research article modeling the dynamic failure of...

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2013 Article ID 815158 11 pageshttpdxdoiorg1011552013815158

Research ArticleModeling the Dynamic Failure of Railroad Tank Cars Usinga Physically Motivated Internal State Variable PlasticityDamageNonlocal Model

Fazle R Ahad1 Koffi Enakoutsa1 Kiran N Solanki2

Yustianto Tjiptowidjojo1 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 enakoutsayahoofr

Received 28 September 2012 Revised 11 December 2012 Accepted 14 January 2013

Academic Editor Chung-Souk Han

Copyright copy 2013 Fazle R Ahad et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We used a physically motivated internal state variable plasticitydamage 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 carThemotivations 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 plasticityfailuremodelThe results of the simulations carriedout with the aid of AbaqusExplicit 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 instancethe numerical prediction of the puncture resistance of hazmattank carsrsquo structural integrity) Furthermore in the fewmodels that do a mathematical length scale aimed at solvingthe postbifurcation problem is absent As a consequencewhen 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

2 Modelling and Simulation in Engineering

0

200

400

600

800

1000

1200

0

200

400

600

800

1000

1400

1200

Stre

ss (M

Pa)

Stre

ss (M

Pa)

0 01 02 03 04 05 06 07 08Strain

0 01 02 03 04 05 06 07Strain

Reload

20∘C

20∘C

800∘C

20∘C calculation800∘C calculation20∘C reload calculation

20∘C experiment800∘C experiment20∘C reload experiment

20∘C

900∘C

Strain rate 10minus3S





10minus1 at 900∘C calculation10minus1 at 900∘C experiment


10minus3 at 20∘C experiment10minus5 at 20∘C calculation10minus5 at 900∘C calculation10minus5 at 900∘C experiment10minus5 at 20∘C experiment

10minus3 at 20∘C calculation

Figure 1 Load versus displacement curves for different temperatures and strain rates Full lines BCJ model predictions point linesexperiments Note that the model satisfactorily captures temperature and strain rate histories effects The curves were extracted fromBammann et al [6]

Alternatively to the shortcomings encountered in hazmattank car impactsrsquo numerical simulations we propose to usea nonlocal version of the Bammann-Chiesa-Johnson (BCJ)model Bammann and Aifantis [5] and Bammann et al [6]a physically motivated internal state variable plasticity anddamage model containing a mathematical length scale Theuse of internal state variables will enable the prediction ofstrain rate and temperature history effects see Figure 1Theseeffects can be quite substantial and therefore difficult to incor-porate into material models which assumes that the stress is(1) a unique function of the current strain strain rate andtemperature and (2) independent of the loading path Theeffects of damage are included in the BCJ model howeverthrough a scalar internal state variablewhich tends to degradethe elastic moduli of the material as well as to concentratethe stress The mathematical length scale is introduced inthe model via the nonlocal damage approach of Pijaudier-Cabot and Bazant [7] In the context of concrete damagethese authors have suggested a formulation in which only thedamage variable is nonlocal while the strain the stress andother variables retain their local definitionTheir formulationhas been applied to creep problems by Saanouni et al [8]and extended to plasticity by among others Leblond et al[9] Tvergaard and Needleman [10] and Enakoutsa et al [11]Following suggestion Pijaudier-Cabot and Bazant [7] a non-local evolution equation for the damage within an otherwiseunmodified BCJ model is adopted in the current studyThe time derivative of the damage is expressed as thespatial convolution of a ldquolocal damage raterdquo and bell-shaped

weighting function The width of this function introduces amathematical length scale

In this study a dynamic nonlinear FE analysis carried outwith Abaqus FE code is used to simulate a moving strikercolliding against a stationary hazmat tank car The struc-ture part of this FE model is represented by Lagrangianelements obeying the nonlocal version of the BCJ modelwhile the fluid part is represented by Eulerian elementsThe objective of this study is twofold (1) use a high fidelitymaterial model to idealize the physics occurring during theimpact accident and (2) rectify the computational drawbacks(postbifurcation mesh dependence issues) for this modelon a large-scale boundary value problem The resultingnumerical simulations of hazmat tank car impact scenarioswhich account for nonlocal damage and temperature historyeffects predict satisfactorily the tank car failure processindependently of the element size Although several otherstudies have addressed successfully the mesh dependenceissues in finite element computations of problems involvingdynamic failure of materials as in for instance Voyiadjis etal [12] and Abu Al-Rub and Kim [13] very few of themdeal with real-world large-scale boundary value problemsThe present work investigates the application of the nonlocalintegral approach to predict the failure of a large-scale vehicleunder impact loading conditions The paper is organized asfollows

(i) The first section provides a summary of the equationsof the BCJ model and its nonlocal extension

Modelling and Simulation in Engineering 3

(ii) The second section discusses several methods tonumerically implement the integral-type nonlocaldamage into existing Abaqus FE BCJ model sub-routines The main difficulty encountered in thisimplementation relates to the double loop over theintegration points required by the calculation of sev-eral convolution integrals which might otherwisedismantle the architecture of the entire code

(iii) The last section is devoted to numerical applicationsof the local and nonlocal versions of the BCJmodel onhazmat tank car shell impact accident simulations

2 The BCJ Model Local andNonlocal Versions

21 The BCJModel TheBCJmodel is a physicallybased plas-ticity model coupled with the Cocks and Ashby [14]rsquos voidgrowth failure modelThe BCJmodel incorporates load pathstrain rate and temperature history effects as well as damagethrough the use of scalar and tensor internal state variables forwhich the evolution equations are motivated by dislocationmechanics and cast in a hardening minus recovery formatThe BCJ model also accounts for deviatoric deformationresulting from the presence of dislocations and dilatationaldeformation

The deformation gradient is multiplicatively decomposedinto terms that account for the elastic deviatoric inelasticdilatational inelastic and thermal inelastic parts of themotion For linearized elasticity the multiplicative decom-position of the deformation gradient results in an additivedecomposition of the Eulerian strain rate into elastic devi-atoric inelastic dilatational inelastic and thermal inelasticparts The constitutive equations of the model are writtenwith respect to the intermediate (stress-free) configurationdefined by the inelastic deformation such that the currentconfiguration variables are corotatedwith the elastic spinThepertinent equations of the BCJmodel are expressed as the rateof change of the observable and internal state variables andconsist of the following elements

