Comp. Part. Mech.DOI 10.1007/s40571-015-0092-1

A comparison of finite element analysis to smooth particlehydrodynamics for application to projectile impacton cementitious material

Nikolas A. Nordendale1 · William F. Heard1,2 ·Jesse A. Sherburn2 · Prodyot K. Basu1

Received: 4 September 2015 / Revised: 23 November 2015 / Accepted: 8 December 2015© OWZ (outside the USA) 2015

Abstract The response of structural components of high-strength cementitious (HSC) materials to projectile impactis characterized by high-rate fragmentation resulting fromstrong compressive shock waves coupled with reflected ten-sile waves. Accurate modeling of armor panels of suchbrittle materials under high-velocity projectile impact is acomplex problem requiring meticulous experimental char-acterization of material properties. In a recent paper by theauthors, an approach to handle such problems based on amodified Advanced Fundamental Concrete (AFC) constitu-tive model was developed. In the HSC panels considered inthis study, an analogous approach is applied, and the predic-tions are verified with ballistic impact test data. TraditionalLagrangian finite element analysis (FEA) of these problemstends to introduce errors and suffers from convergence issuesresulting from large deformations at free surfaces. Also, FEAcannot properly account for the issues of secondary impactof spalled fragments when multiple armor panels are used.Smoothed particle hydrodynamics (SPH) is considered tobe an attractive alternative to resolve these and other issues.However, SPH-based quantitative results have been found tobe less accurate than the FEA-based ones when the deforma-tions are not sufficiently large. This paper primarily focuseson a comparison of FEA and SPH models to predict high-velocity projectile impact on single and stacked HSC panels.Results are compared to recent ballistic experiments per-formed as a part of this research, and conclusions are drawnbased on the findings.

B Jesse A. Sherburnjesse.a.sherburn@usace.army.mil

1 Department of Civil Engineering, Vanderbilt University,2301 Vanderbilt Place, Nashville, TN, USA

2 U.S. Army Engineer Research and Development Center,3909 Halls Ferry Road, Vicksburg, MS 39180-6199, USA

Keywords Brittle · Cementitious · Impact · Experiment ·Finite element · Smooth particle hydrodynamics

1 Introduction

The performance of brittle, cementitious armor panels underhigh-rate impact is dependent on a wide range of variablesincluding but not limited to projectile and target geome-try, impact velocity, type of projectile material, and angleof impact of the projectile. Due to comparatively low cost,ease of rapid on-site manufacture, and high early strength,the use of HSC material as protective armor has beenan attractive option for passive protection from weaponeffects. Understanding the performance of such armor pan-els under high-rate ballistic impact is vital to ensure safetyof personnel in forward combat environments. Accuratesimulation of the structural response of armor panels tosuch dynamic loads is a convenient way to significantlydecrease the cost of R&D efforts related to newmaterials andapplications.

Under ballistic impact, HSC panels experience variousstress states that lead to complex failure modes. In the worstcase scenario for a single-projectile impact event, which cor-responds to an impact orientation normal to the surface, acrater is formed in the front surface followed by a strong com-pressive wave. This wave weakens as it traverses through theplate thickness and eventually reflects off the free back sur-face [1]. This compressive wave reflection generates a tensilewave, which if large enough in magnitude, can lead to frac-ture on the rear face of the panel accompanied by spalling ofthe material behind the point of impact. As the speed of com-pressivewaves propagating thematerial can exceed the speedof the projectile, this fracture takes place before the projectilecan even penetrate through the target panel. This condition is

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Fig. 1 Progress of a planeshock wave (Zukas et al. [5])

further complicated asmilitary personnel have found it logis-tically advantageous to use stacks of two or more thin panelsfor protection, rather than a single panel of comparable thick-ness. By taking two-stacked panels back-to-back instead ofcasting one panel of comparable thickness, impact energycan be transferred to more of the panel area, leading to possi-ble improved penetration resistance. However, improvementin energy dissipation leads to secondary damage, and frac-ture tends to occur over a larger area of the panel(s). Suchfracture and spalling phenomena in brittle materials becomeproblematic when using the traditional Lagrangian finiteelement techniques and methods such as element dele-tion must be employed in order to overcome numericalproblems [2].

By using a dynamic user-defined material model and cou-pling it with an element deletion scheme, the highly distortedproblem elements undergoing large strains and strain ratescan be easily handled obviating any computational obsta-cle. Element deletion, also known as element erosion, is atechnique that removes highly distorted elements once a cri-terion is met in order to avoid numerical instabilities [2].However, it has been found that to efficiently use an ele-ment deletion scheme without loss of accuracy or increasedcomputational time, very large strains can still appear inthe model [3]. Such a model can accurately predict the exitvelocity of the penetrating projectile but cannot track thefree-flying fragmented pieces. As the theoretical underpin-nings of the SPH-based method is improved by using thesame material model and model parameters as for a tra-ditional finite element model, it is expected that an SPHmodel can realistically represent the complex failure statesthat occur during an impact event, including the fragmenta-tion process [4]. The purpose of this study is to extend thework of Nordendale et al. [3] by modeling single-panel anddouble-panel concrete perforation experimentswithFEAandSPH modeling techniques in the commercial code Abaqusv6.14 [4].

