The Bammann, Chiesa, and Johnson (BCJ) material model predicts unlimited localization of strain and damage, resulting in a zero dissipation energy at failure. This difficulty resolves when the BCJ model is modified to incorporate a nonlocal evolution equation for the damage, as proposed by Pijaudier-Cabot and Bazant (1987, “Nonlocal Damage Theory,” ASCE J. Eng. Mech., 113, pp. 1512–1533.). In this work, we theoretically assess the ability of such a modified BCJ model to prevent unlimited localization of strain and damage. To that end, we investigate two localization problems in nonlocal BCJ metals: appearance of a spatial discontinuity of the velocity gradient in any finite, inhomogeneous body, and localization of the dissipation energy into finite bands. We show that in spite of the softening arising from the damage, no spatial discontinuity occurs in the velocity gradient. Also, we find that the dissipation energy is continuously distributed in nonlocal BCJ metals and therefore cannot localize into zones of vanishing volume. As a result, the appearance of any vanishing width adiabatic shear band is impossible in a nonlocal BCJ metal. Finally, we study the finite element (FE) solution of shear banding in a rectangular plate, deformed in plane strain tension and containing an imperfection, thereby illustrating the effects of imperfections and finite size on the localization of strain and damage.

References

References
1.
Muhlhaus
,
H.-B.
,
1986
, “
Shear Band Analysis in Granular Materials by Cosserat Theory
,”
Ing.-Arch.
,
56
, pp.
389
399
.10.1007/BF02570619
2.
Tvergaard
,
V.
, and
Needleman
,
A.
,
1997
, “
Nonlocal Effects on Localization in a Void-Sheet
,”
Int. J. Solids Struct.
,
34
, pp.
2221
2238
.10.1016/S0020-7683(96)00140-0
3.
de Borst
,
R.
,
1993
, “
Simulation of Strain Localization: A Reappraisal of the Cosserat Continuum
,”
Eng. Comput.
,
8
(4), pp.
317
332
.10.1108/eb023842
4.
Ramaswamy
,
S.
, and
Aravas
,
N.
,
1998
, “
Finite Element Implementation of Gradient Plasticity Models Part I: Gradient-Dependent Yield Functions
,”
Comp. Meth. App. Mech. Eng.
,
163
, pp.
11
32
.10.1016/j.bbr.2011.03.031
5.
Bazant
,
Z. P.
, and
Lin
,
F.-B.
,
1989
, “
Stability Against Localization of Softening Into Ellipsoids and Bands: Parameter Study
,”
Int. J. Solids Stuct.
,
28
, pp.
1483
1498
.10.1016/0020-7683(89)90114-5
6.
Aifantis
,
E. C.
,
1995
, “
From Micro-Plasticity to Macro-Plasticity: The Scale-Invariance Approach
,”
Trans. ASME J. Eng. Mater. Technol.
,
117
(
4
), pp.
352
355
.10.1115/1.2804724
7.
Pijaudier-Cabot
,
G.
, and
Bazant
,
Z. P.
,
1987
, “
Nonlocal Damage Theory
,”
ASCE J. Eng. Mech.
,
113
, pp.
1512
1533
.10.1061/(ASCE)0733-9399(1987)113:10(1512)
8.
Leblond
,
J. B.
,
Perrin
,
G.
, and
Devaux
,
J.
,
1994
, “
Bifurcation Effects in Ductile Metals With Nonlocal Damage
,”
ASME J. Appl. Mech.
,
61
, pp.
236
242
.10.1115/1.2901435
9.
Bazant
,
Z. P.
,
Belytschko
,
T. B.
, and
Chang
,
T. P.
,
1984
, “
Continuum Theory for Strain-Softening
,”
ASCE J. Eng. Mech.
,
110
, pp.
1666
1692
.10.1061/(ASCE)0733-9399(1984)110:12(1666)
10.
Bazant
,
Z. P.
, and
Pijaudier-Cabot
,
G.
,
1988
, “
Nonlocal Continuum Damage, Localization Instability and Convergence
,”
ASME J. Appl. Mech.
,
55
, pp.
287
293
.10.1115/1.3173674
11.
Pijaudier-Cabot
,
G.
, and
Bode
,
L.