(i) A hypoelasticity law connecting the elastic strain rateto an objective time derivative Cauchy stress tensor is givenby

= 120582 (1 minus 120601) tr (D119890) I + 2120583 (1 minus 120601)D119890 minus

120601

1 minus 120601

120590 (1)

(strictly speaking additional terms containing the tempera-ture 120579 and its derivatives should be added to (1) the influenceof these terms however is significant only for problemsinvolving very high temperatures such as welding problemstherefore we can safely ignore themhere)where120582 is the Lameconstant 120583 is the shear modulus and 120601 denotes the damagevariableTheCauchy stress120590 is convectedwith the elastic spinW119890 as

= minusW119890120590 + 120590W119890 (2)

where in general for any arbitrary tensor variable X Xrepresents the convective derivative Note that the rigid body

rotation is included in the elastic spin therefore the constitu-tivemodel is expressedwith respect to a set of directorswhosedirections are defined by the plastic deformation

(ii) The decomposition of both the skew symmetric andsymmetric parts of the velocity gradient into elastic andinelastic parts for the elastic stretching rateD119890 and the elasticspinW119890 in the absence of elastic-plastic couplings yields

D119890 = D minusD119901 minusD119889 minusDth

W119890 = W minusW119901(3)

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

(iii) Next the equation for the plastic spin W119901 is intro-duced in addition to the flow rules for D119901 and D119889 and thestretching rate due to the unconstrained thermal expansionDth From the kinematics the dilatational inelastic D119889 flowrule is given as

D119889 =

120601

1 minus 120601

I (4)

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

Dth= 119860

120579I (5)

where 119860 is a linearized expansion coefficientFor the plastic flow rule a deviatoric flow rule [15] is

assumed and defined by

D119901 = 119891 (120579) sin ℎ[

1003817

1003817

1003817

1003817

1003817

1205901015840

minus 120572

1003817

1003817

1003817

1003817

1003817

minus [120581 minus 119884 (120579)] (1 minus 120601)

119881 (120579) (1 minus 120601)

]

1205901015840

minus 120572

1003817

1003817

1003817

1003817

1205901015840minus 120572

1003817

1003817

1003817

1003817

(6)

where 120579 is the temperature 120581 the scalar hardening variable the objective rate of change of 120572 the tensor hardeningvariable and 1205901015840 the deviatoric Cauchy stress

There are several choices for the form of W119901 Theassumption that W119901 = 0 allows recovery of the Jaumannstress rate Alternatively this function can be described by theGreen-Naghdy rate of Cauchy stress We used the Jaumannrate for the numerical applications in this paper

(iv) The evolution equations for the kinematic andisotropic hardening internal state variables are given in ahardening minus recovery format by

= ℎ (120579)D119901 minus [radic

2

3

119903

119889(120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

+ 119903

119904(120579)] 120572120572

= 119867 (120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

minus [radic

2

3

119877

119889(120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

+ 119877

119904(120579)] 120581

2

(7)

where ℎ and119867 are the hardening moduli 119903119904and 119877

119904are scalar

functions of 120579 describing the diffusion-controlled ldquostaticrdquo orldquothermalrdquo recovery and 119903

119889and 119877

119889are the functions of 120579

describing dynamic recovery


(v) To describe the inelastic response the BCJ modelintroduces nine functions which can be separated into threegroupsThe first three are the initial yield the hardening andthe recovery functions defined as

119881 (120579) = 119862

1exp(minus1198622

120579

)

119884 (120579) = 119862

3exp(minus1198624

120579

)

119891 (120579) = 119862

5exp(minus

119862

6

120579

)

(8)

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

119903

119889(120579) = 119862

7exp(minus

119862

8

120579

)

ℎ (120579) = 119862

9exp(minus

119862

10

120579

)

119903

119904(120579) = 119862

11exp(minus11986212

120579

)

(9)

The last group is related to the isotropic hardening processand is composed of

119877

119889(120579) = 119862

13exp(minus11986214

120579

)

119867 (120579) = 119862

15exp(minus

119862

16

120579

)

119877

119904(120579) = 119862

17exp(minus

119862

18

120579

)

(10)

In (8) (9) and (10)119862119894is some parameter of the model which

needs to be determined(vi)The evolution equation for damage credited to Cocks

and Ashby [14] is given by

120601 = [

1

(1 minus 120601)

119899minus (1 minus 120601)] sin ℎ [(1 minus 119899)

(1 + 119899)

P

120590

]

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

(11)

Note that this void growth model displays a ldquosin ℎrdquo depen-dency on the triaxiality factor P120590 as well as an additionalparameter 119899 along with the initial value of the damage 120601

0

required to calculate damage growth(vii) The last equation to complete the description of the

model is one that computes the temperature change duringhigh strain rate deformations such as those encountered inhigh rate impact loadings For these problems a nonconduct-ing (adiabatic) temperature change following the assumptionthat 90 of the plastic work is dissipated as heat is assumedTaylor and Quinney [16] were the first to measure the energydissipation from mechanical work as being between 5and 50 of the total work for various materials and strainlevels Therefore the rate of the change of the temperature isassumed to follow

120579 =

09

120588119862

119907

(120590D119901) (12)

where 120588 and 119862

119907represent the material density and a specific

heat respectivelyThe empirical assumption in (12) has permitted noniso-

thermal solution by FE that is not fully coupled with theenergy balance equation (see [6]) Note that the temperaturerise will induce a profound effect on the constitutive behaviorof thematerial Specifically the temperature increase will leadto thermal softening (adiabatic shear bands) and as a resultshear instabilities may arise The model is also suitable topredict mechanical softening through a gradual increase ofthe damage It is well known that practical F applications ofconstitutive models involving softening like the BCJ modelare strongly mesh-dependent According to Rousselier [17]this problem can be obviated by putting a lower limit onthe element size However this practice is not optimaltheoretically Another more elaborated solution consists ofincluding a mathematical length scale in these constitutivemodels The following section presents a technique to embedthe BCJ model with just such a mathematical length scale

