pueblo conf paper

Proceedings of the 2011 ASME Joint Rail ConferenceJRC2011-56063

March 16-18, 2011, Pueblo, Colorado, USA

JRC2011-56063

A PHYSICALLY MOTIVATED INTERNAL STATE VARIABLE PLASTICITY /DAMAGEMODEL EMBEDDED WITH A LENGTH SCALE FOR HAZMAT TANK CARS’

STRUCTURAL INTEGRITY APPLICATIONS

Fazle R. Ahad ∗

MSU/CAVSStarkville, Mississippi 39759

Email: [email protected]

Koffi EnakoutsaMSU/CAVS

Starkville, Mississippi 39759Email: [email protected]

Kiran N. SolankiMSU/CAVS


Yustianto TjipowidjojoMSU/CAVS


Douglas J. BammannMSU/CAVS


ABSTRACTIn this study, we use a physically-motivated internal state

variable plasticity/damage model containing a mathematicallength scale to represent the material behavior in finite element(FE) simulations of a large scale boundary value problem. Thisproblem consists of a moving striker colliding against a station-ary hazmat tank car. The motivations are (1) to reproduce withhigh fidelity finite deformation and temperature histories,dam-age, and high rate phenomena which arise during the impact and(2) to address the pathological mesh size dependence of the FEsolution in the post-bifurcation regime. We introduce the math-ematical length scale in the model by adopting a nonlocal evo-lution equation for the damage, as suggested by Pijaudier-Cabotand Bazant (1987) in the context of concrete. We implement thisevolution equation into existing implicit and explicit versions ofthe FE subroutines of the plasticity/failure model. The results ofthe FE simulations, carried out with the aid of Abaqus/ExplicitFE code, show that the material model, accounting for tempera-ture histories and nonlocal damage effects, satisfactorily predictsthe damage progression during the tank car impact accident andsignificantly reduces the pathological mesh size effects.

∗Address all correspondence to this author.

1 INTRODUCTION

The design of accident-resistant hazmat tank cars requiresmaterial models which describe the physical mechanisms that oc-cur during the accident. In the case of impact accidents suchascollisions, finite deformation and temperature histories,damageand high rate phenomena are generated in the vicinity of the im-pact region. Unfortunately, the majority of material models usedin the finite element simulation of hazmat tank car impact scenar-ios do not account for such physical features. Furthermore,in thefew models that do, a mathematical length scale aimed at solvingthe post-bifurcation problem is absent. As a consequence, whenone material point fails, the boundary value problem for such ma-terial models changes, from a hyperbolic to an elliptical systemof differential equations in dynamic problems, and the reverse instatics. In both cases, the boundary value problem becomes ill-posed ( Muhlhaus (1986), Tvergaard and Needleman (1997), deBorst (1993), Ramaswamy and Aravas (1998)), as the boundaryand initial conditions for one system of differential equations arenot suitable for the other. As a result, bifurcations with aninfinitenumber of bifurcated branches appear, which raises the problemof selecting the relevant one, especially in numerical computa-tions where this drawback manifests itself as a pathological sen-sitivity of the results to the finite element discretization.

1 Copyright c© 2011 by ASME

Alternatively to the shortcomings encountered in hazmattank car impacts’ numerical simulations, we propose to use anonlocal version of the BCJ1 model ( Bammann and Aifantis(1987) and Bammann et al. (1993)), a physically-motivated in-ternal state variable plasticity and damage model containing amathematical length scale. The use of internal state variableswill enable the prediction of strain rate and temperature historieseffects. These effects can be quite substantial and therefore diffi-cult to incorporate into material models, which assumes that thestress is (1) a unique function of the current strain, strainrate andtemperature and (2) is independent of the loading path. The ef-fects of damage are included in the BCJ model, however, througha scalar internal state variable which tends to degrade the elasticmoduli of the material as well as to concentrate the stress. Themathematical length scale is introduced in the model via thenon-local damage approach of Pijaudier-Cabot and Bazant (1987). Inthe context of concrete damage, these authors suggested a for-mulation in which only the damage variable is nonlocal, whilethe strain, the stress and other variables retain their local def-inition. Their formulation has been applied to creep problemsby Saanouni et al. (1989) and extended to plasticity by, amongothers, Leblond et al. (1994) and Tvergaard and Needleman(1995). Following Pijaudier-Cabot and Bazant (1987)’s sugges-tion, a nonlocal evolution equation for the damage within anoth-erwise unmodified BCJ model is adopted in the current study.The time-derivative of the damage is expressed as the spatialconvolution of a “local damage rate” and bell-shaped weightingfunction. The width of this function introduces a mathematicallength scale.