,
1991
, “
Two-Dimensional Analysis of Strain Localization With Nonlocal Continuum Damage
,” paper presented at the
8th ASCE Engineering Mechanics Speciality Conference
,
Columbus, OH
.
12.
Gurson
,
A. L.
,
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
, pp.
2
15
.10.1115/1.3443401
13.
Bammann
,
D. J.
,
1984
, “
An Internal Variable Model of Viscoplasticity
,”
Int. J. Eng. Sci.
22
,
1041
1053
.10.1016/0020-7225(84)90105-8
14.
Bammann
,
D. J.
, and
Johnson
,
G. C.
,
1984
, “
Developement of a Strain Rate Sensitivity Plasticity Model
,”
Engineering Mechanics in Civil Engineering
,
A. P.
Boresi
, and
K. P.
Chong
, eds., ASCE, New York, pp. 454–457.
15.
Bammann
,
D. J.
,
1985
, “
An Internal State Variable Model for Elastic-Viscoplasticity
,”
The Mechanics of Dislocations: Proceedings of an International Symposium
,
E. C.
Aifantis
, and
J. P.
Hirth
, eds.,
The American Society of Metals
, Metals Park, OH, p.
103
.
16.
Bammann
,
D. J.
, and
Aifantis
,
E. C.
,
1987
, “
A Model for Finite-Deformation Plasticity
,”
Acta Mech.
,
69
, pp.
97
117
.10.1007/BF01175716
17.
Bammann
,
D. J.
, and
Johnson
G. C.
,
1987
, “
On the Kinematics of Finite-Deformation Plasticity
,”
Acta Mech.
,
70
, pp.
1
13
.10.1007/BF01174643
18.
Bammann
,
D. J.
, and
Aifantis
,
E. C.
,
1987
, “
A Model for Finite-Deformation Plasticity
,”
Acta. Mech.
,
69
, pp.
97
117
.10.1007/BF01175716
19.
Bammann
,
D. J.
,
1988
, “
Modelling the Large Strain-High Temperature Response of Metals
,”
Modeling and Control of Casting and Welding Process IV
,
A. F.
Giamei
, and
G. J.
Abbaschian
, eds.,
TMS Publications
,
Warrendale, PA
.
20.
Bammann
,
D. J.
,
Chiesa
,
M. L.
,
McDonald
,
A.
,
Kawahara
,
W. A.
,
Dike
,
J. J.
, and
Revelli
,
V. D.
,
1990b
, “
Prediction of Ductile Failure in Metal Structures
,”
Failure Criteria and Analysis in Dynamic Response
, AMD-Vol.
107
, H. Lindberg, ed., The American Society of Mechanical Engineers, Dallas, TX, November, pp. 7–12.
21.
Bammann
,
D. J.
,
1990a
, “
Modelling Temperature and Strain Rate Dependent Large Deformations of Metals
,”
Appl. Mech. Rev.
,
43
(
5
), p.
S312
–S319.10.1115/1.3120834
22.
Bammann
,
D. J.
,
Chiesa
,
M. L.
,
Horstemeyer
,
M. F.
, and
Weingarten
,
L. I.
,
1993
, “
Failure in Ductile Materials Using Finite Element Methods
,”
Structural Crashworthiness and Failure, Applied Science
,
N.
Jones
and
T.
Weirzbicki
, eds., Elsevier Applied Science, London, pp.
1
52
.
23.
Bammann
,
D. J.
,
Chiesa
,
M. L.
, and
Johnson
,
G. C.
,
1995
, “
An Internal State Variable Model for Temperature and Strain Rate Dependent Metals
,”
Constitutive Laws: Experiments and Numerical Implementation
,
A. M.
Rajendran
and
R. C.
Batra
, eds., CIMNE, Barcelona, pp.
84
97
.
24.
Johnson
,
G. R.
, and
Cook
,
W. H.
,
1983
, “
A Constitutive Model and Data for Metals Subjected to Large Strain, High Strain Rates and High Temperatures
,”
Proceedings of the Seventh International Symposium on Ballistics
,
April 19–21
,
The Hague, The Netherlands
, pp.
541
547
.
25.
Cocks
,
A. C. F.
, and
Ashby
,
M. G.
,
1980
, “
Intergranular Fracture During Power-Law Creep Under Multiaxial Stresses
,”
Met. Sci.
,
14
, pp.
395
402
.10.1016/0036-9748(80)90333-6
26.
