Since the forming limit diagram (FLD) was introduced by Keeler, etc., five decades ago, it has been intensively studied by researchers and engineers. Most work has focused on the in-plane deformation which is considered as the dominant mode of the majority forming processes. However the effect of out-of-plane deformation becomes important in the accurate prediction of formability when thick sheet metals and/or smaller forming radii are encountered. Recent research on the stretch-bending induced FLD (BFLD) has been inconclusive. Some studies indicated that the bending effect will enhance a sheet metal's formability while others suggested otherwise. In this paper, we present an in-depth study of the through-thickness bending effect on the forming limits. The Marciniak–Kuczynski (M–K) analysis is extended to include bending, and models based on both flow theory and deformation theory of plasticity are proposed. The study is limited to the right-hand-side of FLD where the bending is along the major stretch direction. The radial return method is adopted as the framework to integrate constitutive equations. The results show that the bending process decreases the sheet metal formability with the flow-theory based model, while the opposite is true if the deformation theory based analysis is adopted. A detailed examination of the deformation histories from those two models reveals that the loading–unloading-reverse loading process during stretch-bending holds the key to the understanding of the conflicting results. The insight gained from the proposed FLD prediction model in this paper provides a new understanding of how the bending process affects the FLD, which can be used to predict and explain the localized necking phenomenon under the stretch-bending condition.

References

1.
Keeler
,
S.
,
1965
, “
Determination of Forming Limits in Automotive Stampings
,”
SAE
Technical Paper, Paper No. 650535.10.4271/650535
2.
Keeler
,
S. P.
, and
Backofen
,
W. A.
,
1963
, “
Plastic Instability and Fracture in Sheets Stretched Over Rigid Punches
,”
ASM Trans. Q
,
56
(
1
), pp.
25
48
.
3.
Goodwin
,
G. M.
,
1968
, “
Application of Strain Analysis to Sheet Metal Forming Problems in the Press Shop
,” SAE Technical Paper No. 680093.
4.
Hill
,
R.
,
1952
, “
On Discontinuous Plastic States, With Special Reference to Localized Necking in Thin Sheets
,”
J. Mech. Phys. Solids
,
1
(
1
), pp.
19
30
.10.1016/0022-5096(52)90003-3
5.
Marciniak
,
Z.
, and
Kuczynski
,
K.
,
1967
, “
Limit Strains in the Processes of Stretch-Forming Sheet Metal
,”
Int. J. Mech. Sci.
,
9
(
9
), pp.
609
612
, IN1–IN2, 613–620.10.1016/0020-7403(67)90066-5
6.
Storen
,
S.
, and
Rice
,
J.
,
1975
, “
Localized Necking in Thin Sheets
,”
J. Mech. Phys. Solids
,
23
(
6
), pp.
421
441
.10.1016/0022-5096(75)90004-6
7.
Hutchinson
,
J.
, and
Neale
,
K.
,
1978
, “
Sheet Necking-II: Time-Independent Behavior
,”
Proceedings of a Symposium of Mechanics of Sheet Metal Forming
,
D. P.
Koistinen
and
N. M.
Wang
, eds., pp.
127
150
.
8.
Hutchinson
,
J.
, and
Neale
,
K.
,
1978
, “
Sheet Necking-III: Strain-Rate Effects
,”
Proceedings of a Symposium of Mechanics of Sheet Metal Forming
,
D. P.
Koistinen
and
N. M.
Wang
, eds., pp.
269
285
.
9.
Hutchinson
,
J.
,
Neale
,
K.
, and
Needleman
,
A.
,
1978a
, “
Sheet Necking-Validity of Plane Stress Assumptions of the Long-Wavelength Approximation
,”
Proceedings of a Symposium of Mechanics of Sheet Metal Forming
,
D. P.
Koistinen
and
N.M.
Wang
, eds., pp.
111
126
.
10.
Aghaie-Khafri
,
M.
,
Mahmudi
,
R.
, and
Pishbin
,
H.
,
2002
, “
Role of Yield Criteria and Hardening Laws in the Prediction of Forming Limit Diagrams
,”
Metall. Mater. Trans. A
,
33
(
5
), pp.
1363
1371
.10.1007/s11661-002-0061-1
11.
Cao
,
J.
,
Yao
,
H.
,
Karafillis
,
A.
, and
Boyce
,
M. C.
,
2000
, “
Prediction of Localized Thinning in Sheet Metal Using a General Anisotropic Yield Criterion
,”
Int. J. Plast.
,
16
(
9
), pp.
1105
1129
.10.1016/S0749-6419(99)00091-1
12.
Laukonis
,
J. V.
, and
Ghosh
,
A. K.
,
1978
, “
Effects of Strain Path Changes on the Formability of Sheet Metals
,”
Metall. Mater. Trans. A
,
9
(
12
), pp.
1849
1856
.10.1007/BF02663419
13.
Butuc
,
M. C.
,
Da Rocha
,
A. B.
,
Duarte
,
J. F.
,
Barlat
,
F.
, and
Gracio
,
J. J.
,
2002
, “
Forming Limit Diagram for 6016-T4 Aluminium Alloy Deformed along Linear and Complex Strain Paths
,”
Key Eng. Mater.
,
230
, pp.
529
532
.10.4028/www.scientific.net/KEM.230-232.529
14.
Lu
,
Z.
, and
Lee
,
D.
,
1987
, “
Prediction of History-Dependent Forming Limits by Applying Different Hardening Models
,”
Int. J. Mech. Sci.