1.1 Nature of impact of cementitious targets

Predicting the response of a structure due to an intense,impulsive loading can be involved. Over the years, sophis-ticated mathematical solutions have been created for suchproblems mostly in the context of semi-infinite or infinite(unbounded) bodies. The impact of a projectile or blast waveon a solid target in the normal direction is governed byequations of (a) conservation of mass, (b) conservation ofmomentum, and (c) conservation of energy. These equationsare manifested in the Rankine–Hugoniot relations, some-times termed as “jump” equations, which do not describethe properties of a specific material, but relate the changein associated variables across a shock front. This situa-tion can be summarized by a uniform pressure P1 suddenlyapplied to one face of a plate of compressible materialunder an initial pressure P0. The sudden pressure changecauses a pressure pulse to propagate as a wave travelingat velocity, say, Us . The application of P1 compresses thematerial to a new density ρ1 and accelerates the motionof the compressed material to a velocity Up. In Fig. 1, atany instant of time, the shock front is depicted by the lineAA. At time (t + dt), the shock front has moved to positionCC. Across the shock front, mass, momentum, and energymust be conserved [5], and the compact form of resultingRankine-Hugoniot relationswill take the forms shown inEqs.(1)–(3).

For application to a specific material, it is necessary togenerate curves (or Hugoniot curves) depicting the locus ofstates achievable by shock transition fromagiven initial state.For instance, in a particular case, the final pressure and rela-tive volume reached depend on the initial conditions presentwhen the shock arrives [6]. Actually, a Hugoniot curve con-tains the minimally required information about a material toadequately solve the shock-propagation problem, consistentwith a given set of variables. In order to determine all theparameters associated with the shock front, it is sometimes

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necessary to append the stated equations with an experimen-tally determined equation of state.

ρ0Us = ρ1(Us −Up

)(1)

p1 − p0 = ρ0U2s − ρ1

(Us −Up

)2 (2)

E ′1 − E ′

0 = 1

2

(1

ρ0− 1

ρ1

)(p1 + p0) . (3)

Analytical and simulation studies of the nonlinear behav-ior of concrete structures have often been focused on thebehavior of isolated, simple structural components. Avail-ability of meaningful quantitative data on failure processesof concrete and other brittle, geologic materials paved theway to increased effort for accurate prediction of responsesof concrete structures under extreme loading conditions,like blast and impact. Although currently available commer-cial finite element software packages have a wide range ofcapabilities for many areas of mechanical stress analysis,inadequate material models are a major hindrance in thecase of structures made of brittle, geologic materials likeconcrete, ceramics, and geopolymers. The currently incor-porated material models can, in a general way, describe theelastic–plastic behavior of reinforced or unreinforced con-crete under simple quasi-static loading conditions. However,as advances in concrete and ceramic materials are made toimprove performance under extreme loading conditions, thegenerally accepted constitutive equations that describe thebasic characteristics of concrete no longer apply [7].

Complexities of material failure near a free surface somedistance from the localized area of application of an impul-sive load have been studied extensively. In the case ofceramic or cementitious materials that have high compres-sive strengths but relatively weak tensile strengths, spallingat the free surfaces is a phenomenon to be expected, dueto reflection of incident-compressive impulses generated byhigh-velocity ballistic impact [8]. The progressive effectstaking place during high-velocity impact and penetrationevents for a thin panel are illustrated in Fig. 2. In the first fewmicroseconds of impact, local material cratering takes place.Also, directly ahead of the projectile, the material undergoeslocalmaterial compaction.Due to the high rate of loading, thematerial undergoes inertial confinement which would other-wise allow for some of the material in front of the projectileto expand outward laterally. Therefore, the material in frontof the projectile essentially undergoes radially confined com-pression. As reiterated later, this is the rationale for using amaterial model that has been calibrated using triaxial com-pression test results, specifically hydrostatic compression,uniaxial strain compression, and simple triaxial compres-sion with constant radial pressures. After the stated initialeffects, the strong compressive shock wave is weakened as ittraverses through the plate thickness and is then reflected off

Fig. 2 Physical characteristics of high-velocity penetration of brittletarget

the back surface. If the reflected tensile wave is large enough,spalling occurs when the reflected tensile wave is no longeroverdriven by the incoming compressive wave. Apart fromlongitudinal effects in the direction of the path of the projec-tile, the shock wave also travels laterally in the plane of theplate at a slower speed and eventually gets reflected from theperiphery of the plate.

1.2 Traditional Lagrangian FEA versus SPH

SPH is a numerical method that is part of the larger familyof meshless methods. Instead of defining nodes and ele-ments, one only needs a collection of points or particles torepresent a given body. The SPH capability in Abaqus isa fully Lagrangian modeling scheme that permits the dis-cretization of a prescribed set of continuum equations byinterpolating the properties directly at a discrete set of pointsdistributed over the solution domain without the need ofa traditional spatial mesh [9]. By following this approach,the difficulties associated with fragmentation and free-flyingmotion of spalled particles coupled with very large defor-mations of free surfaces are resolved in a realistic manner.Moreover, in a case where it is necessary to consider thesecondary impact of fragmented particles against, say, asecond target a certain distance away from the first tar-get, SPH has no additional computational cost associated