Taylor
,
G. I.
, and
Quinney
,
H.
,
1934
, “
The Latent Energy Remaining in a Metal After Cold Working
,”
Proc. R. Soc. London A
,
143
, pp.
307
326
.10.1098/rspa.1934.0004
27.
Fremond
,
M.
, and
Nedjar
,
B.
,
1993
, “
Endommagement et Principe des Puissances Virtuelles
,”
C. R. Acc. Sci. II
,
317
(
7
), pp.
857
864
.
28.
Lorentz
,
E.
,
1997
, “
Lois de Comportment a Gradients de Variables Internes: Construction, Formulation Variationnelle et Mise en Oeuvre Numerique
,” Ph.D. dissertation, Ecole Polytechnique, Palaisseau, France (in French).
29.
Aravas
,
N.
,
1987
, “
On the Numerical Integration of a Class of Pressure-Dependent Plasticity Models
,”
Int. J. Numer. Methods Eng.
,
24
, pp.
1395
1416
.10.1002/nme.1620240713
30.
Bazant
,
Z. P.
, and
Pijaudier-Cabot
,
G.
,
1989
, “
Measurement of the Characteristic Length for Nonlocal Continuum
,”
J. Eng. Mech.
,
115
, pp.
755
767
.10.1061/(ASCE)0733-9399(1989)115:4(755)
31.
Enakoutsa
,
K.
,
Leblond
,
J. B.
, and
Perrin
,
G.
,
2007
, “
Numerical Implementation and Assessment of the GLPD Micromorphic Model of Ductile Rupture
,”
Eur. J. Mech. A/Solids
,
28
, pp.
445
460
.10.1016/j.euromechsol.2008.11.004
32.
Enakoutsa
,
K.
, and
Leblond
,
J. B.
,
2009
, “
Numerical Implementation and Assessment of a Phenomenological Nonlocal Model of Ductile Rupture
,”
Comput. Methods Appl. Mech. Eng.
,
196
, pp.
1946
1957
.10.1016/j.cma.2006.10.003
33.
Johnson
,
W.
,
1987
, “
Henry Tresca as the Originator of Adiabatic Shear Heat Lines
,”
Int. J. Mech. Sci.
,
29
, pp.
301
310
.
34.
Coutney
,
T. H.
,
2000
,
Mechanical Behavior of Materials
,
McGraw-Hill
,
New York
.
35.
Hutchinson
,
J. W.
,
2000
, “
Plasticity at the Micron Scale
,”
Int. J. Solids Struct.
,
37
, pp.
225
238
.10.1016/S0020-7683(99)00090-6
36.
Lemaitre
,
J.
, and
Chaboche
,
J. L.
,
1989
,
Mechanics of Solid Materials
,
Cambridge University Press
,
Cambridge
, UK.
37.
Horstemeyer
,
M. F.
,
Matalanis
,
M. M.
, and
Siebe
,
M. L.
,
2000
, “
Micromechanical Finite Element Calculations of Temperature and Void Configuration Effects on Void Growth and Coalescence
,”
Int. J. Plast.
,
16
, pp.
979
1015
.10.1016/S0749-6419(99)00076-5
38.
Halphen
,
B.
, and
Nguyen
,
Q. S.
,
1975
, “
Sur les Matériaux Standards Généralisés
,”
J. Mech
,
14
, pp.
39
63
, in French.
39.
Drabek
,
T.
, and
Bohm
,
H. J.
,
2005
, “
Damage Model for Studying Ductile Matrix Failure in Composites
,”
Comput. Mater. Sci.
,
32
, pp.
329
336
.10.1016/j.commatsci.2004.09.035
40.
Baaser
,
H.
, and
Tvergaard
,
V.
,
2003
, “
A New Algorithmic Approach Treating Nonlocal Effects at Finite Rate-Independent Deformation Using Rousselier Damage Model
,”
Comput. Methods Appl. Mech. Eng.
,
192
, pp.
107
124
.10.1016/S0045-7825(02)00535-2
41.
Tvergaard
,
V.
, and
Needleman
,
A.
,
1995
, “
Effects of Nonlocal Damage in Porous Plastic Solids
,”
Int. J. Solids Struct.
,
32
, pp.
1063
1077
.10.1016/0020-7683(94)00185-Y
You do not currently have access to this content.