In this study, a dynamic nonlinear finite element analysis,carried out with Abaqus/Explicit FE code, is used to simulatea moving striker colliding against a stationary hazmat tankcar.The structure part of this finite element model is represented byLagrangian elements obeying the nonlocal version of the BCJmodel, while the fluid part is represented by Eulerian elements.The objective of this study is two-fold: (1) use a high fidelitymaterial model to idealize the physics occurring during theim-pact accident and (2) rectify the computational drawbacks (post-bifurcation mesh dependence issues) for this model on a large-scale boundary value problem. The resulting numerical simu-lations of hazmat tank car impact scenarios, which account fornonlocal damage and temperature history effects predict satisfac-torily the tank car failure process independently of the elementsize. The originality of this work lies in that, prior to thisstudy2,never have the post-bifurcation mesh dependence issues been in-vestigated on large-scale computations problems for steels. Thepaper is organized as follows:

Section 1 describes the physics associated with the failureprocess of a hazmat tank car impact accident.

1BCJ: Bammann-Chiesa-Johnson2To the best of the authors’ knowledge.

Section 2 provides a summary of the equations of the BCJmodel and its nonlocal extension.Section 3 discusses several methods to numerically im-plement the integral-type nonlocal damage into existingAbaqus finite element BCJ model subroutines. The maindifficulty encountered in this implementation relates to thedouble loop over the integration points required by the cal-culation of several convolution integrals, which might other-wise dismantle the architecture of the entire code.Finally Section 4 is devoted to numerical applications of thelocal and nonlocal BCJ model on hazmat tank car shell im-pact accident simulations.

2 PHYSICS OF THE DAMAGE FROM HAZMAT TANKCAR IMPACT

The physical mechanisms responsible for the failure of thehazmat tank car during impact accidents initiates at the time ofthe contact, wherein a local heating arises in the impact region asa result of an intense plastic shear deformation. This heat is gen-erated quickly such that the effects of conduction are negligible;hence, the process can be considered adiabatic. As a result,mate-rial softening occurs in the impact region, while the surroundingmaterial continues to harden. When the load increases, the localregion deforms more than the surrounding material and narrowadiabatic shear bands are formed. While these bands do not dete-riorate the hazmat tank car structural integrity as cracks do, theyare precursors to fracture. Typically, nucleation, growth, and co-alescence and/or cracks may appear in the shear bands, when theimpact load exceeds the local material strength. The rupture pro-cess of the hazmat tank car during the impact accident will thenoccur in two stages: (1) local heating in the impact region lead-ing to the development of adiabatic shear bands and (2) ductilerupture by void nucleation, growth and coalescence in the shearbands. The thermal softening involved in this rupture process isnot only significant for high-velocity impact accidents, but alsofor impact accidents that occur during tank cars collisions. Infact, thermal softening effects can be observed even for a strainrate of the order of 1.0s−1 for a wide number of steels, for exam-ple the 304L stainless steel. It has notably been checked by Bam-mann et al. (1993) on the experimental and theoretical groundsthat thermal softening effects occur in torsion tests of thin-walledtubes of 304L stainless steel at room temperature (see Figure1 forthe details) for strain rate 10.0s−1. The BCJ material model iscapable of representing with high fidelity the strain rate, temper-ature history, load path, and damage effects which arise duringthe impact accident. This model is presented in the next section.


Figure 1. Comparison of experimental and calculation curves for torsion

of thin-walled tubes of 304L stainless steel at 20C for strain rates of 0.01

and 10.0 s−1. Note the thermal softening effects. Calculations are shown

by lines and data by markers. The curves are extracted from Bammann

et al. (1993).

