A bifurcation analysis of forming limits for an orthotropic sheet metal is presented in this paper. The approach extends Stören and Rice's (S–R) bifurcation analysis for isotropic materials, with materials following a vertex theory of plasticity at the onset of localized necking. The sheet orthotropy is represented by the Hill’48 yield criterion with three r-values in the rolling (r0), the transverse (r90) and the diagonal direction (r45). The emphasis of the study is on the examination of r-value effect on the sheet metal forming limit, expressed as a combination of the average r-value raverage and the planar anisotropy (Δr). Forming limits under both zero extension assumption and minimum extension assumption as well as necking band orientation evolution are investigated in detail. The comparison between the experimental result and predicted forming limit diagram (FLD) is presented to validate the extended bifurcation analysis. The r-value effect is observed under uniaxial and equal-biaxial loadings. However, no difference is found under plane strain condition in strain-based FLD which is consistent with Hill's theory. The force maximum criterion is also used to analyze FLD for verification.

References

References
1.
Keeler
,
S.
, 1965, “Determination of Forming Limits in Automotive Stampings,”
SAE
Technical Paper No. 650535.10.4271/650535
2.
Graf
,
A.
, and
Hosford
,
W.
,
1994
, “
The Influence of Strain-Path Changes on Forming Limit Diagrams of A1 6111 T4
,”
Int. J. Mech. Sci.
,
36
(
10
), pp.
897
910
.10.1016/0020-7403(94)90053-1
3.
Yoshida
,
K.
, and
Kuroda
,
M.
,
2012
, “
Comparison of Bifurcation and Imperfection Analyses of Localized Necking in Rate-Independent Polycrystalline Sheets
,”
Int. J. Solids Struct.
,
49
(
15–16
), pp.
2073
2084
.10.1016/j.ijsolstr.2012.04.010
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
.10.1016/0020-7403(67)90066-5
6.
Stören
,
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. W.
, and
Neale
,
K. W.
,
1978
,
“Sheet Necking-II: Time-Independent Behavior
,” Mechanics of Sheet Metal Forming,
D. P.
Koistinen
and
N. M.
Wang
, eds.,
Plenum
,
New York
, pp.
111
126
.10.1007/978-1-4613-2880-3_6
8.
Kuroda
,
M.
, and
Tvergaard
,
V.
,
2000
, “
Forming Limit Diagrams for Anisotropic Metal Sheets With Different Yield Criteria
,”
Int. J. Solids Struct.
,
37
(
37
), pp.
5037
5059
.10.1016/S0020-7683(99)00200-0
9.
Zhu
,
X. H.
,
Weinmann
,
K.
, and
Chandra
,
A.
,
2001
, “
A Unified Bifurcation Analysis of Sheet Metal Forming Limits
,”
ASME J. Eng. Mater. Technol.
,
123
(
3
), pp.
329
333
.10.1115/1.1370397
10.
Chow
,
C. L.
,
Jie
,
M.
, and
Hu
,
S. J.
,
2003
, “
Forming Limit Analysis of Sheet Metals Based on a Generalized Deformation Theory
,”
ASME J. Eng. Mater. Technol.
,
125
(
3
), pp.
260
265
.10.1115/1.1586938
11.
Jaamialahmadi
,
A.
, and
Kadkhodayan
,
M.
,
2011
,“
An Investigation Into the Prediction of Forming Limit Diagrams for Normal Anisotropic Material Based on Bifurcation Analysis
,”
ASME J. Appl. Mech.
,
78
(
3
), p.
031006
.10.1115/1.4003351
12.
Jaamialahmadi
,
A.
, and
Kadkhodayan
,
M.
,
2012
, “
A Modified Storen-Rice Bifurcation Analysis of Sheet Metal Forming Limit Diagrams
,”
ASME J. Appl. Mech.
,
79
(
6
), p.
061004
.10.1115/1.4005538
13.
Dasappa
,
P.
,
K.
,
Inal
, and
Mishra
,
R.
,
2012
, “
The Effects of Anisotropic Yield Functions and Their Material Parameters on Prediction of Forming Limit Diagrams
,”
Int. J. Solids Struct.
,
49
(
25
), pp.
3528
3550
.10.1016/j.ijsolstr.2012.04.021
14.
Li
,
Y.
, and
Karr
,
D. G.
,
2009
, “
Prediction of Ductile Fracture in Tension by Bifurcation, Localization, and Imperfection Analyses
,”
Int. J. Plast.
,
25
(
6
), pp.
1128
1153
.10.1016/j.ijplas.2008.07.001
15.
Hijazi
,
A.
,
Yardi
,
N.
, and
Madhavan
,
V.
,
2004
, “
Determination of Forming Limit Curves Using 3D Digital Image Correlation and In-Situ Observation
,”
International SAMPE Symposium and Exhibition, Long Beach, CA, May 16–20
, pp.
16
20
.
16.
He
,
J.
,
Zeng
,
D.
,
Zhu
,
X. H.
,
Xia
,
Z. C.
, and
Li
,
S. H.
,
2014
, “
Effect of Nonlinear Strain Paths on Forming Limits Under Isotropic and Anisotropic Hardening
,”
Int. J. Solids Struct.
,
51
(
2
), pp.
402
415
.10.1016/j.ijsolstr.2013.10.013
17.
He
,
J.
,
Xia
,
Z. C.
,
Zhu
,
X. H.
,
Zeng
,
D.
, and
Li
,
S. H.
,
2013
, “
Sheet Metal Forming Limits Under Stretch-Bending With Anisotropic Hardening
,”
Int. J. Mech. Sci.
,
75
, pp.
244
256
.10.1016/j.ijmecsci.2013.07.007
18.
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
19.
Hill
,
R.
,
1967
, “
The Essential Structure of Constitutive Laws for Metal Composites and Polycrystals
,”
J. Mech. Phys. Solids
,
15
(
2
), pp.
79
95
.10.1016/0022-5096(67)90018-X
20.
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
21.
Xia
,
Z. C.
,
2001
, “
Failure Analysis of Tubular Hydroforming
,”
ASME J. Eng. Mater. Technol.
,
123
(
4
), pp.
423
429
.10.1115/1.1394966
22.
Xia
,
Z. C.
, and
Zeng
,
D.
,
2009
, “
Sheet Metal Forming Limit Under Stretch-Bending
,”
ASME
Paper No. MSEC_ICMP2008-72555.10.1115/MSEC_ICMP2008-72555
23.
McClintock
,
F. A.
,
Zhou
,
Q.
, and
Wierzbick
,
T.
,
1993
, “
Necking in Plane Strain Under Bending With Constant Tension
,”
J. Mech. Phys. Solids
,
41
(
8
), pp.
1327
1343
.10.1016/0022-5096(93)90082-Q
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