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with tracking these fragments through a large empty vol-ume obviating the need to use the highly computationallyintensive coupled Eulerian–Lagrangian scheme. The SPHscheme incorporated in Abaqus includes an artifact knownas the “domain” representing a rectangular region computedat the beginning of the analysis as the bounding box withinwhich the particles are tracked. This fixed rectangular boxis normally 10% larger than the overall dimensions of thewhole model and is centered at the geometric center ofthe model. As the analysis progresses, if a particle movesoutside the domain, it then behaves like a free-flying pointmass and no longer contributes to the SPH calculations. Asmentioned earlier, SPH interpolates the properties of eachparticle. This is done using what is called a “smoothinglength calculation.” Even though particulate elements aredefined in each model using one node per element, the SPHmethod computes contributions from each element basedon adjacent particles that are within a sphere of influence.This smoothing length governs the interpolation basis of themethod. For every increment, this local connectivity is recal-culated internally, and the kinematic quantities like normaland shear strains as well as deformation gradients are com-puted [4].

However, there are some limitations inherent with SPHmodels that are not present in traditional finite element mod-els. In regions of the model where deformations are not toolarge and elements are not highly distorted, the SPH analysesare found to be less accurate in general than Lagrangian finiteelement analyses. Particle motion may become unstable inthe presence of a tensile stress field. Such tensile instabil-ity is a source of concern with this method. This instabilityis strictly related to the interpolation technique of the stan-dard smoothed particle dynamic method. Because of thisinstability, particles tend to clump together and show pseudofractures and artificial voids. The underlying cause has beenshown to be a lack of formal consistency in SPH. It can-not reproduce exactly any arbitrary class of functions on adefined set of points. Because of this drawback, a numericalclumping instability manifests itself when the particles aremutually attracted. More specifically, the SPH kernel func-tion is incapable of keeping the particles apart once they aresufficiently close to each other [10].

One of the useful features in Abaqus is the ability to auto-matically convert finite elements to SPH particles [4]. Thiseliminated many of the limitations associated with apply-ing loads directly to bodies that could, eventually, becomeparticles. In the Abaqus version used in this study, thisimplementation has the limitation that once the elements areconverted to particles, whether by time-, stress-, or strain-based criterion, they are free-flying particles that no longerobey the laws of symmetric boundary conditions. This leadsto inaccurate fragmentation patterns if a user tries to takeadvantage of the symmetry in the problem.

2 Material model description

Over the past forty years, several material models have beendeveloped to model concrete and other brittle ceramic mate-rials subjected to large strains, high strain rates, and severepressures. Some of the first models that emerged were that ofWillamandWarnke [11] andOttosen [12]. Both of thesewereoriginally developed for static analysis of concrete structures,and the strength failure surfaces included dependency on thetriaxial yield ratio. Holmquist et al. [13] (HJC) developedone of the first models for application to concrete penetrationand blast loading. The original HJC strength failure surfaceincluded a dependency on pressure and the second devia-toric stress, but not on the third invariant of the stress. TheHCJ model also included a pressure–volume definition thatincorporated hydrostatic crushing as well as damage accu-mulation. After the HJC model was developed, many othermodels began to emerge that were similar in structure andapplication. A review of many concrete models is given byTuandLu [14].Gebbeken andRuppert [15] included the thirddeviatoric stress invariant into the failure surface of the HCJmodel as well as other additional changes. Another modelthat emerged was the concrete damage model by Karagozianand Case [16,17]. Around the same time, the Riedel, Her-maier, and Thoma model was developed and improved overa decade [18–20]. One feature that is required in modelingconcrete is the dependence on the third invariant of the stresstensor [18,21–24]. This led some researchers to include thisdependency in a modified HJC model [23]. In this study, aconstitutive model will be chosen that includes dependencyon the third invariant of the stress. In perforation-type prob-lems, it has been shown that the third invariant is importantwhen targets are thin as opposed to deep penetration prob-lems [22,24].

The constitutivemodel used in the present study is a lightlymodified version of the Advanced Fundamental Concrete(AFC) model [25] which was first implemented into Abaqusas a user-defined material by Sherburn et al. [26]. In somerecent studies, the AFC model has been shown to modelperforation experiments. Sherburn et al. [27] used the AFCmodel alongwith the coupledEulerian–Lagrangian approachto model perforation of finite thickness concrete slabs with alarge caliber penetrator. In similar study, Sherburn et al. [28]modeled the identical large caliber perforation experiments,but used the AFC model along with the meshfree methodknown as the reproducing kernel particle method [29].Outlined in the following sections are the three principal com-ponents utilized in the AFC model: (1) an equation of statefor the pressure–volume relation that includes the nonlineareffects of compaction, (2) a representation of the deviatoricstrength of the intact and fractured material in the form of apressure, strain rate, and third invariant-dependent yield sur-face, and (3) a damage model that transitions the material

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from the intact state to the fractured state. Further improve-ments and developments were made by the authors and wereshown to have excellent performance. For a full description ofthese changes and improvements, see the previous article bythe authors [3]. This model accounts for processes like irre-versible hydrostatic crushing, material yielding, plastic flow,and damage evolution. The model is based on a nonlinearpressure–volume relationship and a linear shear relationship(constant shear modulus, G). As with most of the simplisticmodels for geomaterials, the proposed model uncouples thehydrostatic and deviatoric responses so that no volumetricstrain can develop due to deviatoric loading.

