Int J Fract (2019) 216:211–221https://doi.org/10.1007/s10704-019-00354-0

ORIGINAL PAPER

Effect of high-rate dynamic comminution on penetrationof projectiles of various velocities and impact angles intoconcrete

Wen Luo · Viet T. Chau · Zdenek P. Bažant

Received: 27 September 2018 / Accepted: 22 February 2019 / Published online: 5 March 2019© Springer Nature B.V. 2019

Abstract The dynamic ‘overstress’, i.e., the appar-ent increase of strength of concrete at very high strainrates (10–106/s) experienced in projectile impact andpenetration, has recently been explained by a new the-ory with partial analogy to turbulence. The increaseis attributed to comminution of concrete driven by therelease of local kinetic energy of the shear strain ratefield of forming fragments which, in projectile impactproblems, exceeds the strain energy of the fragmentsby orders of magnitude. This theory gives the particlesize distribution and the additional local kinetic energydensity, �K , proportional to the deviatoric strain ratesquare. To match test results, �K must be dissipatedduring finite element simulations of impact. In previoussimulations, �K was, at first, dissipated by an artifi-cial equivalent viscosity (not empirical but predictedby the theory). Later it was found that dissipation byupscaling material tensile strength is equally effective.This theoretically justified upscaling is adopted heresince it is more realistic when microplane constitutivemodel M7 for fracturing damage in concrete is used.All artificial damping is eliminated from the computerprogram. While previous simulations with the com-minution theory were limited to orthogonal impacts,and only the cases of penetration of slabs of various

V. T. ChauLos Alamos National Lab, Los Alamos, USA

W. Luo · Z. P. Bažant (B)Northwestern University, Evanston, Illinois 60208, USAe-mail: z-bazant@northwestern.edu

thickness by projectile of one velocity and penetrationdepths for different velocities, the present study alsoanalyzes further test data on oblique impacts at vari-ous impact angles up to 35◦, and on the exit velocitiesand penetration depths of projectiles of different veloc-ities. For each test series on one and the same concrete,the material parameters are calibrated on one test andthen, using the same parameters, all the other tests arepredicted. All the predictions of exit velocities and pen-etration depths of projectiles, as well as entry and exitcraters, are quite accurate.

Keywords Impact comminution · Microplane model ·Oblique angle

1 Introduction

In the analysis of impact of projectiles into concrete,the material strength and energy dissipation appear toincrease strongly as the strain rate exceeds 10/s, andbecomes major for 104/s. Several continuum Ožboltand Sharma (2011), Travaš et al. (2009) and discreteFeng et al. (2017), Feng et al. (2018) models wereproposed to explain this problem, called the ’dynamicoverstress’. Recently itwas proposed and demonstratedBažant and Caner (2013, 2014), Caner and Bažant(2014) that the physical justification can be found inmaterial fracturing driven not by the release of strainenergy, as in classical fracture mechanics, but by therelease of local kinetic energy of shear strain-rate field

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within the forming particles comminuting concrete intosmall fragments. At the rate of 104s−1, the kineticenergy of a forming particle exceeds themaximumpos-sible strain energy in the particle by two orders of mag-nitude. The comminution process thus dissipates muchof the kinetic energy of projectile impact and deceler-ates the projectile.

This phenomenon of comminution is not coveredby the conventional damage constitutive laws basedon standard quasi-static triaxial testing, which predictmuch larger fragments. The theory proposed in Bažantand Caner (2014), Caner and Bažant (2014), whichbears partial analogy to turbulence, was shown to givea strain-rate-dependent expression for the additionalkinetic energy density�K that is dissipated during pro-jectile penetration by the fracture energy of fine mate-rial fragments.

