diagram of equivalent stress ( eq ) vs. stress ratio (˛). SRFLSD is very similar to ESRFLD, ESRFLD is the diagram of equivalent strain (εeq ) against stress ratio (˛) while SRFLSD is the diagram of equivalent stress ( eq ) against stress ratio (˛). The predictions by the SRFLSD for different pre-straining conditions are demonstrated in Fig. 12 for the same data sets. The predictions by the SRFLSD looks better than ESRFLD, the cause has already discussed in Fig. 6. The advantages and disadvantages of SRFLSD are exactly same as FLSD. 10. Extended Stress Based Forming Limit Diagram (XFLSD) During the onset of necking the stress states are not plane stress in some sheet metal forming operations, such as hydroforming and stretch flange forming. Using finite element computations Simha et al. [40] have demonstrated that three-dimensional loading take place during the onset of necking in hydroforming. For that reason to account the presence of through-thickness components during forming, Gotoh et al. [41] and Smith et al. [42] have proposed analytical modifications to the FLSD. Consequently, Simha et al.

[40] have demanded for a formability criterion that can predict the limit of formability under non-linear strain paths, as well as three-dimensional load paths since the neck forms under threedimensional loading. To take account the through thickness stress effect in strain path independent limiting criteria, Simha et al. [40] proposed XFLSD which is a diagram of equivalent stress ( eq ) vs. hydrostatic/mean stress ( m ). If we look carefully, the XFLSD is the exactly similar version of TFSD. In TFSD the equivalent stress is plotted against stress triaxiality while the XFLSD is the diagram of equivalent stress vs. hydrostatic/mean stress. In TFSD, the x-axis is normalized with equivalent stress while for XFLSD x-axis is not normalized and hydrostatic/mean stress is kept. The predictive competence of XFLSD is shown in Fig. 13 for same data sets. The predictions are exactly similar like TFSD. The XFLSD has identical advantages and limitations like TFSD. 11. Polar Effective Plastic Strain Diagram (PEPSD) The concept of PEPSD has first proposed by Stoughton and Yoon [43]. Apart from strain path independency, the shape of the FLC in the polar diagram is very similar to the shape of the strain FLC for the as-received condition. Both FLD and PEPSD have the same direction that corresponds to uniaxial, plane strain and equal-biaxial strain path. Like FLD, PEPSD also has a cusp at plane-strain mode. Therefore, engineers habituated to FLD can easily understand the PEPSD. The PEPSD is very similar to XFLD, PEPSD is plotted in polar co-ordinate while XFLD is plotted in Cartesian co-ordinate, but the variables are exactly same equivalent strain (εeq ) and strain rate ratio (ˇ). For PEPSD, angle () is “tan−1 (ˇ)” and EPS is the equivalent strain. Fig. 14 represents the PEPSD for various pre-straining conditions of Al alloy A1611T4 and Al alloy AA5083. The strain path

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model JMP-348; No. of Pages 13

ARTICLE IN PRESS

10

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx

Fig. 13. Extended Stress Based Forming Limit Diagram (XFLSD): equivalent stress ( eq ) vs. hydrostatic/mean stress ( m ) plot: (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

so easy to understand, and (c) only yield criterion is essential for calculation. The main limitations of PEPSD are: (a) dependent on yield criterion, (b) no such formulation available to take account the through thickness stress effect.

independency of PEPSD at high biaxial pre-straining is not valid. Like other previous models PEPSD also fails during high biaxial pre-straining condition. The benefits of PEPSD are: (a) less strain path dependent in comparison to FLD, (b) shape is similar to FLD

0.40

(a)

0.35

0.35

0.30

0.30

0.25

0.25

EPS * COS(θ)

EPS * COS(θ)

0.40

0.20 As received Biaxial pre-straining 0.03 0.085 0.13 0.17

0.15 0.10 0.05 0.00 -0.2

-0.1

0.0

0.15 0.10

Al alloy A1 6111 T4

0.05

Tested perpendicular to RD

Al alloy A1 6111 T4 0.00 -0.2 -0.1 0.0

0.1

0.2

0.3

0.4

0.16

0.35

0.14

(c)