2.1 Pressure

The pressure–volume behavior of the model characterizes anonlinear bulk modulus and irreversible volumetric crushingthat contributes to material damage. The compressive pres-sure behavior can be separated into three distinct zones: (1)an initial linear-elastic zone followed by (2) an irreversiblecrushing response zone, as the air voids begin to collapse,represented by a cubic polynomial, and finally (3) a linear-elastic locking zone corresponding to a fully packed materialafter all air voids have been crushed out. Additionally, thismodel treats initial loading, unloading, and reloading dif-ferently. For the HSC material considered in this study,the typical pressure–volume behavior is shown in Fig. 3.As shown in the figure, the initial elastic zone correspondsto volumetric strains below the crushing volumetric strain,μcrush. Initial loading, unloading, and reloading in the elasticzone all follow a linear-elastic behavior defined by the elas-tic bulk modulus, Ke = Pcrush/μcrush, where Pcrush is themaximum attainable pressure in the initial elastic zone. Theirreversible crushing response occurs when the volumetricstrain exceeds the crushing volumetric strain, μcrush but has

Fig. 3 Pressure–volume relation for HSC material characterized by auniaxial strain compressive test

not yet exceeded the locking volumetric strain value, μlock.The crushing region begins at the end point of the initial elas-tic zone (Pcrush, μcrush). This region is characterized by anirreversible volumetric compaction following the third-orderpolynomial representation

P = K1μ + K2μ2 + K3μ

3. (4)

In Eq. (4), K1, K2, and K3 are material constants, P is themean normal stress (pressure), and μ is the measure of vol-umetric strain that is equal to the ratio of the initial volumeminus the current volume to the current volume. In this equa-tion, the traditional soil mechanics sign convention is used(compression> 0, tension< 0), which means that P as com-puted by Eq. (4) is equal to the first invariant of the stresstensor, I1, multiplied by −1 (that is, I1 = −P). In the crush-ing region, unloading and reloading are nonlinear with thebulk modulus varying linearly between Ke and Klock as μ

varies between μcrush and μlock. However, it should be notedthat since the change in μ during a typical unload–reloadcycle in the crush zone is generally only a small percentageof the value of (μlock − μcrush), the response is nearly lin-ear in most cases. Finally, the linear-elastic locking region inthe model is defined by a locking bulk modulus, Klock, andoccurs for volumetric strains above the locking value of vol-umetric strain,μlock. Unloading and reloading in the lockingregion are treated as linear elastic with the bulk modulus,Klock .

2.2 Strength

Shear or distortional behavior of the model is characterizedby plastic flow, material yielding, and damage initiation aswell as evolution. Following the sign convention of contin-uummechanics, the mean normal stress values less than zerodenote compression. Here, the yield surface is represented bythe following two equations, depending on whether the stateof stress is in compression (Eq. 5) or tension (Eq. 6):

σmax =(C1 − (C2 + (C1 − C2) D) eAn I1 − C4 I1

)

×(1 + C3Ln (ε̇) + C11Ln (ε̇)2

)(5)

σmax = (C1 − (C2 + (C1 − C2) D)) (1 + C3Ln (ε̇)

+C11Ln (ε̇)2)

(Tmax − I1) /Tmax, (6)

where C1,C2,C3,C4,C11, and An are constants for a par-ticular material and are nonzero, D is the scalar damageparameter that varies between 0 (intact) and 1 (damaged),ε is the strain rate, Tmax is the limiting (maximum allow-able) tensile pressure, and the value of σmaxis restricted tovalues that are greater than or equal to zero. The originalAFC model proposed a generic Johnson–Cook strain rate

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(a) (b)

Fig. 4 a Triaxial test setup where σ1 = σ2. b Axial stress versus axial strain of triaxial tests at various confining pressures

Fig. 5 Failure surface for HSC material

law in the failure surface, but it has been shown that manymaterials, including concrete and HSC materials, show non-linear behavior in the log-scale of the strain rate [30]. For thisreason, the Huh–Kang strain rate law provides a significant

improvement with no significant degeneration of computa-tional efficiency [31]. To determine the failure surface data,the following system of equations must be solved:

p = −1

3(2σ1 + σ3) (7)

q = σ1 − σ3. (8)

In Eqs. (7) and (8), p is the hydrostatic pressure and q is twotimes the shear stress. In Fig. 4, each curve corresponds toa different value of confining pressure, σ1. One stress datapoint from each stress–strain curve at different levels of con-finement is plotted in the meridional plane (p − q plane).This technique calibrates the shape and position of the yieldsurface, as shown in Fig. 4b, and is adequate to define amodel if it is to be used as a failure surface (perfect plas-ticity), as shown in Fig. 5. The AFC model also considers amodification of the failure surface for behavior in extension.The extension failure surface is determined by using the thirdinvariant of the stress tensor to calculate the Lode angle usingaWillam-Warnke Lode function [21]which gives a reductionfactor that is multiplied by Eq. (5).