In the initial finite element modeling Bažant andCaner (2013, 2014), Caner andBažant (2014),�K wasdissipated by an additional shear viscosity η obtainedfrom the calculated kinetic energy release. This formu-lation, which is natural for rate-dependent energy dis-sipation, led to a model with partial analogy to the the-ory of turbulence Bažant and Caner (2014). In the ini-tial, simplified, approach, η was considered to dependonly on the strain rate. In a subsequent more rigorousapproach Su et al. (2015), Bažant and Su (2015), η wasconsidered to also depend on the strain-rate derivative.Later it was realized that what matters in finite elementmodeling is the magnitude of energy dissipation andnot the precise way of energy dissipation in the com-puter program. In subsequent computer programming,the energy was dissipated by increasing the strengthlimit as a function of the strain rate, which was moreconvenient for the microplane model as a constitutivedamage model for concrete Kirane et al. (2015). Allthese approaches led to a good match of the test dataon the exit velocities of projectiles from concrete wallsof different thicknesses, and the latter also matchedwell the measured penetration depths of projectiles ofdifferent velocities.

In the current study, the effect of �K is introducedsimilarly to Kirane et al. (2015)—by scaling up thematerial strength in the microplane model M7. Thestrain-dependent strength limits (called stress-strainboundaries) of the microplane damage constitutivemodel are raised based on the shear strain rate so asto dissipate the additional kinetic density �K in theforming particles. It is also assumed that the particles

form instantly. This method leads to strength increasedepending on both the strain rate and its derivative.

The present goal is to verify the capability of theM7material model with scaled boundaries to predict: (1)the projectile exit velocity; (2) depth of penetration and(3) the crater shape. To this end, simulations of orthog-onal and oblique projectile impact on concrete wallsare conducted. The results are then compared with thetest data found in the literature. First, orthogonal fixed-velocity impacts into targets of various thicknesses areanalyzed. Then, orthogonal impacts into a concrete slabof fixed thickness, sufficiently thin for perforation, arecomputationally simulated for various striking veloc-ities; the same for the oblique impacts into thin andthick concrete slabs. In the thin slab case, the exit (orresidual) velocities are compared, while in the thickslab case, too thick for perforation, the depths of pene-tration (DOP) are. Aswill be seen, all these simulationsyield good agreement with the experimental data fromthe literature.

In the analysis of impact of projectiles into concrete,the apparent material strength and energy dissipationappear to increase sharply as the strain rate increasesabove 10/s and becomes major for 104/s. To explainand model this phenomenon, called the ’dynamic over-stress’, it was recently proposed and demonstratedBažant and Caner (2013, 2014), Caner and Bažant(2014) that the physical justification must be soughtin material fracturing driven not only by the release ofstrain energy, as in classical fracture mechanics, butalso, and predominantly, by the release of local kineticenergy of shear strain-rate field within forming parti-cles, comminuting concrete into small fragments. Atthe rate of 104s−1, the kinetic energy of the formingparticle exceeds the maximum possible strain energyin the particle by two orders of magnitude. Thus, thecomminution process can dissipate much of the kineticenergy of projectile impact.

This phenomenon is not covered by the conventionaldamage constitutive lawsbasedon standardquasi-statictriaxial testing, which predict much larger fragments.The theory proposed inBažant andCaner (2014),Canerand Bažant (2014), which bears some analogy to tur-bulence, was shown to give a strain-rate-dependentexpression for the additional kinetic energydensity�Kthat is dissipated during projectile penetration by thefracture energy of fine material fragments.

In the initial finite element modeling Bažant andCaner (2013, 2014), Caner andBažant (2014),�K was

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Effect of high-rate dynamic comminution on penetration of projectiles of various velocities 213

dissipated by an additional shear viscosity η obtainedfrom the calculated kinetic energy release. This for-mulation, which is natural for rate-dependent energydissipation, led to a model with partial analogy to thetheoryof turbulence. In the initial, simplified, approach,η was considered to depend only on the strain rate. Ina subsequent more rigorous approach Su et al. (2015),Bažant and Su (2015), η was considered to also dependon the strain-rate derivative. Later it was realized thatwhat matters in finite element modeling is the magni-tude of energy dissipation and not the precise way ofenergy dissipation in the computer program. In sub-sequent computer programming, the energy was dis-sipated by increasing the strength limit as a functionof the strain rate, which was more convenient for themicroplane model as a constitutive damage model forconcrete Kirane et al. (2015). All these approaches ledto a good match of the test data on the exit velocitiesof projectiles from concrete walls of different thick-nesses, and the latter also matched well the measuredpenetration depths of projectiles of different velocities.