3 THE BCJ MODEL CONTAINING A MATHEMATICALLENGTH SCALE

3.1 THE BCJ MODELThe BCJ model is a physically-based plasticity model cou-

pled with the Cocks and Ashby (1980)’s void growth failuremodel. The BCJ model incorporates load path, strain rate, andtemperature history effects, as well as damage through the useof scalar and tensor internal state variables for which the evolu-tion equations are motivated by dislocation mechanics and castin a hardening minus-recovery format. The BCJ model also ac-counts for deviatoric deformation resulting from the presence ofdislocations and dilatational deformation.

The deformation gradient is multiplicatively decomposedinto terms that account for the elastic, deviatoric inelastic, di-latational inelastic, and thermal inelastic parts of the motion. Forlinearized elasticity, the multiplicative decompositionof the de-formation gradient results in an additive decomposition oftheEulerian strain rate into elastic, deviatoric inelastic, dilatationalinelastic, and thermal inelastic parts. The constitutive equationsof the model are written with respect to the intermediate (stressfree) configuration defined by the inelastic deformation such thatthe current configuration variables are co-rotated with theelasticspin. The pertinent equations of the BCJ model are expressedasthe rate of change of the observable and internal state variablesand consist of the following elements.

• A hypoelasticity law connecting the elastic strain rate to an

objective time-derivative Cauchy stress tensor is given by:

σ = λ(1−φ)tr(De)I+2µ(1−φ)De−φ

1−φσ, (1)

3 whereλ is the Lame constant,µ is the shear modulus, andφ denotes the damage variable. The Cauchy stressσ is con-vected with the elastic spinWe as

σ = σ−Weσ+ σWe (2)

where, in general, for any arbitrary tensor variableX, Xrepresents the convective derivative. Note that the rigidbody rotation is included in the elastic spin; therefore,the constitutive model is expressed with respect to a setof directors whose direction are defined by the plasticdeformation.

• The decomposition of both the skew symmetric and sym-metric parts of the velocity gradient into elastic and inelasticparts for the elastic stretching rateDe and the elastic spinWe in the absence of elastic-plastic couplings yields

De = D−Dp−Dd−Dth

We = W−Wp.(3)

Note that for problems in the shock regime, only the devi-atoric elastic strain part is linearized enabling prediction oflarge elastic volume changes.

• Next, the equation for the plastic spinWp is introduced, inaddition to the flow rules forDp andDd, and the stretchingrate due to the unconstrained thermal expansionDth. Fromthe kinematics, the dilatational inelasticDd flow rule is givenas:

Dd =φ

1−φI. (4)

Assuming isotropic thermal expansion, the unconstrainedthermal stretching rateDth can be expressed by

Dth = AθI, (5)

3Strictly speaking, additional terms containing the temperature θ and itsderivatives should be added to Eqn.1. The influence of the additional terms, how-ever, is significative only for problems involving very hightemperatures, such aswelding problems; therefore we can safely ignore them here.


whereA is a linearized expansion coefficient.For the plastic flow rule, a deviatoric flow rule ( Bammann(1988)) is assumed and defined by

Dp = f(θ)sinh

[

||σ′−α||− [κ−Y(θ)](1−φ)

V(θ)(1−φ)

]

σ′−α||σ′−α||

,

(6)whereθ is the temperature,κ the scalar hardening variable,α the objective rate of change ofα, the tensor hardeningvariable, andσ′ the deviatoric Cauchy stress.

There are several choices for the form ofWp. The as-sumptionWp = 0 allows recovery of the Jaumann stressrate. Alternatively, this function can be described by theGreen-Naghdy rate of Cauchy stress. We used the Jaumannrate for the numerical applications in this paper.

• The evolution equations for the kinematic and isotropichardening internal state variables are given in a hardeningminus recovery format by

α = h(θ)Dp−

[

√

23rd(θ)||Dp||+ rs(θ)

]

||α||α

κ = H(θ)||Dp||−

[

√

23Rd(θ)||Dp||+Rs(θ)

]

κ2,(7)

where h and H are the hardening moduli, rs and Rs are scalarfunctions of θ describing the diffusion-controlled “static”or “thermal” recovery, and rd and Rd are the functions ofθdescribing dynamic recovery.

• To describe the inelastic response, the BCJ model in-troduces nine functions which can be separated into threegroups. The first three are the initial yield, the hardening,and the recovery functions, defined as

V(θ) = C1exp(−C2/θ)Y(θ) = C3exp(−C4/θ)f(θ) = C5exp(−C6/θ).