Fig. 6 Ashcrete armor panel test plate: front view (left), side view (center), and front test setup (right)

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Fig. 7 MIL-P-46593AStandard 0.50 Caliber FSP (left)and model (right)

2.3 Damage

This model also accounts for material damage that developsprogressively during the course of stress loading histories.The characterization of material damage effectively providesa reduced failure surface due to excessive plastic shear strainaswell as hydrostatic crushing.Material damage is quantifiedusing a scalar damage parameter, D, that is computed fromthe equation

D =∑(

�εp

−I1D1+ �μp

1.5μlock

), (9)

where D1 is an input parameter greater than zero, values of(−I1D1) are restricted to being greater than 0.01, Δεp is anincrement in the effective deviatoric plastic strain,Δμp is anincrement of volumetric plastic strain, and μlock is the lock-ing volumetric strain described in Sect. 2.1. The computedvalue inside the parenthesis of Eq. (9) is considered the dam-age increment for a given time step. The damage parameter ata given time is the sum total of all the previous damage incre-ments. This damage parameter, D, is included in the failuresurface based on Eqs. (5) and (6). For the damage portionof the AFC model, the same data points used for the fail-ure surface are converted to data points in terms of effectiveplastic strain (nonrecoverable strain) as a function of meannormal stress. These data are used to find the parameter D1

as defined in Eq. (9).

3 Experimental and model setup

The HSC material considered in this study will herein afterbe referred to as “Ashcrete” due to the major contributorymaterial component being fly ash. The panels tested in the

Table 1 Ashcrete material constants for user-defined material model

Variable Description Value Units

ρ Density of material 2276.3 kg/m3

G Shear modulus 18,457 MPa

C1 Failure surface constant 125.85 MPa

C2 Failure surface constant 102.00 MPa

C3 Failure surface constant 0.012089

C4 Failure surface constant 1.0

C11 Failure surface constant −0.000627

An Failure surface constant 0.00754125 1/MPa

Plock Failure surface constant 640.46 MPa

Tmax Maximum allowabletensile pressure

4.3780 MPa

Pcrush Equation of state constant 60.6 MPa

μcrush Equation of state constant 0.00683811

μlock Equation of state constant 0.13814

K1 Equation of state constant 6429.9 MPa

K2 Equation of state constant −47,138.6 MPa

K3 Equation of state constant 255,724.2 MPa

D1 Damage constant 0.000311742 1/MPa

laboratory were approximately 304.8 by 304.8 by 11.9 mmand clamped at the periphery. One of the test panels is shownin Fig. 6 as well as the same test panel in the clampedcondition in the test setup. The projectile used was a MIL-P-46593A Standard fragment simulating projectile (FSP). Itis a 0.50 caliber, 207 grain FSP made of 4340-H Steel witha Rockwell Hardness of C = 30 ± 1. Multiple impact testswere performed with impact velocities varying from 1067m/s to 1097 m/s in a direction normal to the plane of thepanel. A photograph of the FSP is shown in Fig. 7 along withthe projectile model and mesh used for all the simulationsdiscussed in this paper.

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Table 2 4340-H steel material constants for Johnson–Cook materialmodel

Variable Description Value Units

ρ Density of material 7830 kg/m3

E Elastic modulus 205,000 MPa

ν Poisson’s ratio 0.29

A Johnson–Cook plasticity constant 792 MPa

B Johnson–Cook plasticity constant 510 MPa

n Johnson–Cook plasticity constant 0.26

m Johnson–Cook plasticity constant 1.03

θmelt Melting temperature 1793 K

θtrans. Transition temperature 293 K

C Strain rate constant 0.014

ε̇0 Reference strain rate 0.002 1/s

D1 Johnson–Cook damage constant 0.05

D2 Johnson–Cook damage constant 3.44

D3 Johnson–Cook damage constant −2.12

D4 Johnson–Cook damage constant 0.002

D5 Johnson–Cook damage constant 0.61

The FSP model contained 584 linear hexahedral elements(C3D8R) and was always modeled with finite elements inboth the FEAmodels and SPH models. A mesh convergencestudy was undertaken by the authors in a previous studythat showed the appropriate mesh refinement to obtain accu-

Fig. 9 Velocity versus time for FEA and SPH models compared toexperimental average (single-panel case)

rate results, and all subsequent simulations used these meshrefinements [3]. The velocity of the projectile was monitoredwith multiple infrared photoelectric velocity screens andchronographs to record precise entrance and exit velocities.All initial shots were fired to strike the target panel eccentri-cally, near a corner. This enabled remounting of the damagedpanel for further tests by impacting near the opposite corner,and thus determines the residual ballistic resistance of a dam-aged panel. A previous study by the authors demonstrated

Fig. 8 Assembly and mesh ofsingle- and double-panel models

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Fig. 10 Single panel: FEA model (top row), SPH model (middle row), and test panels (bottom row); front face on left and back face on right

that at such high velocities for this material, the location ofimpact in a panel has minimal bearing on the exit velocity ofthe projectile [3]. The present study will verify if the resultsof the previous study are valid in the case of damaged panelsor multiple panels undergoing projectile impact as well.

The user-defined material constants used for Ashcrete arelisted in Table 1. The AFC material constants for Ashcretewere taken from previous work [3]. For the projectile, theJohnson–Cook plasticity and damagemodels were used. The

model constants for 4340-H steel are given in Table 2. Theconstants were taken directly from two studies completed byJohnson and Cook [32,33].