0.25 As received Uniaxial pre-straining 0.05 0.095 0.14 Plane strain pre-straining 0.05 0.12

0.15 0.10

Al alloy A1 6111 T4 -0.1

0.0

EPS * SIN(θ)

0.2

0.3

0.4

0.2

0.3

(d)

0.10 0.08 0.06 0.04

Pre-straining parallel to RD Tested perpendicular to RD 0.1

0.1

0.12

EPS * COS(θ)

EPS * COS(θ)

0.30

0.20

Pre-straining perpendicular to RD Tested perpendicular to RD

EPS * SIN(θ)

0.40

0.00 -0.2

As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

0.20

EPS * SIN(θ)

0.05

(b)

0.4

0.02

Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03 Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

Al alloy AA5083 0.00 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12

EPS * SIN(θ)

Fig. 14. Polar Effective Plastic Strain Diagram (PEPSD): (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model

ARTICLE IN PRESS

JMP-348; No. of Pages 13

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx 500

450

(a)

450

400

400

350

(b)

300

300 250 As received Biaxial pre-straining 0.03 0.085 0.13 0.17

200 150 100

PESS * COS(ψ)

PESS * COS (ψ)

350

50

100

150

200

250

300

As received Uniaxial pre-straining 0.05 0.095 0.14 0.18 Plane strain pre-straining 0.05 0.11

200 150

50 Al alloy A1 6111 T4 0 -50 0 50 100

Tested perpendicular to RD 0

250

100

Al alloy A1 6111 T4

50 0 -50

350

400

PESS * SIN (ψ )

200

250

300

350

400

350

350

300

300

250 As received Uniaxial pre-straining 0.05 0.095 0.14 Plane strain pre-straining 0.05 0.12

200 150 100

0

50

100

150

200

250

300

350

250 200 Experimental data points As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

150 100

Pre-straining parallel to RD Tested perpendicular to RD

Al alloy A1 6111 T4

(d)

400

PESS * COS(ψ)

PESS * COS(ψ)

150

450

(c)

400

0 -50

Pre-straining perpendicular to RD Tested perpendicular to RD

PESS * SIN (ψ )

450

50

11

400

PESS * SIN (ψ)

Experimental curve fit As received Uniaxial pre-strain 0.04 Biaxial pre-strain 0.03

50 Al alloy AA5083 0 -50

0

50

100

150

200

250

300

PESS * SIN (ψ )

Fig. 15. Polar Effective Stress Diagram (PESSD): (a) biaxial pre-strain paths of Al alloy A1611T4 (b) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to rolling direction (RD) of Al alloy A1611T4 (c) uniaxial and plane strain pre-strain paths; pre-strain given in parallel to transverse direction (TD) of Al alloy A1611T4 (d) uniaxial and biaxial pre-strain paths of Al alloy AA5083.

12. Polar Effective Stress Diagram (PESSD) It is observed from all previous path independent limiting criteria that forming limiting curves lie in a narrow band in stress space. By considering the fact, PESSD is calculated in this work. For PESSD, angle ( ) is “tan−1 (˛)” and PESS is the equivalent stress ( eq ). The PESSD is very similar to SRFLSD, PESSD is plotted in polar coordinate while SRFLSD is plotted in Cartesian co-ordinate, but the variables are exactly same equivalent stress ( eq ) and stress ratio (˛). For PESSD, angle ( ) is “tan−1 (˛)” and PESS is the equivalent stress. For various pre-straining conditions of Al alloy A1611T4 and Al alloy AA5083, the PESSDs are illustrated in Fig. 15. The predictions by PESSD looks better than PEPSD, however similar types deviation is also noticed by zooming the PESSD for equi-biaxial pre-straining condition. The reason is already discussed in Fig. 6. It should be noted that there is no physical significance to the meaning of the variables (PESS*SIN ( ), PESS*COS ( )). The physically meaningful variables with respect to sheet metal formability are the angle ( ) and the equivalent stress ( eq ). The direction reveals the direction of the current in-plane stress rates and equivalent stress ( eq ) is the radial variable in the polar diagram. As PEPSD is the polar coordinate version of XFLD, PESSD is also the polar coordinate version of SRFLSD. All advantages and disadvantages of SRFLSD will also be applicable for PESSD. 13. Discussion and conclusions The effects of strain path alterations on forming limits have been studied by number of researchers considering the non-proportional loading histories which are combinations of two linear deformation paths. The selected results from literatures have indicated that the

