Abstract

The onset of necking in dynamically expanding ductile rings is delayed due to the stabilizing effect of inertia, and with increasing expansion velocity, both the number of necks incepted and the number of fragments increase. In general, neck retardation is expected to delay fragmentation as necking is often the precursor to fracture. However, in porous ductile materials, it is possible that fracture can occur without significant necking. Thus, the objective of this work is to unravel the complex interaction of initial porosity and inertia on the onset of necking and fracture. To this end, we have carried out a series of finite element calculations of unit cells with sinusoidal geometric perturbations and varying levels of initial porosity under a wide range of dynamic loading conditions. In the calculations, the material is modeled using a constitutive framework that includes many of the hardening and softening mechanisms that are characteristics of ductile metallic materials, such as strain hardening, strain rate hardening, thermal softening, and damage-induced softening. The contribution of the inertia effect on the loading process is evaluated through a dimensionless parameter that combines the effects of loading rate, material properties, and unit cell size. Our results show that low initial porosity levels favor necking before fracture, and high initial porosity levels favor fracture before necking, especially at high loading rates where inertia effects delay the onset of necking. The finite element results are also compared with the predictions of linear stability analysis of necking instabilities in porous ductile materials.

References

References
1.
Niordson
,
F. L.
,
1965
, “
A Unit for Testing Materials At High Strain Rates
,”
Exp. Mech.
,
5
(
1
), pp.
29
32
. 10.1007/BF02320901
2.
Grady
,
D. E.
, and
Benson
,
D. A.
,
1983
, “
Fragmentation of Metal Rings by Electromagnetic Loading
,”
Exp. Mech.
,
12
(
4
), pp.
393
400
. 10.1007/BF02330054
3.
Grady
,
D. E.
,
Kipp
,
M. E.
, and
Benson
,
D. A.
,
1984
, “Energy and Statistical Effects in the Dynamic Fragmentation of Metal Rings,”
Mechanical Properties At High Strain Rates
,
Harding
,
J.
, ed.,
Institute of Physics
, pp.
315
320
.
4.
Altynova
,
M.
,
Hu
,
X.
, and
Daehn
,
G. S.
,
1996
, “
Increased Ductility in High Velocity Electromagnetic Ring Expansion
,”
Metall. Trans. A
,
27
(
7
), pp.
1837
1844
. 10.1007/BF02651933
5.
Pandolfi
,
A.
,
Krysl
,
P.
, and
Ortiz
,
M.
,
1999
, “
Finite Element Simulation of Ring Expansion and Fragmentation: The Capturing of Length and Time Scales Through Cohesive Models of Fracture
,”
Int. J. Fract.
,
95
(
1/4
), pp.
279
297
. 10.1023/A:1018672922734
6.
Sørensen
,
N. J.
, and
Freund
,
L. B.
,
2000
, “
Unstable Neck Formation in a Ductile Ring Subjected to Impulsive Radial Loading
,”
Int. J. Solids Struct.
,
37
(
16
), pp.
2265
2283
. 10.1016/S0020-7683(98)00315-1
7.
Becker
,
R.
,
2002
, “
Ring Fragmentation Predictions Using the Gurson Model With Material Stability Conditions As Failure Criterion
,”
Int. J. Solids Struct.
,
39
(
13–14
), pp.
3555
3580
. 10.1016/S0020-7683(02)00170-1
8.
Mercier
,
S.
, and
Molinari
,
A.
,
2004
, “
Analysis of Multiple Necking in Rings Under Rapid Radial Expansion
,”
Int. J. Impact Eng.
,
30
(
4
), pp.
403
419
. 10.1016/S0734-743X(03)00063-0
9.
Triantafyllidis
,
N.
, and
Waldenmyer
,
J. R.
,
2004
, “
Onset of Necking in Electro-Magnetically Formed Rings
,”
J. Mech. Phys. Solids
,
52
(
9
), pp.
2127
2148
. 10.1016/j.jmps.2004.02.009
10.