(8)

The second group is related to the kinematic hardening pro-cess and consists of the following functions:

rd(θ) = C7exp(−C8/θ)h(θ) = C9exp(−C10/θ)rs(θ) = C11exp(−C12/θ).

(9)

The last group is related to the isotropic hardening process

and is composed of

Rd(θ) = C13exp(−C14/θ)H(θ) = C15exp(−C16/θ)Rs(θ) = C17exp(−C18/θ).

(10)

In Eqns. (8, 9, 10),Ci is some parameter of the model whichneed to be determined.

• The evolution equation for damage, credited to Cocks andAshby (1980), is given by

φ =

[

1(1−φ)n − (1−φ)

]

sinh

[

(1−n)

(1+n)

P

σ

]

‖ Dp ‖ . (11)

Note that this void growth model displays a “sinh”-dependence on the triaxiality factorP /σ, as well as anadditional parameter n along with the initial value of thedamageφ0 required to calculate damage growth.

• The last equation to complete the description of the modelis one that computes the temperature change during highstrain rate deformations, such as those encountered in highrate impact loadings. For these problems, a non-conducting(adiabatic) temperature change following the assumptionthat 90% of the plastic work is dissipated as heat is as-sumed. Taylor and Quinney (1934) were the first to measurethe energy dissipation from mechanical work as being be-tween 5− 50% of the total work for various materials andstrain levels. Therefore, the rate of the change of the tem-perature is assumed to follow

θ =0.9ρCv

(σDp), (12)

whereρ and Cv represent the material density and a specificheat, respectively.

The empirical assumption in Eqn.(12) has permitted non-isothermal solution by finite element that is not fully coupledwith the energy balance equation (see Bammann et al. (1993)).Note that the temperature rise will induce a profound effectonthe constitutive behavior of the material. Specifically, the tem-perature increase will lead to thermal softening (adiabatic shearbands), and as a result shear instabilities may arise. The modelis also suitable to predict mechanical softening through a grad-ual increase of the damage. It is well known that practical finiteelement applications of constitutive models involving softening,like the BCJ model, are strongly mesh-dependent. Accordingto Rousselier (1981), this problem can be obviated by putting a


lower limit on the element size. However, this practice is notoptimal theoretically. Another, more elaborated, solution con-sists of including a mathematical length scale in these constitu-tive models. The following section presents a technique to embedthe BCJ model with just such a mathematical length scale.

3.2 EMBEDDING A LENGTH SCALE IN THE BCJMODEL

Following Pijaudier-Cabot and Bazant (1987)’s suggestionin the context of concrete damage, we propose to delocalize thevariable(s) responsible for softening. In the BCJ model, soft-ening may arise from two mechanisms: a gradual increase ofthe damage (under isothermal conditions) or a temperature rise(in adiabatic conditions) followed by an increase of the damage.While temperature and damage parameters seem to govern soft-ening in adiabatic conditions, review of the model’s constitutiveequations provided in the previous section reveals that these twovariables are related. We choose to introduce the length scaleon the damage evolution equation. This choice appears quiteappealing from the physical point of view. Indeed, in the caseof heterogeneous materials, for example, the damage can onlybe defined by considering “elementary” volumes of size greaterthan the voids spacing4 and is therefore a nonlocal quantity.

The evolution equation of this variable is given by a convo-lution integral including a bell-weighting function the width ofwhich introduces a mathematical length scale:

φ(x) =1

B(x)

Z

Ωφloc(y)A(x−y)dΩy. (13)

In this equation,Ω denotes the volume studied, and A the Bellweight function defined as

A(x) = exp(− ‖ x ‖2 /l2), (14)

where l is the mathematical length scale. The factor B(x) and the“local damage rateφloc” are given by

B(x) =Z

ΩA(x−y)dΩy (15)

and Eqn.(11), respectively. The function A is indefinitely dif-ferentiable and does not introduce any Dirac’sδ-distribution atthe point 0. This means that the functionφ is not partially lo-cal but entirely nonlocal. The function A is also isotropic andnormalized. The point here is thatφ must be equal toφloc if thelatter variable is spatially uniform. This would not be the case

4The Cocks and Ashby (1980) void growth model is based on a cylinder con-taining a spherical void.

near the boundary ofΩ in the absence of the normalization fac-tor 1/B. The presence of this term allows for the coincidenceeverywhere.