FLD is extremely sensitive to the alterations of strain paths. The strain limits in FLD could be either elevated or dropped depending on the type of strain path alteration. More explicitly, uniaxial pre-straining elevates the limit strains for subsequent in-plane plane strain and biaxial stretching, while plane strain pre-straining increases the limit strains for all subsequent strain combinations, and equi-biaxial pre-straining decreases the limit strains for all strain combinations. The present investigation has confirmed that, at least in comparison to the FLD, Traditional Failure Diagram (TFD), Traditional Stress based Failure Diagram (TFSD), Extended Forming Limit Diagram (XFLD), Extended Stress Ratio Based Forming Limit Diagram (ESRFLD), Stress Based Forming Limit Diagram (FLSD), Stress Ratio and Stress Based Forming Limit Diagram (SRFLSD), Extended Stress Based Forming Limit Diagram (XFLSD), Polar Effective Stress Diagram (PESSD), and Polar Effective Plastic Strain Diagram (PEPSD) are approximately not sensitive to strain path changes for small pre-staring condition. More precisely, all the stress/strain based limiting criteria discussed in this article except FLD are almost path-independent when the pre-straining is in smaller amount. Since prediction of localization at corners is a case where plane stress condition is violated, as a consequence prediction using FLD and other criteria derived from plane stress condition becomes difficult. The choice of an appropriate yield criterion shows significant impact on the quality of the results for all strain path independent strain based limiting criteria (TFD, XFLD, ESRFLD and PEPSD). Whereas, the selection of suitable both yield criterion and strain-hardening law show substantial influence on the quality of the results for all strain path independent stress based limiting criteria (TFSD, FLSD, SRFLSD, XFLSD and PESSD). Selection of

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model JMP-348; No. of Pages 13

ARTICLE IN PRESS

12

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx

strain-hardening law should be preferred to base on the flow curve gained during a biaxial bulge test, as the stress–strain curve obtained from uniaxial tensile test only provide flow stresses for a small range of strain. By selecting proper yield criterion and strainhardening law the calculation errors can be minimized in some extent, but complete removal of errors are not possible. The TFD, TFSD and XFLSD have inherent variables (e.g. hydrostatic stress) to take care the through thickness stress effect, however in-depth experimental verification needed before use it for industrial purpose. All the strain path independent strain based limiting criteria (TFD, XFLD, ESRFLD and PEPSD) assume that for a particular final strain path the cumulative equivalent strain is not altering with different pre-straining paths. Similarly all the strain path independent stress based limiting criteria (TFSD, FLSD, SRFLSD, XFLSD and PESSD) assume that for a particular final strain path the cumulative equivalent stress is constant with various pre-straining paths. The cumulative equivalent stress is determined from cumulative equivalent strain by considering suitable strain hardening law. Therefore, all the strain path independent strain/stress based limiting criteria are based on the assumption that the cumulative equivalent strain is not changing with diverse pre-straining paths. However, unfortunately this assumption is not valid for high biaxial prestraining condition. For that reason, all the strain path independent strain/stress based limiting criteria show large systematic deviation for large biaxial pre-straining conditions. Accurate determination of local strains near necked area are possible today due to the availability of advance image correlation technique. But, still now no experimental technique is available to determine directly the local stresses near necked area. Normally we determine local stresses from local strains by using strain hardening law and yield criteria. Therefore, slight errors in experimental data fitting into equations of strain hardening law and yield criteria result cumulative error in computation of local stresses. Generation of proper experimental database of local stresses and strains near necked area for various loading path will help researcher to understand the loading path dependency of sheet metal forming limiting criteria and approach to resolve it.