,
29
(
2
), pp.
123
137
.10.1016/0020-7403(87)90047-6
15.
Shi
,
M.
, and
Gerdeen
,
J.
,
1991
, “
Effect of Strain Gradient and Curvature on Forming Limit Diagrams for Anisotropic Sheets
,”
J. Mater. Shaping Technol.
,
9
(
4
), pp.
253
268
.10.1007/BF02833650
16.
Assempour
,
A.
,
Nejadkhaki
,
H. K.
, and
Hashemi
,
R.
,
2010
, “
Forming Limit Diagrams With the Existence of Through-Thickness Normal Stress
,”
Comput. Mater. Sci.
,
48
(
3
), pp.
504
508
.10.1016/j.commatsci.2010.02.013
17.
Eyckens
,
P.
,
Van Bael
,
A.
, and
Van Houtte
,
P.
,
2011
, “
An Extended Marciniak-Kuczynski Model for Anisotropic Sheet Subjected to Monotonic Strain Paths with through-Thickness Shear
,”
Int. J. Plast.
,
27
(
10
), pp.
1577
1597
.10.1016/j.ijplas.2011.03.008
18.
Triantafyllidis
,
N.
,
1980
, “
Bifurcation Phenomena in Pure Bending
,”
J. Mech. Phys. Solids
,
28
(
3-4
), pp.
221
245
.10.1016/0022-5096(80)90005-8
19.
Triantafyllidis
,
N.
,
Needleman
,
A.
, and
Tvergaard
,
V.
,
1982
, “
On the Development of Shear Bands in Pure Bending
,”
Int. J. Solids Struct.
,
18
(
2
), pp.
121
138
.10.1016/0020-7683(82)90021-X
20.
Sriram
,
S.
,
Yao
,
H.
, and
Ramisetti
,
N.
,
2012
, “
Development of an Empirical Model to Characterize Fracture Behavior During Forming of Advanced High Strength Steels Under Bending Dominated Conditions
,”
ASME J. Manuf. Sci. Eng.
,
134
(
3
), p.
031003
.10.1115/1.4006092
21.
Simha
,
C. H.
,
Grantab
,
R.
, and
Worswick
,
M. J.
,
2008
, “
Application of an Extended Stress-Based Forming Limit Curve to Predict Necking in Stretch Flange Forming
,”
ASME J. Manuf. Sci. Eng.
,
130
(
5
), p.
051007
.10.1115/1.2844593
22.
Lin
,
G.
,
Hu
,
S. J.
, and
Cai
,
W.
,
2009
, “
Evaluation of Formability in Bending/Hemming of Aluminum Alloys Using Plane-Strain Tensile Tests
,”
ASME J. Manuf. Sci. Eng.
,
131
(
5
), p.
051009
.10.1115/1.3123316
23.
Salandro
,
W. A.
,
Jones
,
J. J.
,
McNeal
,
T. A.
,
Roth
,
J. T.
,
Hong
,
S. T.
, and
Smith
,
M. T.
,
2010
, “
Formability of Al 5xxx Sheet Metals Using Pulsed Current for Various Heat Treatments
,”
ASME J. Manuf. Sci. Eng.
,
132
(
5
), p.
051016
.10.1115/1.4002185
24.
Xia
,
Z.
,
Cedric
, and
Danielle Zeng
,
2008
, “
Sheet Metal Forming Limit Under Stretch-Bending
,”
Proceedings of the ASME International Manufacturing Science and Engineering Conference
.
25.
Tharrett
,
M.
, and
Stoughton
,
T.
,
2003
, “
Stretch-Bend Forming Limits of 1008 Ak Steel, 70/30 Brass, and 6010 Aluminum
,”
Dislocations Plast. Met. Form.
, pp.
199
201
.
26.
Tharrett
,
M. R.
, and
Stoughton
,
T. B.
,
2003
, “
Stretch-Bend Forming Limits of 1008 Ak Steel
,” SAE Paper No. 01-1157.
27.
Kitting
,
D.
,
Koplenig
,
M.
,
Ofenheimer
,
A.
,
Pauli
,
H.
, and
Till
,
E.
,
2009
, “
Application of a “Concave-Side Rule” Approach for Assessing Formability of Stretch-Bent Steel Sheets
,”
Int. J. Mater. Form.
, pp.
427
430
.10.1007/s12289-009-0483-z
28.
Kitting
,
D.
,
Ofenheimer
,
A.
,
Pauli
,
H.
, and
Till
,
E.
,
2010
, “
A Phenomenological Concept to Predict Formability in Stretch-Bending Forming Operations
,”
Int. J. Mater. Form.
, pp.
1163
1166
.10.1007/s12289-010-0979-6
29.
Hutchinson
,
J.
,
1970
, “
Elastic-Plastic Behaviour of Polycrystalline Metals and Composite
,”
Proc. R. Soc. London, Ser. A
,
319
(
1537
), pp.
247
272
.10.1098/rspa.1970.0177
30.
Xia
,
Z. C.
,
2001
, “
Failure Analysis of Tubular Hydroforming
,”
ASME J. Eng. Mater. Technol.
,
123
(
4
),
423
429
.10.1115/1.1394966
31.
He
,
J.
,
Xia
,
Z. C.
,
Zeng
,
D.
, and
Li
,
S. H.
,
2013
, “
Forming Limits of a Sheet Metal After Continuous-Bending-Under-Tension Loading
,”
ASME J. Eng. Mater. Technol.
,
135
(
3
), p.
031009
.10.1115/1.4023676
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