Zhang
,
H.
, and
Ravi-Chandar
,
K.
,
2006
, “
On the Dynamics of Necking and Fragmentation—I. Real-Time and Post-mortem Observations in Al 6061-O
,”
Int. J. Fract.
,
142
(
3–4
), pp.
183
217
. 10.1007/s10704-006-9024-7
11.
Zhou
,
F.
,
Molinari
,
J. F.
, and
Ramesh
,
K. T.
,
2006
, “
An Elasto-Visco-Plastic Analysis of Ductile Expanding Ring
,”
Int. J. Impact Eng.
,
33
(
1–12
), pp.
880
891
. 10.1016/j.ijimpeng.2006.09.070
12.
Grady
,
D.
,
2006
,
Fragmentation of Rings and Shells the Legacy of N. F. Mott
, 1 ed.
, Springer-Verlag
,
Berlin, Heidelberg
,
13.
Rusinek
,
A.
, and
Zaera
,
R.
,
2007
, “
Finite Element Simulation of Steel Ring Fragmentation Under Radial Expansion
,”
Int. J. Impact Eng.
,
34
(
4
), pp.
799
822
. 10.1016/j.ijimpeng.2006.01.003
14.
Zhang
,
H.
, and
Ravi-Chandar
,
K.
,
2008
, “
On the Dynamics of Necking and Fragmentation—II. Effect of Material Properties Geometrical Constraints and Absolute Size
,”
Int. J. Fract.
,
150
(
1–2
), pp.
3
36
. 10.1007/s10704-008-9233-3
15.
Zhang
,
H.
, and
Ravi-Chandar
,
K.
,
2010
, “
On the Dynamics of Localization and Fragmentation-IV. Expansion of Al 6061-O Tubes
,”
Int. J. Fract.
,
163
(
1–2
), pp.
41
65
. 10.1007/s10704-009-9441-5
16.
Janiszewski
,
J.
,
2012
, “
Ductility of Selected Metals Under Electromagnetic Ring Test Loading Conditions
,”
Int. J. Solids Struct.
,
49
(
7–8
), pp.
1001
1008
. 10.1016/j.ijsolstr.2012.01.005
17.
Rodríguez-Martínez
,
J. A.
,
Vadillo
,
G.
,
Fernández-Sáez
,
J.
, and
Molinari
,
A.
,
2013
, “
Identification of the Critical Wavelength Responsible for the Fragmentation of Ductile Rings Expanding At Very High Strain Rates
,”
J. Mech. Phys. Solids
,
61
(
6
), pp.
1357
1376
. 10.1016/j.jmps.2013.02.003
18.
Cliche
,
N.
, and
Ravi-Chandar
,
K.
,
2018
, “
Dynamic Strain Localization in Magnesium Alloy AZ31B-O
,”
Mech. Mater.
,
116
, pp.
189
201
. 10.1016/j.mechmat.2017.09.008
19.
Hu
,
X.
, and
Daehn
,
G. S.
,
1996
, “
Effect of Velocity on Flow Localization in Tension
,”
Acta Mater.
,
44
(
3
), pp.
1021
1033
. 10.1016/1359-6454(95)00228-6
20.
Guduru
,
P. R.
, and
Freund
,
L. B.
,
2002
, “
The Dynamics of Multiple Neck Formation and Fragmentation in High Rate Extension of Ductile Materials
,”
Int. J. Solids Struct.
,
39
(
21–22
), pp.
5615
5632
. 10.1016/S0020-7683(02)00367-0
21.
N’souglo
,
K. E.
,
Srivastava
,
A.
,
Osovski
,
S.
, and
Rodríguez-Martínez
,
J. A.
,
2018
, “
Random Distributions of Initial Porosity Trigger Regular Necking Patterns At High Strain Rates
,”
Proc. R. Soc. A: Math. Phys. Eng. Sci.
,
474
(
2211
), p.
20170575
. 10.1098/rspa.2017.0575
22.
Han
,
J. B.
, and
Tvergaard
,
V.