consequences in a final strain state (ε1f , ε2f ), then the strain rate ratio (ˇ) will be

Appendix A. Transformation relations

Triaxiality () is the ratio of mean stress ( m ) and equivalent stress ( eq ), and it can be expressed as

The strain rate ratio (ˇ) is the ratio of minor strain rate (dε2 ) and major strain rate (dε1 ) and defined as ˇ=

dε2 dε1

2 1

(A2)

The hydrostatic/mean stress ( m ) can be written as m =

ε2f − ε2i

(A6)

ε1f − ε1i

Similarly, the stress ratio (˛) will be ˛=

2f − 2i

(A7)

1f − 1i

The cumulative equivalent strain (εeq ) can be determined for pre-straining path as follows εeq = εeqi + εeqf

(A8)

where εeqi and εeqf are the equivalent strains for initial pre-straining and final deformation path, respectively. εeqi can be determined from strain state (ε1i , ε2i ), while the εeqf can be determined from strain state (ε1f − ε1i , ε2f − ε2i ). Appendix B. von Mises yield criterion Von Mises yield criterion can be defined for plane stress condition as



f = eq =

12 + 22 − 1 2

(B1)

The relation between equivalent stress ( eq ) and major stress ( 1 ) can be written as eq = 1



1 − ˛ + ˛2

(B2)

Similarly, the relation between equivalent strain rate (dεeq ) and major strain rate (dε1 ) can be written as

 2 dεeq = √ dε1 1 + ˇ + ˇ2 3

(B3)

This relation between ˛ and ˇ which can be expressed as ˇ=

2˛ − 1 2−˛

(B4)

˛=

1 + 2ˇ ˛+ˇ

(B5)

=

m 1+ˇ = √  eq 3 1 + ˇ + ˇ2

(B6)

(A1)

Similarly, the stress ratio (˛) is the ratio of minor stress ( 2 ) and major stress ( 1 ) and defined as ˛=

ˇ=

1 (1 + 2 + 3 ) 3

Hill’s quadratic normal anisotropic yield criterion is a special case of Hill general anisotropic yield criterion. In this yield criterion the equivalent stress ( eq ) is a function of the principal stresses ( 1 and  2 ) and the normal anisotropy coefficient (r). The equivalent stress ( eq ) for plane stress condition can be defined by:



(A3) eq =

The power law equation can be written as eq = y + Kεneq

(A4)



eq = y + Aεeq + C 1 − exp −bεeq



12 + 22 −

(A5)

where, A, b and C are material constants,  y is the yield stress. If the pre-straining consequences in a strain state (ε1 , ε2 ) = (ε1i , ε2i ), where the index i denotes initial stage, and the secondary stage

2r 1 2 1+r

(C1)

Similarly the equivalent strain rate (dεeq ) can be defined by 1+r dεeq = √ 1 + 2r

where, K and n are material constants,  y is the yield stress. The El-Magd law can be written as



Appendix C. Hill quadratic normal anisotropic yield criterion



dε21 + dε22 +

2r dε1 dε2 1+r

(C2)

The relationship between equivalent stress ( eq ) and major stress ( 1 ) can be given as



eq = 1

1 + ˛2 −

2r ˛ 1+r

(C3)

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025

G Model

ARTICLE IN PRESS

JMP-348; No. of Pages 13

S.K. Paul / Journal of Manufacturing Processes xxx (2015) xxx–xxx

Similarly the relationship between equivalent strain rate (dεeq ) and major strain rate (dε1 ) can be given as dεeq

1+r = ε1 √ 1 + 2r



1 + ˇ2 +

2r ˇ 1+r

(C4)

The relationship between strain rate ratio (ˇ) and stress ratio (˛) is ˇ=

(1 + r) ˛ − r 1 + r − r˛

(C5)