22 Embedding a Length Scale in the BCJ Model Followingsuggestion Pijaudier-Cabot and Bazant [7] in the context ofconcrete damage we propose to delocalize the variable(s)responsible for softening In the BCJ model softening mayarise from twomechanisms a gradual increase of the damage(under isothermal conditions) or a temperature rise (inadiabatic conditions) followed by an increase of the damageWhile temperature and damage parameters seem to governsoftening in adiabatic conditions review of the modelrsquos con-stitutive equations provided in the previous section revealsthat these two variables are related We choose to introducethe length scale on the damage evolution equation Thischoice appears quite appealing from the physical point ofview Indeed in the case of heterogeneous materials forexample the damage can only be defined by consideringldquoelementaryrdquo volumes of size greater than the voids spacing(the Cocks and Ashby [14] void growth model is based ona cylinder containing a spherical void) and is therefore anonlocal quantity

The evolution equation of this variable is given by aconvolution integral including a bell-weighting function thewidth of which introduces a mathematical length scale

120601 (x) = 1

119861 (x)int

Ω

120601

loc(y) 119860 (x minus y) 119889Ωy (13)

In this equationΩ denotes the volume studied and119860 the bellweight function defined as

119860 (x) = exp(minusx2

119897

2) (14)

where 119897 is the mathematical length scale (note that the phys-ical length scale available in the model due to rate sensitivity(see [18] for more details) is so small that it is impossible tosolve a boundary value problem since this will require finiteelements of size smaller than that physical length scale which


is already very small) The factor 119861(x) and the ldquolocal damagerate

120601

locrdquo are given by

119861 (x) = int

Ω

119860 (x minus y) 119889Ωy (15)

and (11) respectively The function 119860 is indefinitely differen-tiable and does not introduce anyDiracrsquos 120575-distribution at thepoint 0 This means that the function

120601 is not partially localbut entirely nonlocal The function 119860 is also isotropic andnormalized The point here is that

120601 must be equal to

120601

loc

if the latter variable is spatially uniform This would notbe the case near the boundary of Ω in the absence of thenormalization factor 1119861 The presence of this term allowsfor the coincidence everywhere

The new evolution equation for the damage (13) alongwith the equation for the temperature rise (12) shouldpredict satisfactorily the dynamic failure process of the tankcar impact independently of the mesh size Indeed the localdamage rate (13) implicitly depends on the temperature risethrough the strain rate sensitivity parameter (see (6) and[19])When the temperature gradually rises which is the casein high-velocity impact loadings the damage rate in this zonequickly climbs to a high value as a result the damage growsrapidly to reach the critical failure damage The convolutionintegral in (13) enhances the rapid damage increase sinceit involves the sum of several positive terms each of whichcontains a local damage velocity

Nonetheless the numerical implementation of the newevolution equation for the damage rate into an existing FEcode is not an easy task because of the double loop overintegration points required by the calculation of several con-volution integrals which may potentially compromise theentire architecture of the existing codeThe following sectionis devoted to explaining this implementation

3 Numerical Treatment ofNonlocal Damage Rate

Thenumerical implementation of the original BCJmodel intoan FE code such as Abaqus has been extensively addressedin Bammann et al [6] consequently it will not be repeatedhere Recall however that this implementation is based onradial return method of R D Krieg and D B Krieg [20]to solve numerically the equations presented in Section 21for the deviatoric stresses equivalent plastic strain pressuretemperature and damage at each new time step The algo-rithm is available in both the implicit and explicit versions ofABAQUS

To implement the nonlocal extension of the BCJ modelin the implicit version of the code we have computed thenonlocal damage increment at elastic-plastic convergence bymeans of the built-in subroutineURDFIL which stores all thevariables necessary for the convolution operation performsthis operationThismethod involves all the integration pointsof the finite element model considered However since theGaussian influence weighting function vanishes for integra-tions points that are outside the region limited by the char-acteristic length scale only the integration points inside that

zonematter in the calculation of the convolution integralThenonlocal damage increment are stored for all the integrationpoints involved in the FE model and are used to calculate thedamage at time 119905 + Δ119905 for the next time-step following theformula

120601 (119905 + Δ119905) asymp 120601 (119905) +

120601 (119905) Δ119905 (16)

This updated value is an explicit estimation of the damageat 119905 + Δ119905 and is not used to repeat the whole process ofsolution between times 119905 and 119905 + Δ119905 Thus the algorithm isnot fully implicit but mixed implicitexplicit implicit withrespect to all parameters (displacements stresses hardeningparameters etc) except the damage parameter this preservesthe compatibility with the implicit schemes used for usualnonsoftening models Enakoutsa et al [11] used the samenumerical technique to implement a nonlocal version ofGurson [21] model following a slightly modified Aravas [22]algorithm the so-called ldquoprojection into the yield surfacerdquoalgorithm The explicit nature of Enakoutsa et alrsquos [11]algorithm with respect to the damage allowed these authorsto prove that the projection problem associated with Gursonrsquosnonlocal model has a unique solution The latter property isa direct consequence of the fact that the constitutive equa-tions of Gursonrsquos nonlocal model belong to the class of gene-ralized standardmaterials as defined byHalphen andNguyen[23] We apply this numerical treatment of nonlocal damagerate on a double-notched edge plane-strain specimen underuniaxial tension problem to illustrate the validity of themethod to avoid ill-posed issues in this laboratory-orientedboundary value problem The tests have been run using twomeshes with element sizes 15 and 3mm in the notched regionof the specimen respectively The value of the characteristiclength scale was 9mm in the two cases considered it canbe considered as a parameter that determined the regionin which the interactions between neighboring integrationpoints are nonzero In this work the length scale introducedin themodel is not a physical length scale but amathematicallength scale aimed at eradicating the pathological mesh sizedependence issues predicted by the local model Hence theonly requirement it must satisfy is to be greater than the finiteelement size considered so that interactions of neighboringpoints can be accounted for The specimen was made of304L stainless steel and the material parameters of whichare provided in Horstemeyer et al [24] The major remarkthat must be made about the simulation results for the plateproblem is that while the local BCJ model concentrates thedamage within a layer of width of one element size (seeFigure 2) the presence of the delocalization term spreads thedamage region to a zone beyond this layer and allows for theelimination of the mesh size effects see Figure 3

