1. Nonlocal modeling in high-velocity impact failure of 6061-T6
aluminum
F.R. Ahad a
, K. Enakoutsa a,⇑
, K.N. Solanki b,*, D.J. Bammann c
a
Center for Advanced Vehicular Systems, Mississippi State University, 200 Research Boulevard, Mississippi State, MS 39762, USA
b
School of Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287, USA
c
Mechanical Engineering Department, Mississippi State University, Mississippi State, MS 39762, USA
a r t i c l e i n f o
Article history:
Received 7 August 2012
Received in final revised form 19 September
2013
Available online 24 October 2013
Keywords:
Mesh dependence
Dynamic failure
Damage delocalization
BCJ model
Nonlocal modeling
a b s t r a c t
In this paper, we present numerical simulations with local and nonlocal models under
dynamic loading conditions. We show that for finite element (FE) computations of high-
velocity, impact problems with softening material models will result in spurious post-
bifurcation mesh dependency solutions. To alleviate numerical instabilities associated
within the post-bifurcation regime, a characteristic length scale was added to the constitu-
tive relations and calibrated through a series of different notch specimen tests. This work
aims to assess the practical elevance of the modified model to yield mesh independent
results in the numerical simulations of high-velocity impact problems. To this end, we con-
sider the problem of a rigid projectile moving at a range of velocities between 89 and
107 m/s, colliding against a 6061-T6 Aluminum disk. A material model embedded with a
characteristic length scale in the manner proposed by Pijaudier-Cabot and Bazant
(1987), but in the context of concrete damage, was utilized to describe the damage
response of the disk. The numerical result shows that the addition of a characteristic length
scale to the constitutive model does eliminate the pathological mesh dependency and
shows excellent agreements between numerical and experimental results. Furthermore,
the application of a nonlocal model for higher strain rate behavior shows the ability of
the model to address intense localized deformations, irreversible flow, softening, and final
failure. Finally, we show that the length scale introduced in the model can be calibrated
using a series of tensile notch specimen tests.
Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Recently, efforts to introduce a numerical length scale into continuum models have led to a resurgence of research in the
area of generalized continua (e.g., see Dillon and Kratochvill, 1970; Nunziato and Cowin, 1979; Bammann and Aifantis, 1982;
Aifantis, 1984; Bammann, 1988; Brown et al., 1989; McDowell et al., 1992; Zbib et al., 1992; Fleck and Hutchinson, 1993;
Tvergaard and Needleman, 1995; Gurtin, 1996; Nix and Gao, 1998; Ramaswamy and Aravas, 1998; Gurtin, 2000; Regueiro
et al., 2002; Solanki et al., 2010). This is partially due to the fact that the local theory treats a body as a ‘‘continuum’’ of par-
ticles or points, the only geometrical property being that of position. A closer look at materials reveals a complex microstruc-
ture of grains, subgrains, shear bands and other topological features of the distribution of mass that are not taken into
account by classical local theories. If the observer is far enough removed from a grain, he will see only a point. But a theory
that strips away all of the geometrical properties of a grain except for the position of its center of mass will certainly fail to
0749-6419/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijplas.2013.10.001
⇑ Corresponding authors.
E-mail addresses: enakoutsa@yahoo.fr (K. Enakoutsa), kiran.solanki@asu.edu (K.N. Solanki).