And its inverse relationship is ˛=

(1 + r) ˇ + r 1 + r + rˇ

(C6)

References [1] Chandra Sekhar K, Narayanasamy R, Venkateswarlu K. Formability, fracture and void coalescence analysis of a cryorolled Al Mg Si alloy. Mater Des 2014;57:351–9. [2] Narayanasamy R, Narayanan CS. Forming limit diagram for Indian interstitial free steels. Mater Des 2006;27:882–99. [3] Graf AF, Hosford WF. Effect of changing stain path and forming limit diagram of Al 2008-T4. Metall Trans A 1993;24:2503–12. [4] Graf AF, Hosford WF. The influence of strain path changes on forming limit diagram. Int J Mech Sci 1994;36:897–910. [5] Hariharan K, Balaji C. Material optimization: a case study using sheet metalforming analysis. J Mater Process Technol 2009;209:324–31. [6] Paul SK, Manikandan G, Verma RK. Prediction of entire forming limit diagram from simple tensile material properties. J Strain Anal Eng Des 2013;48(6):386–94. [7] Zhalehfar F, Hosseinipour SJ, Nourouzi S, Gorji AH. A different approach for considering the effect of non-proportional loading path on the forming limit diagram of AA5083. Mater Des 2013;50:165–73. [8] Zhalehfar F, Hashemi R, Hosseinipour SJ. Experimental and theoretical investigation of strain path change effect on forming limit diagram of AA5083. Int J Adv Manuf Technol 2015;76:1343–52. [9] Assempour A, Hashemi R, Abrinia K, Ganjiani M, Masoumi E. A methodology for prediction of forming limit stress diagrams considering the strain path effect. Comput Mater Sci 2009;45:195–204. [10] Narayanasamy R, Narayanan CS. Forming, fracture and wrinkling limit diagram for if steel sheets of different thickness. Mater Des 2008;29:1467–75. [11] Jeyakrishnan PR, Chockalingam KKSK, Narayanasamy R, Venkateswarlu K. Study on the wrinkling of bulged AA 5052 alloy sheet metal during restoration. Mater Des 2013;52:541–6. [12] Centeno G, Bagudanch I, Martínez-Donaire AJ, García-Romeu ML, Vallellano C. Critical analysis of necking and fracture limit strains and foming forces in single-point incremental forming. Mater Des 2014;63:20–9. [13] Chung K, Ahn K, Yoo DH, Chung KH, Seo MH, Park SH. Formability of TWIP (twinning induced plasticity) automotive sheets. Int J Plast 2011;27:52–81. [14] Paul SK, Mukherjee M, Kundu S, Chandra S. Prediction of hole expansion ratio for automotive grade steels. Comput Mater Sci 2014;89:189–97. [15] Paul SK. Non-linear correlation between uniaxial tensile properties and shearedge hole expansion ratio. J Mater Eng Perform 2014;23(9):3100–7. [16] Keeler SP. Determination of forming limits in automotive stampings. Sheet Met Ind 1965;42:683–91. [17] Goodwin GM. Application of strain analysis to sheet metal forming problems in the press shop. Metall Ital 1968;60:764–74. [18] Wang K, Carsley JE, He B, Li J, Zhang L. Measuring forming limit strains with digital image correlation analysis. J Mater Process Technol 2014;214:1120–30.