To assess the validity of the method to regularize haz-mat tank car impact boundary value problems requires theimplementation of the nonlocal damage rate in the explicitversion of the BCJ model subroutine One option of doingso consists of branching outside the ldquonblock looprdquo in thesubroutine VUMAT which allows the computation of thenonlocal damage rate at each integration point using thecoordinates the local damage velocities and the weights of


(a) (b)

Figure 2 Comparison of the damage distribution of a double-notched edge specimen numerical tensile tests using the local BCJ model fortwo different mesh sizes coarse (a) and fine (b) meshes Note the localization of the damage in a layer of width one element size for the twomeshes

(a) (b)

Figure 3 Comparison of the damage distribution of double-notched edge specimen numerical tensile tests using the nonlocal BCJ modelcoarse (a) and fine (b) meshes the value of the length scale used in the simulations is 9mm The nonlocal damage method has spread thedamage to neighboring regions also note the similarity between the damage pattern in the two cases

ldquonblockrdquo integration points This method also successfullyavoids the localization problems arising in the numericalsimulations of double-notched edge specimen tensile testsand therefore is excepted to reduce if not completely removethe mesh sensitivity issues arising in the hazmat tank carimpacts numerical simulations

4 Hazmat Tank Car Shell ImpactFE Simulations

The goal of the hazmat tank car shell impact FE simula-tions section is to predict numerically the hazmat tank car

structural failure process independently of the mesh sizeeffects as encountered in numerical computations basedon material models involving softening just like the BCJmaterial model No comparison with experimental resultsis needed for this purpose We describe later the impactaccident physical problem and the associated FE model inAbaqus andwe present different numerical predictions of thedamage created by the impact

41 Problem Definition The problem considered here isschematized in Figure 5 it consists of a ram car weighing


Figure 4 Geometry of the 6 times 6 inch impact face impactor used inthe hazmat tank car impact numerical simulations

286000 pounds and a protruding beam towhich an impactorwas attached moving horizontally at 10ms into an immobilehazmat tank car Ninety percent of the tank car is filledwith water mixed with clay slurry which together has theapproximate density of liquid chlorine The tank is subjectedto 100 psi internal pressureThe impactor used in the problemhas a 6 times 6 inch impact face see Figure 4

42 FE Model of the Problem The FE model of the problemconsists of an Eulerianmesh representing the fluid in the tankcar and a Lagrangianmesh idealizing the tank the jacket andthe impactor The impactor consists of R3D4 rigid elementswhile the tank and the jacket are meshed using solid C3D8Relements For the sake of simplicity the space between thetank and the jacket is assumed to be empty The two legs arerigid and are idealized with squared analytical surfaces onwhich reference points located below the jacket are assignedEach reference point is kinematically coupled with a definiteset of nodes on the jacket and the tank car The contactalgorithm available in AbaqusExplicit FE code is used toaccount for all possible contacts between the different partsin the model

The Eulerianmesh is based on the volume-of-fluid (VOF)method The VOF method widely used in computationalfluid dynamics problems tracks and locates the fluid-freesurface it belongs to the class of Eulerian methods that arecharacterized by either a stationary or moving mesh TheVOF method suitably captures the change of the fluid inter-face topology In this method the material in each elementis tracked as it flows through the mesh using the elementvolume fraction (EVF) a unique parameter for each ele-ment and each material The material parameters used inthe simulations are determined from tension compressionand torsion tests under different constant strain rate andtemperatures We used the material parameters provided inHorstemeyer et al [24] Although these constants are based

on isothermal constant strain rate tests the model can beapplied to arbitrary temperature and strain rate histories

The stationary part of the problem consists of a tank carsurrounded by a jacket The dimensions of the tank car andthe jacket are the same as in the works of Tang et al [25] andTang et al [26] The tank is a 0777-inch-thick cylinder andis closed at its two ends with elliptical caps of aspect ratio2 The tank body material consists of 304L stainless steelthe jacket is a 0119-inch-thick made one of the same steelA 4-inch-thick layer separates the jacket from the tank carThis layer is introduced to account for insulation and thermalprotection between the tank car and the jacket The entireassembly (tanklayerjacket) is supported by two rigid legsand this assembly is placed with one side against a rigid walland the other side exposed to impact from the impactor

43 Results and Discussions This section presents evidenceof the integral nonlocal damage method to reduce meshdependence issues which arise in the FE solution of hazmattank car impact accident problems To that end the calcu-lations were performed on three different meshes coarsemedium and fine meshes (see Figure 6) for the local andnonlocal versions of the BCJ model in adiabatic condi-tions Under isothermal conditions the BCJ model difficultlypredicts failure Bammann et al [6] therefore isothermalconditions are not considered here The element sizes in theimpact region are 75mm 50mm and 35mm for the coarsemedium and fine meshes respectivelyThe nonlocal damageeffects are introduced at each integration point by takingthe convolution integral over ldquonblockrdquo integration pointsThis introduces in the model some weak nonlocality definedby the interactions between each integration point andldquonblockrdquo other integration points not necessary located in theimmediate vicinity of the latter The length scale associatedwith such nonlocality can be considered of measure ldquonblockintegration pointsrdquo

Due to the calculation of the integral nonlocal evolutionequation of the damage the CPU time required by the nonlo-cal calculations is higher than the case with the local model Itusually takes up to 12 hours to complete the simulation of theimpact problemwith nonlocal damage using the coarse mesh(59049 integration points for the structure part of themodel)The medium (80373 integration points for the structure partof the model) and fine meshes (158037 integration pointsfor the structure part) simulation times are longer 15 hoursfor the medium mesh and almost 20 hours for the refinemesh This long time period is also due to the presence ofthe fluid which is modeled by the element volume fraction(EVF) technique available in ABAQUS

