International Journal of Plasticity 55 (2014) 108–132

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International Journal of Plasticity

journal homepage: www.elsevier .com/locate / i jp las

Nonlocal modeling in high-velocity impact failure of 6061-T6aluminum

0749-6419/$ - see front matter � 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).

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, USAb School of Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287, USAc 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 2012Received in final revised form 19 September2013Available online 24 October 2013

Keywords:Mesh dependenceDynamic failureDamage delocalizationBCJ modelNonlocal modeling

a b s t r a c t

In this paper, we present numerical simulations with local and nonlocal models underdynamic 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 associatedwithin 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 workaims to assess the practical elevance of the modified model to yield mesh independentresults 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 and107 m/s, colliding against a 6061-T6 Aluminum disk. A material model embedded with acharacteristic length scale in the manner proposed by Pijaudier-Cabot and Bazant(1987), but in the context of concrete damage, was utilized to describe the damageresponse of the disk. The numerical result shows that the addition of a characteristic lengthscale to the constitutive model does eliminate the pathological mesh dependency andshows excellent agreements between numerical and experimental results. Furthermore,the application of a nonlocal model for higher strain rate behavior shows the ability ofthe model to address intense localized deformations, irreversible flow, softening, and finalfailure. Finally, we show that the length scale introduced in the model can be calibratedusing a series of tensile notch specimen tests.

� 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 thearea 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; Regueiroet 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 intoaccount by classical local theories. If the observer is far enough removed from a grain, he will see only a point. But a theorythat strips away all of the geometrical properties of a grain except for the position of its center of mass will certainly fail to

Koffi Enakoutsa1

e-mail: koffi.enakoutsa@msstate.edu

Fazle R. Ahade-mail: fra11@cavs.msstate.edu

MSU/CAVS,

Starkville, MS 39759

Kiran N. SolankiASU/SEMTE,

Tempe, AZ 85287

e-mail: kiran.solanki@asu.edu

Yustianto Tjiptowidjojoe-mail: yusti@cavs.msstate.edu

Douglas J. Bammanne-mail: bammann@cavs.msstate.edu

MSU/CAVS,

Starkville, MS 39759

Using Damage Delocalization toModel Localization Phenomenain Bammann-Chiesa-JohnsonMetalsThe Bammann, Chiesa, and Johnson (BCJ) material model predicts unlimited localiza-tion of strain and damage, resulting in a zero dissipation energy at failure. This difficultyresolves when the BCJ model is modified to incorporate a nonlocal evolution equationfor the damage, as proposed by Pijaudier-Cabot and Bazant (1987, “Nonlocal DamageTheory,” ASCE J. Eng. Mech., 113, pp. 1512–1533.). In this work, we theoretically assessthe ability of such a modified BCJ model to prevent unlimited localization of strain anddamage. 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 inspite 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 innonlocal BCJ metals and therefore cannot localize into zones of vanishing volume. As aresult, the appearance of any vanishing width adiabatic shear band is impossible in anonlocal BCJ metal. Finally, we study the finite element (FE) solution of shear bandingin 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 strainand damage. [DOI: 10.1115/1.4007352]

1 Introduction

The numerical applications of the BCJ constitutive law, justlike all constitutive laws for such materials as porous metals orbrittle concrete or rocks, exhibit pathological mesh sensitivity inthe postfailure initiation and poststrain and damage localizationregimes. This issue originates, for dynamic problems, in thechange of the system of differential equations from hyperbolic toelliptic, and for static problems, in the reverse. In either case, theproblem becomes ill-posed [1–4,41], as the boundary and initialconditions for hyperbolic system of equations are not suitablewith the other one. Accordingly, discontinuities in the strain anddamage distribution occur, and the strain and damage have thetendency 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 eliminatesthese difficulties. In particular, in problems involving strain local-ization, the addition of a length scale introduces a “localizationlimiter” which prevents unlimited localization of the strain anddamage. For example, Bazant and Lin [5] used this method to pre-scribe a minimum dimension for the strain localization region tostudy the stability against localization into ellipsoids and planarbands. Another motivation for the introduction of length scalesinto constitutive laws stems from an attempt to capture more ofthe underlying physics of the materials, while still utilizing contin-uum models. A complete review of this type of modeling is givenin Ref. [6] but is not duplicated here.