,
1995
, “
Effect of Inertia on the Necking Behaviour of Ring Specimens Under Rapid Axial Expansion
,”
Eur. J. Mech. A/Solids
,
14
(
2
), pp.
287
307
.
23.
ABAQUS/Explicit
,
2010
,
Abaqus Explicit V6.10 User’s Manual
, version 6.10,
ABAQUS Inc.
,
Richmond, USA
,
24.
Xue
,
Z.
,
Vaziri
,
A.
, and
Hutchinson
,
J.
,
2008
, “
Material Aspects of Dynamic Neck Retardation
,”
J. Mech. Phys. Solids
,
56
(
1
), pp.
93
113
. 10.1016/j.jmps.2007.04.003
25.
Gurson
,
A.
,
1977
, “
Continuum Theory of Ductile Rupture by Void Nucleation and Growth. Part I: Yield Criteria and Flow Rules for Porous Ductile Media
,”
ASME J. Eng. Mater. Technol.
,
99
(
1
), pp.
2
15
. 10.1115/1.3443401
26.
Tvergaard
,
V.
, and
Needleman
,
A.
,
1984
, “
Analysis of the Cup-Cone Fracture in a Round Tensile Bar
,”
Acta Metall.
,
32
(
1
), pp.
157
169
. 10.1016/0001-6160(84)90213-X
27.
Tvergaard
,
V.
,
1981
, “
Influence of Voids on Shear Band Instabilities Under Plane Strain Conditions
,”
Int. J. Fract.
,
17
(
4
), pp.
389
407
. 10.1007/BF00036191
28.
Tvergaard
,
V.
,
1982
, “
On Localization in Ductile Materials Containing Spherical Voids
,”
Int. J. Fract.
,
18
(
4
), pp.
237
252
.
29.
Pan
,
J.
,
Saje
,
M.
, and
Needleman
,
A.
,
1983
, “
Localization of Deformation in Rate Sensitive Porous Plastic Solids
,”
Int. J. Fract.
,
21
(
4
), pp.
261
278
. 10.1007/BF00942345
30.
Srivastava
,
A.
,
Ponson
,
L.
,
Osovski
,
S.
,
Bouchaud
,
E.
,
Tvergaard
,
V.
, and
Needleman
,
A.
,
2014
, “
Effect of Inclusion Density on Ductile Fracture Toughness and Roughness
,”
J. Mech. Phys. Solids
,
63
, pp.
62
79
. 10.1016/j.jmps.2013.10.003
31.
Srivastava
,
A.
,
Osovski
,
S.
, and
Needleman
,
A.
,
2017
, “
Engineering the Crack Path by Controlling the Microstructure
,”
J. Mech. Phys. Solids
,
100
, pp.
1
20
. 10.1016/j.jmps.2016.12.006
32.
Osovski
,
S.
,
Srivastava
,
A.
,
Ponson
,
L.
,
Bouchaud
,
E.
,
Tvergaard
,
V.
,
Ravi-Chandar
,
K.
, and
Needleman
,
A.
,
2015
, “
The Effect of Loading Rate on Ductile Fracture Toughness and Fracture Surface Roughness
,”
J. Mech. Phys. Solids
,
76
, pp.
20
46
. 10.1016/j.jmps.2014.11.007
33.
Osovski
,
S.
,
Srivastava
,
A.
,
Williams
,
J.
, and
Needleman
,
A.
,
2015
, “
Grain Boundary Crack Growth in Metastable Titanium β Alloys
,”
Acta Mater.
,
82
, pp.
167
178
. 10.1016/j.actamat.2014.08.062
34.
Belytschko
,
T.
,
Chiapetta
,
R. L.
, and
Bartel
,
H. D.
,
1976
, “
Efficient Large Scale Non-Linear Transient Analysis by Finite Elements
,”
Int. J. Numer. Methods Eng.
,
10
(
3
), pp.
579
596
. 10.1002/nme.1620100308
35.
Peirce
,
D.