Figure 7 illustrates the repartition of damage on the sur-face opposite to the impacted surface of the tank for all threemeshes at the same time in the case of the local BCJ modelThe results of the simulations using the local BCJ model aremesh dependent the pattern of the damage is determinedby the size of the elements Therefore decreasing of FEsize will modify the global response of the structure (whichdepends explicitly on the number of elements) Furthermorethe energy generated during the impact tends to zero when


Striker(119881119910 = 119881119911 = 0)

Fixed wall

Tank car

Rigid wheel-1119880119910 = 119880119911 = 0

Rigid wheel-2

119880119903119909 = 119880119903119910 = 119880119903119911 = 0

119880119910 = 119880119911 = 0

119880119903119909 = 119880119903119910 = 119880119903119911 = 0

Figure 5 Schematization of the hazmat tank car impact problem

(a) (b)

(c)

Figure 6 Different mesh discretizations of the tank car (a) coarse mesh (b) mediummesh and (c) fine meshThe meshes are made of solidC3D8R elements There are four C3D8R elements in the thickness of the tank to develop a bending resistance The size of the elements in theimpact zone is 75mm 50mm and 35mm for the coarse medium and fine meshes respectively

the size of the FE approaches zero This leads to the mean-ingless conclusion that the tank car fails during the accidentwith zero energy dissipated In fact the FE solutions dependnot only on the size of the elements but also on their natureorientation degree of interpolation function as well as on theFE approximation space as presented inDarve et al [27]Themesh size effects encountered in the tank car FE simulationswith the local BCJ model are more emphasized in Figure 8which illustrates the damage pattern on the outer surface ofthe tank car In this specific case the damage pattern recoversthe whole deformation region of the tank car this zoneshrinks with reducing mesh size which is not realistic fromthe physical point of view

Figure 9 is the analogous of Figure 7 Here the pattern ofthe damage for the threemeshes is relatively similar the sameobservation holds for a latter time The similarity would bemore significant if a very refined mesh was used because asin the case of local BCJ model the element size determines

the damage pattern Thus the nonlocal damage rate anda fortiori the damage itself has a significant influence onthe response of the hazmat tank car to the impact load-ing

Also remarkable from Figure 9 are the bands which doappear in the local simulations they seem to have almostcompletely disappeared In fact the nonlocal damage hasdelayed their apparition to latter times where these bandsare extended over several elementsThis postponement arisesbecause the damage in the bands ismuch larger at latter stagesof the impact loading due to the progressive developmentof considerable stress in this zone Figure 10 analogous ofFigure 8 presents the damage pattern obtained with thenonlocal BCJ model on the outer surface of the tank car bodyin the impact region for two different mesh sizes Like inFigure 8 the damage patterns predicted with the nonlocalBCJ model for the two mesh sizes considered are quasisimilar


(a) (b)

(c)

Figure 7 An illustration of the mesh size effects in the FE simulations of hazmat tank car using the local BCJ model The figures show thedamage pattern on the inner surface of the tank located behind the impact region for a coarse (a) medium (b) and fine (c) mesh at the sametime Note the reduction of the damage pattern size with decreasing mesh element size

(a) (b)

Figure 8 Illustration of themesh size effects in the FE simulations of hazmat tank car using the local BCJmodelThe figures show the damagepattern on the outer surface of the tank located in the impact region for a coarse (a) and fine (b) mesh at the same time Note the differencebetween the damage pattern for the meshes

5 Summary

With the aid of AbaqusExplicit FE code we have presenteda dynamic nonlinear FE simulation of a hazmat tank car shellimpact In these simulations the tank car body material wasidealized with a physically motivated internal state variablemodel containing a mathematical length scale the nonlo-cal BCJ model This model consists of all the constitutiveequations of the original BCJ model except for the evolutionequation for the damage which is modified into a nonlocal

one Numerical methods to implement this equation inexisting BCJ model subroutines were also presented

The results of the simulations show the ability of thedamage delocalization technique to significantly reduce thepathological mesh dependency issues while predicting sat-isfactory hazmat tank car impact failure provided thatnonlocal damage effects are coupledwith temperature historyeffects (adiabatic effects) However the performance of ourapproach will fully demonstrate its power when the damageprogression it predicts reproduces the real-world hazmat


(a) (b) (c)

Figure 9 An illustration of the introduction of nonlocal damage effects in the BCJmodel to reduce themesh size effects in the FE simulationsof 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 inFigure 7 coarse (a) medium (b) and fine (c) Note the quasisimilarity of the damage pattern size for the three meshes

(a) (b)

Figure 10 Another illustration of the mitigation of mesh size effects in the FE simulations of hazmat tank car using nonlocal BCJ modelThefigures show the damage pattern on the outer surface of the tank body located in the impact region for coarse (a) and fine (b) meshes at thesame time Note the similarity between the damage patterns

tank car impact damage progression This comparison withexperimental results will be investigated in the near future

Acknowledgments

The study was supported by the US Department of Trans-portation Office of the Secretary Grant no DTOS59-08-G-00103 Dr Esteban Marin and Mrs Clemence Bouvardare gratefully thanked for their insightful discussions on thetopic

References

[1] H B Muhlhaus ldquoShear band analysis in granular materials bycosserat theoryrdquo Ingenieur-Archiv vol 56 no 5 pp 389ndash3991986

[2] V Tvergaard and A Needleman ldquoNonlocal effects on local-ization in a void-sheetrdquo International Journal of Solids andStructures vol 34 no 18 pp 2221ndash2238 1997

[3] R de Borst ldquoSimulation of strain localization A reappraisal ofthe Cosserat continuumrdquo Engineering Computations vol 8 no4 pp 317ndash332 1991

[4] S Ramaswamy and N Aravas ldquoFinite element implementationof gradient plasticity models part I gradient-dependent yieldfunctionsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 163 no 1-4 pp 11ndash32 1998