In the current study, the effect of �K is introducedsimilarly to Kirane et al. (2015)—by scaling up thematerial strength in the microplane model M7. Thestrain-dependent strength limits (called stress-strainboundaries) of the microplane damage constitutivemodel are raised based on the shear strain rate so asto dissipate the additional kinetic density �K in theforming particles. It is also assumed that the particlesform instantly. This method leads to strength increasedepending on both the strain rate and its derivative.

The present goal is to verify the capability of theM7material model with scaled boundaries to predict: (1)the projectile exit velocity; (2) depth of penetration and(3) the crater shape. To this end, simulations of orthog-onal and oblique projectile impact on concrete wallsare conducted. The results are then compared with thetest data found in the literature. First, orthogonal fixed-velocity impacts into targets of various thicknesses areanalyzed. Then, orthogonal impacts into a concrete slabof fixed thickness, sufficiently thin for perforation, arecomputationally simulated for various striking veloc-ities; the same for the oblique impacts into thin andthick concrete slabs. In the thin slab case, the exit (orresidual) velocities are compared, while, in the case ofa slab too thick for perforation, the depths of penetra-tion (DOP) are. As will be seen, all these simulationsyield good agreement with the experimental data fromthe literature.

2 Review of microplane model M7 and scaling ofits boundaries

The microplane model M7 Caner and Bažant (2012) isthe latest version in a series of progressively improvedmicroplane models labelled M0, M1, M2,…, M6,developed first for concrete (and then extended to otherquasibrittlematerials). The basic idea of themicroplanemodel is to express the constitutive law not in terms oftensors, but in terms of the vectors of stress and strainacting on a generic plane of any orientation in themate-rial microstructure, called the microplane. The use ofvectors is analogous to the Taylor models used for plas-ticity of polycrystallinemetals, but there aremajor con-ceptual differences Bažant et al. (2000b).

To avoid model instability in postpeak softening, akinematic constraint is used instead of a static one. Thatis, the strain vector on each microplane is obtainedby projecting the macroscopic strain tensor onto themicroplane. So we have,

εN = εi j Ni j , εM = εi j Mi j and εN = εi j Li j (1)

where εN , εM and εL are the magnitudes of the threestrain vectors corresponding to each microplane, andNi j = nin j , Mi j = (nim j + min j )/2 and Li j =(ni l j + li n j )/2 are the projection tensors; n, m andl are the three mutually orthogonal normal and tan-gential unit vectors characterizing that microplane, andthe subscripts i (1 = 1, 2, 3) denote cartesian compo-nents. Instead of defining the constitutive law in termsof the stress and strain tensors, it is defined on eachmicroplane, which allows calculating the microplanestress increments from the given microplane strainincrements. Finally, to ensure internal equilibrium, avariational principle (principle of virtual work) is usedto relate the stresses on the microplanes (σN , σM andσL ) to the macro-continuum stress tensor σi j , which isexpressed as

2π

3σi jδεi j =

∫�

(σN εN + σMδεM + σLδεL)d� (2)

In the microplane model M7, the microstresses aresubjected to strain-dependent boundaries (or strengthlimits) of four types: (1) the tensile normal boundarycaptures the progressive tensile fracturing; (2) the com-pressive volumetric boundary captures phenomenonsuch as pore collapse under extreme pressures while

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softening is absent; (3) the compressive deviatoricboundary captures softening in compression; and (4)the shear boundary captures friction.

During each loading step, if any microplane stressexceeds one of the boundaries, it is returned tothe boundary with displacement being fixed. Themicroplane model automatically captures the signifi-cant suppression of crack formation by confining pres-sure Caner and Bažant (2014). The failure envelopescan be calibrated by sequential tuning of 5 free param-eters of the microplane model k1 ∼ k5) according toseveral characteristic test data.