International Journal of Plasticity 55 (2014) 108–132
Contents lists available at ScienceDirect
International Journal of Plasticity
journal homepage: www.elsevier.com/locate/ijplas

2. Koffi Enakoutsa1
e-mail: koffi.enakoutsa@msstate.edu
Fazle R. Ahad
e-mail: fra11@cavs.msstate.edu
MSU/CAVS,
Starkville, MS 39759
Kiran N. Solanki
ASU/SEMTE,
Tempe, AZ 85287
e-mail: kiran.solanki@asu.edu
Yustianto Tjiptowidjojo
e-mail: yusti@cavs.msstate.edu
Douglas J. Bammann
e-mail: bammann@cavs.msstate.edu
MSU/CAVS,
Starkville, MS 39759
Using Damage Delocalization to
Model Localization Phenomena
in Bammann-Chiesa-Johnson
Metals
The Bammann, Chiesa, and Johnson (BCJ) material model predicts unlimited localiza-
tion 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, inhomogene-
ous 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 ve-
locity 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. [DOI: 10.1115/1.4007352]
1 Introduction
The numerical applications of the BCJ constitutive law, just
like all constitutive laws for such materials as porous metals or
brittle concrete or rocks, exhibit pathological mesh sensitivity in
the postfailure initiation and poststrain and damage localization
regimes. This issue originates, for dynamic problems, in the
change of the system of differential equations from hyperbolic to
elliptic, and for static problems, in the reverse. In either case, the
problem becomes ill-posed [1–4,41], as the boundary and initial
conditions for hyperbolic system of equations are not suitable
with the other one. Accordingly, discontinuities in the strain and
damage distribution occur, and the strain and damage have the
tendency to concentrate into localized bands of vanishing volume,
resulting in a spurious zero dissipation energy at failure. The addi-
tion of a mathematical length scale to the BCJ model eliminates
these difficulties. In particular, in problems involving strain local-
ization, the addition of a length scale introduces a “localization
limiter” which prevents unlimited localization of the strain and
damage. For example, Bazant and Lin [5] used this method to pre-
scribe a minimum dimension for the strain localization region to
study the stability against localization into ellipsoids and planar
bands. Another motivation for the introduction of length scales
into constitutive laws stems from an attempt to capture more of
the underlying physics of the materials, while still utilizing contin-
uum models. A complete review of this type of modeling is given
in Ref. [6] but is not duplicated here.
One method to add a mathematical length scale to the BCJ
model, based on a previous suggestion made by Pijaudier-Cabot
and Bazant [7] in the context of concrete damage and extended by
Leblond et al. [8] to plasticity, consists of adopting a nonlocal
evolution equation for the damage that involves a spatial convolu-
tion of some “local damage rate” and a Gaussian weighting func-
tion. The width of this function introduces a characteristic length
scale. In this formulation, the damage parameter is the only nonlo-
cal variable, because a formulation in which all state variables are
nonlocal will lead to serious complications, especially in the
expression of the equilibrium equations [9]. This assumption
appears quite attractive from the physical point of view, because
in the case of porous metals, the porosity (damage variable) is
essentially a nonlocal quantity and there is no reason why the
stress, the strain and other similar variables should be nonlocal.
Theoretical studies of the properties of nonlocal models have
focused, for the most part, on instabilities, bifurcations, and local-
ization related problems, since it is in these contexts that the non-
local concept is supposed to bring significant contributions. We
will mention the works of Bazant and Pijaudier-Cabot [10] and
Bazant and Lin [5] on strain localization in one-dimensional rods
and three-dimensional solids, the bifurcation studies of Pijaudier-
Cabot and Bode [11] and Leblond et al. [8] that are pertinent for
the development of the present work. The works of Leblond et al.
[8] deal with the appearance of bifurcation phenomena in a porous
ductile material obeying the delocalized version of the classical
Gurson [12] model. It was notably checked by Leblond et al. [8]
that no bifurcation of the first type can occur if the hardening
slope of the sound matrix is positive; however, bifurcations of sec-
ond type are possible. Note that the works of Bazant and
Pijaudier-Cabot [10], Bazant and Lin [5], Pijaudier-Cabot and
Bode [11], and Leblond et al. [8] on the properties of delocalized
models have focused on constitutive models that are rate-
independent and do not contain any temperature history.
In the present paper, we intent to follow up the study of the
properties of damage delocalization for models involving temper-
ature and rate sensitivity effects. The model considered will be
that of Bammann-Chiesa-Johnson [13–23], the delocalization
being introduced using the approach suggested by Pijaudier-Cabot
and Bazant [7]. There are several reasons to choose the BCJ
model over its competitors such as the Johnson and Cook’s [24]
model for problems involving temperature and rate sensitivity
1
Corresponding author.
Contributed by the Materials Division of ASME for publication in the JOURNAL OF
ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received February 14, 2012;
final manuscript received August 1, 2012; published online September 6, 2012.
Assoc. Editor: Irene Beyerlein.