One method to add a mathematical length scale to the BCJmodel, based on a previous suggestion made by Pijaudier-Cabotand Bazant [7] in the context of concrete damage and extended byLeblond et al. [8] to plasticity, consists of adopting a nonlocalevolution 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 lengthscale. In this formulation, the damage parameter is the only nonlo-cal variable, because a formulation in which all state variables arenonlocal will lead to serious complications, especially in theexpression of the equilibrium equations [9]. This assumptionappears quite attractive from the physical point of view, becausein the case of porous metals, the porosity (damage variable) isessentially a nonlocal quantity and there is no reason why thestress, the strain and other similar variables should be nonlocal.

Theoretical studies of the properties of nonlocal models havefocused, 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. Wewill mention the works of Bazant and Pijaudier-Cabot [10] andBazant and Lin [5] on strain localization in one-dimensional rodsand three-dimensional solids, the bifurcation studies of Pijaudier-Cabot and Bode [11] and Leblond et al. [8] that are pertinent forthe development of the present work. The works of Leblond et al.[8] deal with the appearance of bifurcation phenomena in a porousductile material obeying the delocalized version of the classicalGurson [12] model. It was notably checked by Leblond et al. [8]that no bifurcation of the first type can occur if the hardeningslope of the sound matrix is positive; however, bifurcations of sec-ond type are possible. Note that the works of Bazant andPijaudier-Cabot [10], Bazant and Lin [5], Pijaudier-Cabot andBode [11], and Leblond et al. [8] on the properties of delocalizedmodels 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 theproperties of damage delocalization for models involving temper-ature and rate sensitivity effects. The model considered will bethat of Bammann-Chiesa-Johnson [13–23], the delocalizationbeing introduced using the approach suggested by Pijaudier-Cabotand Bazant [7]. There are several reasons to choose the BCJmodel over its competitors such as the Johnson and Cook’s [24]model for problems involving temperature and rate sensitivity

1Corresponding 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 / 041014-1Copyright VC 2012 by ASME

Downloaded From: http://materialstechnology.asmedigitalcollection.asme.org/ on 08/20/2013 Terms of Use: http://asme.org/terms

THEORETICAL & APPLIED MECHANICS LETTERS 2, 051005 (2012)

Damage smoothing effects in a delocalized rate sensitivity model formetals

K. Enakoutsa,1, a) K. N. Solanki,2 F. R. Ahad,1 Y. Tjiptowidjojo,1 and D. J. Bammann11)Center for Advanced Vehicular Systems, Mississippi State University, Mississippi 39762, USA2)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 methodto softening constitutive models yields numerical results that are independent of the size of thefinite element. However, the prediction of real-world large and small scale problems using thedelocalization method remains in its infancy. One of the drawbacks encountered is that the predictedload 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 metallicmaterials. We use some theoretical arguments to explain the failure of the delocalized modelconsidered, following which a simple remedy is proposed to deal with it. Future works involve thenumerical implementation of the new version of the delocalized model in order to assess its ability toreproduce 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 damagedelocalization method is very well known to removethe pathological mesh size effects in the finite element(FE) solution of problems involving these constitutivemodels.1–4 Another closely related technique which con-sists of incorporating gradient terms in the evolutionequation of the parameter(s) governing softening yieldsthe same conclusions, although its numerical implemen-tation into FE codes is not an easy task compared tothat of the delocalization technique. A complete reviewof 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 reacheda situation where they are applicable to small or largescale structure problems. For example, Enakoutsa etal.1 have demonstrated that the use of nonlocal Gur-son model6 does eliminate spurious mesh size effects inFE simulations of ductile rupture of typical pre-crackedTa specimens, but fails to reproduce the experimen-tal load versus displacement curve, i.e., the predictedload-displacement curve remains quasi-stationary forsome time and decreases abruptly. According to theseauthors, this undesirable feature is due to excessivesmoothing of the damage distribution in the ligamentahead of the crack tip of the specimen. They provideda theoretical explanation of this phenomenon based onsuch as crude assumptions as unboundness of the bodyconsidered 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 adoptingthe nonlocal concept for the logarithm of the damageinstead of the damage itself; this has the avantage toeliminate the analogy between the nonlocal evolutionequation and a diffusion equation. Good agreementsbetween experimental and numerical results were thenobtained.