[5] D J Bammann and E C Aifantis ldquoA model for finite-defor-mation plasticityrdquo Acta Mechanica vol 69 no 1ndash4 pp 97ndash1171987

[6] D J Bammann M L Chiesa M F Horstemeyer and L IWeingarten ldquoFailure in ductile materials using finite elementmethodsrdquo in Structural Crashworhiness and Failure N Jonesand TWierzbicki Eds pp 1ndash54 Elsevier Applied Science NewYork NY USA 1993

[7] G Pijaudier-Cabot and Z P Bazant ldquoNonlocal damage theoryrdquoJournal of Engineering Mechanics vol 113 no 10 pp 1512ndash15331987

[8] K Saanouni J L Chaboche and P M Lesne ldquoOn the creepCrack542 growth prediction by a nonlocal damage formula-tionrdquo European Journal ofMechanics A vol 8 pp 437ndash459 1989

[9] J B Leblond G Preein and J Devaux ldquoBifurcation effects inductile metals with nonlocal damagerdquo ASME Journal of AppliedMechanics vol 61 no 2 pp 236ndash242 1994

[10] V Tvergaard and A Needleman ldquoEffects of nonlocal damagein porous plastic solidsrdquo International Journal of Solids andStructures vol 32 no 8-9 pp 1063ndash1077 1995

[11] K Enakoutsa J B Leblond and G Perrin ldquoNumerical imple-mentation and assessment of a phenomenological nonlocalmodel of ductile rupturerdquoComputerMethods in AppliedMecha-nics and Engineering vol 196 no 13ndash16 pp 1946ndash1957 2007

[12] G Z Voyiadjis R K A Al-Rub and A N Palazotto ldquoCon-stitutive modeling and simulation of perforation of targets byprojectilesrdquo AIAA Journal vol 46 no 2 pp 304ndash316 2008

[13] R K Abu Al-Rub and S-M Kim ldquoPredicting mesh-indepen-dent ballistic limits for heterogeneous targets by a nonlocaldamage computational frameworkrdquo Composites B vol 40 no6 pp 495ndash510 2009


[14] A C F Cocks and M F Ashby ldquoIntergranular fracture duringpower-law creep under multiaxial stressesrdquo Metal Science vol14 no 8-9 pp 395ndash402 1980

[15] J D Bammann ldquoModelling the large strain-high temperatureresponse of metalsrdquo in Modeling and Control of Casting andWelding Process IV A F Giamei and G J Abbaschian EdsTMS Publications Warrendale Pa USA 1988

[16] G I Taylor and H Quinney ldquoThe latent energy remaining ina metal after cold workingrdquo Proceedings of the Royal Society Avol 143 no 849 1934

[17] G Rousselier Finite Deformation Constitutive Relations andDuctile Fracture S Nemat-Nasser Ed North-Holland Ams-terdam The Netherlands 1981

[18] A Needleman ldquoMaterial rate dependence and mesh sensitiv-ity in localization problemsrdquo Computer Methods in AppliedMechanics and Engineering vol 67 no 1 pp 69ndash86 1988

[19] K Yamaguchi S Ueda F S Jan and N Takakura ldquoEffectsof strain rate and temperature on deformation resistance ofstainless steelrdquo in Mechanical Behavior of Materials-VI vol 3pp 805ndash810 Pergamon Press Oxford UK 1992

[20] R D Krieg and D B Krieg ldquoAccuracies of numerical solutionmethods for the elastic-perfectly plastic modelrdquo ASME Journalof Pressure Vessel Technology vol 99 no 4 pp 510ndash515 1977

[21] A L Gurson ldquoContinuum theory of ductile rupture by voidnucle ation and growth part I Yield criteria and flow rules forporous ductile mediardquo ASME Journal of Engineering Materialsand Technology vol 99 no 1 pp 2ndash15 1977

[22] N Aravas ldquoOn the numerical integration of a class of pressuredependent plasticity modelsrdquo International Journal for Numeri-cal Methods in Engineering vol 24 no 7 pp 1395ndash1416 1987

[23] B Halphen and Q S Nguyen ldquoSur les materiaux standardsgeneralisesrdquo Journal de Mecanique vol 14 pp 39ndash63 1975(French)

[24] M F Horstemeyer M M Matalanis A M Sieber and M LBotos ldquoMicromechanical finite element calculations of tem-perature and void configuration effects on void growth andcoalescencerdquo International Journal of Plasticity vol 16 no 7 pp979ndash1015 2000

[25] Y H Tang H Yu J E Gordon D Y Jeong and A BPerlman ldquoAnalysis of railroad tank car shell impacts usingfinite element methodrdquo in Proceedings of the ASMEIEEEASCEJoint Rail Conference (JRC rsquo08) no JRC2008-63014 pp 61ndash70Wilmington Del USA April 2008

[26] Y H Tang H Yu J E Gordon and D Y Jeong ldquoFinite elementanalyses of railroad tank car head impactsrdquo in Proceedings oftheASMERail TransportationDivision Fall Technical Conference(RTDF rsquo08) no RTDF2008-74022 Chicago Ill USA Septem-ber 2008

[27] F Darve P Y Hicher and J M Reynouard Mecanique desGeomateriaux Herms Paris France 1995

International Journal of

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RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Journal ofEngineeringVolume 2014

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VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



0

200

400

600

800

1000

1200

0

200

400

600

800

1000

1400

1200

Stre

ss (M

Pa)

Stre

ss (M

Pa)

0 01 02 03 04 05 06 07 08Strain

0 01 02 03 04 05 06 07Strain

Reload

20∘C

20∘C

800∘C

20∘C calculation800∘C calculation20∘C reload calculation

20∘C experiment800∘C experiment20∘C reload experiment

20∘C

900∘C








10minus3 at 20∘C experiment10minus5 at 20∘C calculation10minus5 at 900∘C calculation10minus5 at 900∘C experiment10minus5 at 20∘C experiment