The new evolution equation for the damage, Eqn.(13), alongwith the equation for the temperature rise, Eqn.(12), shouldpredict satisfactorily the failure process of the tank car im-pact independently of the mesh size. Indeed, the local damagerate Eqn.(13) implicitly depends on the temperature through thestrain rate sensitivity parameter (see Eqn.(6) and Yamaguchi etal. (1992)). When the temperature gradually rises, which isthecase in high-velocity impact loadings, the damage rate in thiszone quickly climbs to a high value; as a result, the damagegrowths rapidly to reach the critical failure damage. The convo-lution integral in Eqn.(13) enhances the rapid damage increase,since it involves the sum of several positive terms each of whichcontains a local damage velocity. The mathematical length scalein the convolution integral eliminates the mesh sensitivity effects.

Nonetheless, the numerical implementation of the new evo-lution equation for the damage rate into an existing finite elementcode is not an easy task because of the double loop over inte-gration points required by the calculation of several convolutionintegrals, which may potentially compromise the entire architec-ture of the existing code. The following section is devoted toexplaining this implementation.

4 NUMERICAL TREATMENT OF THE NONLOCALDAMAGE RATEThe numerical implementation of the original BCJ model

into a finite element code such as ABAQUS has been extensivelyaddressed in Bammann et al. (1993); consequently, it will notbe repeated here. Recall, however, that this implementation isbased on Krieg and Krieg (1977)’s radial return method to solvenumerically the equations presented in Section 3.1 for the de-viatoric stresses, equivalent plastic strain, pressure, temperature,and damage at each new time-step. The algorithm is availableinboth the implicit and explicit versions of ABAQUS.

To implement the nonlocal damage rate in the implicit ver-sion of the code, we have computed the nonlocal damage incre-ment at convergence by means of the subroutine URDFIL, whichstores all the variables necessary for the convolution operation,performs this operation, and stores the nonlocal damage incre-ment for all the integration points involved in the finite elementmodel. The nonlocal damage increment is used to calculate thedamage at timet + ∆t for the next time-step following the for-mula:

φ(t+ ∆t)≈ φ(t)+ φ(t)∆t. (16)

This updated value is an explicit estimation of the damage att + ∆t and is not used to repeat the whole process of solution


between times t and t+ ∆t. Thus, the algorithm is not fully im-plicit, but mixed implicit/explicit. Enakoutsa et al. (2007) usedthe same technique to implement a nonlocal version of Gurson(1977)’s model following a slightly modified Aravas (1987)’s al-gorithm, the so-called “projection into the yield surface”algo-rithm. The explicit nature of Enakoutsa et al. (2007)’s algo-rithm with respect to the damage allowed these authors to provethat theprojection problemassociated with Gurson’s nonlocalmodel has a unique solution. The latter property is a direct con-sequence of the fact that the constitutive equations of Gurson’snonlocal model belong to the class of Generalized Standard Ma-terials of Halphen and Nguyen (1975). We apply this numeri-cal treatment of nonlocal damage rate on a double-notched edgeplane-strain specimen in tension problem to illustrate thevalidityof the method to avoid ill-posed issues in this laboratory-orientedboundary value problem. The tests have been run on two mesheswith element sizes 1.5 and 3mm, respectively; the value of thematerial model length scale was 9mmin the two cases. The spec-imen was made of 304L stainless steel which material parametersare provided in Horstemeyer et al. (2000). While the local dam-age model concentrates the damage within a layer of width oneelement (see Fig.2), the nonlocal damage spreads the damagere-gion to a zone beyond this layer and allows for the elimination ofthe mesh size effects (see Fig.3).

To assess the validity of the method to regularize hazmattank car impact boundary value problems requires implementa-tion of the nonlocal damage rate in the explicit version of the BCJmodel subroutine. One option of doing so consists of branchingoutside the “nblock loop” in the subroutine VUMAT, which al-lows the computation of the nonlocal damage rate at each inte-gration point using the coordinates, the local damage velocities,and the weights of “Nblock” integration points. This methodalso successfully avoids the localization problems arising in thenumerical simulations of double-notched edge specimen tensiletests and therefore is excepted to reduce, if not completelyre-move, the mesh-sensitivity issues arising in the hazmat tank carimpacts numerical simulations.