4 Single-panel results

A single Ashcrete panel was simulated at an impact velocityof 1076.8 m/s, fired eccentrically. Two models were built:

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Fig. 11 Velocity versus time for FEA and SPH models compared toexperiment (stacked panel case)

(1) a traditional finite element model and (2) a model incor-porating the SPH capabilities of Abaqus. Both models areidentical in assembly, boundary conditions, material prop-erties, and number of elements. In both models, explicittime integration is used. For the SPH model, all elementsin the Ashcrete panel were converted to particles at thebeginning of the analysis. Each element was converted toa single particle located at the element center. Figure 8shows the assembly and mesh of both the single- paneland double-panel models. Both the FEA and SPH modelscontained 2,177,415 elements in the panels (linear hexahe-dral elements for the FEA model and particles for the SPHmodel). The mesh was both structured and uniform in alldirections. A similar convergence study was performed onthe SPH model as the FEA model, and the results indicatedthe particle resolution adequately converged for this impactproblem.

A number of test panels were shot, and the results ofthe models are compared to the average result of thesetests. Two metrics were used in comparing the performanceof the two models with the experimental result. The firstmetric is the residual velocity of the FSP after full pene-tration. The second metric is the damage pattern seen inthe panel. Due to the heterogeneous nature of HSC mate-rials, there was a wide range of values that were seen inthe experiments. The average exit velocity in the experi-ments was 829 m/s with a standard deviation of 6.5 m/s.The exit velocity seen in the FEA model was 824 m/s, andthe exit velocity in the SPH model was 832 m/s. Thesevalues have an error that is only 0.67 and 0.28% off ofthe average experimental value, respectively. The velocityhistories of both models are compared in Fig. 9 with thebaseline experimental average shown in the dotted line. Inaddition, some other model runs were performed slightly

varying the FSP position by approximately 1–2 diametersof the nodal spacing. The slight variations did not changethe residual velocity much (less than 1 m/s). These wereperformed to test the sensitivity of the FSP initial impactlocation.

Figure 10 shows the damage pattern from the models,both the front (entrance) and back (exit), as compared tothe experiment. The experiment shows a few radial cracksoriginating from the impact point. Radial cracks are cap-tured by the model as well as the peripheral cracks dueto the boundary conditions of the test. The hole in thepanel was slightly larger than the FSP, with a slightly largercrater on the front and back sides. There was peripherycracking that occurred near the supports directly aboveand along the side of the impact location. One differ-ence between the FEA and SPH case is the FEA damagezone on the front face that is visibly larger than the SPHcase.

5 Stacked panel results

Two Ashcrete panels, stacked back-to-back, were simulatedat an impact velocity of 1074 m/s, aimed eccentrically. Twomodels were built: (1) a traditional finite element model and(2) a model incorporating the SPH capabilities of Abaqus.Both models are identical in assembly, boundary conditions,and material properties. For the FEA model, 4,645,152 ele-ments were used, while the SPH model could only achievea mesh refinement of 3,122,368 particles. The memory con-straints on the available computers prevented the SPHmodelsfrom attaining a similar mesh refinement. The convergencestudy was limited due to this memory restriction, but theresolution used in the two-panel SPH model was shown tobe adequately converged compared to an analogous coarsermesh. Due to the smoothing length calculations required bySPH in conjunction with discretizing two panels with par-ticles instead of one panel as in the previous section, anincredibly large amount of computer memory is required tocomplete these analyses. For the SPH model, all elementsin the Ashcrete panels were converted to particles prior torunning the model in the same manner as the single-panelmodel. To ensure no problems ensued in the SPH model as aresult of the particles in contact with each other from oppos-ing panels occupying the same 3D space, the panels wereplaced a distance of 0.75 mm apart, which corresponds tothe characteristic length of the particles. For consistency, theFEAmodel also had a gap of 0.75mmbetween the two targetpanels. The contact for both the FEAmodel and the SPHwashandled by Abaqus’ general contact algorithm [4].

A number of these pairs of test panels were shot, and theresults from the models are compared to the average resultof these tests. Similar to the results in Sect. 4, the same

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Fig. 12 Stacked panels: FEA model (top row), SPH model (middle row), and test panels (bottom row); front face on left and back face on right

two metrics are used in comparing the performance of thetwo models with the experimental results. The average exitvelocity seen in the experiments was 637 m/s with a stan-dard deviation of 9.6 m/s. The exit velocity seen in the FEAmodel was 539 m/s, and the exit velocity in the SPH modelwas 540 m/s. These values have an error that is 15.5 and15.4% off of the average experimental value, respectively.The velocity histories of bothmodels are compared in Fig. 11with the baseline experimental average shown in the dottedline.

Figure 12 shows the damage pattern from the modelsof the stacked panels, both the front (entrance) and back(exit), as compared to the experiment. It can be seen thatthe fracture pattern in both models shows a large amountof damage in the corner of the panel where the projectilehits. Radial cracks exist in both the models and the exper-iment. The experiment indicates not only a large damagedregion, but also long cracks that almost traverse the entirepanel. Periphery cracks were present on all four sides of thepanel.