10minus3 at 20∘C calculation

Figure 1 Load versus displacement curves for different temperatures and strain rates Full lines BCJ model predictions point linesexperiments Note that the model satisfactorily captures temperature and strain rate histories effects The curves were extracted fromBammann et al [6]

Alternatively to the shortcomings encountered in hazmattank car impactsrsquo numerical simulations we propose to usea nonlocal version of the Bammann-Chiesa-Johnson (BCJ)model Bammann and Aifantis [5] and Bammann et al [6]a physically motivated internal state variable plasticity anddamage model containing a mathematical length scale Theuse of internal state variables will enable the prediction ofstrain rate and temperature history effects see Figure 1Theseeffects can be quite substantial and therefore difficult to incor-porate into material models which assumes that the stress is(1) a unique function of the current strain strain rate andtemperature and (2) independent of the loading path Theeffects of damage are included in the BCJ model howeverthrough a scalar internal state variablewhich tends to degradethe elastic moduli of the material as well as to concentratethe stress The mathematical length scale is introduced inthe model via the nonlocal damage approach of Pijaudier-Cabot and Bazant [7] In the context of concrete damagethese authors have suggested a formulation in which only thedamage variable is nonlocal while the strain the stress andother variables retain their local definitionTheir formulationhas been applied to creep problems by Saanouni et al [8]and extended to plasticity by among others Leblond et al[9] Tvergaard and Needleman [10] and Enakoutsa et al [11]Following suggestion Pijaudier-Cabot and Bazant [7] a non-local evolution equation for the damage within an otherwiseunmodified BCJ model is adopted in the current studyThe time derivative of the damage is expressed as thespatial convolution of a ldquolocal damage raterdquo and bell-shaped

weighting function The width of this function introduces amathematical length scale

In this study a dynamic nonlinear FE analysis carried outwith Abaqus FE code is used to simulate a moving strikercolliding against a stationary hazmat tank car The struc-ture part of this FE model is represented by Lagrangianelements obeying the nonlocal version of the BCJ modelwhile the fluid part is represented by Eulerian elementsThe objective of this study is twofold (1) use a high fidelitymaterial model to idealize the physics occurring during theimpact accident and (2) rectify the computational drawbacks(postbifurcation mesh dependence issues) for this modelon a large-scale boundary value problem The resultingnumerical simulations of hazmat tank car impact scenarioswhich account for nonlocal damage and temperature historyeffects predict satisfactorily the tank car failure processindependently of the element size Although several otherstudies have addressed successfully the mesh dependenceissues in finite element computations of problems involvingdynamic failure of materials as in for instance Voyiadjis etal [12] and Abu Al-Rub and Kim [13] very few of themdeal with real-world large-scale boundary value problemsThe present work investigates the application of the nonlocalintegral approach to predict the failure of a large-scale vehicleunder impact loading conditions The paper is organized asfollows

(i) The first section provides a summary of the equationsof the BCJ model and its nonlocal extension









120601

1 minus 120601

120590 (1)


= minusW119890120590 + 120590W119890 (2)





W119890 = W minusW119901(3)



D119889 =

120601

1 minus 120601

I (4)


Dth= 119860

120579I (5)



D119901 = 119891 (120579) sin ℎ[

1003817

1003817

1003817

1003817

1003817

1205901015840

minus 120572

1003817

1003817

1003817

1003817

1003817


119881 (120579) (1 minus 120601)

]

1205901015840

minus 120572

1003817

1003817

1003817

1003817

1205901015840minus 120572

1003817

1003817

1003817

1003817

(6)




= ℎ (120579)D119901 minus [radic

2

3

119903

119889(120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

+ 119903

119904(120579)] 120572120572

= 119867 (120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

minus [radic

2

3

119877

119889(120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

+ 119877

119904(120579)] 120581

2

(7)


119904are scalar


119889and 119877





119881 (120579) = 119862

1exp(minus1198622

120579

)

119884 (120579) = 119862

3exp(minus1198624

120579

)

119891 (120579) = 119862

5exp(minus

119862

6

120579

)

(8)


119903

119889(120579) = 119862

7exp(minus

119862

8

120579

)

ℎ (120579) = 119862

9exp(minus

119862

10

120579

)

119903

119904(120579) = 119862

11exp(minus11986212

120579

)

(9)


119877

119889(120579) = 119862

13exp(minus11986214

120579

)

119867 (120579) = 119862

15exp(minus

119862

16

120579

)

119877

119904(120579) = 119862

17exp(minus

119862

18

120579

)

(10)




120601 = [

1

(1 minus 120601)


(1 + 119899)

P

120590

]

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

(11)


0



120579 =

09

120588119862

119907

(120590D119901) (12)

where 120588 and 119862






120601 (x) = 1

119861 (x)int

Ω

120601

loc(y) 119860 (x minus y) 119889Ωy (13)



119897

2) (14)




120601


119861 (x) = int

Ω

119860 (x minus y) 119889Ωy (15)




120601

loc








120601 (119905 + Δ119905) asymp 120601 (119905) +

120601 (119905) Δ119905 (16)




(a) (b)


(a) (b)


















Striker(119881119910 = 119881119911 = 0)

Fixed wall

Tank car

Rigid wheel-1119880119910 = 119880119911 = 0

Rigid wheel-2

119880119903119909 = 119880119903119910 = 119880119903119911 = 0

119880119910 = 119880119911 = 0

119880119903119909 = 119880119903119910 = 119880119903119911 = 0


(a) (b)

(c)







(a) (b)

(c)


(a) (b)


5 Summary





(a) (b) (c)


(a) (b)



Acknowledgments


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















120601

1 minus 120601

120590 (1)


= minusW119890120590 + 120590W119890 (2)





W119890 = W minusW119901(3)



D119889 =

120601

1 minus 120601

I (4)


Dth= 119860

120579I (5)



D119901 = 119891 (120579) sin ℎ[

1003817

1003817

1003817

1003817

1003817

1205901015840

minus 120572

1003817

1003817

1003817

1003817

1003817


119881 (120579) (1 minus 120601)

]