,
Shih
,
C. F.
, and
Needleman
,
A.
,
1984
, “
A Tangent Modulus Method for Rate Dependent Solids
,”
Comput. Struct.
,
18
(
5
), pp.
875
887
. 10.1016/0045-7949(84)90033-6
36.
Tvergaard
,
V.
,
1982
, “
Influence of Void Nucleation on Ductile Shear Fracture At a Free Surface
,”
J. Mech. Phys. Solids
,
30
(
6
), pp.
399
425
. 10.1016/0022-5096(82)90025-4
37.
Fressengeas
,
C.
, and
Molinari
,
A.
,
1985
, “
Inertia and Thermal Effects on the Localization of Plastic Flow
,”
Acta Metall.
,
33
(
3
), pp.
387
396
. 10.1016/0001-6160(85)90081-1
38.
Fressengeas
,
C.
, and
Molinari
,
A.
,
1994
, “
Fragmentation of Rapidly Stretching Sheets
,”
Eur. J. Mech. A/Solids
,
13
(
2
), pp.
251
268
.
39.
Mercier
,
S.
, and
Molinari
,
A.
,
2003
, “
Predictions of Bifurcations and Instabilities During Dynamic Extensions
,”
Int. J. Solids Struct.
,
40
(
8
), pp.
1995
2016
. 10.1016/S0020-7683(03)00020-9
40.
Mercier
,
S.
,
Granier
,
N.
,
Molinari
,
A.
,
Llorca
,
F.
, and
Buy
,
F.
,
2010
, “
Multiple Necking During the Dynamic Expansion of Hemispherical Metallic Shells, From Experiments to Modelling
,”
J. Mech. Phys. Solids
,
58
(
7
), pp.
955
982
. 10.1016/j.jmps.2010.05.001
41.
Godinger
,
A.
,
Rotbaum
,
Y.
,
Vaz-Romero
,
A.
,
Rodríguez-Martínez
,
J. A.
, and
Rittel
,
D.
,
2017
, “
On the Relation Between Shape Imperfections of a Specimen and Necking Growth Rate Under Dynamic Conditions
,”
Int. J. Eng. Sci.
,
119
, pp.
278
287
. 10.1016/j.ijengsci.2017.06.020
42.
El Maï
,
S.
,
Mercier
,
S.
,
Petit
,
J.
, and
Molinari
,
A.
,
2014
, “
An Extension of the Linear Stability Analysis for the Prediction of Multiple Necking During Dynamic Extension of Round Bar
,”
Int. J. Solids Struct.
,
51
(
21–22
), pp.
3491
3507
. 10.1016/j.ijsolstr.2014.05.019
43.
Vaz-Romero
,
A.
,
Rodríguez-Martínez
,
J. A.
,
Mercier
,
S.
, and
Molinari
,
A.
,
2017
, “
Multiple Necking Pattern in Nonlinear Elastic Bars Subjected to Dynamic Stretching: The Role of Defects and Inertia
,”
Int. J. Solids Struct.
,
125
, pp.
232
243
. 10.1016/j.ijsolstr.2017.07.001
44.
Knoche
,
P.
, and
Needleman
,
A.
,
1993
, “
The Effect of Size on the Ductility of Dynamically Loaded Tensile Bars
,”
Eur. J. Mech. A/Solids
,
12
(
4
), pp.
585
601
.
45.
Dudzinski
,
D.
, and
Molinari
,
A.
,
1991
, “
Perturbation Analysis of Thermoviscoplastic Instabilities in Biaxial Loading
,”
Int. J. Solids Struct.
,
27
(
5
), pp.
601
628
. 10.1016/0020-7683(91)90216-3
46.
N’souglo
,
K. E.
,
Rodríguez-Martínez
,
J. A.
, and
Cazacu
,
O.
,
2020
, “
The Effect of Tension–Compression Asymmetry on the Formation of Dynamic Necking Instabilities Under Plane Strain Stretching
,”
Int. J. Plast.
,
in press
.
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