The objective of the present paper is to follow up thestudy of the applicability of the delocalization methodin numerical simulations of material behavior. The mo-tivation is to predict accurately the post-bifurcationregime of metals as the design of metal structures re-quires to understand more and more physics of metalsin this particular regime. The model considered will bethat proposed by Bammann-Chiesa-Johnson (BCJ)7–16

but with a modified, delocalized evolution equation forthe damage following an earlier suggestion of Bijaudier-Cabot and Bazant.17 The introduction of the convolu-tion integral of the evolution equation of the damagein the BCJ model incorporates a diffusive effect in theconstitutive model, which prevents the nonlocal dam-age variable to spuriously localize into vanishing bands.However, the diffusive effect unavoidably leads to anunwanted excessive smoothing of the damage. Just likethat in Ref. 1, we provide a theoretical explanation ofthis shortcoming, following which a simple remedy isproposed to deal with it. The rest of the paper pro-vides the constitutive relations of the BCJ model andits nonlocal extension. then it is devoted to a theoret-ical explanation of the unwanted excessive smoothingof the damage. Finally, it presents a simple solution toovercome the excessive damage smoothing shortcoming.

The BCJ model is a physically-based plastic-ity/damage model which incorporates load path, strainrate, temperature, and history effects through the useof internal state variables (ISVs); it also accounts forboth deviatoric deformation resulting from the presence

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2013, Article ID 815158, 11 pageshttp://dx.doi.org/10.1155/2013/815158

Research ArticleModeling the Dynamic Failure of Railroad Tank Cars Usinga Physically Motivated Internal State Variable Plasticity/DamageNonlocal 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, USA2 School of Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287, USA3Mechanical 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 idealizethe material response in finite element simulations of a large-scale boundary value problem. The problem consists of a movingstriker colliding against a stationary hazmat tank car.Themotivations are (1) to reproduce with high fidelity finite deformation andtemperature histories, damage, and high rate phenomena that may arise during the impact accident and (2) to address the materialpostbifurcation regime pathological mesh size issues. We introduce the mathematical length scale in the model by adopting anonlocal evolution equation for the damage, as suggested by Pijaudier-Cabot and Bazant in the context of concrete. We implementthis evolution equation into existing finite element subroutines of the plasticity/failuremodel.The results of the simulations, carriedout with the aid of Abaqus/Explicit finite element code, show that the material model, accounting for temperature histories andnonlocal damage effects, satisfactorily predicts the damage progression during the tank car impact accident and significantly reducesthe pathological mesh size effects.

1. Introduction

The design of accident-resistant hazmat tank cars or theimprovement of existing ones requires material models thatdescribe the physical mechanisms that occur during the acci-dent. In the case of impact accidents such as collisions, finitedeformation, and temperature histories, damage and highrate phenomena are generated in the vicinity of the impactregion. Unfortunately, the majority of material models usedin the finite element (FE) simulation of hazmat tank carimpact scenarios do not account for such physical features(unavoidably, this will under- or overestimate, for instance,the numerical prediction of the puncture resistance of hazmattank cars’ structural integrity). Furthermore, in the fewmodels that do, a mathematical length scale aimed at solvingthe 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 materialmodels changes, from a hyperbolic to an elliptical system ofdifferential equations in dynamic problems, and the reverse instatics. In both cases, the boundary value problem becomes illposed, Muhlhaus [1], Tvergaard and Needleman [2], de Borst[3], and Ramaswamy and Aravas [4], as the boundary andinitial conditions for one system of differential equations arenot suitable for the other. Consequently, discontinuities in thestrain and damage distribution appear, and all the strain anddamage have the tendency to localize in a zone of vanishingvolumes; this results in zero dissipated energy at failure of thematerial. Also, bifurcations with an infinite number of bifur-cated branches appear, which raises the problem of selectingthe relevant one, especially in numerical computations wherethis drawback manifests itself as a pathological sensitivity ofthe results to the FE discretization.