To capture the material comminution effect, micro-plane stress boundaries �σD , �σM and �σL areadjusted in each load step as a function of the deviatoricstrain rate εD . For simplicity, �σD , �σM and �σL areassumed to be the same and are denoted as � f . In thespirit of turbulence analogy Bažant and Caner (2013),Kirane et al. (2015), for a givendeviatoric strain rate εD ,the drop in kinetic energy per unit volume is obtainedas:

−�K = ckρh2ε2D (3)

where h = average particle size, ck = Ip/(2hVp), ρ =mass density, Vp and Ip are the average volume andpolar moment of inertia of each comminuted prismabout its axis, respectively. Combining the expressionof �K with the equation of the principle of virtualwork, one can get the increment of microplane stressboundaries Kirane et al. (2015):

� f = �σD = �σM = �σL = A1ε−4/3D εD, (4)

where A1 is a constant to be calibrated. This expressionshows that, to capture the energy dissipation due tocomminution, it is necessary to make the boundary afunction of both the first and second time derivativesof the effective deviatoric strain.

Apart from the additional parameter A1 for the effectof impact comminution and elastic modulus E controlsthe vertical scaling of stress-strain curves, microplanemodel M7 has 5 major free parameters to be calibratedbased on quasi-static tests: k1 · · · k5. Based on Canerand Bažant (2012), their meanings are: k1: implementsa radial scaling of stress-strain curves (which, in combi-nationwith vertical scaling by E , can also give horizon-tal scaling); k2: controls the height of horizontal asymp-tote of the frictional boundary; k3: controls the shape of

the volumetric boundary; k4: controls the shape of thevolumetric boundary; and k5: controls the volumetric-deviatoric coupling at lowpressures. Since themechan-ical properties of materials found in the literature areusually limited toYoung’smodulus, Poisson’s ratio andcompressive strength, we mainly rely on k1 and E incalibration to fit the compressive strength while the restof the parameters are set to default values. It turns outthat this calibration suffices for acceptable predictionsin impact problems.

3 Model calibration and verification

In commercial finite element software, the constitutivemodel is inserted into the user’s material subroutinesuch as VUMAT in ABAQUS. The microplane con-stitutive model M7 is written in an explicit form, todeliver the stress tensor from the given strain tensor ateach numerical integration point of each finite elementin each time step.

To capture the interaction between the projectile andthe target during impact, both the surface and internalnodes of the target are included in the potential regionof contact with the projectile. To reduce the compu-tational cost, the numerical model simulates only onehalf of the target geometry, by imposing the boundarycondition of symmetry on the central plane. Differentfrom the target, the complete geometry is consideredfor the projectile. Therefore, themass density of projec-tile is reduced by one half to ensure correct simulationresults.

For orthogonal perforation, it would have been evenmore cost-effective to model only a quarter of the geo-metrical domain. But this was not done, so as to havebetter comparability with the cases of oblique impactfor which at least one half of the geometry must bemodeled.

For all the analyses in current study, element dele-tions from the target have been used to avoid unreason-able extreme element deformations. The element dele-tion criterion consists of the maximum (tensile) andminimum (compressive) acceptable principal strains;in other words, on the two extreme eigenvalues of thestrain tensor. The element deletion thresholds were setto values such that a further increase of the thresholdwould not make a significant change to the projectileexit velocities.

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Different from quasistatic processes, damages in anelement do not have enough time to localize in the high-rate dynamical impact problem. Thus, the critical strainthresholds used in the current study are much largerthan in the static or low-rate analysis with model M7.The limits ranged from 0.2 ∼ 0.6 for tensile strain andfrom −0.8 ∼ −1.2 for compressive strain.

In all the explicit Abaqus computations Smith(2009), a certain amount of damping is, by default,introduced in the form of bulk viscosity, to mitigatehigh frequency oscillations. There are two types ofdamping used in Abaqus: linear and quadratic. Thelinear damping generates a bulk viscosity pressure pro-portional to the volumetric strain rate:

pb1 = b1ρcd Leεvol (5)

where b1 is a damping coefficient with the default valueof 0.06, ρ is the current material density, cd is the cur-rent dilatational wave speed, Le is an element charac-teristic length, and εvol is the volumetric strain rate. Forsolid continuum elements, a quadratic damping is alsoused in Abaqus, in the form:

pb2 = ρ(b2Leεvol)2 (6)

in which the default value for b2 is 1.2. Since bothdampings mentioned above are purely numerical andhave huge effects in the high-rate impact problems, theyhave been turned off by setting b1 and b2 to 0 for allthe simulations. To this end, the effect of material com-minution is captured completely by the parameter A1

introduced in the scaled M7 model.The only damping that must be kept is that due to

viscoelasticity of concrete between the cracks and tothe rate dependence of crack growth, as described inBažant et al. (2000a).