Journal of Engineering Materials and Technology OCTOBER 2012, Vol. 134 / 041014-1
Copyright VC 2012 by ASME
Downloaded From: http://materialstechnology.asmedigitalcollection.asme.org/ on 08/20/2013 Terms of Use: http://asme.org/terms

3. THEORETICAL & APPLIED MECHANICS LETTERS 2, 051005 (2012)
Damage smoothing effects in a delocalized rate sensitivity model for
metals
K. Enakoutsa,1, a)
K. N. Solanki,2
F. R. Ahad,1
Y. Tjiptowidjojo,1
and D. J. Bammann1
1)
Center for Advanced Vehicular Systems, Mississippi State University, Mississippi 39762, USA
2)
School of Engineering of Matter, Transport and Energy, Arizona State University, Tempe 85287, USA
(Received 7 May 2012; accepted 30 July 2012; published online 10 September 2012)
Abstract It has been long time established that application of damage delocalization method
to softening constitutive models yields numerical results that are independent of the size of the
finite element. However, the prediction of real-world large and small scale problems using the
delocalization method remains in its infancy. One of the drawbacks encountered is that the predicted
load versus displacement curve suddenly drops, as a result of excessive smoothing of the damage.
The present paper studies this unwanted effect for a delocalized plasticity/damage model for metallic
materials. We use some theoretical arguments to explain the failure of the delocalized model
considered, following which a simple remedy is proposed to deal with it. Future works involve the
numerical implementation of the new version of the delocalized model in order to assess its ability to
reproduce real-world problems. c⃝ 2012 The Chinese Society of Theoretical and Applied Mechanics.
[doi:10.1063/2.1205105]
Keywords Bammann-Chiesa-Johnson model, damage smoothing, Fourier transform, softening
The addition of characteristic length scales to con-
stitutive models involving softening through damage
delocalization method is very well known to remove
the pathological mesh size effects in the finite element
(FE) solution of problems involving these constitutive
models.1–4
Another closely related technique which con-
sists of incorporating gradient terms in the evolution
equation of the parameter(s) governing softening yields
the same conclusions, although its numerical implemen-
tation into FE codes is not an easy task compared to
that of the delocalization technique. A complete review
of this technique and its associated numerical imple-
mentation can be found in Ref. 5. Despite these suc-
cesses, nonlocal or gradient models have not yet reached
a situation where they are applicable to small or large
scale structure problems. For example, Enakoutsa et
al.1
have demonstrated that the use of nonlocal Gur-
son model6
does eliminate spurious mesh size effects in
FE simulations of ductile rupture of typical pre-cracked
Ta specimens, but fails to reproduce the experimen-
tal load versus displacement curve, i.e., the predicted
load-displacement curve remains quasi-stationary for
some time and decreases abruptly. According to these
authors, this undesirable feature is due to excessive
smoothing of the damage distribution in the ligament
ahead of the crack tip of the specimen. They provided
a theoretical explanation of this phenomenon based on
such as crude assumptions as unboundness of the body
considered and homogeneity of the mechanical fields.
Namely, they showed that the nonlocal evolution equa-
tion for the damage is qualitatively similar to some dif-
fusion equations which result in an excessive smooth-
ing of the damage. Following this theoretical analysis,
they proposed a simple remedy to deal with the execes-
a)Corresponding author. Email: koffi@cavs.msstate.edu.
sive smoothing of the damage. It consists of adopting
the nonlocal concept for the logarithm of the damage
instead of the damage itself; this has the avantage to
eliminate the analogy between the nonlocal evolution
equation and a diffusion equation. Good agreements
between experimental and numerical results were then
obtained.
The objective of the present paper is to follow up the
study of the applicability of the delocalization method
in numerical simulations of material behavior. The mo-
tivation is to predict accurately the post-bifurcation
regime of metals as the design of metal structures re-
quires to understand more and more physics of metals
in this particular regime. The model considered will be
that proposed by Bammann-Chiesa-Johnson (BCJ)7–16
but with a modified, delocalized evolution equation for
the damage following an earlier suggestion of Bijaudier-
Cabot and Bazant.17
The introduction of the convolu-
tion integral of the evolution equation of the damage
in the BCJ model incorporates a diffusive effect in the
constitutive model, which prevents the nonlocal dam-
age variable to spuriously localize into vanishing bands.