5 HAZMAT TANK CAR SHELL IMPACT FE SIMULA-TIONS

The goal of the hazmat tank car shell impact FE simula-tions section is to predict numerically the hazmat tank car struc-tural’s failure process independently of the mesh size effects, asencountered in numerical computations based on material mod-els involving softening, just like the BCJ material model. Nocomparison with experimental results is needed for this purpose.We describe below the impact accident physical problem, theas-sociated FE model in Abaqus, and we present different numericalpredictions of the damage created by the impact.

Figure 2. Comparison of the damage distribution of a double-notched

edge specimen numerical tensile tests using the local BCJ model for two

different mesh sizes, coarse (top) and fine (bottom) meshes. Note the

localization of the damage in a layer of width one element for the two

meshes.

5.1 PROBLEM DEFINITIONThe problem considered here is a ram car weighing 286,000

pounds and with a protruding beam to which an impactor wasattached, moving horizontally at 10 m/s into an immobile hazmattank car. Ninety percent of the tank car is filled with water mixedwith clay slurry, which together has the approximate density ofliquid chlorine. The tank is subjected to 100 psi internal pressure.The impactor used in the problem has a 6X6 inch impact face(see Fig.4). The stationary part of the problem consists of atankcar surrounded by a jacket. The dimensions of the tank car andthe jacket are the same as in the work of Tang et al. (2008a) andTang et al. (2008b). The tank is a 0.777-inch-thick cylinder, andclosed at its two ends with elliptical caps of aspect ratio 2.Thetank body material consists of 304L stainless steel; the jacket is a0.119-inch-thick made of the same steel. A 4-inches-thick layerseparates the jacket from the tank car. This layer is introduced toaccount for insulation and thermal protection between the tankcar and the jacket. The entire assembly (Tank/Layer/Jacket) issupported by two rigid legs and this assembly is placed with oneside against a rigid wall and the other side exposed to impactfrom the impactor.


Figure 3. Comparison of the damage distribution of a double-notched

edge specimen numerical tensile tests using the nonlocal BCJ model,

coarse (top) and fine (bottom) meshes; the value of the length scale used

in the simulations is 9 mm. The nonlocal damage method has spread

the damage to neighboring regions; also, note the similarity between the

damage pattern in the two cases.

Figure 4. Geometry of the 6X6 inch impact face impactor used in the

hazmat tank car impact numerical simulations.

5.2 FE MODEL OF THE PROBLEM IN ABAQUSThe FE model of the problem consists of an Eulerian mesh

representing the fluid in the tank car and a Lagrangian mesh ide-alizing the tank, the jacket, and the impactor. The impactorcon-sists of R3D4 rigid elements, while the tank and the jacket aremeshed using solid C3D8R elements. For the sake of simplicity,the space between the tank and the jacket is assumed to be empty.

The two legs are rigid and are idealized with squared analyticalsurfaces on which reference points located below the jacketareassigned. Each reference point is kinematically coupled with adefinite set of nodes on the jacket and the tank car. The contactalgorithm available in the 6.10 version of Abaqus/Explicitfiniteelement code is used to account for all possible contacts betweendifferent parts in the model.

The Eulerian mesh is based on the volume-of-fluid (VOF)method. The VOF method (widely used in Computational FluidDynamics) tracks and locates the fluid free surface; it belongs tothe class of Eulerian methods that are characterized by either astationary or moving mesh. The VOF method suitably capturesthe change of the fluid interface topology. In this method, thematerial in each element is tracked as it flows through the meshusing the Eulerian volume fraction (EVF), a unique parameterfor each element and each material.

The material parameters are determined from tension, com-pression and torsion tests under different constant strainrateand temperatures. We used the material parameters providedin Horstemeyer et al. (2000).

5.3 SIMULATIONS RESULTS: VERIFICATION OF THEREDUCTION OF MESH SIZE EFFECTS

This section presents evidence of the integral-nonlocal dam-age method’s to reduce mesh-dependence issues which arise inthe FE solution of hazmat tank car impact accident problems.To that end, the calculations were performed on three differentmeshes, a coarse, medium and fine mesh (see Fig.5), for the localand nonlocal versions of the BCJ model in adiabatic conditions.Under isothermal conditions, the BCJ model difficultly predictsfailure (see Bammann et al. (1993)); therefore, isothermalcon-ditions are not considered here. The element size in the impactregion is 75mm, 50mmand 35mmfor the coarse, medium andfine meshes, respectively. The nonlocal damage effects are intro-duced at each integration point by taking the convolution integralover “nblock” integration points. This introduces in the modelsome weak nonlocality which consists of interactions betweeneach integration point and “nblock” other integration points notnecessary located in the vicinity of the latter.