3.1 Orthogonal perforation with various targetthicknesses

The M7 model with scaled boundaries was first eval-uated using the published tests Cargile (1999), Franket al. (2012) of orthogonal projectile perforation, per-formed at the Geotechnical and Structure Laboratoryof the US Army Engineer Research and DevelopmentCenter (ERDC) in Vicksburg. These tests used circularslabs of four thicknesses, 127, 216, 254 and 280 mm,

madeof concreteWES-5000,whose standard compres-sion strength was 48 MPa. The slabs were cast in steelculvert pipes of diameter 1.52 m.

The projectiles, which were hollow and made ofsteel, had an ogival-nose (caliber radius head 3.0,length/diameter ratio 7.0 and diameter 50.8 mm) andmass 2.3 kg.Theprojectiles impacted the concrete slabswith the velocity of 310 m/s at the angle of 90◦. Theperforation tests were carried out two or three times foreach thickness of the slab. First, theM7model was cal-ibrated to fit the test data for concrete type WES-5000,used in these tests. This concrete had Young’s modu-lus E = 25 GPa and Poisson’s ratio ν = 0.18. Theoptimum values of the M7 parameters, which achievedvery good fits of quasi-static uni-, bi- and tri-axial tests,were k1 = 11 × 10−5, k2 = 110, k3 = 30, k4 = 100and k5 = 1.

The mesh used to discretize the slabs was uniform,and consisted of cubic elements of average size 7.5mm.The projectile was treated as rigid body since no exces-sive damage, melt or distortion was observed after thetest. To prevent spuriousmesh sensitivity, themodelingwas performed in the sense of the crack band model, inwhich the finite-element size (or mesh size) should beequal to the material characteristic length, which char-acterizes the size of the representative volume element(RVE) of the material and is used as the localizationlimiter.

The element size was considered as 7.5 mm, whichis about one to two times the maximum aggregate size.Note that if, in quasi-static problems, the element sizeis changed, the crack band model requires adjustingthe post-peak softening of the damage constitutive law.This guarantees that energy dissipated in the crackband, which would always localize into one elementwidth, would not change. This method has also beenadopted by commercial softwares such as ATENA andOOFEM. However, in projectile impact problems, thedeformation is generally so fast that there is not enoughtime for the cracking damage to localize, thus the post-peak of the damage law need not, and should not, beadjusted.

Figure 1 shows the evolution of logarithmic straincontour during the impact of projectile into 127 mmplain concrete target. Since predicted the crater shapesfor various thicknesses are similar, the results for thisparticular case only are shown.As seen, the crater shapeand size are reasonable and realistic.

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Fig. 1 Strain (maximumprincipal logarithmic strain)contour evolution duringorthogonal projectile impactinto 127mm plain concretetarget (v0 = 310 m/s)

(a) (b)

(d)(c)

(e) (f)

(g)

The relation between the exit (or residual) velocityand target thickness is shown Fig. 1a. The test data aremarked in the figurewith large circular points. The caseof target thickness 254mmwas used for calibration, forwhich parameter A1 is 2.5 × 10−5 resulted. Then theexit velocities for various target thicknesses (127 mm,160 mm, 216 mm and 270 mm) were predicted usingthe calibrated parameter. As can be seen, the predictedvalues (thick curve) show good agreement to the testdata (large circles).

Figure 2b shows the evolution of projectile veloci-ties during impact. After the projectile nose enters thetarget, its velocity decreases at a constant rate duringthe tunneling stage. The deceleration of the projectileis given by the slope of the common tangent of thevelocity curves and equals−2.4×104g, where g is theacceleration of gravity.