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Fig. 13 Cross section of panel showing effective plastic strain for both FEA (left column) and SPH (right column) through time

6 Discussion

As shown in the previous section, results of the SPH modelshow a similar performance compared to the traditionalFEA models in residual velocity prediction. However, animportant aspect of modeling high-rate impact of a brit-tle target is the issue of spalling and fragmentation whichis where FEA models struggle. It has been shown by theauthors in a previous article that FEA models are good atpredicting exit velocities of projectiles if the impact veloc-ity is near the ballistic limit of the target, but without anelement deletion scheme, the elements will continue todeform until excessive distortion causes instability in the

model. The standard metric for element deletion is effec-tive plastic strain, εmax

pl . The previous article [3] showed thatdeleting an element when εmax

pl is below 1.5 (150% strain)causes inaccuracies in the model. However, a value thislarge negates the model’s ability to show accurate spallingbehavior. To illustrate the spalling phenomena, Fig. 13 showszoomed-in cross-sectional views of both the FEA and SPHsingle-panel models, stepped through time, for compari-son.

As shown, both models perform well in predicting theregions of largest effective plastic strain, but only the SPHmodel can handle the transition of particles from being partof the original target to being free-flying fragments. This

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Fig. 14 Cross section of stacked panels showing effective plastic strain for both FEA (left column) and SPH (right column) through time

ability to model impact ejecta and debris field formation isone of the advantages of SPH over FEA. As shown around80 μs in the FEA model in Fig. 13, much higher plasticstrains developed in the target elements surrounding the pro-jectile. An element deletion scheme can handle this issue,but it still will not give results as seen in the SPH models.This behavior becomes even more pronounced in the stackedpanel case. Figure 14 also shows the effective plastic straincontours through the zoomed-in cross section of both the

FEA and SPH models, stepped through time. While the SPHmodel does show an appropriate fragmentation and spalledbehavior, some local grouping of particles near the rear freesurface was observed. This may be a result of the tensileinstability noted in SPH formulations. In addition to accu-racy, there is also the issue of computation time. While theSPH models show similar performance in residual velocityprediction and fragmentation, there is a significant cost inthe time that it takes for the model to run. On average, the

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Fig. 15 Stress contours of single-panel models, FEA (left column) and SPH (right column), through time

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Fig. 16 Comparison of experimentally deformedFSP and undeformedFSP

SPH models in this study took twice as long to run as theanalogous FEA models.

Another metric for comparison between these two mod-els is the propagation of a compressive (and subsequently,tensile) wave. This phenomenon is better seen throughtime in the single-panel case due to the highly local-ized damage pattern (deviatoric stress cannot pass throughfully damaged elements/particles). Figure 15 shows thevon Mises stress contours through time on the entranceside of the single-panel models. For easier viewing, onlystress values between zero and 150 MPa are shown, sincemuch higher stress states are achieved in the much stifferFSP.

The final major difference noted in the performance of thetwo model types is the deformation of the FSP. Figure 16

shows a comparison of one of the two-panel experimen-tally deformed FSPs to an undeformed FSP. Although theprojectiles go through relatively little permanent deforma-tion (given the extremely high rate of impact), the deformedshapes between the two models are very distinct. Figure 17shows the undeformed and deformed shapes of the FSP forthe FEA and SPH stacked models. As seen by the FSP on theleft, the FEAmodel seems to give amore smoothed deformedshape, while the SPH model causes jagged edges on the pro-jectile nose, likely caused by the type of contact seen betweenthe regular finite elements of the FSP and the particles of thetarget. This difference in the deformation of the projectilemay be responsible for the difference in residual velocity,since permanent plastic strain in the projectile is a form ofabsorbing kinetic energy. It can be expected that, since theprediction of residual velocity decreased in accuracy fromthe single-panel case to the stacked panel case, additionaldecrease in accuracywould be seen for simulationswithmorepanels.

7 Conclusions

The robust and efficient AFC model with modificationswas implemented in a dynamic user-defined material withinAbaqus, and thismodelwas validated against ballistic impactexperiments. These experiments involved high-velocityimpact of an FSP against single panels and stacked dou-

Fig. 17 FSP before (top) and after (bottom) impact (stacked models)

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ble panels. The same target panel material model was usedfor both the traditional FEA and SPH. The results of thesemodels were compared to the experimental results and toeach other. For this class of problem, the ability of theSPH models to capture residual velocity proved to be justas accurate as traditional FEA models, but they are signif-icantly more computationally expensive. It was shown thatthe strength of SPH is in its ability to model debris fieldformation where the traditional FEA cannot. Possible issuesdealing with tensile instability were identified, and furtherstudy into their implications as it pertains to this problem isneeded.

Acknowledgments Permission to publish was granted by the Direc-tor, Geotechnical and Structures Laboratory. Simulations were partlyperformed on the Department of Defense Super Computing Resourcehigh performance computers.

References

1. Nordendale N, HeardW, Ahn J, Basu P (2012)Mixed-dimensionalandmulti-scalemodeling in computationalmechanics. In: Proceed-ings of the 2012EngineeringMechanics InstituteConferenceNotreDame, TN

2. Zukas JA (2004) Introduction to hydrocodes. Studies in appliedmechanics, vol 49. Elsevier, Amsterdam

3. Nordendale N, Heard W, Hickman M, Zhang B, Basu P (2013)Cementitious material models for simulating projectile impacteffects. J Comput Mater Sci 79:745–758

4. Simulia (2014) Abaqus analysis user’s manual. Dassault Systemes,Providence

5. Zukas JA, Nicholas T, Swift HF, Greszczuk LB, Curran DR (1982)Impact dynamics. Wiley, New York

6. GathersGR (1994) Selected topics in shockwave physics and equa-tion of state modeling. World Scientific Publishing Co., Livermore