Figure 6 illustrates the repartition of damage on the surfaceopposite to the impacted surface of the tank for all three meshesat the same time, in the case of the local BCJ model. The re-sults of the simulations using the local BCJ model are mesh-dependent: the pattern of the damage is determined by the sizeof the elements. Therefore, decreasing of FE size will modifythe global response of the structure (which depends explicitly onthe number of elements). Furthermore, the energy generateddur-ing the impact tends to zero when the size of the FE approacheszero. This leads to the meaningless conclusion that the tankcarfails during the accident with zero energy dissipated. In fact, theFE solutions depend not only on the size of the elements, but


Figure 5. Meshes of the tank car. Top: coarse mesh, middle: medium

mesh and bottom: fine mesh. The meshes are made of solid C3D8R

elements. There are four C3D8R elements in the thickness of the tank to

develop a bending resistance. The size of the elements in the impact zone

is 75 mm, 50 mmand 35 mmfor the coarse, medium and fine meshes,

respectively.

on their nature, orientation, degree of interpolation function-inshort, on the finite elements approximation space, as presentedin Darve et al. (1995).

Figure.7 is the analogous of Fig.6. Here, the pattern of thedamage for the three meshes is relatively similar; this observationholds for a latter time. The similarity would be more significantif a very refined mesh were used because, as in the case of lo-cal BCJ model, the element size determines the damage pattern.Thus, the nonlocal damage rate, anda fortiori the damage itself,has a significant influence on the response of the hazmat tank carto the impact loading.

Also remarkable from Figs.(7) are the bands, which do ap-

Figure 6. An illustration of the mesh size effects in the FE simulations of

hazmat tank car using the local BCJ model. The figures show the damage

pattern on the inner surface of the tank located behind the impact region

for a coarse (top), medium (middle) and fine (bottom) mesh at the same

time. Note the reduction of the damage pattern size with decreasing mesh

element size.

pear in the local simulations, seem to have almost completelydisappeared. In fact, the nonlocal damage has delayed theirap-parition to latter times where these bands are extended oversev-eral elements. This postponement arises because the damageinthe bands is much larger at latter stages of the impact loading,due to the progressive development of considerable stress in thiszone.


Figure 7. An illustration of the introduction of nonlocal damage effects in

the BCJ model to reduce the mesh size effects in the FE simulations of

hazmat tank car. The contour plots of the damage on the inner face of the

tank are shown for three different meshes at the same time as in Fig.6:

coarse (top), medium (middle) and fine (bottom). Note the quasi-similarity

of the damage pattern size for the three meshes.

6 SUMMARY AND PERSPECTIVESWith the aid of Abaqus/Explicit FE code, we have presented

a dynamic nonlinear FE simulations of a hazmat tank car shellimpact. In these simulations, the tank car body material waside-alized with a physically-motivated internal state variable modelcontaining a mathematical length scale, the nonlocal BCJ model.This model consists of all the constitutive equations of theorig-inal BCJ model, except the evolution equation for the damage,

which is modified into a nonlocal one. Numerical methods toimplement this equation in existing BCJ model subroutines werealso presented.

The results of the simulations show the ability of the damagedelocalization technique to significantly reduce the pathologi-cal mesh-dependency issues while predicting satisfactoryhazmattank car impact failure, provided that nonlocal damage effectsare coupled with temperature history effects (adiabatic effects).However the performance of our approach will fully demonstrateits power when the damage progression it predicts reproducesthe real world hazmat tank car impact damage progression. Thiscomparison with experimental results will be investigatedin thenear future.

AKNOWLEDGMENTSThe study was supported by the U.S. Department of Trans-

portation, Office of the Secretary, Grant No. DTOS59-08-G-00103. Dr. Esteban Marin and Mrs. Cl ´emence Bouvard aregratefully thanked for their insightful discussions on thetopic.

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