Also shown in Fig. 2 (in gray) is the curve ofexit velocities predicted by the M7 model alone, withno effect of material comminution but includes the

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Effect of high-rate dynamic comminution on penetration of projectiles of various velocities 217

(a) (b)

Fig. 2 a Comparison of simulated exit velocities with test data for various target thicknesses b Evolution of simulated vertical velocityduring impact into targets of various thicknesses (v0 = 310 m/s)

physical damping due to the viscoelasticity of con-crete between the cracks and to rate dependence ofcrack growth Bažant et al. (2000a), Bažant and Jirásek(2018), Bažant and Li (1997). Comparing the twocurves of exit velocities, one can see that the decreaseof exit velocity is almost halved if the comminution isnot considered. In addition, the effect of comminutionbecomes more significant as the target thickness getscloser to the depth of penetration, for which case thedifference between two exit velocities could be as largeas 150 m/s. In this regard it should warned that seem-ingly good matches of the observed exit velocities andpenetration depths can be obtained even without thecomminution if one introduces suitably high artificialdamping. Such simulations (sometimes seen at confer-ences and in journals) have, of course, no predictivevalue and are, in fact, misleading.

3.2 Orthogonal impacts with various strikingvelocities

The microplane model M7 with scaled boundaries pre-viously used for constant striking velocities is nowusedto predict the exit velocity for various striking veloci-ties as well. To this end, the simulations are comparedwith the impact experiments reported in Hanchak et al.(1992). In these experiments Hanchak et al. (1992),projectiles of 0.50 kg mass, with ogival nose, werelaunched by a 30-mm smooth-bore powder gun at strik-ing velocities Vs between 300 and 1100 m/s. The tar-get was located 3.66 m from the end of the gun barrel.

The residual velocities were measured by X-ray pho-tography. They showed that the projectiles were notfractured by the targets, and the recovered projectilesshowed only minor nose erosion. It is therefore reason-able for computations to treat the projectiles as rigidbodies.

Both normal strength concrete (NSC) and highstrength concrete (HSC) targetswere tested. Their com-pression strengths and densities were: fc = 48 MPaand ρ = 2440 kg/m3 for NSC target, fc = 140 MPaand ρ = 2520 kg/m3 for HSC target.

The fixture allowed the target to be positioned withrespect to the prescribed impact location. By changingthe location of impact point, one could control whetherthe projectile would hit the reinforcement. Test dataHanchak et al. (1992) shows that the effect of strikingthe reinforcement at Vs = 750 m/s had a negligibleeffect. So it suffices to model the target using plainconcrete instead.

The microplane model M7 with scaled boundarieswas first calibrated using the material properties andone exit velocity in the test data. For the NSC target,the parameters were: k1 = 7.5 × 10−5, k2 = 110,k3 = 30, k4 = 100, k5 = 1 and A1 = 1 × 10−6. Forthe HSC target, k1 = 1.3 × 10−4, k2 = 110, k3 = 30,k4 = 100, k5 = 1 and A1 = 2.9×10−6. Then the resid-ual (exit) velocities Vr were predicted by simulationswith various striking velocities Vs . Comparisons of thetest data and model predictions are shown in Fig. 3a,b. As can be seen, the present results agree with theexperiments very well.

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(a) (b)

Fig. 3 Comparison of numerically predicted residual velocities with test data: a normal strength concrete (NSC) targets; b high strengthconcrete (HSC) targets

3.3 Oblique perforation of plain concrete targets

The microplane model M7 has then be used to extendthe analysis to oblique impacts and perforations. It isagain found that the predictions of the residual veloc-ities are consistent with experiments. The test reportFan and Zhou (2003) the projectile velocity along itsaxis, and the impact angle β (0 < β < 90◦) is definedas the acute angle between the projectile velocity andthe outward normal of the impacted surface. Obviously,the orthogonal impact studied previously is the specialcase where β = 0.

The sharp ogive-nose projectiles of the same geom-etry (269.4 mm long and 50 mm in diameter) wereused in all the tests. The thickness of the target wasH = 250 mm, the obliqueness was β = 30◦ and otherrelevant parameters were: fc = 40 MPa, ρ = 2336kg/m3. Besides, three different striking velocities wereused: 315, 407 and 610 m/s. In computations, linearhexahedron elements of size approximately 7.5 mm,with reduced integration, were used. Figure 4 showsthe typical strain contour evolution during the processof projectile penetration, with the initial striking veloc-ity being 600 m/s. One can see that the bottom crateris much larger than the top one, which is realistic.