However, the diffusive effect unavoidably leads to an
unwanted excessive smoothing of the damage. Just like
that in Ref. 1, we provide a theoretical explanation of
this shortcoming, following which a simple remedy is
proposed to deal with it. The rest of the paper pro-
vides the constitutive relations of the BCJ model and
its nonlocal extension. then it is devoted to a theoret-
ical explanation of the unwanted excessive smoothing
of the damage. Finally, it presents a simple solution to
overcome the excessive damage smoothing shortcoming.
The BCJ model is a physically-based plastic-
ity/damage model which incorporates load path, strain
rate, temperature, and history effects through the use
of internal state variables (ISVs); it also accounts for
both deviatoric deformation resulting from the presence

4. Hindawi Publishing Corporation
Modelling and Simulation in Engineering
Volume 2013, Article ID 815158, 11 pages
http://dx.doi.org/10.1155/2013/815158
Research Article
Modeling the Dynamic Failure of Railroad Tank Cars Using
a Physically Motivated Internal State Variable Plasticity/Damage
Nonlocal Model
Fazle R. Ahad,1
Koffi Enakoutsa,1
Kiran N. Solanki,2
Yustianto Tjiptowidjojo,1
and Douglas J. Bammann3
1
Center for Advanced Vehicular Systems, Mississippi State University, 200 Research Boulevard, Mississippi State, MS 39762, USA
2
School of Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287, USA
3
Mechanical Engineering Department, Mississippi State University, Mississippi State, MS 39762, USA
Correspondence should be addressed to Koffi Enakoutsa; enakoutsa@yahoo.fr
Received 28 September 2012; Revised 11 December 2012; Accepted 14 January 2013
Academic Editor: Chung-Souk Han
Copyright © 2013 Fazle R. Ahad et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We used a physically motivated internal state variable plasticity/damage model containing a mathematical length scale to idealize
the material response in finite element simulations of a large-scale boundary value problem. The problem consists of a moving
striker colliding against a stationary hazmat tank car. The motivations are (1) to reproduce with high fidelity finite deformation and
temperature histories, damage, and high rate phenomena that may arise during the impact accident and (2) to address the material
postbifurcation regime pathological mesh size issues. We introduce the mathematical length scale in the model by adopting a
nonlocal evolution equation for the damage, as suggested by Pijaudier-Cabot and Bazant in the context of concrete. We implement
this evolution equation into existing finite element subroutines of the plasticity/failure model. The results of the simulations, carried
out with the aid of Abaqus/Explicit finite element code, show that the material model, accounting for temperature histories and
nonlocal damage effects, satisfactorily predicts the damage progression during the tank car impact accident and significantly reduces
the pathological mesh size effects.
1. Introduction
The design of accident-resistant hazmat tank cars or the
improvement of existing ones requires material models that
describe the physical mechanisms that occur during the acci-
dent. In the case of impact accidents such as collisions, finite
deformation, and temperature histories, damage and high
rate phenomena are generated in the vicinity of the impact
region. Unfortunately, the majority of material models used
in the finite element (FE) simulation of hazmat tank car
impact scenarios do not account for such physical features
(unavoidably, this will under- or overestimate, for instance,
the numerical prediction of the puncture resistance of hazmat
tank cars’ structural integrity). Furthermore, in the few
models that do, a mathematical length scale aimed at solving
the postbifurcation problem is absent. As a consequence,
when one material point fails in the course of the numerical
simulations, the boundary value problem for such material
models changes, from a hyperbolic to an elliptical system of
differential equations in dynamic problems, and the reverse in
statics. In both cases, the boundary value problem becomes ill
posed, Muhlhaus [1], Tvergaard and Needleman [2], de Borst
[3], and Ramaswamy and Aravas [4], as the boundary and
initial conditions for one system of differential equations are
not suitable for the other. Consequently, discontinuities in the
strain and damage distribution appear, and all the strain and
damage have the tendency to localize in a zone of vanishing
volumes; this results in zero dissipated energy at failure of the
material. Also, bifurcations with an infinite number of bifur-
cated branches appear, which raises the problem of selecting
the relevant one, especially in numerical computations where
this drawback manifests itself as a pathological sensitivity of
the results to the FE discretization.