7. Chen W (1982) Plasticity in Reinforced Concrete. McGraw-HillBook Co., New York

8. ReinhartW, Chhabildas L, KippM,Wilson L (1999) Spall strengthmeasurement of concrete for varying aggregate sizes. In: 15th USArmy Symposium, Myrtle Beach, SC

9. Johnson GR, Stryk RA, Beissel SR (1996) SPH for high velocityimpact computations. Comput Methods in Appl Mech Eng 139(1–4):347–373

10. Mehra V, Sijoy CD, Mishra V, Chaturvedi S (2012) Tensile insta-bility and artificial stresses in impact problems in SPH. J Phys ConfSer 377:1–5

11. Willam KG, Warnke EP (1974) Constitutive model for the triaxialbehavior of concrete. In: IABSE Seminar on Concrete StructureSubjected to Triaxial Stresses, Bergamo, Italy

12. Ottosen NS (1977) A failure criterion for concrete. J Eng MechDiv ASCE 103(EM4):527–535

13. Holmquist TJ, Johnson GR, Cook WH (1993) A computationalconstitutive model for concrete subjected to large strains, highstrain rates, and high pressures. In: Proceedings of the 14th Inter-national Symposium on Ballistics, Quebec City, Canada

14. Tu Z, Lu Y (2009) Evaluation of typical concrete material modelsused in hydrocodes for high dynamic response simulations. Int JImpact Eng 36:132–146

15. Gebbeken N, RuppertM (2000) A newmaterial model for concretein high-dynamic hydrocode simulations. Arch ApplMech 70:463–478

16. Malvar LJ, Crawford JE,Wesevich JW (1997)A plasticity concretematerial model for Dyna3D. Int J Impact Eng 19(9–10):847–873

17. MalvarLJ,Crawford JE,MorrillKB (2000)K&Cconcretematerialmodel release III-automated generation of material model input.K&C Technical Report TR-99-24-B1, Glendale, CA

18. Riedel W (2000) Beton unter dynamischen Lasten Meso- undmakromechanische Modelle und ihre Parameter. Doctoral Thesis,Institut Kurzzeitdynamik, Ernst-Mach-Institut, der BundeswehrMunchen, Freiburg [in German]

19. Riedel W, Kawai N, Kondo K (2009) Numerical assessment forimpact strength measurements in concrete materials. Int J ImpactEng 36:283–293

20. Tu Z, Lu Y (2010) Modifications of RHT material model forimproved numerical simulation of dynamic response of concrete.Int J Impact Eng 37:1072–1082

21. FossumAF, Brannon RM (2006) On a viscoplastic model for rockswith mechanism-dependent characteristic times. Acta Geotech1:89–106

22. Frank A, Adley M (2007) On the importance of a three-invariantmodel for simulating the perforation of concrete slabs. In: Proceed-ings of 78th shock and vibration symposium. Philadelphia, PA

23. Polonco-Loria M, Hopperstad OS, Borvik T, Berstad T (2008)Numerical predictions of ballistic limits for concrete slabs usinga modified version of the HJC concrete model. Int J Impact Eng35:290–303

24. Beissel SR, Homquist TJ, JohnsonGR (2012) Influence of the thirdinvariant in the ballistic impact of silicon carbide. Int J Impact Eng45:52–59

25. Adley M, Frank A, Danielson K, Akers S, O’Daniel J (2010) Theadvanced fundamental concrete (AFC) Model: Technical ReportERDC/GSL TR-10-51. U.S. Army Engineer Research and Devel-opment Center, Vicksburg, MS

26. Sherburn JA, Heard W, Park H, Chen W (2011) Modeling of thesplit Hopkinson pressure bar: application to cortuf. In: 82nd Shockand Vibration Symposium, Baltimore, MD

27. Sherburn JA, Hammons MI, Roth MJ (2014) Modeling finitethickness slab perforation using a coupled Eulerian-Lagrangianapproach. Int J Solids Struct 51:4406–4413

28. Sherburn JA, Roth MJ, Chen JS, Hillman M (2015) Meshfreemodeling of concrete slab perforation using a reproducing ker-nel particle impact and penetration formulation. Int J Impact Eng86:96–110. doi:10.1016/j.ijimpeng.2015.07.009

29. Chen JS, Pan C,Wu CT, LiuWK (1996) Reproducing kernel parti-cle methods for large deformation analysis of nonlinear structures.Comput Methods App Mech Eng 139:49–74

30. Schwer L (2007) Optional strain-rate forms for the Johnson Cookconstitutivemodel and the role of the parameter Epsilon_01. In: 6thEur LS-DYNA Users’ Conference Gothenburg, Sweden: ERAB

31. Huh H, Kang W (2002) Crash-worthiness assessment of thin-walled structures with the high-strength steel sheet. Int J Veh Des30(1/2):1–21

32. Johnson GR, Cook WH (1983) A constitutive model and data formetals subjected to large strains, high strain rates and high tempera-ture. In: Proceedings of 7th International Symposium on Ballistics.The Gague, Netherlands

33. Johnson GR, Cook WH (1985) Fracture characteristics of threemetals subjected to various strain, strain rates, temperature andpressures. Eng Fract Mech 21(1):31–48

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