First, the model is calibrated using the case of strik-ing velocity 600 m/s. For the normal strength concreteslab the parameters are: k1 = 7.5 × 10−5, k2 = 110,k3 = 30, k4 = 100, k5 = 1 and A1 = 4.5×10−6. Thenthe relations between the residual and striking veloci-ties are obtained by interpolation of several predictions(see Fig. 5). It is seen that the predictions (thick lines)match the test data (large points) quite well.

3.4 Oblique impact into thick concrete slabs withdifferent angles

Next our model is used to match the tests of thedepth of penetration of projectile into thick plain con-crete, reported in Liu et al. (2009). The velocity direc-tion and angle β are defined the same way as before;0 < β < 90◦). The density of concrete target is 2280kg/m3, the compressive strength is 30 MPa, and thethickness is 800 mm. The projectile has a density of7800 kg/m3, length of 180 mm and diameter of 20 mm.The projectile material is an alloy of 35 CrMnSiA andis modeled as a rigid body. The calibrated parametersare: k1 = 7.5 × 10−5, k2 = 110, k3 = 30, k4 = 100,k5 = 1 and A1 = 1.2 × 10−6.

The comparisons of the depth of penetration (DOP)fromsimulations and experiments are shown inTable. 1.From the table, one can see that when the oblique angleβ is smaller than 35◦, the predictions agree very wellwith the experiment and the relative error is always lessthan 3.6%.

When β > 35◦, however, the predicted DOPs aremuch larger than those measured. This increase of rel-ative error arises mainly from the rigid-body assump-tion for the projectile. Different from normal perfo-ration tests, steel projectiles could endure significantinelastic deflection when the impact inclination angleis large, with the nose bending upward. In that case,the projectile trajectory is no longer a straight line. Theprojectile becomes curved, with the nose up, whichcan greatly reduce the DOP. To model this deviation,it would obviously be necessary to include a realistic

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Effect of high-rate dynamic comminution on penetration of projectiles of various velocities 219

Fig. 4 Strain contourevolution during obliqueprojectile impact into250mm plain concretetarget (v0 = 600 m/s)

(a) (b)

(c) (d)

(e) (f)

Fig. 5 Comparison of numerically predicted residual velocitiesof projectile with test data (β = 30◦)

Table 1 Comparison of depth of penetrations from test resultsand simulations for obliqueprojectile impact into concrete targets

Impactvelocity (m/s)

Obliqueangle (◦)

TestedDOP (cm)

SimulatedDOP (cm)

545 28 22.5 23.1

528 30 21.9 22.3

525 30 23 23.8

545 33 21.6 22.4

552 33 20 20.7

537 35 25.2 26

548 35 23 23.8

541 41 15.1 18.9

545 41 16 19.2

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constitutive model for the projectile (which is beyondthe scope of this study).

4 Conclusions

1. ThemicroplanemodelM7with stress-strain bound-aries scaled according the deviatoric strain rate pro-vides an effective method for simulating and pre-dicting the projectile impact into concrete slabs. Itcaptures the phenomenon of dynamic ‘overstress’by taking account of the local kinetic energy dis-sipation. It is shown that the model predicts accu-rately the exit velocities of the projectiles, cratershapes and penetration depths, for both orthogonaland oblique impacts.

2. The local kinetic energy dissipation is calculatedby a previously proposed theory in which concretecomminution is driven by the kinetic energy of thestrain-rate field of formingparticles,which suppliesthe fracture energy of the fragments about to form.In this theory, bearing partial analogy to the the-ory of turbulence, the stress boundaries (or strengthlimits) scale with the −4/3 power of the effectivedeviatoric strain rate and the time derivative of thatrate.

3. The rigid-body assumption for the projectiles isvalid only when the impact angle deviates from thenormal by less than 35◦. For larger angles the pathbecomes curved and the projectiles would have tobe treated as deformable.

4. To get physically correct predictions, the defaultartificial damping of explicit dynamic finite ele-ment programs should be eliminated. The damp-ing should consist only of the material comminu-tion, the viscoelasticity of thematerials between thecracks, and the crack growth rate effect.

Acknowledgements Funding under AROGrant W911NF-15-101240 to Northwestern University is gratefully acknowledged.

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