13

[19] Hariharan K, Nguyen N-T, Barlat F, Lee M-G, Kim JH. A pragmatic approach to accommodate in-plane anisotropy in forming limit diagrams. Mech Res Commun 2014;62:5–17. [20] Assempour A, Nejadkhaki HK, Hashemi R. Forming limit diagrams with the existence of through-thickness normal stress. Comput Mater Sci 2010;48: 504–8. [21] Hashemi R, Abrinia K. Analysis of the extended stress-based forming limit curve considering the effects of strain path and through-thickness normal stress. Mater Des 2014;54:670–7. [22] Chung K, Kim H, Lee C. Forming limit criterion for ductile anisotropic sheets as a material property and its deformation path insensitivity, Part I: Deformation path insensitive formula based on theoretical models. Int J Plast 2014;58: 3–34. [23] Chung K, Kim H, Lee C. Forming limit criterion for ductile anisotropic sheets as a material property and its deformation path insensitivity, Part II: Boundary value problems. Int J Plast 2014;58:35–65. [24] Werber A, Liewald M, Nester W, Grünbaum M, Wiegand K, Simon J, Timm J, Bassi C, Hotz W. Influence of different pre-stretching modes on the Forming Limit Diagram of AA 6014. Key Eng Mater 2012;504–506:71–6. [25] Werber A, Liewald M, Nester W, Grünbaum M, Wiegand K, Simon J, Timm J, Hotz W. Assessment of forming limit stress curves as failure criterion for nonproportional forming processes. Prod Eng Res Dev 2013;7:213–21. [26] Hill R. A general theory of uniqueness and stability in elastic/plastic solids. J Mech Phys Solids 1958;6:236–49. [27] Paul SK. Predicting the flow behavior of metals under different strain rate and temperature through phenomenological modeling. Comput Mater Sci 2012;65:91–9. [28] Abbassi F, Mistou S, Zghal A. Failure analysis based on microvoid growth for sheet metal during uniaxial and biaxial tensile tests. Mater Des 2013;49:638–46. [29] Pirondi A, Bonora N, Steglich D, Brocks W, Hellmann D. Simulation of failure under cyclic plastic loading by damage models. Int J Plast 2006;22:2146–70. [30] Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 1985;21(1):31–48. [31] Wierzbicki T, Xue L. On the effect of the third invariant of the stress deviator on ductile fracture. In: Impact and Crashworthiness Lab report no. 136; 2005. [32] Wierzbicki T, Bao Y, Lee Y, Bai Y-WY. Calibration and evaluation of seven fracture models. Int J Mech Sci 2005;47:719–43. [33] Paul SK. Theoretical analysis of strain- and stress-based forming limit diagrams. J Strain Anal Eng Des 2013;48(3):177–88. [34] Zeng D, Chappuis L, Xia ZC, Zhu X. A path independent forming limit criterion for sheet metal forming simulations. SAE 2008;01:1445. [35] Yoshida K, Kuwabara T, Kuroda M. Path-dependence of the forming limit stresses in a sheet metal. Int J Plast 2007;23:361–84. [36] Kleemola HJ, Kumpulainen JO. Factors influencing the forming limit diagram: Part I—The experimental determination of the forming limits of sheet steel. J Mech Work Technol 1980;3:289–302. [37] Arrieux R, Bedrin C, Bovin M. Determination of an intrinsic forming limit stress diagram for isotropic sheets. In: Proceedings of the 12th IDDRG congress. 1982. p. 61–71, vol. 2. [38] Stoughton TB. A general forming limit criterion for sheet metal forming. Int J Mech Sci 2000;42:1–27. [39] Yoshida K, Kuwabara T. Effect of strain hardening behavior on forming limit stresses of steel tube subjected to nonproportional loading paths. Int J Plast 2007;23:1260–84. [40] Simha CHM, Grantab R, Worswick MJ. Computational analysis of stress-based forming limit curves. Int J Solids Struct 2007;44:8663–84. [41] Gotoh M, Chung C, Iwata N. Effect of out-of-plane stress on the forming limit strain of sheet metals. JSME Int J 1995;38(1):123–32. [42] Smith LM, Averill RC, Lucas JP, Stoughton TB, Matin PH. Influence of transverse normal stress on sheet metal formability. Int J Plast 2003;19(10):1567–83. [43] Stoughton TB, Yoon JW. Path independent forming limits in strain and stress spaces. Int J Solids Struct 2012;49:3616–25.

Please cite this article in press as: Paul SK. Path independent limiting criteria in sheet metal forming. J Manuf Process (2015), http://dx.doi.org/10.1016/j.jmapro.2015.06.025