1205901015840

minus 120572

1003817

1003817

1003817

1003817

1205901015840minus 120572

1003817

1003817

1003817

1003817

(6)




= ℎ (120579)D119901 minus [radic

2

3

119903

119889(120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

+ 119903

119904(120579)] 120572120572

= 119867 (120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

minus [radic

2

3

119877

119889(120579)

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

+ 119877

119904(120579)] 120581

2

(7)


119904are scalar


119889and 119877





119881 (120579) = 119862

1exp(minus1198622

120579

)

119884 (120579) = 119862

3exp(minus1198624

120579

)

119891 (120579) = 119862

5exp(minus

119862

6

120579

)

(8)


119903

119889(120579) = 119862

7exp(minus

119862

8

120579

)

ℎ (120579) = 119862

9exp(minus

119862

10

120579

)

119903

119904(120579) = 119862

11exp(minus11986212

120579

)

(9)


119877

119889(120579) = 119862

13exp(minus11986214

120579

)

119867 (120579) = 119862

15exp(minus

119862

16

120579

)

119877

119904(120579) = 119862

17exp(minus

119862

18

120579

)

(10)




120601 = [

1

(1 minus 120601)


(1 + 119899)

P

120590

]

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

(11)


0



120579 =

09

120588119862

119907

(120590D119901) (12)

where 120588 and 119862






120601 (x) = 1

119861 (x)int

Ω

120601

loc(y) 119860 (x minus y) 119889Ωy (13)



119897

2) (14)




120601


119861 (x) = int

Ω

119860 (x minus y) 119889Ωy (15)




120601

loc








120601 (119905 + Δ119905) asymp 120601 (119905) +

120601 (119905) Δ119905 (16)




(a) (b)


(a) (b)


















Striker(119881119910 = 119881119911 = 0)

Fixed wall

Tank car

Rigid wheel-1119880119910 = 119880119911 = 0

Rigid wheel-2

119880119903119909 = 119880119903119910 = 119880119903119911 = 0

119880119910 = 119880119911 = 0

119880119903119909 = 119880119903119910 = 119880119903119911 = 0


(a) (b)

(c)







(a) (b)

(c)


(a) (b)


5 Summary





(a) (b) (c)


(a) (b)



Acknowledgments


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119881 (120579) = 119862

1exp(minus1198622

120579

)

119884 (120579) = 119862

3exp(minus1198624

120579

)

119891 (120579) = 119862

5exp(minus

119862

6

120579

)

(8)


119903

119889(120579) = 119862

7exp(minus

119862

8

120579

)

ℎ (120579) = 119862

9exp(minus

119862

10

120579

)

119903

119904(120579) = 119862

11exp(minus11986212

120579

)

(9)


119877

119889(120579) = 119862

13exp(minus11986214

120579

)

119867 (120579) = 119862

15exp(minus

119862

16

120579

)

119877

119904(120579) = 119862

17exp(minus

119862

18

120579

)

(10)




120601 = [

1

(1 minus 120601)


(1 + 119899)

P

120590

]

1003817

1003817

1003817

1003817

D1199011003817100381710038171003817

(11)


0



120579 =

09

120588119862

119907

(120590D119901) (12)

where 120588 and 119862






120601 (x) = 1

119861 (x)int

Ω

120601

loc(y) 119860 (x minus y) 119889Ωy (13)



119897

2) (14)




120601


119861 (x) = int

Ω

119860 (x minus y) 119889Ωy (15)




120601

loc








120601 (119905 + Δ119905) asymp 120601 (119905) +

120601 (119905) Δ119905 (16)




(a) (b)


(a) (b)


















Striker(119881119910 = 119881119911 = 0)

Fixed wall

Tank car

Rigid wheel-1119880119910 = 119880119911 = 0

Rigid wheel-2

119880119903119909 = 119880119903119910 = 119880119903119911 = 0

119880119910 = 119880119911 = 0

119880119903119909 = 119880119903119910 = 119880119903119911 = 0


(a) (b)

(c)







(a) (b)

(c)


(a) (b)


5 Summary





(a) (b) (c)


(a) (b)



Acknowledgments


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120601


119861 (x) = int

Ω

119860 (x minus y) 119889Ωy (15)




120601

loc








120601 (119905 + Δ119905) asymp 120601 (119905) +

120601 (119905) Δ119905 (16)




(a) (b)


(a) (b)


















Striker(119881119910 = 119881119911 = 0)

Fixed wall

Tank car

Rigid wheel-1119880119910 = 119880119911 = 0

Rigid wheel-2

119880119903119909 = 119880119903119910 = 119880119903119911 = 0

119880119910 = 119880119911 = 0

119880119903119909 = 119880119903119910 = 119880119903119911 = 0


(a) (b)

(c)







(a) (b)

(c)


(a) (b)


5 Summary





(a) (b) (c)


(a) (b)



Acknowledgments


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)


(a) (b)


















Striker(119881119910 = 119881119911 = 0)

Fixed wall

Tank car

Rigid wheel-1119880119910 = 119880119911 = 0

Rigid wheel-2

119880119903119909 = 119880119903119910 = 119880119903119911 = 0

119880119910 = 119880119911 = 0

119880119903119909 = 119880119903119910 = 119880119903119911 = 0


(a) (b)

(c)







(a) (b)

(c)


(a) (b)


5 Summary





(a) (b) (c)


(a) (b)



Acknowledgments


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















Striker(119881119910 = 119881119911 = 0)

Fixed wall

Tank car

Rigid wheel-1119880119910 = 119880119911 = 0

Rigid wheel-2

119880119903119909 = 119880119903119910 = 119880119903119911 = 0

119880119910 = 119880119911 = 0

119880119903119909 = 119880119903119910 = 119880119903119911 = 0


(a) (b)

(c)







(a) (b)

(c)


(a) (b)


5 Summary





(a) (b) (c)


(a) (b)



Acknowledgments


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)

(c)


(a) (b)


5 Summary





(a) (b) (c)


(a) (b)



Acknowledgments


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b) (c)


(a) (b)



Acknowledgments


References































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article modeling the dynamic failure of...

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